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    \textbf{GRP004-1.ax} Subgroup (equality) axioms






$\mbox{subgroup\_member}(x) \ \Rightarrow \ \mbox{subgroup\_member}(x')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse},\mbox{axiom})$

$( \mbox{subgroup\_member}(x)\mbox{ and }\mbox{subgroup\_member}(y)\mbox{ and }x\cdot y = z ) \ \Rightarrow \ \mbox{subgroup\_member}(z)\qquad\mbox{cnf}(\mbox{closure\_of\_multiply},\mbox{axiom})$

\medskip\textbf{GRP003+0.ax} Group theory axioms






$\forall x{:}\;\mbox{identity}\cdot x{=}x\qquad\mbox{fof}(\mbox{left\_identity},\mbox{axiom})$

$\forall x{:}\;x\cdot \mbox{identity}{=}x\qquad\mbox{fof}(\mbox{right\_identity},\mbox{axiom})$

$\forall x{:}\;x'\cdot x{=}\mbox{identity}\qquad\mbox{fof}(\mbox{left\_inverse},\mbox{axiom})$

$\forall x{:}\;x\cdot x'{=}\mbox{identity}\qquad\mbox{fof}(\mbox{right\_inverse},\mbox{axiom})$

$\forall x,y{:}\;x\cdot y{=}x\cdot y\qquad\mbox{fof}(\mbox{total\_function}_1,\mbox{axiom})$

$\forall w,x,y,z{:}\;( ( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z = w )\qquad\mbox{fof}(\mbox{total\_function}_2,\mbox{axiom})$

$\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w )\qquad\mbox{fof}(\mbox{associativity}_1,\mbox{axiom})$

$\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w )\qquad\mbox{fof}(\mbox{associativity}_2,\mbox{axiom})$

\medskip\textbf{GRP004-2.ax} Lattice ordered group (equality) axioms






$\mbox{greatest\_lower\_bound}(x,y) = \mbox{greatest\_lower\_bound}(y,x)\qquad\mbox{cnf}(\mbox{symmetry\_of\_glb},\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,y) = \mbox{least\_upper\_bound}(y,x)\qquad\mbox{cnf}(\mbox{symmetry\_of\_lub},\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,\mbox{greatest\_lower\_bound}(y,z)) = \mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(x,y),z)\qquad\mbox{cnf}(\mbox{associativity\_of\_glb},\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,\mbox{least\_upper\_bound}(y,z)) = \mbox{least\_upper\_bound}(\mbox{least\_upper\_bound}(x,y),z)\qquad\mbox{cnf}(\mbox{associativity\_of\_lub},\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,x) = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_lub},\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,x) = x\qquad\mbox{cnf}(\mbox{idempotence\_of\_gld},\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,\mbox{greatest\_lower\_bound}(x,y)) = x\qquad\mbox{cnf}(\mbox{lub\_absorbtion},\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,\mbox{least\_upper\_bound}(x,y)) = x\qquad\mbox{cnf}(\mbox{glb\_absorbtion},\mbox{axiom})$

$x\cdot \mbox{least\_upper\_bound}(y,z) = \mbox{least\_upper\_bound}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{monotony\_lub}_1,\mbox{axiom})$

$x\cdot \mbox{greatest\_lower\_bound}(y,z) = \mbox{greatest\_lower\_bound}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{monotony\_glb}_1,\mbox{axiom})$

$\mbox{least\_upper\_bound}(y,z)\cdot x = \mbox{least\_upper\_bound}(y\cdot x,z\cdot x)\qquad\mbox{cnf}(\mbox{monotony\_lub}_2,\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(y,z)\cdot x = \mbox{greatest\_lower\_bound}(y\cdot x,z\cdot x)\qquad\mbox{cnf}(\mbox{monotony\_glb}_2,\mbox{axiom})$

\medskip\textbf{GRP006-0.ax} Group theory (Named groups) axioms






$\mbox{group\_member}(\mbox{identity\_for}(\mbox{xg}),\mbox{xg})\qquad\mbox{cnf}(\mbox{identity\_in\_group},\mbox{axiom})$

$\mbox{xg}\cdot \mbox{identity\_for}(\mbox{xg}){=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\mbox{xg}\cdot x{=}\mbox{identity\_for}(\mbox{xg})\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$\mbox{group\_member}(x,\mbox{xg}) \ \Rightarrow \ \mbox{group\_member}(\mbox{xg}',\mbox{xg})\qquad\mbox{cnf}(\mbox{inverse\_in\_group},\mbox{axiom})$

$\mbox{xg}\cdot \mbox{xg}'{=}x\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\mbox{xg}\cdot x{=}\mbox{xg}'\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$( \mbox{group\_member}(x,\mbox{xg})\mbox{ and }\mbox{group\_member}(y,\mbox{xg}) ) \ \Rightarrow \ \mbox{xg}\cdot x{=}y\qquad\mbox{cnf}(\mbox{total\_function1}_1,\mbox{axiom})$

$( \mbox{group\_member}(x,\mbox{xg})\mbox{ and }\mbox{group\_member}(y,\mbox{xg}) ) \ \Rightarrow \ \mbox{group\_member}(m(\mbox{xg},x,y),\mbox{xg})\qquad\mbox{cnf}(\mbox{total\_function1}_2,\mbox{axiom})$

$( \mbox{xg}\cdot x{=}y\mbox{ and }\mbox{xg}\cdot x{=}y ) \ \Rightarrow \ w = z\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( \mbox{xg}\cdot x{=}y\mbox{ and }\mbox{xg}\cdot y{=}z\mbox{ and }\mbox{xg}\cdot \mbox{xy}{=}z ) \ \Rightarrow \ \mbox{xg}\cdot x{=}\mbox{yz}\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( \mbox{xg}\cdot x{=}y\mbox{ and }\mbox{xg}\cdot y{=}z\mbox{ and }\mbox{xg}\cdot x{=}\mbox{yz} ) \ \Rightarrow \ \mbox{xg}\cdot \mbox{xy}{=}z\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

\medskip\textbf{GRP004-0.ax} Group theory (equality) axioms






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x'\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

\medskip\textbf{GRP008-1.ax} Cancellative semigroups axioms






$a\cdot b = c\cdot b \ \Rightarrow \ a = c\qquad\mbox{cnf}(\mbox{right\_cancellation},\mbox{axiom})$

$a\cdot b = a\cdot c \ \Rightarrow \ b = c\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

\medskip\textbf{GRP008-0.ax} Semigroups axioms






$( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{cnf}(\mbox{associativity\_of\_multiply},\mbox{axiom})$

\medskip\textbf{GRP007+0.ax} Group theory (Named Semigroups) axioms






$\forall g,x,y{:}\;( ( \mbox{group\_member}(x,g)\mbox{ and }\mbox{group\_member}(y,g) ) \ \Rightarrow \ \mbox{group\_member}(m(g,x,y),g) )\qquad\mbox{fof}(\mbox{total\_function},\mbox{axiom})$

$\forall g,x,y,z{:}\;( ( \mbox{group\_member}(x,g)\mbox{ and }\mbox{group\_member}(y,g)\mbox{ and }\mbox{group\_member}(z,g) ) \ \Rightarrow \ m(g,m(g,x,y),z) = m(g,x,m(g,y,z)) )\qquad\mbox{fof}(\mbox{associativity},\mbox{axiom})$

\medskip\textbf{GRP003-0.ax} Group theory axioms






$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$x\cdot x'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z = w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

\medskip\textbf{GRP003-2.ax} Subgroup axioms for the GRP003 group theory axioms






$( \mbox{subgroup\_member}(a)\mbox{ and }\mbox{subgroup\_member}(b)\mbox{ and }a\cdot b'{=}c ) \ \Rightarrow \ \mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_product\_and\_inverse},\mbox{axiom})$

\medskip\textbf{GRP005-0.ax} Group theory axioms






$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z{=}w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }w\cdot z{=}x ) \ \Rightarrow \ w\cdot z{=}y\qquad\mbox{cnf}(\mbox{product\_substitution}_3,\mbox{axiom})$

\medskip\textbf{GRP002-0.ax} Semigroup axioms






$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z = w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

\medskip\textbf{GRP004+0.ax} Group theory (equality) axioms






$\forall x{:}\;\mbox{identity}\cdot x = x\qquad\mbox{fof}(\mbox{left\_identity},\mbox{axiom})$

$\forall x{:}\;x'\cdot x = \mbox{identity}\qquad\mbox{fof}(\mbox{left\_inverse},\mbox{axiom})$

$\forall x,y,z{:}\;( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{fof}(\mbox{associativity},\mbox{axiom})$

\medskip\textbf{GRP001-0.ax} Monoid axioms






$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z = w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

\medskip\textbf{GRP003-1.ax} Subgroup axioms for the GRP003 group theory axioms






$\mbox{subgroup\_member}(x) \ \Rightarrow \ \mbox{subgroup\_member}(x')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse},\mbox{axiom})$

$( \mbox{subgroup\_member}(a)\mbox{ and }\mbox{subgroup\_member}(b)\mbox{ and }a\cdot b{=}c ) \ \Rightarrow \ \mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_product},\mbox{axiom})$

\medskip\textbf{GRP181-4.p} Distributivity of a lattice






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p12x}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p12x}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p12x}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,c) = \mbox{greatest\_lower\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12x}_4,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,c) = \mbox{least\_upper\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12x}_5,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(x,y)' = \mbox{least\_upper\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p12x}_6,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(x,y)' = \mbox{greatest\_lower\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p12x}_7,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_p12x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP543-1.p} Axiom for Abelian group theory, in division and identity, part 3






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),a)),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-1.005.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP200-1.p} In Loops, Moufang-1 =$>$ Moufang-2.






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot \mbox{left\_division}(x,y) = y\qquad\mbox{cnf}(\mbox{multiply\_left\_division},\mbox{axiom})$

$\mbox{left\_division}(x,x\cdot y) = y\qquad\mbox{cnf}(\mbox{left\_division\_multiply},\mbox{axiom})$

$\mbox{right\_division}(x,y)\cdot y = x\qquad\mbox{cnf}(\mbox{multiply\_right\_division},\mbox{axiom})$

$\mbox{right\_division}(x\cdot y,y) = x\qquad\mbox{cnf}(\mbox{right\_division\_multiply},\mbox{axiom})$

$x\cdot \mbox{right\_inverse}(x) = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( x\cdot ( y\cdot z ) )\cdot x = ( x\cdot y )\cdot ( z\cdot x )\qquad\mbox{cnf}(\mbox{moufang}_1,\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b\ne a\cdot ( b\cdot ( c\cdot b ) )\qquad\mbox{cnf}(\mbox{prove\_moufang}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP033-3.p} In subgroups, the identity is the group identity






$x{=}x\qquad\mbox{cnf}(\mbox{reflexivity},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetry},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$x{=}y \ \Rightarrow \ x'{=}y'\qquad\mbox{cnf}(\mbox{inverse\_substitution},\mbox{axiom})$

$x{=}y \ \Rightarrow \ x\cdot w{=}y\cdot w\qquad\mbox{cnf}(\mbox{multiply\_substitution}_1,\mbox{axiom})$

$x{=}y \ \Rightarrow \ w\cdot x{=}w\cdot y\qquad\mbox{cnf}(\mbox{multiply\_substitution}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }x\cdot w{=}z ) \ \Rightarrow \ y\cdot w{=}z\qquad\mbox{cnf}(\mbox{product\_substitution}_1,\mbox{axiom})$

$( x{=}y\mbox{ and }w\cdot x{=}z ) \ \Rightarrow \ w\cdot y{=}z\qquad\mbox{cnf}(\mbox{product\_substitution}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }w\cdot z{=}x ) \ \Rightarrow \ w\cdot z{=}y\qquad\mbox{cnf}(\mbox{product\_substitution}_3,\mbox{axiom})$

$( a{=}b\mbox{ and }\mbox{subgroup\_member}(a) ) \ \Rightarrow \ \mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{subgroup\_member\_substitution},\mbox{axiom})$

$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$x\cdot x'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z{=}w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$( \mbox{subgroup\_member}(a)\mbox{ and }\mbox{subgroup\_member}(b)\mbox{ and }a\cdot b'{=}c ) \ \Rightarrow \ \mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_product\_and\_inverse},\mbox{axiom})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(j(a))\qquad\mbox{cnf}(\mbox{subgr2\_clause}_1,\mbox{hypothesis})$

$( j(a)\cdot a{=}j(a)\mbox{ and }a\cdot j(a){=}j(a) ) \ \Rightarrow \ \neg\,\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{prove\_subgr}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP556-1.p} Axiom for Abelian group theory, in division and inverse, part 4






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(a,c))'),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP527-1.p} Axiom for Abelian group theory, in division, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(c,b))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP299-1.p} An identity generated by HR, number 03746






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_3' = x_4 ) \ \Rightarrow \ x_4\cdot \mbox{sk\_c}_7\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP193-1.p} A combination of distributivity and monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p8\_9a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p8\_9a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},c) = c\qquad\mbox{cnf}(\mbox{p8\_9a}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p8\_9a}_4,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b\cdot c),\mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c)) = \mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{p8\_9a}_5,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b\cdot c)\ne \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{prove\_p8\_9a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP670-1.p} RIF loops are ARIF - a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot ( a\cdot b ) ) = ( ( a\cdot ( b\cdot c ) )\cdot a )\cdot b\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c )\ne ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP029-1.p} In semigroups, left id and inverse =$>$ right id exists

If there are a left identity and left inverse, then there
 is a right identity element.




$\mbox{include}(\mbox{'Axioms/GRP002-0.ax'})$

$\mbox{identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$a'\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\neg\,\mbox{not\_right\_identity}(a)\cdot a{=}\mbox{not\_right\_identity}(a)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP584-1.p} Axiom for Abelian group theory, in double div and id, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},b),\mbox{double\_divide}(c,\mbox{double\_divide}(b,a)))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP460-1.p} Axiom for group theory, in division and identity, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{identity},b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP679-1.p} Commutants in Bol loops 1

If Q is a Bol loop, and if a,b in C(Q), then so are a$\wedge$2, b$\wedge$-1,
 and a$\wedge$2b.




$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot a\ne a\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP720-2.p} In commutative A-loops, squares form a subloop






$a\cdot 1 = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( \mbox{op\_a}\cdot \mbox{op\_a} )\cdot ( \mbox{op\_b}\cdot \mbox{op\_b} )\ne a\cdot a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

\medskip\textbf{GRP667-4.p} 2-divisible ARIF loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a)\cdot f(a) = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-2.005.p} a.(b.a) = (b.a).a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP455-1.p} Axiom for group theory, in division and identity, part 2






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP724-1.p} Loops with abelian inner mapping group - associativity

Uniquely 2-divisible loops with abelian inner mapping group of
 exponent 2 are associative.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$s(a)\cdot s(a) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$s(a\cdot a) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = \mbox{ld}(c\cdot b,c\cdot ( b\cdot a ))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_t}(a,b) = \mbox{ld}(b,a\cdot b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),b) = a\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),b,c) = a\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),b,c) = a\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP566-1.p} Axiom for Abelian group theory, in double div and id, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(\mbox{identity},c))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP495-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,c),\mbox{double\_divide}(\mbox{identity},\mbox{identity})),\mbox{double\_divide}(a,c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP163-1.p} Prove transitivity axiom using the GLB transformation

This problem proves the original transitivity axiom from
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_transb}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,c) = b\qquad\mbox{cnf}(\mbox{ax\_transb}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,c)\ne a\qquad\mbox{cnf}(\mbox{prove\_ax\_transb},\mbox{negated\_conjecture})$

\medskip\textbf{GRP691-1.p} In a power associative conjugacy closed loop, c$\wedge$3 is WIP






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) = \mbox{op\_d}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_d}\cdot \mbox{op\_d} = \mbox{op\_e}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_e}\cdot \mbox{op\_e} = \mbox{op\_f}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_d}\cdot i(a\cdot \mbox{op\_d})\ne i(a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP732-1.p} Basarab's theorem on CC loops






$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,a)\cdot ( a\cdot c ) = a\cdot ( b\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot \mbox{ld}(b,c\cdot b) = ( a\cdot c )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{ld}(c\cdot d,d\cdot c) )\ne ( a\cdot b )\cdot \mbox{ld}(c\cdot d,d\cdot c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP023-1.p} The inverse of the identity is the identity






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{identity}'\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_id\_is\_id},\mbox{negated\_conjecture})$

\medskip\textbf{GRP183-4.p} Orthogonal elements form a subgroup with orthogonal parts






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p20x}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p20x}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p20x}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,\mbox{identity}),\mbox{least\_upper\_bound}(a',\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_20x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP398-3.p} Subgroup (equality) axioms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-1.ax'})$

\medskip\textbf{GRP563-1.p} Axiom for Abelian group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b'),c),\mbox{divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP166-3.p} Multiplication with a positive element increases a value






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{identity}) = a\qquad\mbox{cnf}(\mbox{lat3a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,\mbox{identity}) = b\qquad\mbox{cnf}(\mbox{lat3a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,b\cdot a)\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_lat3a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP611-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b)',c)',\mbox{double\_divide}(a,c))' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP155-1.p} Prove monotonicity axiom using the GLB transformation

This problem proves the original monotonicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_mono1b},\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a\cdot c,b\cdot c)\ne a\cdot c\qquad\mbox{cnf}(\mbox{prove\_ax\_mono1b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP161-1.p} Prove reflexivity axiom using the GLB transformation

This problem proves the original reflexivity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,a)\ne a\qquad\mbox{cnf}(\mbox{prove\_ax\_reflb},\mbox{negated\_conjecture})$

\medskip\textbf{GRP070-1.p} Single axiom for group theory, in division and inverse

This is a single axiom for group theory, in terms of division
 and inverse.




$\mbox{divide}(\mbox{divide}(x,x),\mbox{divide}(y,\mbox{divide}(\mbox{divide}(z,\mbox{divide}(u,y)),u'))) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP353-1.p} An identity generated by HR, number 22878






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_3' = x_4 ) \ \Rightarrow \ x_4\cdot \mbox{sk\_c}_7\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP305-1.p} An identity generated by HR, number 05617






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_3\cdot \mbox{sk\_c}_4\ne \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-1.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-1.ax'})$

$\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(a,b))\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$a\cdot \mbox{element\_in\_O}_2(a,b){=}b\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{negated\_conjecture})$

$b\cdot a'{=}c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP363-1.p} An identity generated by HR, number 24611






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_7\cdot x_3 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = x_3 ) \ \Rightarrow \ x_4'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP025-2.p} All groups of order 2 are isomorphic

If G1 has exactly two elements and G2 has exactly two
 elements, then there exists an isomorphism [a one-to-one and
 onto homomorphism] between them.




$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

$g_1\ne g_2\qquad\mbox{cnf}(\mbox{two\_groups},\mbox{hypothesis})$

$\mbox{group\_member}(a,g_1)\qquad\mbox{cnf}(\mbox{a\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(b,g_1)\qquad\mbox{cnf}(\mbox{b\_member\_of\_group}_1,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$\mbox{group\_member}(c,g_2)\qquad\mbox{cnf}(\mbox{c\_member\_of\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(d,g_2)\qquad\mbox{cnf}(\mbox{d\_member\_of\_group}_2,\mbox{hypothesis})$

$c\ne d\qquad\mbox{cnf}(\mbox{c\_not\_d},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne d\qquad\mbox{cnf}(\mbox{a\_not\_d},\mbox{hypothesis})$

$b\ne c\qquad\mbox{cnf}(\mbox{b\_not\_c},\mbox{hypothesis})$

$b\ne d\qquad\mbox{cnf}(\mbox{b\_not\_d},\mbox{hypothesis})$

$\mbox{group\_member}(x,g_1) \ \Rightarrow \ ( x = a\mbox{ or }x = b )\qquad\mbox{cnf}(\mbox{a\_and\_b\_only\_members\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_2) \ \Rightarrow \ ( x = c\mbox{ or }x = d )\qquad\mbox{cnf}(\mbox{c\_and\_d\_only\_members\_of\_group}_2,\mbox{hypothesis})$

$\mbox{identity\_for}(g_1) = a\qquad\mbox{cnf}(\mbox{a\_identity\_of\_group}_1,\mbox{hypothesis})$

$\mbox{identity\_for}(g_2) = c\qquad\mbox{cnf}(\mbox{c\_identity\_of\_group}_2,\mbox{hypothesis})$

$g_1\cdot a{=}x\qquad\mbox{cnf}(\mbox{a\_left\_identity},\mbox{hypothesis})$

$g_1\cdot x{=}a\qquad\mbox{cnf}(\mbox{a\_right\_identity},\mbox{hypothesis})$

$g_2\cdot c{=}x\qquad\mbox{cnf}(\mbox{c\_left\_identity},\mbox{hypothesis})$

$g_2\cdot x{=}c\qquad\mbox{cnf}(\mbox{c\_right\_identity},\mbox{hypothesis})$

$\mbox{isomorphism}_1(a) = c\qquad\mbox{cnf}(\mbox{a\_maps1\_to\_c},\mbox{hypothesis})$

$\mbox{isomorphism}_1(b) = d\qquad\mbox{cnf}(\mbox{b\_maps1\_to\_d},\mbox{hypothesis})$

$\mbox{isomorphism}_2(a) = d\qquad\mbox{cnf}(\mbox{a\_maps2\_to\_d},\mbox{hypothesis})$

$\mbox{isomorphism}_2(b) = c\qquad\mbox{cnf}(\mbox{b\_maps2\_to\_c},\mbox{hypothesis})$

$\mbox{group\_member}(d_1,g_1)\qquad\mbox{cnf}(\mbox{d1\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$\mbox{group\_member}(d_2,g_1)\qquad\mbox{cnf}(\mbox{d2\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$\mbox{group\_member}(d_3,g_1)\qquad\mbox{cnf}(\mbox{d3\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$g_1\cdot d_1{=}d_2\qquad\mbox{cnf}(\mbox{d1\_times\_d2\_is\_d}_3,\mbox{negated\_conjecture})$

$g_2\cdot \mbox{isomorphism}_1(d_1){=}\mbox{isomorphism}_1(d_2) \ \Rightarrow \ \neg\,g_2\cdot \mbox{isomorphism}_2(d_1){=}\mbox{isomorphism}_2(d_2)\qquad\mbox{cnf}(\mbox{prove\_one\_product\_holds\_in\_group}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP195-1.p} In semigroups, xyy=yyx $\to$ (uv)$\wedge$4 = u$\wedge$4v$\wedge$4.

In semigroups, xyy=yyx $\to$ uvuvuvuuv=uuuuvvvv.




$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$x\cdot ( y\cdot y ) = y\cdot ( y\cdot x )\qquad\mbox{cnf}(\mbox{condition},\mbox{hypothesis})$

$a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot b ) ) ) ) ) )\ne a\cdot ( a\cdot ( a\cdot ( a\cdot ( b\cdot ( b\cdot ( b\cdot b ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{GRP441-1.p} Axiom for group theory, in product \& inverse, part 3






$( a\cdot ( b\cdot ( ( b'\cdot c )\cdot ( d\cdot ( a\cdot c ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP186-1.p} Application of distributivity and group theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\ne a\cdot \mbox{greatest\_lower\_bound}(a,b')'\qquad\mbox{cnf}(\mbox{prove\_p}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{GRP089-1.p} Single axiom for Abelian group theory, in division

This is a single axiom for Abelian group theory, in terms
 of division




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(x,y),\mbox{divide}(z,y))) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP559-1.p} Axiom for Abelian group theory, in division and inverse, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(b,c),\mbox{divide}(a,c))') = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP166-1.p} Multiplication with a positive element increases a value






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{identity}) = a\qquad\mbox{cnf}(\mbox{lat2a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,\mbox{identity}) = b\qquad\mbox{cnf}(\mbox{lat2a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,a\cdot b)\ne a\cdot b\qquad\mbox{cnf}(\mbox{prove\_lat2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP553-1.p} Axiom for Abelian group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(a,c))'),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP778+1.p} Napoleon's quasigroups: Gruenbaum's theorem 1






$\forall b,a{:}\;a\setminus a\cdot b = b\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( a\setminus b ) = b\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;( a\cdot b )\cdot ( c\cdot d ) = ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall b,a{:}\;( ( a\cdot b )\cdot b )\cdot ( b\cdot ( b\cdot a ) ) = b\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( d(x_0,x_1,x_2)\iff x_0\cdot x_1 = x_1\cdot x_2 )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_3,x_4,x_5{:}\;( m(x_3,x_4,x_5)\iff ( x_3\cdot x_4 )\cdot ( x_4\cdot x_5 ) = x_3\cdot x_5 )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$d(a_1,b,c)\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$d(a,b_1,c)\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$d(a,b,c_1)\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$d(a_2,b_1,c_1)\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$d(a_1,b_2,c_1)\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$d(a_1,b_1,c_2)\qquad\mbox{fof}(\mbox{sos}_{15},\mbox{axiom})$

$m(b_1,b,b_2)\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP300-1.p} An identity generated by HR, number 03748






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_4\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6' = \mbox{sk\_c}_4\mbox{ and }x_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_3'\ne \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP185-1.p} Application of monotonicity and distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity}),\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity}))\ne \mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p22a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP102-1.p} Single axiom for Abelian group theory, in double div and id

This is a single axiom for Abelian group theory, in terms
 of double division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(y,x),z),\mbox{double\_divide}(y,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP147-1.p} Prove least upper-bound axiom using the GLB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = a\qquad\mbox{cnf}(\mbox{ax\_lub1b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,c) = b\qquad\mbox{cnf}(\mbox{ax\_lub1b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,b),c)\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_lub1b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP355-1.p} An identity generated by HR, number 22926






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\mbox{ and }x_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_8\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_8 = x_4 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP402-1.p} Prove nilpotent implies associativity in cancellative semigroup

Assume a cancellative semigroup admits a commutator operation.
 Then the following three properties are equivalent:
 (1) commutator is associative;
 (2) commutator distributes over product;
 (3) the semigroup is nilpotent of class 2.
 This is a generalization of the corresponding theorem for group
 theory. The problem here is to prove (1) implies (2).




$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$a\cdot b = b\cdot ( a\cdot \mbox{commutator}(a,b) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$\mbox{commutator}(a,b)\cdot c = c\cdot \mbox{commutator}(a,b)\qquad\mbox{cnf}(\mbox{nilpotency},\mbox{axiom})$

$\mbox{commutator}(\mbox{commutator}(a,b),c)\ne \mbox{commutator}(a,\mbox{commutator}(b,c))\qquad\mbox{cnf}(\mbox{prove\_commutator\_is\_associative},\mbox{negated\_conjecture})$

\medskip\textbf{GRP712-1.p} In Buchsteiner loops fourth powers are nuclear - a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( a\cdot b )\cdot c) = \mbox{rd}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot ( a\cdot ( a\cdot a ) ) )\cdot ( b\cdot c )\ne ( ( a\cdot ( a\cdot ( a\cdot a ) ) )\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP524-1.p} Axiom for Abelian group theory, in division, part 4






$\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,\mbox{divide}(a,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP268-1.p} An identity generated by HR, number 01106






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( x_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_8\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot x_4 = \mbox{sk\_c}_7\mbox{ and }x_5\cdot \mbox{sk\_c}_6 = x_4\mbox{ and }x_5' = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8 ) \ \Rightarrow \ \mbox{sk\_c}_6'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP004-2.p} Left inverse and identity =$>$ Right inverse exists

In a group with left inverses and left identity every element
 has a right inverse.




$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$\neg\,a\cdot a'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_right\_inverse},\mbox{negated\_conjecture})$

\medskip\textbf{GRP183-2.p} Orthogonal elements form a subgroup with orthogonal parts






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p20}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p20}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p20}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,\mbox{identity}),\mbox{greatest\_lower\_bound}(a,\mbox{identity})')\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{GRP481-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity})),\mbox{double\_divide}(\mbox{double\_divide}(c,\mbox{double\_divide}(d,\mbox{double\_divide}(d,\mbox{identity}))),\mbox{double\_divide}(a,\mbox{identity}))),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP667-5.p} 2-divisible ARIF loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a)\cdot f(a) = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP467-1.p} Axiom for group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(b,\mbox{divide}(\mbox{divide}(c,\mbox{divide}(d,b)),d'))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP085-1.p} Single axiom for Abelian group theory, in product and inverse

This is a single axiom for Abelian group theory, in terms
 of product and inverse.




$( ( x\cdot y )\cdot z )\cdot ( x\cdot z )' = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP657+1.p} A quasigroup satisfying Moufang 4 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot ( c\cdot a ) = a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP291-1.p} An identity generated by HR, number 03387






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP658+1.p} Bol-Moufang identity 1 implies the existence of a unit element






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot ( b\cdot b ) )\cdot c = ( a\cdot b )\cdot ( b\cdot c )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP008-1.p} Unknown meaning






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$( q(a)\mbox{ and }a\cdot b{=}c ) \ \Rightarrow \ b\cdot a{=}c\qquad\mbox{cnf}(\mbox{unknown\_meaning}_2,\mbox{axiom})$

$j(a)\cdot a{=}h(a)\mbox{ or }a\cdot j(a){=}h(a)\mbox{ or }q(a)\qquad\mbox{cnf}(\mbox{unknown\_meaning}_3,\mbox{axiom})$

$( j(a)\cdot a{=}h(a)\mbox{ and }a\cdot j(a){=}h(a) ) \ \Rightarrow \ q(a)\qquad\mbox{cnf}(\mbox{unknown\_meaning}_4,\mbox{axiom})$

$\neg\,q(\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_q},\mbox{negated\_conjecture})$

\medskip\textbf{GRP589-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,c')')',b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP264-1.p} An identity generated by HR, number 01099






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{27},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{28},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ and }x_3' = \mbox{sk\_c}_9\mbox{ and }x_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ and }x_1' = \mbox{sk\_c}_9\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_8\cdot x_5 = \mbox{sk\_c}_7\mbox{ and }x_6\cdot \mbox{sk\_c}_8 = x_5 ) \ \Rightarrow \ x_6'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{GRP487-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{identity}),\mbox{double\_divide}(b,c))),b),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP484-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP539-1.p} Axiom for Abelian group theory, in division, part 3






$\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(\mbox{divide}(a,c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP164-1.p} The lattice of each LOG is distributive






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{greatest\_lower\_bound}(b,c))\ne \mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,b),\mbox{least\_upper\_bound}(a,c))\qquad\mbox{cnf}(\mbox{prove\_distrnu},\mbox{negated\_conjecture})$

\medskip\textbf{GRP714-1.p} In Buchsteiner loops fourth powers are nuclear - c






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( a\cdot b )\cdot c) = \mbox{rd}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot ( c\cdot ( c\cdot c ) ) ) )\ne ( a\cdot b )\cdot ( c\cdot ( c\cdot ( c\cdot c ) ) )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP234-1.p} An identity generated by HR, number 00574






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_2 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = x_2 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP515-1.p} Axiom for Abelian group theory, in product and inverse, part 3






$a\cdot ( ( b\cdot c )\cdot ( a\cdot c )' ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP754-1.p} A new basis for trimedial quasigroups: part 2b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot ( b\cdot c ) = ( \mbox{rd}(a,a)\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot ( b\cdot \mbox{ld}(c,c) )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c )\ne ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP707-1.p} A C-loop of exponent four with central squares is flexible






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( a\cdot ( a\cdot a ) ) = 1\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot a )\cdot b = b\cdot ( a\cdot a )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot a\ne a\cdot ( b\cdot a )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP674+1.p} Extra loop commutation property 3

In an extra loop, z commutes with [x;y; t] if and only if t
 commutes with [x;y; z] if and only if [x;y; z][x;y; t] =
 [x;y; zt].




$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot a ) ) = ( ( a\cdot b )\cdot c )\cdot a\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}\cdot \mbox{op\_t})\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\mbox{op\_t}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{op\_t}\mbox{ and }\mbox{op\_z}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t})\cdot \mbox{op\_z}\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP167-5.p} Product of positive and negative parts

Each element in a lattice ordered group can be stated as a
 product of it's positive and it's negative part.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b)' = \mbox{greatest\_lower\_bound}(a',b')\qquad\mbox{cnf}(p_{10},\mbox{axiom})$

$\mbox{positive\_part}(x) = \mbox{least\_upper\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_1,\mbox{axiom})$

$\mbox{negative\_part}(x) = \mbox{greatest\_lower\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_2,\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,\mbox{greatest\_lower\_bound}(y,z)) = \mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(x,y),\mbox{least\_upper\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_3,\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,\mbox{least\_upper\_bound}(y,z)) = \mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(x,y),\mbox{greatest\_lower\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_4,\mbox{axiom})$

$a\ne \mbox{positive\_part}(a)\cdot \mbox{negative\_part}(a)\qquad\mbox{cnf}(\mbox{prove\_lat}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP662-1.p} Conjugacy closed with ab = 1 implies ba is in the nucleus - b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d} = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot ( \mbox{op\_d}\cdot \mbox{op\_c} ) )\cdot b\ne a\cdot ( ( \mbox{op\_d}\cdot \mbox{op\_c} )\cdot b )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP612-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b)',c)',\mbox{double\_divide}(a,c))' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP398-2.p} Subgroup axioms for the GRP003 group theory axioms






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

\medskip\textbf{GRP617-1.p} PQEx lemma

Proves commutativity of mutliplication across two trivially
 intersecting subgroups.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{subgroup1\_member}(x) \ \Rightarrow \ \mbox{subgroup1\_member}(x')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse}_1,\mbox{axiom})$

$( \mbox{subgroup1\_member}(a)\mbox{ and }\mbox{subgroup1\_member}(b)\mbox{ and }a\cdot b{=}c ) \ \Rightarrow \ \mbox{subgroup1\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_product}_1,\mbox{axiom})$

$\mbox{subgroup2\_member}(x) \ \Rightarrow \ \mbox{subgroup2\_member}(x')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse}_2,\mbox{axiom})$

$( \mbox{subgroup2\_member}(a)\mbox{ and }\mbox{subgroup2\_member}(b)\mbox{ and }a\cdot b{=}c ) \ \Rightarrow \ \mbox{subgroup2\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_product}_2,\mbox{axiom})$

$\mbox{subgroup1\_member}(x) \ \Rightarrow \ \mbox{subgroup1\_member}(a\cdot ( x\cdot a' ))\qquad\mbox{cnf}(\mbox{normality}_1,\mbox{hypothesis})$

$\mbox{subgroup2\_member}(x) \ \Rightarrow \ \mbox{subgroup2\_member}(a\cdot ( x\cdot a' ))\qquad\mbox{cnf}(\mbox{normality}_2,\mbox{hypothesis})$

$( \mbox{subgroup1\_member}(x)\mbox{ and }\mbox{subgroup2\_member}(x) ) \ \Rightarrow \ x = \mbox{identity}\qquad\mbox{cnf}(\mbox{trivial\_intersection},\mbox{hypothesis})$

$\mbox{subgroup1\_member}(v)\qquad\mbox{cnf}(\mbox{v\_in\_G}_1,\mbox{hypothesis})$

$\mbox{subgroup2\_member}(u)\qquad\mbox{cnf}(\mbox{u\_in\_G}_2,\mbox{hypothesis})$

$v\cdot u\ne u\cdot v\qquad\mbox{cnf}(\mbox{prove\_vu\_equals\_uv},\mbox{negated\_conjecture})$

\medskip\textbf{GRP749-1.p} Simplifying a basis for trimedial quasigroups: part 1






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$( a\cdot ( a\cdot a ) )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( ( a\cdot a )\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot c )\ne ( a\cdot c )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP060-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$x\cdot ( y\cdot ( z\cdot ( ( z'\cdot ( u\cdot y )' )\cdot x ) ) )' = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP082-1.p} Single axiom for group theory, in double division and inverse

This is a single axiom for group theory, in terms of double
 division and inverse.




$\mbox{double\_divide}(x',\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(y,z))',\mbox{double\_divide}(u,\mbox{double\_divide}(y,u)))') = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP729-1.p} Bruck loops that are centrally nilpotent - 2nd easy part b

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - 2nd easy part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot b = ( b\cdot a )\cdot \mbox{op\_k}(a,b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e) = 1\qquad\mbox{cnf}(c_{21},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{asoc}(c,d,e)) = 1\qquad\mbox{cnf}(c_{22},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{op\_k}(c,d))\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP558-1.p} Axiom for Abelian group theory, in division and inverse, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(b,c),\mbox{divide}(a,c))') = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP054-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( ( z\cdot ( x'\cdot ( y\cdot ( y'\cdot y )' )' )' )'\cdot ( z\cdot y ) )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP548-1.p} Axiom for Abelian group theory, in division and identity, part 4






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,b)),\mbox{divide}(\mbox{divide}(b,c),a)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP040-4.p} In subgroups of order 2, inverse is an involution






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_is\_in\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(a')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }a\cdot d{=}c ) \ \Rightarrow \ d = b\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot b{=}c ) \ \Rightarrow \ d = a\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(a,b))\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$a\cdot \mbox{element\_in\_O}_2(a,b){=}b\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\neg\,\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{d\_in\_subgroup},\mbox{hypothesis})$

$b\cdot a'{=}c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{hypothesis})$

$a\cdot c{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{hypothesis})$

$( a' )' = a\qquad\mbox{cnf}(\mbox{prove\_inverse\_is\_self\_cancelling},\mbox{negated\_conjecture})$

\medskip\textbf{GRP176-1.p} General form of distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$c\cdot ( \mbox{least\_upper\_bound}(a,b)\cdot d )\ne \mbox{least\_upper\_bound}(c\cdot ( a\cdot d ),c\cdot ( b\cdot d ))\qquad\mbox{cnf}(\mbox{prove\_p}_{07},\mbox{negated\_conjecture})$

\medskip\textbf{GRP041-2.p} Reflexivity is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$\neg\,a{=}a\qquad\mbox{cnf}(\mbox{prove\_reflexivity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP503-1.p} Axiom for group theory, in double division and inverse, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c)'),\mbox{double\_divide}(b',\mbox{double\_divide}(d,\mbox{double\_divide}(a,d))')) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP094-1.p} Single axiom for Abelian group theory, in division and identity

This is a single axiom for Abelian group theory, in terms
 of division and identity




$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(x,y)),\mbox{divide}(\mbox{divide}(y,z),x)) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP522-1.p} Axiom for Abelian group theory, in division, part 2






$\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,\mbox{divide}(a,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP104-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y),z')',y)') = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP110-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y)',z)',\mbox{double\_divide}(x,z))' = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP004-1.p} Left inverse and identity =$>$ Right inverse exists

In a group with left inverses and left identity every element
 has a right inverse.




$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\neg\,a\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_inverse},\mbox{negated\_conjecture})$

\medskip\textbf{GRP403-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b\cdot ( b'\cdot b ) )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP326-1.p} An identity generated by HR, number 18694






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_7\cdot x_3 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = x_3\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_1\cdot \mbox{sk\_c}_6\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP747+1.p} Right alternative loop rings: the abelian group case






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( ( x_0\cdot x_1 )\cdot x_2 = x_0\cdot ( x_2\cdot x_1 )\mbox{ and }( x_0\cdot x_2 )\cdot x_1 = x_0\cdot ( x_1\cdot x_2 ) )\mbox{ or }( ( x_0\cdot x_1 )\cdot x_2 = ( x_0\cdot x_2 )\cdot x_1\mbox{ and }x_0\cdot ( x_1\cdot x_2 ) = x_0\cdot ( x_2\cdot x_1 ) ) )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$( a\cdot b )\cdot c = a\cdot ( b\cdot c )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP535-1.p} Axiom for Abelian group theory, in division, part 3






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),c)),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP596-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c,b)')')' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP128-4.003.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( x\cdot z_1{=}z_2\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( x\cdot z_1{=}z_2\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-9.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP187-1.p} Orthogonal elements commute






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,a'),\mbox{least\_upper\_bound}(b,b')) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p33}_1,\mbox{hypothesis})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_p}_{33},\mbox{negated\_conjecture})$

\medskip\textbf{GRP061-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( x\cdot ( y\cdot ( ( y'\cdot z )\cdot ( u\cdot ( x\cdot z ) )' ) ) )' = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP171-2.p} Positive elements form a semigroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p04c}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p04c}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a\cdot b)\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p04c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP480-1.p} Axiom for group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),\mbox{divide}(c,\mbox{divide}(b,d)))',d) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP172-1.p} Negative elements form a semigroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p04b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p04b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a\cdot b)\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p04b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP401-1.p} Prove distributivity implies nilpotent in cancellative semigroup

Assume a cancellative semigroup admits a commutator operation.
 Then the following three properties are equivalent:
 (1) commutator is associative;
 (2) commutator distributes over product;
 (3) the semigroup is nilpotent of class 2.
 This is a generalization of the corresponding theorem for group
 theory. The problem here is to prove (1) implies (2).




$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$a\cdot b = b\cdot ( a\cdot \mbox{commutator}(a,b) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$\mbox{commutator}(a\cdot b,c) = \mbox{commutator}(a,c)\cdot \mbox{commutator}(b,c)\qquad\mbox{cnf}(\mbox{commutator\_distributes\_over\_product},\mbox{axiom})$

$\mbox{commutator}(a,b)\cdot c\ne c\cdot \mbox{commutator}(a,b)\qquad\mbox{cnf}(\mbox{prove\_nilpotency},\mbox{negated\_conjecture})$

\medskip\textbf{GRP207-1.p} Single non-axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$u\cdot ( y\cdot ( ( ( z\cdot z' )\cdot ( u\cdot y )' )\cdot u ) )' = u\qquad\mbox{cnf}(\mbox{single\_non\_axiom},\mbox{axiom})$

$x\cdot ( y\cdot ( ( ( z\cdot z' )\cdot ( u\cdot y )' )\cdot x ) )'\ne u\qquad\mbox{cnf}(\mbox{try\_prove\_this\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{GRP186-3.p} Application of distributivity and group theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\ne a\cdot \mbox{least\_upper\_bound}(a',b)\qquad\mbox{cnf}(\mbox{prove\_p23x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP304-1.p} An identity generated by HR, number 05204






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_3 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = x_3 ) \ \Rightarrow \ x_4'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP699-1.p} Variety of power associative, WIP conjugacy closed loops - 2b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot a )\cdot ( a\cdot c ) = a\cdot ( ( ( b\cdot a )\cdot a )\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot ( b\cdot ( c\cdot b ) ) = ( a\cdot ( b\cdot ( b\cdot c ) ) )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot c\ne ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-2.004.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP050-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$z\cdot ( ( ( z\cdot y )'\cdot x )'\cdot ( y'\cdot ( y'\cdot y ) ) )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP759+1.p} A 4-element non-abelian group






$\forall a{:}\;a\cdot i(a) = e\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot e = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a,c{:}\;a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\mbox{op\_a}\cdot \mbox{op\_b}\ne \mbox{op\_b}\cdot \mbox{op\_a}\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall x{:}\;( x = c_1\mbox{ or }x = c_2\mbox{ or }x = c_3\mbox{ or }x = c_4 )\qquad\mbox{fof}(a,\mbox{axiom})$

$c_1\ne c_2\qquad\mbox{fof}(\mbox{c1\_not\_c}_2,\mbox{axiom})$

$c_1\ne c_3\qquad\mbox{fof}(\mbox{c1\_not\_c}_3,\mbox{axiom})$

$c_1\ne c_4\qquad\mbox{fof}(\mbox{c1\_not\_c}_4,\mbox{axiom})$

$c_2\ne c_3\qquad\mbox{fof}(\mbox{c2\_not\_c}_3,\mbox{axiom})$

$c_2\ne c_4\qquad\mbox{fof}(\mbox{c2\_not\_c}_4,\mbox{axiom})$

$c_3\ne c_4\qquad\mbox{fof}(\mbox{c3\_not\_c}_4,\mbox{axiom})$

\medskip\textbf{GRP416-1.p} Axiom for group theory, in product \& inverse, part 2






$( a\cdot ( ( ( b\cdot a )'\cdot ( b\cdot c' ) )'\cdot ( a'\cdot a )' )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP133-2.004.p} (a.b).(b.a) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-2.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-1.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(x,y))\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }x\cdot \mbox{element\_in\_O}_2(x,y) = y\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_in\_O}_2,\mbox{negated\_conjecture})$

$b\cdot a' = c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c = d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_in\_O}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP493-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,c),\mbox{double\_divide}(\mbox{identity},\mbox{identity})),\mbox{double\_divide}(a,c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP066-1.p} Single axiom for group theory, in division and identity

This is a single axiom for group theory, in terms of division
 and identity




$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),x),z)))),z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP175-2.p} Positivity is preserved under inner automorphisms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p06b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a'\cdot ( b\cdot a ))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p06b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP073-1.p} Single axiom for group theory, in division and inverse

This is a single axiom for group theory, in terms of division
 and inverse.




$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(x,y),z),\mbox{divide}(u,z))',\mbox{divide}(y,x)) = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP777+1.p} Napoleon's quasigroups: the centroid relation






$\forall b,a{:}\;a\setminus a\cdot b = b\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( a\setminus b ) = b\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;( a\cdot b )\cdot ( c\cdot d ) = ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall b,a{:}\;( ( a\cdot b )\cdot b )\cdot ( b\cdot ( b\cdot a ) ) = b\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{bigC}(a,b,c) = ( a\cdot b )\cdot ( c\cdot a )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$( a\cdot c )\cdot ( c\cdot b ) = a\cdot b\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_0{:}\;\mbox{bigC}(a,b,x_0) = \mbox{bigC}(c,c,x_0)\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP454-1.p} Axiom for group theory, in division and identity, part 1






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP443-1.p} Axiom for group theory, in product \& inverse, part 2






$( a\cdot ( b\cdot ( ( c\cdot c' )\cdot ( d\cdot ( a\cdot b ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP394-3.p} Group theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

\medskip\textbf{GRP663-1.p} Conjugacy closed with ab = 1 implies ba is in the nucleus - c






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d} = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( \mbox{op\_d}\cdot \mbox{op\_c} )\ne a\cdot ( b\cdot ( \mbox{op\_d}\cdot \mbox{op\_c} ) )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP040-3.p} In subgroups of order 2, inverse is an involution






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(a,b))\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$a\cdot \mbox{element\_in\_O}_2(a,b){=}b\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\neg\,\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{d\_in\_subgroup},\mbox{hypothesis})$

$b\cdot a'{=}c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{hypothesis})$

$a\cdot c{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{hypothesis})$

$( a' )' = a\qquad\mbox{cnf}(\mbox{prove\_inverse\_is\_self\_cancelling},\mbox{negated\_conjecture})$

\medskip\textbf{GRP745+1.p} Right alternative loop rings: the extra case






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( ( x_0\cdot x_1 )\cdot x_2 = x_0\cdot ( x_1\cdot x_2 )\mbox{ and }( x_0\cdot x_2 )\cdot x_1 = x_0\cdot ( x_2\cdot x_1 ) )\mbox{ or }( ( x_0\cdot x_1 )\cdot x_2 = x_0\cdot ( x_2\cdot x_1 )\mbox{ and }( x_0\cdot x_2 )\cdot x_1 = x_0\cdot ( x_1\cdot x_2 ) ) )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot a ) ) = ( ( a\cdot b )\cdot c )\cdot a\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP236-1.p} An identity generated by HR, number 00602






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_8' = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_5\cdot x_6 = \mbox{sk\_c}_7\mbox{ and }x_5' = x_6\mbox{ and }x_6\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_2 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_8 = x_2 ) \ \Rightarrow \ x_3'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{GRP024-5.p} Levi commutator problem.

In group theory, if the commutator [x,y] is associative,
 then x*[y,z] = [y,z]*x.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{commutator}(x,y) = x'\cdot ( y'\cdot ( x\cdot y ) )\qquad\mbox{cnf}(\mbox{name},\mbox{axiom})$

$\mbox{commutator}(\mbox{commutator}(x,y),z) = \mbox{commutator}(x,\mbox{commutator}(y,z))\qquad\mbox{cnf}(\mbox{associativity\_of\_commutator},\mbox{hypothesis})$

$a\cdot \mbox{commutator}(b,c)\ne \mbox{commutator}(b,c)\cdot a\qquad\mbox{cnf}(\mbox{prove\_center},\mbox{negated\_conjecture})$

\medskip\textbf{GRP310-1.p} An identity generated by HR, number 17399






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_3 = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_8 = x_3\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_2 = \mbox{sk\_c}_7\mbox{ and }x_5\cdot \mbox{sk\_c}_8 = x_2 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{GRP020-1.p} Inverse of X times X is the identity






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a'\cdot a\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_inverse\_X\_times\_X\_is\_id},\mbox{negated\_conjecture})$

\medskip\textbf{GRP447-1.p} Axiom for group theory, in division, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP096-1.p} Single axiom for Abelian group theory, in division and inverse

This is a single axiom for Abelian group theory, in terms
 of division and inverse




$\mbox{divide}(\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(x,z))'),z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP551-1.p} Axiom for Abelian group theory, in division and identity, part 3






$\mbox{divide}(\mbox{divide}(\mbox{identity},a),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,a),c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP555-1.p} Axiom for Abelian group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(a,c))'),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP052-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$z\cdot ( ( ( y'\cdot y )\cdot ( ( z\cdot y' )'\cdot x )' )\cdot y )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP284-1.p} An identity generated by HR, number 02999






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP202-1.p} In Loops, Moufang-3 =$>$ Moufang-1.






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot \mbox{left\_division}(x,y) = y\qquad\mbox{cnf}(\mbox{multiply\_left\_division},\mbox{axiom})$

$\mbox{left\_division}(x,x\cdot y) = y\qquad\mbox{cnf}(\mbox{left\_division\_multiply},\mbox{axiom})$

$\mbox{right\_division}(x,y)\cdot y = x\qquad\mbox{cnf}(\mbox{multiply\_right\_division},\mbox{axiom})$

$\mbox{right\_division}(x\cdot y,y) = x\qquad\mbox{cnf}(\mbox{right\_division\_multiply},\mbox{axiom})$

$x\cdot \mbox{right\_inverse}(x) = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( ( x\cdot y )\cdot x )\cdot z = x\cdot ( y\cdot ( x\cdot z ) )\qquad\mbox{cnf}(\mbox{moufang}_3,\mbox{axiom})$

$( a\cdot ( b\cdot c ) )\cdot a\ne ( a\cdot b )\cdot ( c\cdot a )\qquad\mbox{cnf}(\mbox{prove\_moufang}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-3.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$( a\cdot b{=}c\mbox{ and }a\cdot d{=}c ) \ \Rightarrow \ d = b\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot b{=}c ) \ \Rightarrow \ d = a\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( a' )' = a\qquad\mbox{cnf}(\mbox{inverse\_is\_self\_cancelling},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_is\_in\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(a')\qquad\mbox{cnf}(\mbox{subgroup\_member\_inverse\_are\_in\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(a,b))\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$a\cdot \mbox{element\_in\_O}_2(a,b){=}b\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{negated\_conjecture})$

$b\cdot a'{=}c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP421-1.p} Axiom for group theory, in product \& inverse, part 1






$( ( a\cdot ( b'\cdot ( c'\cdot ( c'\cdot c )' ) )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP613-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b')',c)',\mbox{double\_divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP781-1.p} Distributivity of commutator in cancellative semigroups






$m(x,m(y,z)) = m(m(x,y),z)\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

$m(x,z) = m(y,z) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{cancellation},\mbox{axiom})$

$m(z,x) = m(z,y) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{cancellation}_{001},\mbox{axiom})$

$m(y,m(x,c(x,y))) = m(x,y)\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$m(x,m(y,m(z,m(y,x)))) = m(y,m(x,m(z,m(x,y))))\qquad\mbox{cnf}(\mbox{assumption},\mbox{axiom})$

$c(m(x,y),z)\ne m(c(x,z),c(y,z))\qquad\mbox{cnf}(\mbox{distributivity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP047-2.p} Multiply substitution 2 is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$\neg\,c\cdot a{=}c\cdot b\qquad\mbox{cnf}(\mbox{prove\_multiply\_substitution}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP593-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c,b)')')' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP199-1.p} Nilpotent CS satisfy the quotient condition.

Nilpotent cancellative semigroups satisfy the quotient condition.




$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$x\cdot ( y\cdot ( z\cdot ( y\cdot x ) ) ) = y\cdot ( x\cdot ( z\cdot ( x\cdot y ) ) )\qquad\mbox{cnf}(\mbox{nilpotency},\mbox{hypothesis})$

$b\cdot b_0 = a\cdot a_0\qquad\mbox{cnf}(\mbox{prove\_quotient}_1,\mbox{negated\_conjecture})$

$d\cdot b_0 = c\cdot a_0\qquad\mbox{cnf}(\mbox{prove\_quotient}_2,\mbox{negated\_conjecture})$

$b\cdot d_0 = a\cdot c_0\qquad\mbox{cnf}(\mbox{prove\_quotient}_3,\mbox{negated\_conjecture})$

$d\cdot d_0\ne c\cdot c_0\qquad\mbox{cnf}(\mbox{prove\_quotient}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP730-1.p} Bruck loops that are centrally nilpotent - 2nd easy part c

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - 2nd easy part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot b = ( b\cdot a )\cdot \mbox{op\_k}(a,b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e) = 1\qquad\mbox{cnf}(c_{21},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{asoc}(c,d,e)) = 1\qquad\mbox{cnf}(c_{22},\mbox{axiom})$

$\mbox{asoc}(\mbox{op\_k}(a,b),c,d)\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP145-1.p} Prove greatest lower-bound axiom using the GLB transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b),b)\ne \mbox{greatest\_lower\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_glb3b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP434-1.p} Axiom for group theory, in product \& inverse, part 2






$( ( ( ( ( a\cdot b )\cdot c )'\cdot a )\cdot b )\cdot ( d\cdot d' ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP149-1.p} Prove least upper-bound axiom using a transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = a\qquad\mbox{cnf}(\mbox{ax\_lub1d}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,c) = b\qquad\mbox{cnf}(\mbox{ax\_lub1d}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{least\_upper\_bound}(a,b),c)\ne c\qquad\mbox{cnf}(\mbox{prove\_ax\_lub1d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP422-1.p} Axiom for group theory, in product \& inverse, part 2






$( ( a\cdot ( b'\cdot ( c'\cdot ( c'\cdot c )' ) )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP428-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( ( ( b'\cdot ( a'\cdot c ) )'\cdot d )\cdot ( b\cdot d )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP394+3.p} Group theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/GRP004+0.ax'})$

\medskip\textbf{GRP472-1.p} Axiom for group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b)',\mbox{divide}(\mbox{divide}(c,d),a)),\mbox{divide}(d,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP189-1.p} Consequence of lattice theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne b\qquad\mbox{cnf}(\mbox{prove\_p38b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP497-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP592-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,c')')',b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP101-1.p} Single axiom for Abelian group theory, in double div and id

This is a single axiom for Abelian group theory, in terms
 of double division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(y,\mbox{double\_divide}(z,x)),\mbox{double\_divide}(z,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP448-1.p} Axiom for group theory, in division, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP197-1.p} In cancellative semigroups, xxxyy=yxxyx $\to$ bbaaa = abaab.






$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$x\cdot ( x\cdot ( x\cdot ( y\cdot y ) ) ) = y\cdot ( x\cdot ( x\cdot ( y\cdot x ) ) )\qquad\mbox{cnf}(\mbox{condition},\mbox{hypothesis})$

$b\cdot ( b\cdot ( a\cdot ( a\cdot a ) ) )\ne a\cdot ( b\cdot ( a\cdot ( a\cdot b ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{GRP135-1.005.p} ((b.a).b).b) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP003-2.p} The left identity is also a right identity






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$\neg\,a\cdot \mbox{identity}{=}a\qquad\mbox{cnf}(\mbox{prove\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP580-1.p} Axiom for Abelian group theory, in double div and id, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,a),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP508-1.p} Axiom for Abelian group theory, in product and inverse, part 4






$( ( ( a\cdot b )'\cdot ( b\cdot a ) )'\cdot ( ( c\cdot d )'\cdot ( c\cdot ( ( e\cdot f' )\cdot d' )' ) ) )'\cdot f = e\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP156-1.p} Prove monotonicity axiom using a transformation

This problem proves the original monotonicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_mono1c}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a\cdot c,b\cdot c)\ne a\cdot c\qquad\mbox{cnf}(\mbox{prove\_ax\_mono1c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP569-1.p} Axiom for Abelian group theory, in double div and id, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP184-3.p} Orthogonal elements commute and form a subgroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\cdot \mbox{least\_upper\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p21x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-9.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,x,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,y,x)\qquad\mbox{cnf}(\mbox{qg2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP471-1.p} Axiom for group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,d)))',\mbox{divide}(\mbox{divide}(d,c),a)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP315-1.p} An identity generated by HR, number 18383






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{GRP716-1.p} Strongly right alternative rings 2a

If a has a 2-sided inverse, then R(a$\wedge$-1) = R(a)$\wedge$-1 and
 L(a)$\wedge$-1 = R(a)L(a$\wedge$-1)R(a$\wedge$-1).




$( a + b ) + c = a + ( b + c )\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a + b = b + a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a + \mbox{op}_0 = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$a + -a = \mbox{op}_0\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot ( b + c ) = a\cdot b + a\cdot c\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot b ) = ( a\cdot b )\cdot b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_a}\cdot \mbox{op\_b} = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_b}\cdot \mbox{op\_a} = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_a}\cdot ( ( \mbox{op\_b}\cdot ( a\cdot \mbox{op\_b} ) )\cdot \mbox{op\_a} )\ne a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP398-1.p} Subgroup axioms for the GRP003 group theory axioms






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-1.ax'})$

\medskip\textbf{GRP772-1.p} Buchsteiner loop lemma 9






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$\mbox{quotient}(j(a),a) = j(a)\cdot i(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$( ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot b )\cdot c = ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{20},\mbox{axiom})$

$i(a\cdot b)\cdot i(i(a)) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{21},\mbox{axiom})$

$j(j(a))\cdot j(b\cdot a) = j(b)\qquad\mbox{cnf}(\mbox{sos}_{22},\mbox{axiom})$

$( a\cdot ( b\cdot c ) )\cdot a(a,b,c) = ( a\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{23},\mbox{axiom})$

$( a\cdot b )\cdot c(b,a) = b\cdot a\qquad\mbox{cnf}(\mbox{sos}_{24},\mbox{axiom})$

$c(a,b)\cdot ( c\cdot d ) = ( c(a,b)\cdot c )\cdot d\qquad\mbox{cnf}(\mbox{sos}_{25},\mbox{axiom})$

$( a\cdot b )\cdot c(c,d) = a\cdot ( b\cdot c(c,d) )\qquad\mbox{cnf}(\mbox{sos}_{26},\mbox{axiom})$

$a(a,b,c)\cdot ( d\cdot e ) = ( a(a,b,c)\cdot d )\cdot e\qquad\mbox{cnf}(\mbox{sos}_{27},\mbox{axiom})$

$( a\cdot b )\cdot a(c,d,e) = a\cdot ( b\cdot a(c,d,e) )\qquad\mbox{cnf}(\mbox{sos}_{28},\mbox{axiom})$

$a(a,b,c)\cdot ( c\setminus a(c,a,b)\cdot c ) = 1\qquad\mbox{cnf}(\mbox{sos}_{29},\mbox{axiom})$

$a(a,i(b),c) = a(a,j(b),c)\qquad\mbox{cnf}(\mbox{sos}_{30},\mbox{axiom})$

$a(i(a),b,c) = a(j(a),b,c)\qquad\mbox{cnf}(\mbox{sos}_{31},\mbox{axiom})$

$a(j(a),b,c) = a(b,c,a)\qquad\mbox{cnf}(\mbox{sos}_{32},\mbox{axiom})$

$a(x_0,x_1,x_1)\ne a(x_1,x_1,x_0)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP550-1.p} Axiom for Abelian group theory, in division and identity, part 2






$\mbox{divide}(\mbox{divide}(\mbox{identity},a),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,a),c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP033-4.p} In subgroups, the identity is the group identity






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(j(a))\qquad\mbox{cnf}(\mbox{subgr2\_clause}_1,\mbox{hypothesis})$

$( j(a)\cdot a{=}j(a)\mbox{ and }a\cdot j(a){=}j(a) ) \ \Rightarrow \ \neg\,\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{prove\_subgr}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP664+1.p} Conjugacy closed with ab = 1 implies ba is in the nucleus - a






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall x_0,x_1,x_2,x_3{:}\;( x_0\cdot x_1 = 1 \ \Rightarrow \ ( ( x_1\cdot x_0 )\cdot ( x_2\cdot x_3 ) = ( ( x_1\cdot x_0 )\cdot x_2 )\cdot x_3\mbox{ and }( x_2\cdot ( x_1\cdot x_0 ) )\cdot x_3 = x_2\cdot ( ( x_1\cdot x_0 )\cdot x_3 )\mbox{ and }( x_2\cdot x_3 )\cdot ( x_1\cdot x_0 ) = x_2\cdot ( x_3\cdot ( x_1\cdot x_0 ) ) ) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP018-1.p} X times identity is X






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot \mbox{identity}\ne a\qquad\mbox{cnf}(\mbox{prove\_X\_times\_id\_is\_X},\mbox{negated\_conjecture})$

\medskip\textbf{GRP086-1.p} Single axiom for Abelian group theory, in product and inverse

This is a single axiom for Abelian group theory, in terms
 of product and inverse.




$x\cdot ( ( y\cdot z )\cdot ( x\cdot z )' ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP766-1.p} Buchsteiner loop lemma 3






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$l(a,b,c) = a\cdot b\setminus a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$l(a,a,b\cdot c) = l(a,a,b)\cdot l(a,a,c)\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$\mbox{eta}(x_0)\cdot ( x_1\cdot x_2 )\ne ( \mbox{eta}(x_0)\cdot x_1 )\cdot x_2\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP675-1.p} In CC-loops, associators are in the center of the nucleus - 1a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{asoc}(a,b,c)\cdot ( d\cdot e )\ne ( \mbox{asoc}(a,b,c)\cdot d )\cdot e\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP706-1.p} Every F-quasigroup is isotopic to a Moufang loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( \mbox{ld}(a,a)\cdot c )\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot \mbox{rd}(c,c) )\cdot ( b\cdot c )\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$f(a,b) = \mbox{rd}(a,\mbox{op\_c})\cdot \mbox{ld}(\mbox{op\_c},b)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$f(a,f(b,f(a,c)))\ne f(f(f(a,b),a),c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP775+1.p} Equivalent definition for Green's relation D






$\forall c,b,a{:}\;( a\cdot b )\cdot c = a\cdot ( b\cdot c )\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall x_0,x_1{:}\;( l(x_0,x_1)\iff ( x_0\cdot x_1 = x_0\mbox{ and }x_1\cdot x_0 = x_1 ) )\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall x_2,x_3{:}\;( r(x_2,x_3)\iff ( x_2\cdot x_3 = x_3\mbox{ and }x_3\cdot x_2 = x_2 ) )\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_4,x_5{:}\;( d(x_4,x_5)\iff \exists x_6{:}\;( r(x_4,x_6)\mbox{ and }l(x_6,x_5) ) )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_7,x_8{:}\;( d(x_7,x_8)\iff ( x_7\cdot ( x_8\cdot x_7 ) = x_7\mbox{ and }x_8\cdot ( x_7\cdot x_8 ) = x_8 ) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP666-3.p} Inverse property A-loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) )\ne ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP044-2.p} Product subsitution 1 is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$a\cdot c{=}\mbox{result}\qquad\mbox{cnf}(\mbox{product\_with\_a},\mbox{hypothesis})$

$\neg\,b\cdot c{=}\mbox{result}\qquad\mbox{cnf}(\mbox{prove\_product\_with\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP760+1.p} A group that must be infinite

A group containing an element of order 2 and having square roots
 must be infinite.




$\forall a{:}\;a\cdot i(a) = e\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot e = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a,c{:}\;a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$a\cdot a = e\mbox{ and }a\ne e\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;b\cdot b = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

\medskip\textbf{GRP122-1.p} Derive associativity from a single axiom for groups order 4






$y\cdot ( ( y\cdot ( ( y\cdot y )\cdot ( x\cdot z ) ) )\cdot ( z\cdot ( z\cdot z ) ) ) = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$\mbox{identity}\cdot \mbox{identity} = \mbox{identity}\qquad\mbox{cnf}(\mbox{single\_axiom}_2,\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP393-2.p} Semigroups axioms






$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

\medskip\textbf{GRP396+1.p} Group theory (Named Semigroups) axioms






$\mbox{include}(\mbox{'Axioms/GRP007+0.ax'})$

\medskip\textbf{GRP773-1.p} Buchsteiner loop problem






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$\mbox{ld}(a,( a\cdot b )\cdot c) = \mbox{rd}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$( ( a\cdot a )\cdot b )\cdot c\ne ( a\cdot a )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-2.005.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP537-1.p} Axiom for Abelian group theory, in division, part 1






$\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(\mbox{divide}(a,c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP433-1.p} Axiom for group theory, in product \& inverse, part 1






$( ( ( ( ( a\cdot b )\cdot c )'\cdot a )\cdot b )\cdot ( d\cdot d' ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP128-4.004.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( x\cdot z_1{=}z_2\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( x\cdot z_1{=}z_2\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP465-1.p} Axiom for group theory, in division and identity, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{identity},a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP411-1.p} Axiom for group theory, in product \& inverse, part 3






$( ( a\cdot ( b\cdot c )' )'\cdot ( a\cdot c' ) )\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-4.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-1.ax'})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_is\_in\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(a,b))\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$a\cdot \mbox{element\_in\_O}_2(a,b){=}b\mbox{ or }\mbox{subgroup\_member}(b)\mbox{ or }\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{negated\_conjecture})$

$b\cdot a'{=}c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-7.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,x,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,y,x)\qquad\mbox{cnf}(\mbox{qg2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP512-1.p} Axiom for Abelian group theory, in product and inverse, part 4






$( ( a\cdot b )\cdot c )\cdot ( a\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-7.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-1.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{inverse\_inverse},\mbox{axiom})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{inverse\_of\_identity},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_in\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(x,y))\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }x\cdot \mbox{element\_in\_O}_2(x,y) = y\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_in\_O}_2,\mbox{negated\_conjecture})$

$b\cdot a' = c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c = d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_in\_O}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP583-1.p} Axiom for Abelian group theory, in double div and id, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},b),\mbox{double\_divide}(c,\mbox{double\_divide}(b,a)))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP752-1.p} A new basis for trimedial quasigroups: part 1b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$( a\cdot ( a\cdot a ) )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( ( a\cdot a )\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot c ) = ( a\cdot c )\cdot ( b\cdot c )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$( a\cdot b )\cdot c\ne ( a\cdot c )\cdot ( b\cdot \mbox{ld}(c,c) )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP461-1.p} Axiom for group theory, in division and identity, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{identity},b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP568-1.p} Axiom for Abelian group theory, in double div and id, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(\mbox{identity},c))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP133-1.003.p} (a.b).(b.a) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-4.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP424-1.p} Axiom for group theory, in product \& inverse, part 1






$( ( a\cdot ( b'\cdot c )' )'\cdot ( a\cdot c' ) )'\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP405-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b\cdot ( b'\cdot b ) )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP560-1.p} Axiom for Abelian group theory, in division and inverse, part 4






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(b,c),\mbox{divide}(a,c))') = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP479-1.p} Axiom for group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),\mbox{divide}(c,\mbox{divide}(b,d)))',d) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-4.004.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot z_2{=}y\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg4}_1,\mbox{negated\_conjecture})$

$( z_1\cdot z_2{=}y\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ y\cdot x{=}z_2\qquad\mbox{cnf}(\mbox{qg4}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP409-1.p} Axiom for group theory, in product \& inverse, part 1






$( ( a\cdot ( b\cdot c )' )'\cdot ( a\cdot c' ) )\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP468-1.p} Axiom for group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(b,\mbox{divide}(\mbox{divide}(c,\mbox{divide}(d,b)),d'))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-1.005.p} a.(b.a) = (b.a).b

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP001-4.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{squareness},\mbox{hypothesis})$

$a\cdot b = c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{hypothesis})$

$b\cdot a\ne c\qquad\mbox{cnf}(\mbox{prove\_b\_times\_a\_is\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP087-1.p} Single axiom for Abelian group theory, in product and inverse

This is a single axiom for Abelian group theory, in terms
 of product and inverse.




$x\cdot ( ( ( x\cdot y )'\cdot z )\cdot y ) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP183-1.p} Orthogonal elements form a subgroup with orthogonal parts

For each X {Y: X orth Y} is a subgroup. Moreover, pp(a)
 is orthogonal to np(a).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,\mbox{identity}),\mbox{greatest\_lower\_bound}(a,\mbox{identity})')\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{GRP215-1.p} An identity generated by HR, number 00396






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_2 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = x_2 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP297-1.p} An identity generated by HR, number 03730






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_3' = x_4 ) \ \Rightarrow \ x_4\cdot \mbox{sk\_c}_7\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP767-1.p} Buchsteiner loop lemma 4






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$l(a,b,c) = a\cdot b\setminus a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$l(a,a,b\cdot c) = l(a,a,b)\cdot l(a,a,c)\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{20},\mbox{axiom})$

$j(j(x_0))\cdot j(x_1\cdot x_0)\ne j(x_1)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP119-1.p} Derive order 4 from a single axiom for groups order 4






$y\cdot ( ( y\cdot ( ( y\cdot y )\cdot ( x\cdot z ) ) )\cdot ( z\cdot ( z\cdot z ) ) ) = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$\mbox{identity}\cdot \mbox{identity} = \mbox{identity}\qquad\mbox{cnf}(\mbox{single\_axiom}_2,\mbox{axiom})$

$a\cdot ( a\cdot ( a\cdot a ) )\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_order}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP432-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( b\cdot ( ( ( c\cdot c' )\cdot ( d\cdot b )' )\cdot a ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP533-1.p} Axiom for Abelian group theory, in division, part 1






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),c)),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP406-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b'\cdot ( b'\cdot b ) ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP098-1.p} Single axiom for Abelian group theory, in division and inverse

This is a single axiom for Abelian group theory, in terms
 of division and inverse




$\mbox{divide}(\mbox{divide}(\mbox{divide}(x,y'),z),\mbox{divide}(x,z)) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP430-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( b\cdot ( ( ( c\cdot c' )\cdot ( d\cdot b )' )\cdot a ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP113-1.p} Lemma for proving all groups of order 4 are cyclic

Prove that any group of order 4 must satisfy one of the
 following relations, where the elements of the group are
 a, b, c, and the identity.
 1) the square of every element is the identity.
 2) the square of a is b, the cube of a is c, and the fourth
 power of a is the identity.
 3) the square of b is c, the cube of b is a, and the fourth
 power of b is the identity.
 4) the square of c is a, the cube of c is b, and the fourth




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x = a\mbox{ or }x = b\mbox{ or }x = c\mbox{ or }x = \mbox{identity}\qquad\mbox{cnf}(\mbox{all\_of\_group}_1,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{a\_not\_identity},\mbox{hypothesis})$

$b\ne c\qquad\mbox{cnf}(\mbox{b\_not\_c},\mbox{hypothesis})$

$b\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{b\_not\_identity},\mbox{hypothesis})$

$c\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{c\_not\_identity},\mbox{hypothesis})$

$( a\cdot a = \mbox{identity}\mbox{ and }b\cdot b = \mbox{identity} ) \ \Rightarrow \ c\cdot c\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{square\_identity},\mbox{negated\_conjecture})$

$( a\cdot a = b\mbox{ and }a\cdot ( a\cdot a ) = c ) \ \Rightarrow \ a\cdot ( a\cdot ( a\cdot a ) )\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{condition\_a},\mbox{negated\_conjecture})$

$( b\cdot b = c\mbox{ and }b\cdot ( b\cdot b ) = a ) \ \Rightarrow \ b\cdot ( b\cdot ( b\cdot b ) )\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{condition\_b},\mbox{negated\_conjecture})$

$( c\cdot c = a\mbox{ and }c\cdot ( c\cdot c ) = b ) \ \Rightarrow \ c\cdot ( c\cdot ( c\cdot c ) )\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{condition\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP205-1.p} In Loops, Moufang-3 =$>$ Moufang-4.






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot \mbox{left\_division}(x,y) = y\qquad\mbox{cnf}(\mbox{multiply\_left\_division},\mbox{axiom})$

$\mbox{left\_division}(x,x\cdot y) = y\qquad\mbox{cnf}(\mbox{left\_division\_multiply},\mbox{axiom})$

$\mbox{right\_division}(x,y)\cdot y = x\qquad\mbox{cnf}(\mbox{multiply\_right\_division},\mbox{axiom})$

$\mbox{right\_division}(x\cdot y,y) = x\qquad\mbox{cnf}(\mbox{right\_division\_multiply},\mbox{axiom})$

$x\cdot \mbox{right\_inverse}(x) = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( ( x\cdot y )\cdot x )\cdot z = x\cdot ( y\cdot ( x\cdot z ) )\qquad\mbox{cnf}(\mbox{moufang}_3,\mbox{axiom})$

$x\cdot ( ( y\cdot z )\cdot x )\ne ( x\cdot y )\cdot ( z\cdot x )\qquad\mbox{cnf}(\mbox{prove\_moufang}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP478-1.p} Axiom for group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),\mbox{divide}(c,\mbox{divide}(b,d)))',d) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP501-1.p} Axiom for group theory, in double division and inverse, part 3






$\mbox{double\_divide}(a',\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c))',\mbox{double\_divide}(d,\mbox{double\_divide}(b,d)))') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP737-1.p} Nonmedial left distributive left 2-symmetric quasigroup






$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$a\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot d )\ne ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP135-2.002.p} ((b.a).b).b) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP189-2.p} Consequence of lattice theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p38b}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p38b}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p38b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne b\qquad\mbox{cnf}(\mbox{prove\_p38b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP178-1.p} A consequence of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p09a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p09a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},c) = c\qquad\mbox{cnf}(\mbox{p09a}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p09a}_4,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b\cdot c)\ne \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{prove\_p09a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP077-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{identity}),\mbox{double\_divide}(y,z))),y),\mbox{identity})) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP088-1.p} Single axiom for Abelian group theory, in division

This is a single axiom for Abelian group theory, in terms
 of division




$\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(z,\mbox{divide}(x,y)))) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-4.004.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot z_2{=}x\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_1\cdot z_2{=}x\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ y\cdot x{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP609-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b)',c)',\mbox{double\_divide}(a,c))' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP764-1.p} Buchsteiner loop lemma 1






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$x_0\cdot ( \mbox{eta}(x_0)\cdot x_1 )\ne j(j(x_0))\cdot x_1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP011-4.p} Left cancellation






$( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x'\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$b\cdot c = d\cdot c\qquad\mbox{cnf}(\mbox{product\_equality},\mbox{hypothesis})$

$b\ne d\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation},\mbox{negated\_conjecture})$

\medskip\textbf{GRP179-1.p} For converting between GLB and LUB






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b)'\ne \mbox{greatest\_lower\_bound}(a',b')\qquad\mbox{cnf}(\mbox{prove\_p}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{GRP776+1.p} A homomorphic mapping between two groups 

A mapping between two groups that respects multiplication is a 
 homomorphism.




$\forall b,a{:}\;( ( g(a)\mbox{ and }g(b) ) \ \Rightarrow \ g(a\cdot b) )\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ g(a^{-1}) )\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$g(\mbox{eh})\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall c,b,a{:}\;( ( g(a)\mbox{ and }g(b)\mbox{ and }g(c) ) \ \Rightarrow \ ( a\cdot b )\cdot c = a\cdot ( b\cdot c ) )\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ \mbox{eh}\cdot a = a )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ a\cdot \mbox{eh} = a )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ a\cdot a^{-1} = \mbox{eh} )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ a^{-1}\cdot a = \mbox{eh} )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall b,a{:}\;( ( h(a)\mbox{ and }h(b) ) \ \Rightarrow \ h(a + b) )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall b,a{:}\;( h(a) \ \Rightarrow \ h(\mbox{opp}(b)) )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$h(\mbox{eg})\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall c,b,a{:}\;( ( h(a)\mbox{ and }h(b)\mbox{ and }h(c) ) \ \Rightarrow \ ( a + b ) + c = a + ( b + c ) )\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;( h(a) \ \Rightarrow \ \mbox{eg} + a = a )\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall a{:}\;( h(a) \ \Rightarrow \ a + \mbox{eg} = a )\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$\forall a{:}\;( h(a) \ \Rightarrow \ a + \mbox{opp}(a) = \mbox{eg} )\qquad\mbox{fof}(\mbox{sos}_{15},\mbox{axiom})$

$\forall a{:}\;( h(a) \ \Rightarrow \ \mbox{opp}(a) + a = \mbox{eg} )\qquad\mbox{fof}(\mbox{sos}_{16},\mbox{axiom})$

$\forall a{:}\;( g(a) \ \Rightarrow \ h(f(a)) )\qquad\mbox{fof}(\mbox{sos}_{17},\mbox{axiom})$

$\forall b,a{:}\;f(a\cdot b) = f(a) + f(b)\qquad\mbox{fof}(\mbox{sos}_{18},\mbox{axiom})$

$\forall x_0{:}\;( f(\mbox{eh}) = \mbox{eg}\mbox{ and }( \neg\,g(x_0)\mbox{ or }f(x_0^{-1}) = \mbox{opp}(f(x_0)) ) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP532-1.p} Axiom for Abelian group theory, in division, part 4






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,c)),\mbox{divide}(a,b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP601-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{double\_divide}(a,c))')',c)' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP055-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( ( z\cdot ( x'\cdot ( y'\cdot ( y'\cdot y )' ) )' )'\cdot ( z\cdot y ) )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP170-1.p} General form of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p03a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(c,d) = d\qquad\mbox{cnf}(\mbox{p03a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a\cdot c,b\cdot d)\ne b\cdot d\qquad\mbox{cnf}(\mbox{prove\_p03a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP598-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c',b)')') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP462-1.p} Axiom for group theory, in division and identity, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{identity},b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP526-1.p} Axiom for Abelian group theory, in division, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(c,b))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP660+1.p} Bol-Moufang identity 3 implies a unit element






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot a = a\cdot ( b\cdot ( c\cdot a ) )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP131-1.005.p} (3,2,1) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP078-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},x),\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y),\mbox{identity}),\mbox{double\_divide}(z,y)))) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP166-4.p} Multiplication with a positive element increases a value






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,\mbox{identity}) = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat3b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,\mbox{identity}) = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat3b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b\cdot a)\ne a\qquad\mbox{cnf}(\mbox{prove\_lat3b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP150-1.p} Prove least upper-bound axiom using the LUB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{least\_upper\_bound}(a,b))\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_lub2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP594-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c,b)')')' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP190-2.p} Something useful for estimations






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{p39c}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a',b')\ne a'\qquad\mbox{cnf}(\mbox{prove\_p39c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP032-3.p} In subgroups, there is an identity






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP492-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{identity}),\mbox{double\_divide}(c,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP751-1.p} A new basis for trimedial quasigroups: part 1a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$( a\cdot ( a\cdot a ) )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( ( a\cdot a )\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot c ) = ( a\cdot c )\cdot ( b\cdot c )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$a\cdot ( b\cdot c )\ne ( \mbox{rd}(a,a)\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP128-1.004.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP118-1.p} Derive associativity from a single axiom for groups order 3






$x\cdot ( ( x\cdot ( ( x\cdot y )\cdot z ) )\cdot ( \mbox{identity}\cdot ( z\cdot z ) ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP445-1.p} Axiom for group theory, in division, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP666+6.p} Inverse property A-loops are Moufang






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall b,a{:}\;i(a)\cdot ( a\cdot b ) = b\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b )\cdot i(b) = a\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;x_2\cdot ( x_0\cdot ( x_2\cdot x_1 ) ) = ( ( x_2\cdot x_0 )\cdot x_2 )\cdot x_1\mbox{ or }\forall x_3,x_4,x_5{:}\;x_3\cdot ( x_5\cdot ( x_4\cdot x_5 ) ) = ( ( x_3\cdot x_5 )\cdot x_4 )\cdot x_5\mbox{ or }\forall x_6,x_7,x_8{:}\;( x_8\cdot x_6 )\cdot ( x_7\cdot x_8 ) = ( x_8\cdot ( x_6\cdot x_7 ) )\cdot x_8\mbox{ or }\forall x_9,x_{10},x_{11}{:}\;( x_{11}\cdot x_9 )\cdot ( x_{10}\cdot x_{11} ) = x_{11}\cdot ( ( x_9\cdot x_{10} )\cdot x_{11} )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP029-2.p} In semigroups, left id and inverse =$>$ right id exists

If there are a left identity and left inverse, then there
 is a right identity element.




$x{=}x\qquad\mbox{cnf}(\mbox{reflexivity},\mbox{axiom})$

$x{=}y \ \Rightarrow \ y{=}x\qquad\mbox{cnf}(\mbox{symmetry},\mbox{axiom})$

$( x{=}y\mbox{ and }y{=}z ) \ \Rightarrow \ x{=}z\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$x{=}y \ \Rightarrow \ x\cdot w{=}y\cdot w\qquad\mbox{cnf}(\mbox{multiply\_substitution}_1,\mbox{axiom})$

$x{=}y \ \Rightarrow \ w\cdot x{=}w\cdot y\qquad\mbox{cnf}(\mbox{multiply\_substitution}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }x\cdot w{=}z ) \ \Rightarrow \ y\cdot w{=}z\qquad\mbox{cnf}(\mbox{product\_substitution}_1,\mbox{axiom})$

$( x{=}y\mbox{ and }w\cdot x{=}z ) \ \Rightarrow \ w\cdot y{=}z\qquad\mbox{cnf}(\mbox{product\_substitution}_2,\mbox{axiom})$

$( x{=}y\mbox{ and }w\cdot z{=}x ) \ \Rightarrow \ w\cdot z{=}y\qquad\mbox{cnf}(\mbox{product\_substitution}_3,\mbox{axiom})$

$x{=}y \ \Rightarrow \ x'{=}y'\qquad\mbox{cnf}(\mbox{inverse\_substitution},\mbox{axiom})$

$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z{=}w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\mbox{identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$a'\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\neg\,\mbox{not\_right\_identity}(a)\cdot a{=}\mbox{not\_right\_identity}(a)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-4.005.p} a.(b.a) = (b.a).a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y\mbox{ or }e_5\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y\mbox{ or }x\cdot e_5{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot y{=}z_2\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ y\cdot x{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_1\cdot y{=}z_2\mbox{ and }y\cdot x{=}z_1 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP170-4.p} General form of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p03d}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(c,d) = c\qquad\mbox{cnf}(\mbox{p03d}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a\cdot c,b\cdot d)\ne b\cdot d\qquad\mbox{cnf}(\mbox{prove\_p03d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP046-2.p} Multiply substitution 1 is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$\neg\,a\cdot c{=}b\cdot c\qquad\mbox{cnf}(\mbox{prove\_multiply\_substitution}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP498-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP168-1.p} Inner group automorphisms are order preserving






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p01a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(c'\cdot ( a\cdot c ),c'\cdot ( b\cdot c ))\ne c'\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{prove\_p01a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP019-1.p} Identity times X is X






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{identity}\cdot a\ne a\qquad\mbox{cnf}(\mbox{prove\_id\_times\_X\_is\_X},\mbox{negated\_conjecture})$

\medskip\textbf{GRP392-1.p} Monoid axioms






$\mbox{include}(\mbox{'Axioms/GRP001-0.ax'})$

\medskip\textbf{GRP079-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},x),\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(y,z),\mbox{double\_divide}(\mbox{identity},\mbox{identity})),\mbox{double\_divide}(x,z))) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP175-1.p} Positivity is preserved under inner automorphisms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p06a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},a'\cdot ( b\cdot a ))\ne a'\cdot ( b\cdot a )\qquad\mbox{cnf}(\mbox{prove\_p06a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP704+1.p} In C-loops the nucleus is normal - c






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_c}\cdot ( a\cdot b ) = ( \mbox{op\_c}\cdot a )\cdot b\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( b\cdot \mbox{op\_c} ) = ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( \mbox{op\_c}\cdot b ) = ( a\cdot \mbox{op\_c} )\cdot b\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall a{:}\;\mbox{op\_d} = \mbox{ld}(a,\mbox{op\_c}\cdot a)\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_e} = ( \mbox{rd}(\mbox{op\_c},a\cdot b)\cdot b )\cdot a\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_f} = a\cdot ( b\cdot \mbox{ld}(a\cdot b,\mbox{op\_c}) )\qquad\mbox{fof}(f_{13},\mbox{axiom})$

$\forall x_4,x_5{:}\;( \mbox{op\_f}\cdot ( x_4\cdot x_5 ) = ( \mbox{op\_f}\cdot x_4 )\cdot x_5\mbox{ and }x_4\cdot ( x_5\cdot \mbox{op\_f} ) = ( x_4\cdot x_5 )\cdot \mbox{op\_f}\mbox{ and }x_4\cdot ( \mbox{op\_f}\cdot x_5 ) = ( x_4\cdot \mbox{op\_f} )\cdot x_5 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP698-1.p} Variety of power associative, WIP conjugacy closed loops - 2a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot a )\cdot ( a\cdot c ) = a\cdot ( ( ( b\cdot a )\cdot a )\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot ( b\cdot ( c\cdot b ) ) = ( a\cdot ( b\cdot ( b\cdot c ) ) )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot ( b\cdot c )\ne \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP022-1.p} Inverse is an involution






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$( a' )'\ne a\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_inverse\_is\_original},\mbox{negated\_conjecture})$

\medskip\textbf{GRP502-1.p} Axiom for group theory, in double division and inverse, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c)'),\mbox{double\_divide}(b',\mbox{double\_divide}(d,\mbox{double\_divide}(a,d))')) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP684-1.p} Axioms of rectangular loops - c






$\mbox{ld}(a,a\cdot a) = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{rd}(a\cdot a,a) = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = \mbox{ld}(a,a\cdot b)\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = \mbox{rd}(a\cdot b,b)\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{ld}(\mbox{ld}(a,b),\mbox{ld}(a,b)\cdot ( c\cdot d )) = \mbox{ld}(a,a\cdot c)\cdot d\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{rd}(( a\cdot b )\cdot \mbox{rd}(c,d),\mbox{rd}(c,d)) = a\cdot \mbox{rd}(b\cdot d,d)\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,a\cdot \mbox{ld}(b,b)) = \mbox{rd}(\mbox{rd}(a,a)\cdot b,b)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a\cdot ( b\cdot c ),b\cdot c)\ne \mbox{rd}(a\cdot c,c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP407-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b'\cdot ( b'\cdot b ) ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP001$\wedge$5.p} TPS problem GRP-COMM2

Group is Abelian iff every element has order 2.




$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$e{:}\;\mbox{\$i}\qquad\mbox{thf}(e,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ e@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy}@ e ) = \mbox{xy}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ \mbox{xz} ) = e\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xa}@ \mbox{xb} ) = ( \mbox{cP}@ \mbox{xb}@ \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_COMM}_2,\mbox{conjecture})$

\medskip\textbf{GRP680-1.p} Commutants in Bol loops 2

If Q is a Bol loop, and if a,b in C(Q), then so are a$\wedge$2, b$\wedge$-1,
 and a$\wedge$2b.




$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(\mbox{op\_c})\cdot a\ne a\cdot i(\mbox{op\_c})\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP181-2.p} Distributivity of a lattice






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p12}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p12}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,c) = \mbox{greatest\_lower\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12}_4,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,c) = \mbox{least\_upper\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12}_5,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_p}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{GRP721-1.p} In commutative A-loops squares form a subloop - a witnessing term






$a\cdot 1 = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$f(a,b) = \mbox{ld}(\mbox{ld}(a\cdot b,a)\cdot \mbox{ld}(a\cdot b,b),1)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot b )\ne f(a,b)\cdot f(a,b)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP303-1.p} An identity generated by HR, number 04703






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP429-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( ( ( b'\cdot ( a'\cdot c ) )'\cdot d )\cdot ( b\cdot d )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP564-1.p} Axiom for Abelian group theory, in division and inverse, part 4






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b'),c),\mbox{divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP117-1.p} Derive right identity from a single axiom for groups order 3






$x\cdot ( ( x\cdot ( ( x\cdot y )\cdot z ) )\cdot ( \mbox{identity}\cdot ( z\cdot z ) ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot \mbox{identity}\ne a\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-4.004.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP736-1.p} Nonmedial left distributive left 3-symmetric quasigroup






$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$a\cdot ( a\cdot ( a\cdot b ) ) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot d )\ne ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP541-1.p} Axiom for Abelian group theory, in division and identity, part 1






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),a)),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP687-1.p} x(y.yz) = (x.yy)z is equivalent to xx.yz = (x.xy)z part 2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c ) = ( a\cdot ( a\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot ( b\cdot c ) )\ne ( a\cdot ( b\cdot b ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP496-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP667-2.p} 2-divisible ARIF loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a)\cdot f(a) = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) )\ne ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-1.003.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP169-2.p} Inverses reverse inequalities






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a',b') = a'\qquad\mbox{cnf}(\mbox{p02b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b)\ne b\qquad\mbox{cnf}(\mbox{prove\_p02b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP028-4.p} In semigroups, left and right solutions =$>$ right id exists

If there are left and right solutions, then there is a right
 identity element.




$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\mbox{left\_solution}(x,y)\cdot x{=}y\qquad\mbox{cnf}(\mbox{left\_soln},\mbox{hypothesis})$

$x\cdot \mbox{right\_solution}(x,y){=}y\qquad\mbox{cnf}(\mbox{right\_soln},\mbox{hypothesis})$

$\neg\,\mbox{not\_identity}(x)\cdot x{=}\mbox{not\_identity}(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP360-1.p} An identity generated by HR, number 23564






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_7\cdot x_3 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = x_3\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP516-1.p} Axiom for Abelian group theory, in product and inverse, part 4






$a\cdot ( ( b\cdot c )\cdot ( a\cdot c )' ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP667+6.p} 2-divisible ARIF loops are Moufang






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a{:}\;f(a)\cdot f(a) = a\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;x_2\cdot ( x_0\cdot ( x_2\cdot x_1 ) ) = ( ( x_2\cdot x_0 )\cdot x_2 )\cdot x_1\mbox{ or }\forall x_3,x_4,x_5{:}\;x_3\cdot ( x_5\cdot ( x_4\cdot x_5 ) ) = ( ( x_3\cdot x_5 )\cdot x_4 )\cdot x_5\mbox{ or }\forall x_6,x_7,x_8{:}\;( x_8\cdot x_6 )\cdot ( x_7\cdot x_8 ) = ( x_8\cdot ( x_6\cdot x_7 ) )\cdot x_8\mbox{ or }\forall x_9,x_{10},x_{11}{:}\;( x_{11}\cdot x_9 )\cdot ( x_{10}\cdot x_{11} ) = x_{11}\cdot ( ( x_9\cdot x_{10} )\cdot x_{11} )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP045-2.p} Product subsitution 2 is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$c\cdot a{=}\mbox{result}\qquad\mbox{cnf}(\mbox{product\_with\_a},\mbox{hypothesis})$

$\neg\,c\cdot b{=}\mbox{result}\qquad\mbox{cnf}(\mbox{prove\_product\_with\_b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP021-1.p} X times inverse of X is the identity






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot a'\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_X\_times\_inverse\_X\_is\_id},\mbox{negated\_conjecture})$

\medskip\textbf{GRP431-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( b\cdot ( ( ( c\cdot c' )\cdot ( d\cdot b )' )\cdot a ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP010-4.p} Inverse is a symmetric relationship

If a is an inverse of b then b is an inverse of a.




$( x\cdot y )\cdot z = x\cdot ( y\cdot z )\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x'\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$c\cdot b = \mbox{identity}\qquad\mbox{cnf}(\mbox{c\_times\_b\_is\_e},\mbox{hypothesis})$

$b\cdot c\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_b\_times\_c\_is\_e},\mbox{negated\_conjecture})$

\medskip\textbf{GRP735-1.p} Nonmedial left distributive quasigroup






$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot d )\ne ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP182-1.p} Positive part of the negative part is identity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},\mbox{greatest\_lower\_bound}(a,\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p17a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP093-1.p} Single axiom for Abelian group theory, in division and identity

This is a single axiom for Abelian group theory, in terms
 of division and identity




$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(\mbox{divide}(\mbox{divide}(x,y),z),x)),z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP186-4.p} Application of distributivity and group theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p23x}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p23x}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p23x}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\ne a\cdot \mbox{least\_upper\_bound}(a',b)\qquad\mbox{cnf}(\mbox{prove\_p23x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP152-1.p} Prove least upper-bound axiom using the LUB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_lub3a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP170-2.p} General form of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{p03b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(c,d) = c\qquad\mbox{cnf}(\mbox{p03b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a\cdot c,b\cdot d)\ne a\cdot c\qquad\mbox{cnf}(\mbox{prove\_p03b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP678-1.p} In CC-loops, associators are in the center of the nucleus - 2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_c}\cdot ( a\cdot b ) = ( \mbox{op\_c}\cdot a )\cdot b\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot ( \mbox{op\_c}\cdot b ) = ( a\cdot \mbox{op\_c} )\cdot b\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{op\_c} ) = ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{asoc}(a,b,c)\cdot \mbox{op\_c}\ne \mbox{op\_c}\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP344-1.p} An identity generated by HR, number 19587






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_6\ne \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{GRP399-1.p} Lattice ordered group (equality) axioms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

\medskip\textbf{GRP134-2.003.p} (a.b).(b.a) = b, no idempotence

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP012-3.p} Inverse of products = Product of inverses

The inverse of products equals the product of the inverse,
 in opposite order




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$( a\cdot b )'\ne b'\cdot a'\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_product\_is\_product\_of\_inverses},\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-4.005.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y\mbox{ or }e_5\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y\mbox{ or }x\cdot e_5{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot z_2{=}y\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg4}_1,\mbox{negated\_conjecture})$

$( z_1\cdot z_2{=}y\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ y\cdot x{=}z_2\qquad\mbox{cnf}(\mbox{qg4}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP042-2.p} Symmetry is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$\neg\,b{=}a\qquad\mbox{cnf}(\mbox{prove\_symmetry},\mbox{negated\_conjecture})$

\medskip\textbf{GRP001-1.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$x\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{square\_element},\mbox{hypothesis})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{negated\_conjecture})$

$\neg\,b\cdot a{=}c\qquad\mbox{cnf}(\mbox{prove\_b\_times\_a\_is\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP534-1.p} Axiom for Abelian group theory, in division, part 2






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),c)),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP453-1.p} Axiom for group theory, in division, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-2.005.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP091-1.p} Single axiom for Abelian group theory, in division

This is a single axiom for Abelian group theory, in terms
 of division




$\mbox{divide}(\mbox{divide}(x,\mbox{divide}(\mbox{divide}(x,y),z)),y) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP253-1.p} An identity generated by HR, number 00722






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$( x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7 ) \ \Rightarrow \ \mbox{sk\_c}_5'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-6.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP446-1.p} Axiom for group theory, in division, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP669-1.p} Moufang loops are RIF






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot ( a\cdot b ) )\ne ( ( a\cdot ( b\cdot c ) )\cdot a )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP175-3.p} Positivity is preserved under inner automorphisms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p06c}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a'\cdot ( b\cdot a ))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p06c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP127-4.004.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_2\cdot y{=}x\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ y\cdot x{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_2\cdot y{=}x\mbox{ and }y\cdot x{=}z_1 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP278-1.p} An identity generated by HR, number 02518






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_3\cdot \mbox{sk\_c}_4\ne \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP540-1.p} Axiom for Abelian group theory, in division, part 4






$\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(\mbox{divide}(a,c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP025-4.p} All groups of order 2 are isomorphic

If G1 has exactly two elements and G2 has exactly two
 elements, then there exists an isomorphism [a one-to-one and
 onto homomorphism] between them.




$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

$( \mbox{xg}\cdot x{=}z\mbox{ and }\mbox{xg}\cdot x{=}w ) \ \Rightarrow \ w = z\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{xg}\cdot z{=}y\mbox{ and }\mbox{xg}\cdot w{=}y ) \ \Rightarrow \ w = z\qquad\mbox{cnf}(\mbox{right\_cancellation},\mbox{axiom})$

$g_1\ne g_2\qquad\mbox{cnf}(\mbox{two\_groups},\mbox{hypothesis})$

$\mbox{group\_member}(a,g_1)\qquad\mbox{cnf}(\mbox{a\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(b,g_1)\qquad\mbox{cnf}(\mbox{b\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(c,g_2)\qquad\mbox{cnf}(\mbox{c\_member\_of\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(d,g_2)\qquad\mbox{cnf}(\mbox{d\_member\_of\_group}_2,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{a\_not\_b},\mbox{hypothesis})$

$c\ne d\qquad\mbox{cnf}(\mbox{c\_not\_d},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{a\_not\_c},\mbox{hypothesis})$

$a\ne d\qquad\mbox{cnf}(\mbox{a\_not\_d},\mbox{hypothesis})$

$b\ne c\qquad\mbox{cnf}(\mbox{b\_not\_c},\mbox{hypothesis})$

$b\ne d\qquad\mbox{cnf}(\mbox{b\_not\_d},\mbox{hypothesis})$

$\mbox{group\_member}(x,g_1) \ \Rightarrow \ ( x = a\mbox{ or }x = b )\qquad\mbox{cnf}(\mbox{a\_and\_b\_only\_members\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_2) \ \Rightarrow \ ( x = c\mbox{ or }x = d )\qquad\mbox{cnf}(\mbox{c\_and\_d\_only\_members\_of\_group}_2,\mbox{hypothesis})$

$\mbox{identity\_for}(g_1) = a\qquad\mbox{cnf}(\mbox{a\_identity\_of\_group}_1,\mbox{hypothesis})$

$\mbox{identity\_for}(g_2) = c\qquad\mbox{cnf}(\mbox{c\_identity\_of\_group}_2,\mbox{hypothesis})$

$g_1\cdot a{=}x\qquad\mbox{cnf}(\mbox{a\_left\_identity},\mbox{hypothesis})$

$g_1\cdot x{=}a\qquad\mbox{cnf}(\mbox{a\_right\_identity},\mbox{hypothesis})$

$g_2\cdot c{=}x\qquad\mbox{cnf}(\mbox{c\_left\_identity},\mbox{hypothesis})$

$g_2\cdot x{=}c\qquad\mbox{cnf}(\mbox{c\_right\_identity},\mbox{hypothesis})$

$\mbox{isomorphism}_1(a) = c\qquad\mbox{cnf}(\mbox{a\_maps1\_to\_c},\mbox{hypothesis})$

$\mbox{isomorphism}_1(b) = d\qquad\mbox{cnf}(\mbox{b\_maps1\_to\_d},\mbox{hypothesis})$

$\mbox{isomorphism}_2(a) = d\qquad\mbox{cnf}(\mbox{a\_maps2\_to\_d},\mbox{hypothesis})$

$\mbox{isomorphism}_2(b) = c\qquad\mbox{cnf}(\mbox{b\_maps2\_to\_c},\mbox{hypothesis})$

$\mbox{group\_member}(d_1,g_1)\qquad\mbox{cnf}(\mbox{d1\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$\mbox{group\_member}(d_2,g_1)\qquad\mbox{cnf}(\mbox{d2\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$\mbox{group\_member}(d_3,g_1)\qquad\mbox{cnf}(\mbox{d3\_member\_of\_group}_1,\mbox{negated\_conjecture})$

$g_1\cdot d_1{=}d_2\qquad\mbox{cnf}(\mbox{d1\_times\_d2\_is\_d}_3,\mbox{negated\_conjecture})$

$g_2\cdot \mbox{isomorphism}_1(d_1){=}\mbox{isomorphism}_1(d_2) \ \Rightarrow \ \neg\,g_2\cdot \mbox{isomorphism}_2(d_1){=}\mbox{isomorphism}_2(d_2)\qquad\mbox{cnf}(\mbox{prove\_one\_product\_holds\_in\_group}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP469-1.p} Axiom for group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,d)))',\mbox{divide}(\mbox{divide}(d,c),a)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP491-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{identity}),\mbox{double\_divide}(c,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-1.003.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP179-2.p} For converting between GLB and LUB






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a',\mbox{identity})\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\qquad\mbox{cnf}(\mbox{prove\_p}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{GRP132-2.002.p} (3,1,2) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP603-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{double\_divide}(a,c))')',c)' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP685+1.p} Axioms of rectangular loops - d






$\forall a{:}\;\mbox{ld}(a,a\cdot a) = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;\mbox{rd}(a\cdot a,a) = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = \mbox{ld}(a,a\cdot b)\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = \mbox{rd}(a\cdot b,b)\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(\mbox{ld}(a,b),\mbox{ld}(a,b)\cdot ( c\cdot d )) = \mbox{ld}(a,a\cdot c)\cdot d\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{rd}(( a\cdot b )\cdot \mbox{rd}(c,d),\mbox{rd}(c,d)) = a\cdot \mbox{rd}(b\cdot d,d)\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot \mbox{ld}(b,b)) = \mbox{rd}(\mbox{rd}(a,a)\cdot b,b)\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_6,x_7,x_8{:}\;( \mbox{rd}(x_6\cdot \mbox{ld}(x_7,x_8),\mbox{ld}(x_7,x_8)) = \mbox{rd}(x_6\cdot x_8,x_8)\mbox{ and }\mbox{rd}(x_6\cdot \mbox{rd}(x_7,x_8),\mbox{rd}(x_7,x_8)) = \mbox{rd}(x_6\cdot x_8,x_8) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP746+1.p} Right alternative loop rings: the group case






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( ( x_0\cdot x_1 )\cdot x_2 = x_0\cdot ( x_1\cdot x_2 )\mbox{ and }( x_0\cdot x_2 )\cdot x_1 = x_0\cdot ( x_2\cdot x_1 ) )\mbox{ or }( ( x_0\cdot x_1 )\cdot x_2 = ( x_0\cdot x_2 )\cdot x_1\mbox{ and }x_0\cdot ( x_1\cdot x_2 ) = x_0\cdot ( x_2\cdot x_1 ) ) )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$( a\cdot b )\cdot c = a\cdot ( b\cdot c )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP288-1.p} An identity generated by HR, number 03168






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-1.005.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP726-1.p} Bruck loops that are centrally nilpotent - hard part

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - the hard part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e)\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP499-1.p} Axiom for group theory, in double division and inverse, part 1






$\mbox{double\_divide}(a',\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c))',\mbox{double\_divide}(d,\mbox{double\_divide}(b,d)))') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP713-1.p} In Buchsteiner loops fourth powers are nuclear - b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( a\cdot b )\cdot c) = \mbox{rd}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( ( b\cdot ( b\cdot ( b\cdot b ) ) )\cdot c )\ne ( a\cdot ( b\cdot ( b\cdot ( b\cdot b ) ) ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP683+1.p} Axioms of rectangular loops - b






$\forall a{:}\;\mbox{ld}(a,a\cdot a) = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;\mbox{rd}(a\cdot a,a) = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = \mbox{ld}(a,a\cdot b)\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = \mbox{rd}(a\cdot b,b)\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(\mbox{ld}(a,b),\mbox{ld}(a,b)\cdot ( c\cdot d )) = \mbox{ld}(a,a\cdot c)\cdot d\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{rd}(( a\cdot b )\cdot \mbox{rd}(c,d),\mbox{rd}(c,d)) = a\cdot \mbox{rd}(b\cdot d,d)\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot \mbox{ld}(b,b)) = \mbox{rd}(\mbox{rd}(a,a)\cdot b,b)\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_3,x_4,x_5{:}\;( x_3\cdot \mbox{ld}(x_4,x_4\cdot x_5) = x_3\cdot x_5\mbox{ and }\mbox{rd}(x_3\cdot x_4,x_4)\cdot x_5 = x_3\cdot x_5 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP069-1.p} Single axiom for group theory, in division and identity

This is a single axiom for group theory, in terms of division
 and identity




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),y),z),\mbox{divide}(\mbox{divide}(\mbox{identity},x),z))) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP474-1.p} Axiom for group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b)',\mbox{divide}(\mbox{divide}(c,d),a)),\mbox{divide}(d,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-4.005.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 5 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y\mbox{ or }e_5\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y\mbox{ or }x\cdot e_5{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP141-1.p} Prove greatest lower-bound axiom using a transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,c) = a\qquad\mbox{cnf}(\mbox{ax\_glb1d}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,c) = b\qquad\mbox{cnf}(\mbox{ax\_glb1d}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b),c)\ne c\qquad\mbox{cnf}(\mbox{prove\_ax\_glb1d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP174-1.p} Each subgroup of positive elements is trivial






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p05b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a') = a'\qquad\mbox{cnf}(\mbox{p05b}_2,\mbox{hypothesis})$

$\mbox{identity}\ne a\qquad\mbox{cnf}(\mbox{prove\_p05b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP574-1.p} Axiom for Abelian group theory, in double div and id, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP500-1.p} Axiom for group theory, in double division and inverse, part 2






$\mbox{double\_divide}(a',\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c))',\mbox{double\_divide}(d,\mbox{double\_divide}(b,d)))') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP194+1.p} In semigroups, a surjective homomorphism maps the zero

If (F,*) and (H,+) are two semigroups, phi is a surjective
 homomorphism from F to H, and id is a left zero for F,
 then phi(id) is a left zero for H.




$\mbox{include}(\mbox{'Axioms/GRP007+0.ax'})$

$\forall x{:}\;( \mbox{group\_member}(x,f) \ \Rightarrow \ \mbox{group\_member}(\mbox{phi}(x),h) )\qquad\mbox{fof}(\mbox{homomorphism}_1,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{group\_member}(x,f)\mbox{ and }\mbox{group\_member}(y,f) ) \ \Rightarrow \ m(h,\mbox{phi}(x),\mbox{phi}(y)) = \mbox{phi}(m(f,x,y)) )\qquad\mbox{fof}(\mbox{homomorphism}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{group\_member}(x,h) \ \Rightarrow \ \exists y{:}\;( \mbox{group\_member}(y,f)\mbox{ and }\mbox{phi}(y) = x ) )\qquad\mbox{fof}(\mbox{surjective},\mbox{axiom})$

$\forall g,x{:}\;( \mbox{left\_zero}(g,x)\iff ( \mbox{group\_member}(x,g)\mbox{ and }\forall y{:}\;( \mbox{group\_member}(y,g) \ \Rightarrow \ m(g,x,y) = x ) ) )\qquad\mbox{fof}(\mbox{left\_zero},\mbox{axiom})$

$\mbox{left\_zero}(f,\mbox{f\_left\_zero})\qquad\mbox{fof}(\mbox{left\_zero\_for\_f},\mbox{hypothesis})$

$\mbox{left\_zero}(h,\mbox{phi}(\mbox{f\_left\_zero}))\qquad\mbox{fof}(\mbox{prove\_left\_zero\_h},\mbox{conjecture})$

\medskip\textbf{GRP656+1.p} A quasigroup satisfying Moufang 3 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot ( c\cdot a ) = ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP463-1.p} Axiom for group theory, in division and identity, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{identity},a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP397-1.p} Cancellative semigroups axioms






$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

\medskip\textbf{GRP739-1.p} Proper commutative A-loop of odd order.






$a\cdot 1 = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$s(a)\cdot s(a) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$s(a\cdot a) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP153-1.p} Prove least upper-bound axiom using the GLB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne b\qquad\mbox{cnf}(\mbox{prove\_ax\_lub3b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP134-2.005.p} (a.b).(b.a) = b, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP016-1.p} There is a homomorphism from a group to itself






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/ALG001-0.ax'})$

$\mbox{group}(f_{69},f_{70})\qquad\mbox{cnf}(\mbox{a\_group},\mbox{negated\_conjecture})$

$\neg\,\mbox{homomorphism}(y,f_{69},f_{70},f_{69},f_{70})\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_homomorphism},\mbox{negated\_conjecture})$

\medskip\textbf{GRP444-1.p} Axiom for group theory, in product \& inverse, part 3






$( a\cdot ( b\cdot ( ( c\cdot c' )\cdot ( d\cdot ( a\cdot b ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP517-1.p} Axiom for Abelian group theory, in product and inverse, part 1






$a\cdot ( ( ( a\cdot b )'\cdot c )\cdot b ) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP748-2.p} Right alternative loop rings: a lemma






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(f_{08},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(f_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(f_{10},\mbox{axiom})$

$a\cdot b = b\cdot a\mbox{ or }i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(f_{11},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) )\ne ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP520-1.p} Axiom for Abelian group theory, in product and inverse, part 4






$a\cdot ( ( ( a\cdot b )'\cdot c )\cdot b ) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP043-2.p} Transitivity is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$b{=}c\qquad\mbox{cnf}(\mbox{b\_equals\_c},\mbox{hypothesis})$

$\neg\,a{=}c\qquad\mbox{cnf}(\mbox{prove\_transitivity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-4.004.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_2\cdot y{=}x\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_2\cdot y{=}x\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-1.005.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP565-1.p} Axiom for Abelian group theory, in double div and id, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(\mbox{identity},c))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP505-1.p} Axiom for Abelian group theory, in product and inverse, part 1






$( ( ( a\cdot b )'\cdot ( b\cdot a ) )'\cdot ( ( c\cdot d )'\cdot ( c\cdot ( ( e\cdot f' )\cdot d' )' ) ) )'\cdot f = e\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP561-1.p} Axiom for Abelian group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b'),c),\mbox{divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP352-1.p} An identity generated by HR, number 22712






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP528-1.p} Axiom for Abelian group theory, in division, part 4






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(c,b))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP081-1.p} Single axiom for group theory, in double division and inverse

This is a single axiom for group theory, in terms of double
 division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(y,z),\mbox{double\_divide}(y,\mbox{double\_divide}(u',z'))')),x)' = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP570-1.p} Axiom for Abelian group theory, in double div and id, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP703+1.p} In C-loops the nucleus is normal - b






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_c}\cdot ( a\cdot b ) = ( \mbox{op\_c}\cdot a )\cdot b\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( b\cdot \mbox{op\_c} ) = ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( \mbox{op\_c}\cdot b ) = ( a\cdot \mbox{op\_c} )\cdot b\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall a{:}\;\mbox{op\_d} = \mbox{ld}(a,\mbox{op\_c}\cdot a)\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_e} = ( \mbox{rd}(\mbox{op\_c},a\cdot b)\cdot b )\cdot a\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_f} = a\cdot ( b\cdot \mbox{ld}(a\cdot b,\mbox{op\_c}) )\qquad\mbox{fof}(f_{13},\mbox{axiom})$

$\forall x_2,x_3{:}\;( \mbox{op\_e}\cdot ( x_2\cdot x_3 ) = ( \mbox{op\_e}\cdot x_2 )\cdot x_3\mbox{ and }x_2\cdot ( x_3\cdot \mbox{op\_e} ) = ( x_2\cdot x_3 )\cdot \mbox{op\_e}\mbox{ and }x_2\cdot ( \mbox{op\_e}\cdot x_3 ) = ( x_2\cdot \mbox{op\_e} )\cdot x_3 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP427-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( ( ( b'\cdot ( a'\cdot c ) )'\cdot d )\cdot ( b\cdot d )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP768-1.p} Buchsteiner loop lemma 5






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$\mbox{quotient}(j(a),a) = j(a)\cdot i(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$( ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot b )\cdot c = ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(x_0),x_1\cdot x_2)\ne t(\mbox{eta}(x_0),x_1)\cdot t(\mbox{eta}(x_0),x_2)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP292-1.p} An identity generated by HR, number 03445






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2\cdot x_1 = \mbox{sk\_c}_6\mbox{ and }x_2' = x_1 ) \ \Rightarrow \ x_1\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP184-4.p} Orthogonal elements commute and form a subgroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p21x}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p21x}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p21x}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(x,y)' = \mbox{least\_upper\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p21x}_4,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(x,y)' = \mbox{greatest\_lower\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p21x}_5,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\cdot \mbox{least\_upper\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p21x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP655+3.p} A quasigroup satisfying Moufang 2 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{ld}(x_1,x_1) = x_0\mbox{ and }\mbox{ld}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP586-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c')',b)') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP511-1.p} Axiom for Abelian group theory, in product and inverse, part 3






$( ( a\cdot b )\cdot c )\cdot ( a\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP459-1.p} Axiom for group theory, in division and identity, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{identity},a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP164-2.p} The lattice of each LOG is distributive






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,\mbox{least\_upper\_bound}(b,c))\ne \mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b),\mbox{greatest\_lower\_bound}(a,c))\qquad\mbox{cnf}(\mbox{prove\_distrun},\mbox{negated\_conjecture})$

\medskip\textbf{GRP521-1.p} Axiom for Abelian group theory, in division, part 1






$\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,\mbox{divide}(a,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP028-2.p} In semigroups, left and right solutions =$>$ right id exists

If there are left and right solutions, then there is a right
 identity element.




$\mbox{include}(\mbox{'Axioms/GRP002-0.ax'})$

$\mbox{left\_solution}(x,y)\cdot x{=}y\qquad\mbox{cnf}(\mbox{left\_soln},\mbox{hypothesis})$

$x\cdot \mbox{right\_solution}(x,y){=}y\qquad\mbox{cnf}(\mbox{right\_soln},\mbox{hypothesis})$

$\neg\,\mbox{not\_identity}(x)\cdot x{=}\mbox{not\_identity}(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP780+1.p} Napoleon's quasigroups: Lamoen's theorem






$\forall b,a{:}\;a\setminus a\cdot b = b\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( a\setminus b ) = b\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;( a\cdot b )\cdot ( c\cdot d ) = ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall b,a{:}\;( ( a\cdot b )\cdot b )\cdot ( b\cdot ( b\cdot a ) ) = b\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( d(x_0,x_1,x_2)\iff x_0\cdot x_1 = x_1\cdot x_2 )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{bigC}(a,b,c) = ( a\cdot b )\cdot ( c\cdot a )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$d(a_1,a_2,a_3)\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$d(b_1,b_2,b_3)\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$d(c_1,c_2,c_3)\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$d(\mbox{bigC}(a_1,b_3,c_2),\mbox{bigC}(a_2,b_1,c_3),\mbox{bigC}(a_3,b_2,c_1))\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP748-4.p} Right alternative loop rings: a lemma






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(f_{08},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(f_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(f_{10},\mbox{axiom})$

$a\cdot b = b\cdot a\mbox{ or }i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(f_{11},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP509-1.p} Axiom for Abelian group theory, in product and inverse, part 1






$( ( a\cdot b )\cdot c )\cdot ( a\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP116-1.p} Derive left identity from a single axiom for groups order 3






$x\cdot ( ( x\cdot ( ( x\cdot y )\cdot z ) )\cdot ( \mbox{identity}\cdot ( z\cdot z ) ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$\mbox{identity}\cdot a\ne a\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP002-2.p} Commutator equals identity in groups of order 3

In a group, if (for all x) the cube of x is the identity
 (i.e. a group of order 3), then the equation [[x,y],y]=
 identity holds, where [x,y] is the product of x, y, the
 inverse of x and the inverse of y (i.e. the commutator
 of x and y).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x\cdot ( x\cdot x ) = \mbox{identity}\qquad\mbox{cnf}(\mbox{x\_cubed\_is\_identity},\mbox{hypothesis})$

$a\cdot b = c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{negated\_conjecture})$

$c\cdot a' = d\qquad\mbox{cnf}(\mbox{c\_times\_inverse\_a\_is\_d},\mbox{negated\_conjecture})$

$d\cdot b' = h\qquad\mbox{cnf}(\mbox{d\_times\_inverse\_b\_is\_h},\mbox{negated\_conjecture})$

$h\cdot b = j\qquad\mbox{cnf}(\mbox{h\_times\_b\_is\_j},\mbox{negated\_conjecture})$

$j\cdot h' = k\qquad\mbox{cnf}(\mbox{j\_times\_inverse\_h\_is\_k},\mbox{negated\_conjecture})$

$k\cdot b'\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_k\_times\_inverse\_b\_is\_e},\mbox{negated\_conjecture})$

\medskip\textbf{GRP178-2.p} A consequence of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p09b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p09b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},c) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p09b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p09b}_4,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b\cdot c)\ne \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{prove\_p09b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP136-1.p} Prove anti-symmetry axiom using the LUB transformation

This problem proves the original anti-symmetry axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_antisyma}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_antisyma}_2,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_ax\_antisyma},\mbox{negated\_conjecture})$

\medskip\textbf{GRP302-1.p} An identity generated by HR, number 04549






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_8\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_2' = x_5\mbox{ and }x_5' = \mbox{sk\_c}_8 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_8\ne x_5\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP159-1.p} Prove monotonicity axiom using a transformation

This problem proves the original monotonicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_mono2c}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(c\cdot a,c\cdot b)\ne c\cdot b\qquad\mbox{cnf}(\mbox{prove\_ax\_mono2c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP026-1.p} All groups of order 3 are isomorphic

If G1 and G2 each have exactly three elements, then there
 exists an isomorphism [a one-to-one and onto homomorphism]
 between them.




$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

$\mbox{group\_member}(a,g_1)\qquad\mbox{cnf}(\mbox{a\_in\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(b,g_1)\qquad\mbox{cnf}(\mbox{b\_in\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(c,g_1)\qquad\mbox{cnf}(\mbox{c\_in\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(f,g_2)\qquad\mbox{cnf}(\mbox{f\_in\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(g,g_2)\qquad\mbox{cnf}(\mbox{g\_in\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(h,g_2)\qquad\mbox{cnf}(\mbox{h\_in\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_1) \ \Rightarrow \ ( x = a\mbox{ or }x = b\mbox{ or }x = c )\qquad\mbox{cnf}(\mbox{all\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_2) \ \Rightarrow \ ( x = f\mbox{ or }x = g\mbox{ or }x = h )\qquad\mbox{cnf}(\mbox{all\_of\_group}_2,\mbox{hypothesis})$

$g_1\cdot a{=}a\qquad\mbox{cnf}(\mbox{a\_times\_a\_is\_a},\mbox{hypothesis})$

$g_1\cdot a{=}b\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_b},\mbox{hypothesis})$

$g_1\cdot b{=}a\qquad\mbox{cnf}(\mbox{b\_times\_a\_is\_b},\mbox{hypothesis})$

$g_1\cdot a{=}c\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_c},\mbox{hypothesis})$

$g_1\cdot c{=}a\qquad\mbox{cnf}(\mbox{c\_times\_a\_is\_c},\mbox{hypothesis})$

$g_1\cdot b{=}b\qquad\mbox{cnf}(\mbox{b\_times\_b\_is\_c},\mbox{hypothesis})$

$g_1\cdot b{=}c\qquad\mbox{cnf}(\mbox{b\_times\_c\_is\_a},\mbox{hypothesis})$

$g_1\cdot c{=}b\qquad\mbox{cnf}(\mbox{c\_times\_b\_is\_a},\mbox{hypothesis})$

$g_1\cdot c{=}c\qquad\mbox{cnf}(\mbox{c\_times\_c\_is\_b},\mbox{hypothesis})$

$g_2\cdot f{=}f\qquad\mbox{cnf}(\mbox{f\_times\_f\_is\_f},\mbox{hypothesis})$

$g_2\cdot f{=}g\qquad\mbox{cnf}(\mbox{f\_times\_g\_is\_g},\mbox{hypothesis})$

$g_2\cdot g{=}f\qquad\mbox{cnf}(\mbox{g\_times\_f\_is\_g},\mbox{hypothesis})$

$g_2\cdot f{=}h\qquad\mbox{cnf}(\mbox{f\_times\_h\_is\_h},\mbox{hypothesis})$

$g_2\cdot h{=}f\qquad\mbox{cnf}(\mbox{h\_times\_f\_is\_h},\mbox{hypothesis})$

$g_2\cdot g{=}g\qquad\mbox{cnf}(\mbox{g\_times\_g\_is\_h},\mbox{hypothesis})$

$g_2\cdot g{=}h\qquad\mbox{cnf}(\mbox{g\_times\_h\_is\_f},\mbox{hypothesis})$

$g_2\cdot h{=}g\qquad\mbox{cnf}(\mbox{h\_times\_g\_is\_f},\mbox{hypothesis})$

$g_2\cdot h{=}h\qquad\mbox{cnf}(\mbox{h\_times\_h\_is\_g},\mbox{hypothesis})$

$\mbox{an\_isomorphism}(a) = f\qquad\mbox{cnf}(\mbox{a\_maps\_to\_f},\mbox{hypothesis})$

$\mbox{an\_isomorphism}(b) = g\qquad\mbox{cnf}(\mbox{b\_maps\_to\_g},\mbox{hypothesis})$

$\mbox{an\_isomorphism}(c) = h\qquad\mbox{cnf}(\mbox{c\_maps\_to\_h},\mbox{hypothesis})$

$\mbox{group\_member}(d_1,g_1)\qquad\mbox{cnf}(\mbox{d1\_in\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(d_2,g_1)\qquad\mbox{cnf}(\mbox{d2\_in\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(d_3,g_1)\qquad\mbox{cnf}(\mbox{d3\_in\_group}_1,\mbox{hypothesis})$

$g_1\cdot d_1{=}d_2\qquad\mbox{cnf}(\mbox{d1\_times\_d2\_is\_d}_3,\mbox{hypothesis})$

$\neg\,g_2\cdot \mbox{an\_isomorphism}(d_1){=}\mbox{an\_isomorphism}(d_2)\qquad\mbox{cnf}(\mbox{prove\_product\_holds\_in\_group}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP755-1.p} In char$>$2, right alternative loop rings are left alternative






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\mbox{ or }a\cdot ( b\cdot c ) = ( a\cdot c )\cdot b\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\mbox{ or }a\cdot ( c\cdot b ) = ( a\cdot b )\cdot c\qquad\mbox{cnf}(f_{08},\mbox{axiom})$

$i(a) = \mbox{ld}(a,1)\qquad\mbox{cnf}(f_{09},\mbox{axiom})$

$i(a\cdot b)\ne i(b)\cdot i(a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP314-1.p} An identity generated by HR, number 18186






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\mbox{ and }x_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_8\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_5' = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_2' = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_6\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP001-3.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/ALG001-0.ax'})$

$\mbox{group}(f_{71},f_{72})\qquad\mbox{cnf}(\mbox{a\_group},\mbox{hypothesis})$

$\mbox{identity}(f_{71},f_{72},f_{73})\qquad\mbox{cnf}(\mbox{f73\_is\_the\_identity},\mbox{hypothesis})$

$x\in f_{71} \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(f_{72},x,x) = f_{73}\qquad\mbox{cnf}(\mbox{x\_squared\_is\_identity},\mbox{hypothesis})$

$\neg\,\mbox{commutes}(f_{71},f_{72})\qquad\mbox{cnf}(\mbox{prove\_the\_group\_is\_commutative},\mbox{negated\_conjecture})$

\medskip\textbf{GRP001-2.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{squareness},\mbox{hypothesis})$

$a\cdot b = c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{hypothesis})$

$b\cdot a\ne c\qquad\mbox{cnf}(\mbox{prove\_b\_times\_a\_is\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP185-3.p} Application of monotonicity and distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity}),\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity}))\ne \mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p22b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP440-1.p} Axiom for group theory, in product \& inverse, part 2






$( a\cdot ( b\cdot ( ( b'\cdot c )\cdot ( d\cdot ( a\cdot c ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP103-1.p} Single axiom for Abelian group theory, in double div and id

This is a single axiom for Abelian group theory, in terms
 of double division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},y),\mbox{double\_divide}(z,\mbox{double\_divide}(y,x)))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP719-1.p} In a commutative RIF loop, all cubes are C-elements






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot ( a\cdot b ) ) = ( ( a\cdot ( b\cdot c ) )\cdot a )\cdot b\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot ( ( b\cdot ( b\cdot b ) )\cdot ( ( b\cdot ( b\cdot b ) )\cdot c ) )\ne ( ( a\cdot ( b\cdot ( b\cdot b ) ) )\cdot ( b\cdot ( b\cdot b ) ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-1.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP616-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b')',c)',\mbox{double\_divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP097-1.p} Single axiom for Abelian group theory, in division and inverse

This is a single axiom for Abelian group theory, in terms
 of division and inverse




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(y,z),\mbox{divide}(x,z))') = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP753-1.p} A new basis for trimedial quasigroups: part 2a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot ( b\cdot c ) = ( \mbox{rd}(a,a)\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot ( b\cdot \mbox{ld}(c,c) )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot ( a\cdot a ) )\cdot ( b\cdot c )\ne ( a\cdot b )\cdot ( ( a\cdot a )\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP525-1.p} Axiom for Abelian group theory, in division, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(c,b))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-2.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP529-1.p} Axiom for Abelian group theory, in division, part 1






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,c)),\mbox{divide}(a,b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP689-1.p} Bruck loop elements of order 2$\wedge$2 commute with elems of order 3$\wedge$2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) ) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot \mbox{op\_d} ) ) ) ) ) ) ) = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d}\ne \mbox{op\_d}\cdot \mbox{op\_c}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP144-1.p} Prove greatest lower-bound axiom using the LUB transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b),b)\ne b\qquad\mbox{cnf}(\mbox{prove\_ax\_glb3a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP172-2.p} Negative elements form a semigroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p04d}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p04d}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},a\cdot b)\ne a\cdot b\qquad\mbox{cnf}(\mbox{prove\_p04d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP181-3.p} Distributivity of a lattice






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = \mbox{greatest\_lower\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12x}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,c) = \mbox{least\_upper\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12x}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(x,y)' = \mbox{least\_upper\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p12x}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(x,y)' = \mbox{greatest\_lower\_bound}(x',y')\qquad\mbox{cnf}(\mbox{p12x}_4,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_p12x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP494-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,c),\mbox{double\_divide}(\mbox{identity},\mbox{identity})),\mbox{double\_divide}(a,c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP017-1.p} The inverse of each element is unique

i.e., if ab=ba=identity and ac=ca=identity then b=c




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot b{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_identity},\mbox{hypothesis})$

$b\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{b\_times\_a\_is\_identity},\mbox{hypothesis})$

$a\cdot c{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_identity},\mbox{hypothesis})$

$c\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{c\_times\_a\_is\_identity},\mbox{hypothesis})$

$b\ne c\qquad\mbox{cnf}(\mbox{prove\_b\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP062-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( x\cdot ( y\cdot ( ( z\cdot z' )\cdot ( u\cdot ( x\cdot y ) )' ) ) )' = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP742-1.p} Proper power associative CC loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot \mbox{rd}(1,a) = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP659+1.p} Bol-Moufang identity 2 implies the existence of a unit element






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot ( b\cdot c ) )\cdot b = ( a\cdot b )\cdot ( c\cdot b )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP063-1.p} Single axiom for group theory, in division

This is a single axiom for group theory, in terms of division




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),y),z),\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),x),z))) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP214-1.p} An identity generated by HR, number 00387






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = x_1\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }x_5\cdot x_1 = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_5'\ne x_1\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

\medskip\textbf{GRP138-1.p} Prove greatest lower-bound axiom using the LUB transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,c) = a\qquad\mbox{cnf}(\mbox{ax\_glb1a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,c) = b\qquad\mbox{cnf}(\mbox{ax\_glb1a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b),c)\ne \mbox{greatest\_lower\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_glb1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP267-1.p} An identity generated by HR, number 01105






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( x_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_8\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_7\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = x_4\mbox{ and }x_5' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8 ) \ \Rightarrow \ \mbox{sk\_c}_6'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-3.004.p} a.(b.a) = (b.a).a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP196-1.p} In semigroups, xyyy=yyyx $\to$ (uy)$\wedge$9 = u$\wedge$9v$\wedge$9.






$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$x\cdot ( y\cdot ( y\cdot y ) ) = y\cdot ( y\cdot ( y\cdot x ) )\qquad\mbox{cnf}(\mbox{condition},\mbox{hypothesis})$

$a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot ( b\cdot ( a\cdot b ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\ne a\cdot ( a\cdot ( a\cdot ( a\cdot ( a\cdot ( a\cdot ( a\cdot ( a\cdot ( a\cdot ( b\cdot ( b\cdot ( b\cdot ( b\cdot ( b\cdot ( b\cdot ( b\cdot ( b\cdot b ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{GRP162-1.p} Prove transitivity axiom using the LUB transformation

This problem proves the original transitivity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_transa}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,c) = c\qquad\mbox{cnf}(\mbox{ax\_transa}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,c)\ne c\qquad\mbox{cnf}(\mbox{prove\_ax\_transa},\mbox{negated\_conjecture})$

\medskip\textbf{GRP412-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( ( ( b'\cdot b )\cdot ( ( a\cdot b' )'\cdot c )' )\cdot b )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP185-2.p} Application of monotonicity and distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p22a}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p22a}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p22a}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity}),\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity}))\ne \mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p22a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP048-2.p} Inverse substitution is dependent






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

$a{=}b\qquad\mbox{cnf}(\mbox{a\_equals\_b},\mbox{hypothesis})$

$\neg\,a'{=}b'\qquad\mbox{cnf}(\mbox{prove\_inverse\_substitution},\mbox{negated\_conjecture})$

\medskip\textbf{GRP134-1.003.p} (a.b).(b.a) = b, no idempotence

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP666-5.p} Inverse property A-loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP655+1.p} A quasigroup satisfying Moufang 2 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP124-4.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP395-1.p} Group theory (Named groups) axioms






$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

\medskip\textbf{GRP191-2.p} Something useful for estimations






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p39d}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a',b')\ne b'\qquad\mbox{cnf}(\mbox{prove\_p39d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP084-1.p} Single axiom for Abelian group theory, in product and inverse

This is a single axiom for Abelian group theory, in terms
 of product and inverse.




$( ( ( x\cdot y )'\cdot ( y\cdot x ) )'\cdot ( ( z\cdot u )'\cdot ( z\cdot ( ( v\cdot w' )\cdot u' )' ) ) )'\cdot w = v\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP697-1.p} Variety of power associative, WIP conjugacy closed loops - 1b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot i(b\cdot a) = i(b)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( a\cdot b )\cdot ( b\cdot ( c\cdot b ) )\ne ( a\cdot ( b\cdot ( b\cdot c ) ) )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP289-1.p} An identity generated by HR, number 03169






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP575-1.p} Axiom for Abelian group theory, in double div and id, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP030-1.p} In semigroups, left id and inverse =$>$ left id=right id

If there are a left identity and left inverse, then the left
 identity is also a right identity.




$\mbox{include}(\mbox{'Axioms/GRP002-0.ax'})$

$\mbox{identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{hypothesis})$

$a'\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{hypothesis})$

$\neg\,a\cdot \mbox{identity}{=}a\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP506-1.p} Axiom for Abelian group theory, in product and inverse, part 2






$( ( ( a\cdot b )'\cdot ( b\cdot a ) )'\cdot ( ( c\cdot d )'\cdot ( c\cdot ( ( e\cdot f' )\cdot d' )' ) ) )'\cdot f = e\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP184-2.p} Orthogonal elements commute and form a subgroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p21}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p21}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p21}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\cdot \mbox{least\_upper\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP513-1.p} Axiom for Abelian group theory, in product and inverse, part 1






$a\cdot ( ( b\cdot c )\cdot ( a\cdot c )' ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP435-1.p} Axiom for group theory, in product \& inverse, part 3






$( ( ( ( ( a\cdot b )\cdot c )'\cdot a )\cdot b )\cdot ( d\cdot d' ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP488-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{identity}),\mbox{double\_divide}(b,c))),b),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP718-1.p} In a commutative RIF loop, all squares are Moufang elements






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot ( a\cdot b ) ) = ( ( a\cdot ( b\cdot c ) )\cdot a )\cdot b\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot a )\cdot ( ( b\cdot c )\cdot ( a\cdot a ) )\ne ( ( a\cdot a )\cdot b )\cdot ( c\cdot ( a\cdot a ) )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP607-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b',\mbox{double\_divide}(a,c))')',c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP037-3.p} In subgroups, the inverse of an element is unique






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{another\_identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{another\_left\_identity},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ a\cdot \mbox{another\_identity}{=}a\qquad\mbox{cnf}(\mbox{another\_right\_identity},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ a\cdot \mbox{another\_inverse}(a){=}\mbox{another\_identity}\qquad\mbox{cnf}(\mbox{another\_right\_inverse},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{another\_inverse}(a)\cdot a{=}\mbox{another\_identity}\qquad\mbox{cnf}(\mbox{another\_left\_inverse},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(\mbox{another\_inverse}(a))\qquad\mbox{cnf}(\mbox{another\_inverse\_in\_subgroup},\mbox{hypothesis})$

$( a\cdot b{=}c\mbox{ and }a\cdot d{=}c ) \ \Rightarrow \ d = b\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{hypothesis})$

$( a\cdot b{=}c\mbox{ and }d\cdot b{=}c ) \ \Rightarrow \ d = a\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(\mbox{another\_identity})\qquad\mbox{cnf}(\mbox{another\_identity\_in\_subgroup},\mbox{hypothesis})$

$a'\ne \mbox{another\_inverse}(a)\qquad\mbox{cnf}(\mbox{prove\_two\_inverses\_are\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{GRP139-1.p} Prove greatest lower-bound axiom using the GLB transformation

This problem proves the original axiom of anti-symmetry from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = c\qquad\mbox{cnf}(\mbox{ax\_glb1b\_1}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,c) = c\qquad\mbox{cnf}(\mbox{ax\_glb1b\_2}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b),c)\ne c\qquad\mbox{cnf}(\mbox{prove\_ax\_glb1b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP075-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(y,\mbox{identity})),\mbox{double\_divide}(\mbox{double\_divide}(z,\mbox{double\_divide}(u,\mbox{double\_divide}(u,\mbox{identity}))),\mbox{double\_divide}(x,\mbox{identity}))),y) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP158-1.p} Prove monotonicity axiom using the GLB transformation

This problem proves the original monotonicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_mono2b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(c\cdot a,c\cdot b)\ne c\cdot a\qquad\mbox{cnf}(\mbox{prove\_ax\_mono2b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP188-1.p} Consequence of lattice theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_p38a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP415-1.p} Axiom for group theory, in product \& inverse, part 1






$( a\cdot ( ( ( b\cdot a )'\cdot ( b\cdot c' ) )'\cdot ( a'\cdot a )' )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP582-1.p} Axiom for Abelian group theory, in double div and id, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},b),\mbox{double\_divide}(c,\mbox{double\_divide}(b,a)))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP600-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c',b)')') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP542-1.p} Axiom for Abelian group theory, in division and identity, part 2






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),a)),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP692-1.p} In a power associative conjugacy closed loop, c$\wedge$6 is extra






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) = \mbox{op\_d}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_d}\cdot \mbox{op\_d} = \mbox{op\_e}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_e}\cdot \mbox{op\_e} = \mbox{op\_f}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_e}\cdot ( a\cdot ( b\cdot \mbox{op\_e} ) )\ne ( ( \mbox{op\_e}\cdot a )\cdot b )\cdot \mbox{op\_e}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP167-3.p} Product of positive and negative parts






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$a\ne \mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{GRP710+1.p} A magma with 2-sided inverses satisfying the C-law is a loop - 1a

In a Bol loop, if c is a commutant element, then c$\wedge$2 is in the
 left nucleus if and only if c is in the right nucleus.




$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a{:}\;a\cdot i(a) = 1\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;i(a)\cdot a = 1\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;\exists x_2{:}\;x_0\cdot x_2 = x_1\mbox{ and }\forall x_3,x_4{:}\;\exists x_5{:}\;x_5\cdot x_4 = x_3\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP562-1.p} Axiom for Abelian group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b'),c),\mbox{divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP128-2.004.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP034-4.p} In subgroups, inverse is closed






$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{closure},\mbox{axiom})$

$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$( \mbox{subgroup\_member}(a)\mbox{ and }\mbox{subgroup\_member}(b)\mbox{ and }b\cdot a'{=}c ) \ \Rightarrow \ \mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{closure\_of\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(a')\qquad\mbox{cnf}(\mbox{prove\_inverse\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP180-2.p} Consequence of converting between GLB and LUB






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p11}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p11}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p11}_3,\mbox{hypothesis})$

$a\cdot ( \mbox{greatest\_lower\_bound}(a,b)'\cdot b )\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_p}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{GRP351-1.p} An identity generated by HR, number 22711






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_6\ne \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP067-1.p} Single axiom for group theory, in division and identity

This is a single axiom for group theory, in terms of division
 and identity




$\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(\mbox{divide}(\mbox{identity},x),z)))),z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP143-1.p} Prove greatest lower-bound axiom using the GLB transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b),a)\ne \mbox{greatest\_lower\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_glb2b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-4.004.p} a.(b.a) = (b.a).a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot y{=}z_2\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ y\cdot x{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_1\cdot y{=}z_2\mbox{ and }y\cdot x{=}z_1 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-3.004.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP181-1.p} Distributivity of a lattice






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = \mbox{greatest\_lower\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,c) = \mbox{least\_upper\_bound}(b,c)\qquad\mbox{cnf}(\mbox{p12}_2,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_p}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-2.003.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP129-2.004.p} a.(b.a) = (b.a).a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP723-1.p} In commutative A-loops of exp 2 square-subloop is associative






$a\cdot 1 = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot a = 1\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a,b) = \mbox{ld}(b,\mbox{ld}(a\cdot b,b))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$f(a,f(b,c))\ne f(f(a,b),c)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP531-1.p} Axiom for Abelian group theory, in division, part 3






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,c)),\mbox{divide}(a,b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP012-1.p} Inverse of products = Product of inverses

The inverse of products equals the product of the inverse,
 in opposite order.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_multiply\_b\_is\_c},\mbox{hypothesis})$

$b'\cdot a'{=}d\qquad\mbox{cnf}(\mbox{inverse\_b\_multiply\_inverse\_a\_is\_d},\mbox{hypothesis})$

$\neg\,c\cdot d{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_c\_multiply\_d\_is\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP058-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$x\cdot ( y\cdot ( ( ( z\cdot z' )\cdot ( u\cdot y )' )\cdot x ) )' = u\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP002-3.p} Commutator equals identity in groups of order 3

In a group, if (for all x) the cube of x is the identity
 (i.e. a group of order 3), then the equation [[x,y],y]=
 identity holds, where [x,y] is the product of x, y, the
 inverse of x and the inverse of y (i.e. the commutator
 of x and y).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{commutator}(x,y) = x\cdot ( y\cdot ( x'\cdot y' ) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$x\cdot ( x\cdot x ) = \mbox{identity}\qquad\mbox{cnf}(\mbox{x\_cubed\_is\_identity},\mbox{hypothesis})$

$\mbox{commutator}(\mbox{commutator}(a,b),b)\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_commutator},\mbox{negated\_conjecture})$

\medskip\textbf{GRP597-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c',b)')') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-4.003.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_2\cdot y{=}x\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_2\cdot y{=}x\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP733+1.p} Non-flexible non-commutative DTS loop.






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(c_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(c_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(c_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(c_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(c_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(c_{06},\mbox{axiom})$

$\forall a{:}\;a\cdot a = 1\qquad\mbox{fof}(c_{07},\mbox{axiom})$

$c\cdot d\ne d\cdot c\qquad\mbox{fof}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot a\ne a\cdot ( b\cdot a )\qquad\mbox{fof}(c_{09},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( x_0\cdot x_1 = x_2 \ \Rightarrow \ ( ( x_0\cdot x_2 = x_1\mbox{ and }x_1\cdot x_2 = x_0 )\mbox{ or }( x_0\cdot x_2 = x_1\mbox{ and }x_2\cdot x_1 = x_0 )\mbox{ or }( x_2\cdot x_0 = x_1\mbox{ and }x_2\cdot x_1 = x_0 ) ) )\qquad\mbox{fof}(c_{10},\mbox{axiom})$

\medskip\textbf{GRP673+1.p} Extra loop commutation property 2

In an extra loop, z commutes with [x;y; t] if and only if t
 commutes with [x;y; z] if and only if [x;y; z][x;y; t] =
 [x;y; zt].




$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot a ) ) = ( ( a\cdot b )\cdot c )\cdot a\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\mbox{op\_t}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{op\_t}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\mbox{op\_z}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t})\cdot \mbox{op\_z}\mbox{ and }\mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}\cdot \mbox{op\_t})\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP554-1.p} Axiom for Abelian group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(a,c))'),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP682+1.p} Axioms of rectangular loops - a






$\forall a{:}\;\mbox{ld}(a,a\cdot a) = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;\mbox{rd}(a\cdot a,a) = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = \mbox{ld}(a,a\cdot b)\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = \mbox{rd}(a\cdot b,b)\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(\mbox{ld}(a,b),\mbox{ld}(a,b)\cdot ( c\cdot d )) = \mbox{ld}(a,a\cdot c)\cdot d\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{rd}(( a\cdot b )\cdot \mbox{rd}(c,d),\mbox{rd}(c,d)) = a\cdot \mbox{rd}(b\cdot d,d)\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot \mbox{ld}(b,b)) = \mbox{rd}(\mbox{rd}(a,a)\cdot b,b)\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( \mbox{ld}(\mbox{ld}(x_0,x_1),\mbox{ld}(x_0,x_1)\cdot x_2) = \mbox{ld}(x_0,x_0\cdot x_2)\mbox{ and }\mbox{ld}(\mbox{rd}(x_0,x_1),\mbox{rd}(x_0,x_1)\cdot x_2) = \mbox{ld}(x_0,x_0\cdot x_2) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP606-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b',\mbox{double\_divide}(a,c))')',c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP120-1.p} Derive left identity from a single axiom for groups order 4






$y\cdot ( ( y\cdot ( ( y\cdot y )\cdot ( x\cdot z ) ) )\cdot ( z\cdot ( z\cdot z ) ) ) = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$\mbox{identity}\cdot \mbox{identity} = \mbox{identity}\qquad\mbox{cnf}(\mbox{single\_axiom}_2,\mbox{axiom})$

$\mbox{identity}\cdot a\ne a\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP668-1.p} Flexible C-loops are ARIF






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c )\ne ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP206-1.p} In Loops, Moufang-4 =$>$ Moufang-1.






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot \mbox{left\_division}(x,y) = y\qquad\mbox{cnf}(\mbox{multiply\_left\_division},\mbox{axiom})$

$\mbox{left\_division}(x,x\cdot y) = y\qquad\mbox{cnf}(\mbox{left\_division\_multiply},\mbox{axiom})$

$\mbox{right\_division}(x,y)\cdot y = x\qquad\mbox{cnf}(\mbox{multiply\_right\_division},\mbox{axiom})$

$\mbox{right\_division}(x\cdot y,y) = x\qquad\mbox{cnf}(\mbox{right\_division\_multiply},\mbox{axiom})$

$x\cdot \mbox{right\_inverse}(x) = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$x\cdot ( ( y\cdot z )\cdot x ) = ( x\cdot y )\cdot ( z\cdot x )\qquad\mbox{cnf}(\mbox{moufang}_4,\mbox{axiom})$

$( a\cdot ( b\cdot c ) )\cdot a\ne ( a\cdot b )\cdot ( c\cdot a )\qquad\mbox{cnf}(\mbox{prove\_moufang}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP193-2.p} A combination of distributivity and monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p8\_9b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p8\_9b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},c) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p8\_9b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p8\_9b}_4,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b\cdot c),\mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c)) = \mbox{greatest\_lower\_bound}(a,b\cdot c)\qquad\mbox{cnf}(\mbox{p8\_9b}_5,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b\cdot c)\ne \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{prove\_p8\_9b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP425-1.p} Axiom for group theory, in product \& inverse, part 2






$( ( a\cdot ( b'\cdot c )' )'\cdot ( a\cdot c' ) )'\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP709-1.p} Bol loop commutant element squared in left and right nucleus - 2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{op\_c} ) = ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot ( a\cdot b )\ne ( ( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot a )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-6.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,x,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,y,x)\qquad\mbox{cnf}(\mbox{qg2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP485-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP667+1.p} 2-divisible ARIF loops are Moufang






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a{:}\;f(a)\cdot f(a) = a\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( x_0\cdot ( x_1\cdot ( x_2\cdot x_1 ) ) = ( ( x_0\cdot x_1 )\cdot x_2 )\cdot x_1 \ \Rightarrow \ x_1\cdot ( x_0\cdot ( x_1\cdot x_2 ) ) = ( ( x_1\cdot x_0 )\cdot x_1 )\cdot x_2 )\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall x_3,x_4,x_5{:}\;( ( x_3\cdot x_4 )\cdot ( x_5\cdot x_3 ) = ( x_3\cdot ( x_4\cdot x_5 ) )\cdot x_3 \ \Rightarrow \ x_3\cdot ( x_4\cdot ( x_3\cdot x_5 ) ) = ( ( x_3\cdot x_4 )\cdot x_3 )\cdot x_5 )\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall x_6,x_7,x_8{:}\;( ( x_6\cdot x_7 )\cdot ( x_8\cdot x_6 ) = x_6\cdot ( ( x_7\cdot x_8 )\cdot x_6 ) \ \Rightarrow \ x_6\cdot ( x_7\cdot ( x_6\cdot x_8 ) ) = ( ( x_6\cdot x_7 )\cdot x_6 )\cdot x_8 )\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP124-2.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP182-2.p} Positive part of the negative part is identity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p17a}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p17a}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p17a}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},\mbox{greatest\_lower\_bound}(a,\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p17a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP192-1.p} Even elements implies trivial group

The assumption all(X,1 =$<$ X) even implies that the group is
 trivial, i.e., all(X, X = 1).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},x) = x\qquad\mbox{cnf}(\mbox{p40a}_1,\mbox{hypothesis})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_p40a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP660+3.p} Bol-Moufang identity 3 implies a unit element






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot a = a\cdot ( b\cdot ( c\cdot a ) )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{ld}(x_1,x_1) = x_0\mbox{ and }\mbox{ld}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP744-1.p} Biassociative non-associative commutative loop of exponent 2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot a = 1\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot ( ( a\cdot ( b\cdot c ) )\cdot c ) = ( a\cdot ( ( a\cdot b )\cdot c ) )\cdot c\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

\medskip\textbf{GRP213-1.p} An identity generated by HR, number 00385






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( x_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }x_5\cdot x_6 = \mbox{sk\_c}_8\mbox{ and }x_5' = x_6\mbox{ and }\mbox{sk\_c}_8' = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_2 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_8 = x_2 ) \ \Rightarrow \ x_3'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP112-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for groups in which the square
 of every element is the identity, in terms of product and
 inverse.




$( ( x\cdot y )\cdot z )\cdot ( x\cdot z ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot a_4\ne b_4\cdot b_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP567-1.p} Axiom for Abelian group theory, in double div and id, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(\mbox{identity},c))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP333-1.p} An identity generated by HR, number 19058






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{GRP577-1.p} Axiom for Abelian group theory, in double div and id, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,a),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP654+2.p} A quasigroup satisfying Moufang 1 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{rd}(x_1,x_1) = x_0\mbox{ and }\mbox{rd}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP549-1.p} Axiom for Abelian group theory, in division and identity, part 1






$\mbox{divide}(\mbox{divide}(\mbox{identity},a),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,a),c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP557-1.p} Axiom for Abelian group theory, in division and inverse, part 1






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(b,c),\mbox{divide}(a,c))') = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP451-1.p} Axiom for group theory, in division, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP100-1.p} Single axiom for Abelian group theory, in double div and id

This is a single axiom for Abelian group theory, in terms
 of double division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(y,\mbox{double\_divide}(x,z)),\mbox{double\_divide}(z,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP154-1.p} Prove monotonicity axiom using the LUB transformation

This problem proves the original mononicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_mono1a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a\cdot c,b\cdot c)\ne b\cdot c\qquad\mbox{cnf}(\mbox{prove\_ax\_mono1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP423-1.p} Axiom for group theory, in product \& inverse, part 3






$( ( a\cdot ( b'\cdot ( c'\cdot ( c'\cdot c )' ) )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP615-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b')',c)',\mbox{double\_divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP127-3.004.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP743-1.p} Biassociative non-associative Steiner loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot ( ( a\cdot ( b\cdot c ) )\cdot c ) = ( a\cdot ( ( a\cdot b )\cdot c ) )\cdot c\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

\medskip\textbf{GRP270-1.p} An identity generated by HR, number 01179






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP137-1.p} Prove anti-symmetry axiom using the GLB transformation

This problem proves the original anti-symmetry axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{ax\_antisymb}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_antisymb}_2,\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_ax\_antisymb},\mbox{negated\_conjecture})$

\medskip\textbf{GRP124-6.005.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4)\mbox{ or }\mbox{product}_1(x,y,e_5) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4)\mbox{ or }\mbox{product}_2(x,y,e_5) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,x,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,y,x)\qquad\mbox{cnf}(\mbox{qg2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP700+1.p} Variety of power associative, WIP conjugacy closed loops - 2c






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot a )\cdot ( a\cdot c ) = a\cdot ( ( ( b\cdot a )\cdot a )\cdot c )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot ( b\cdot ( c\cdot b ) ) = ( a\cdot ( b\cdot ( b\cdot c ) ) )\cdot b\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall x_0{:}\;\exists x_1{:}\;( x_1\cdot x_0 = 1\mbox{ and }x_0\cdot x_1 = 1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP661-1.p} Conjugacy closed with ab = 1 implies ba is in the nucleus - a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d} = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( \mbox{op\_d}\cdot \mbox{op\_c} )\cdot ( a\cdot b )\ne ( ( \mbox{op\_d}\cdot \mbox{op\_c} )\cdot a )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP133-1.004.p} (a.b).(b.a) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP748+1.p} Right alternative loop rings: a lemma






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b )\cdot i(b) = a\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a{:}\;a\cdot i(a) = 1\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall a{:}\;i(a)\cdot a = 1\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b = b\cdot a\mbox{ or }i(a)\cdot ( a\cdot b ) = b )\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;x_2\cdot ( x_0\cdot ( x_2\cdot x_1 ) ) = ( ( x_2\cdot x_0 )\cdot x_2 )\cdot x_1\mbox{ or }\forall x_3,x_4,x_5{:}\;x_3\cdot ( x_5\cdot ( x_4\cdot x_5 ) ) = ( ( x_3\cdot x_5 )\cdot x_4 )\cdot x_5\mbox{ or }\forall x_6,x_7,x_8{:}\;( x_8\cdot x_6 )\cdot ( x_7\cdot x_8 ) = ( x_8\cdot ( x_6\cdot x_7 ) )\cdot x_8\mbox{ or }\forall x_9,x_{10},x_{11}{:}\;( x_{11}\cdot x_9 )\cdot ( x_{10}\cdot x_{11} ) = x_{11}\cdot ( ( x_9\cdot x_{10} )\cdot x_{11} )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP274-1.p} An identity generated by HR, number 01608






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP023-2.p} The inverse of the identity is the identity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{identity}'\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_id\_is\_id},\mbox{negated\_conjecture})$

\medskip\textbf{GRP142-1.p} Prove greatest lower-bound axiom using the LUB transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b),a)\ne a\qquad\mbox{cnf}(\mbox{prove\_ax\_glb2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-2.005.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP074-1.p} Single axiom for group theory, in division and inverse

This is a single axiom for group theory, in terms of division
 and inverse.




$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),y),\mbox{divide}(z,\mbox{divide}(y,u)))',u) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP001+6.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$\forall e{:}\;( ( \forall x,y{:}\;\exists z{:}\;x\cdot y{=}z\mbox{ and }\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w )\mbox{ and }\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w )\mbox{ and }\forall x{:}\;x\cdot e{=}x\mbox{ and }\forall x{:}\;e\cdot x{=}x\mbox{ and }\forall x{:}\;x\cdot x'{=}e\mbox{ and }\forall x{:}\;x'\cdot x{=}e ) \ \Rightarrow \ ( \forall x{:}\;x\cdot x{=}e \ \Rightarrow \ \forall u,v,w{:}\;( u\cdot v{=}w \ \Rightarrow \ v\cdot u{=}w ) ) )\qquad\mbox{fof}(\mbox{commutativity},\mbox{conjecture})$

\medskip\textbf{GRP022-2.p} Inverse is an involution






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$( a' )'\ne a\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_inverse\_is\_original},\mbox{negated\_conjecture})$

\medskip\textbf{GRP588-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c')',b)') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP151-1.p} Prove least upper-bound axiom using the GLB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,\mbox{least\_upper\_bound}(a,b))\ne a\qquad\mbox{cnf}(\mbox{prove\_ax\_lub2b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP201-1.p} In Loops, Moufang-2 =$>$ Moufang-3.






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot \mbox{left\_division}(x,y) = y\qquad\mbox{cnf}(\mbox{multiply\_left\_division},\mbox{axiom})$

$\mbox{left\_division}(x,x\cdot y) = y\qquad\mbox{cnf}(\mbox{left\_division\_multiply},\mbox{axiom})$

$\mbox{right\_division}(x,y)\cdot y = x\qquad\mbox{cnf}(\mbox{multiply\_right\_division},\mbox{axiom})$

$\mbox{right\_division}(x\cdot y,y) = x\qquad\mbox{cnf}(\mbox{right\_division\_multiply},\mbox{axiom})$

$x\cdot \mbox{right\_inverse}(x) = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( ( x\cdot y )\cdot z )\cdot y = x\cdot ( y\cdot ( z\cdot y ) )\qquad\mbox{cnf}(\mbox{moufang}_2,\mbox{axiom})$

$( ( a\cdot b )\cdot a )\cdot c\ne a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(\mbox{prove\_moufang}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP765-1.p} Buchsteiner loop lemma 2






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(x_0))\cdot x_1\ne \mbox{eta}(x_0)\cdot ( x_0\cdot x_1 )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP027-1.p} All groups of order 5 are cyclic

There exists an element in G that generates all other
 elements by taking powers of that element.




$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

$\mbox{group\_member}(a,g)\qquad\mbox{cnf}(\mbox{a\_in\_group},\mbox{hypothesis})$

$\mbox{group\_member}(b,g)\qquad\mbox{cnf}(\mbox{b\_in\_group},\mbox{hypothesis})$

$\mbox{group\_member}(c,g)\qquad\mbox{cnf}(\mbox{c\_in\_group},\mbox{hypothesis})$

$\mbox{group\_member}(d,g)\qquad\mbox{cnf}(\mbox{d\_in\_group},\mbox{hypothesis})$

$\mbox{group\_member}(i,g)\qquad\mbox{cnf}(\mbox{i\_in\_group},\mbox{hypothesis})$

$\mbox{identity\_for}(g) = i\qquad\mbox{cnf}(\mbox{i\_is\_identity},\mbox{hypothesis})$

$\mbox{group\_member}(x,g) \ \Rightarrow \ ( x = a\mbox{ or }x = b\mbox{ or }x = c\mbox{ or }x = d\mbox{ or }x = i )\qquad\mbox{cnf}(\mbox{all\_of\_group},\mbox{hypothesis})$

$m(g,x,m(g,x,m(g,x,m(g,x,x)))) = i\qquad\mbox{cnf}(\mbox{multiplication\_to\_identity},\mbox{hypothesis})$

$\mbox{not\_power\_of}(g,x)\ne x\qquad\mbox{cnf}(\mbox{all\_multiply\_to\_identity},\mbox{hypothesis})$

$\neg\,g\cdot x{=}x\qquad\mbox{cnf}(\mbox{x2\_is\_not\_power},\mbox{negated\_conjecture})$

$\neg\,g\cdot x{=}m(g,x,x)\qquad\mbox{cnf}(\mbox{x3\_is\_not\_power},\mbox{negated\_conjecture})$

$\neg\,g\cdot x{=}m(g,x,m(g,x,x))\qquad\mbox{cnf}(\mbox{x4\_is\_not\_power},\mbox{negated\_conjecture})$

$\neg\,g\cdot x{=}m(g,x,m(g,x,m(g,x,x)))\qquad\mbox{cnf}(\mbox{x5\_is\_not\_power},\mbox{negated\_conjecture})$

\medskip\textbf{GRP701-1.p} Variety of power associative, WIP conjugacy closed loops - 3






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot a )\cdot ( a\cdot c ) = a\cdot ( ( ( b\cdot a )\cdot a )\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot ( b\cdot ( c\cdot b ) ) = ( a\cdot ( b\cdot ( b\cdot c ) ) )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot i(b\cdot a)\ne i(b)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP131-2.002.p} (3,2,1) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP049-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$z\cdot ( ( ( z\cdot y )'\cdot x )'\cdot ( y\cdot ( y'\cdot y ) )' )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP489-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{identity}),\mbox{double\_divide}(b,c))),b),\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP442-1.p} Axiom for group theory, in product \& inverse, part 1






$( a\cdot ( b\cdot ( ( c\cdot c' )\cdot ( d\cdot ( a\cdot b ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP686-1.p} x(y.yz) = (x.yy)z is equivalent to xx.yz = (x.xy)z part 1






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( b\cdot c ) ) = ( a\cdot ( b\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c )\ne ( a\cdot ( a\cdot b ) )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP667-3.p} 2-divisible ARIF loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot b )\cdot c ) = ( a\cdot ( ( b\cdot c )\cdot b ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot a = a\cdot ( b\cdot a )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a)\cdot f(a) = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) )\ne ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP166-2.p} Multiplication with a positive element increases a value






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,\mbox{identity}) = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat2b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,\mbox{identity}) = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat2b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,a\cdot b)\ne a\qquad\mbox{cnf}(\mbox{prove\_lat2b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-8.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_3,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP436-1.p} Axiom for group theory, in product \& inverse, part 1






$a\cdot ( b\cdot ( c\cdot ( ( c'\cdot ( d\cdot b )' )\cdot a ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP092-1.p} Single axiom for Abelian group theory, in division

This is a single axiom for Abelian group theory, in terms
 of division




$\mbox{divide}(\mbox{divide}(x,y),\mbox{divide}(\mbox{divide}(x,z),y)) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP546-1.p} Axiom for Abelian group theory, in division and identity, part 2






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,b)),\mbox{divide}(\mbox{divide}(b,c),a)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP002-1.p} Commutator equals identity in groups of order 3

In a group, if (for all x) the cube of x is the identity
 (i.e. a group of order 3), then the equation [[x,y],y]=
 identity holds, where [x,y] is the product of x, y, the
 inverse of x and the inverse of y (i.e. the commutator
 of x and y).




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$x\cdot x{=}y \ \Rightarrow \ x\cdot y{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{x\_cubed\_is\_identity}_1,\mbox{hypothesis})$

$x\cdot x{=}y \ \Rightarrow \ y\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{x\_cubed\_is\_identity}_2,\mbox{hypothesis})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{negated\_conjecture})$

$c\cdot a'{=}d\qquad\mbox{cnf}(\mbox{c\_times\_inverse\_a\_is\_d},\mbox{negated\_conjecture})$

$d\cdot b'{=}h\qquad\mbox{cnf}(\mbox{d\_times\_inverse\_b\_is\_h},\mbox{negated\_conjecture})$

$h\cdot b{=}j\qquad\mbox{cnf}(\mbox{h\_times\_b\_is\_j},\mbox{negated\_conjecture})$

$j\cdot h'{=}k\qquad\mbox{cnf}(\mbox{j\_times\_inverse\_h\_is\_k},\mbox{negated\_conjecture})$

$\neg\,k\cdot b'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_k\_times\_inverse\_b\_is\_e},\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-1.005.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP544-1.p} Axiom for Abelian group theory, in division and identity, part 4






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),a)),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP741-1.p} Proper Moufang loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP761+1.p} Non-discrete partially ordered group






$\forall a{:}\;a\cdot i(a) = e\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot e = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall a,b,c{:}\;a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a{:}\;o(a,a)\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a,b{:}\;( ( a\ne b\mbox{ and }o(a,b) ) \ \Rightarrow \ \neg\,o(b,a) )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a,b,c{:}\;( ( o(a,b)\mbox{ and }o(b,c) ) \ \Rightarrow \ o(a,c) )\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( o(a,b)\mbox{ and }o(c,d) ) \ \Rightarrow \ o(a\cdot c,b\cdot d) )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$a\ne e\mbox{ and }o(e,a)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

\medskip\textbf{GRP111-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y')',z)',\mbox{double\_divide}(x,z)) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP107-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,y),\mbox{double\_divide}(x,\mbox{double\_divide}(z',y)')') = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP578-1.p} Axiom for Abelian group theory, in double div and id, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,a),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP036-3.p} In subgroups, the identity element is unique






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{another\_identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{another\_left\_identity},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ a\cdot \mbox{another\_identity}{=}a\qquad\mbox{cnf}(\mbox{another\_right\_identity},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ a\cdot \mbox{another\_inverse}(a){=}\mbox{another\_identity}\qquad\mbox{cnf}(\mbox{another\_right\_inverse},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{another\_inverse}(a)\cdot a{=}\mbox{another\_identity}\qquad\mbox{cnf}(\mbox{another\_left\_inverse},\mbox{hypothesis})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(\mbox{another\_inverse}(a))\qquad\mbox{cnf}(\mbox{another\_inverse\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(\mbox{another\_identity})\qquad\mbox{cnf}(\mbox{another\_identity\_in\_subgroup},\mbox{hypothesis})$

$\mbox{identity}\ne \mbox{another\_identity}\qquad\mbox{cnf}(\mbox{prove\_identity\_equals\_another\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP071-1.p} Single axiom for group theory, in division and inverse

This is a single axiom for group theory, in terms of division
 and inverse.




$\mbox{divide}(\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(z,u)))',\mbox{divide}(\mbox{divide}(u,z),x)) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP006-1.p} Inverse is in this group

If S is a non-empty subset of a group such that if X, Y belong
 to S, the XY$\wedge$-1 belongs to S, then S contains X$\wedge$-1 whenever it
 contains X.




$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( \mbox{an\_element}(x)\mbox{ and }\mbox{an\_element}(y)\mbox{ and }x\cdot y'{=}z ) \ \Rightarrow \ \mbox{an\_element}(z)\qquad\mbox{cnf}(\mbox{condition},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\mbox{an\_element}(\mbox{the\_element})\qquad\mbox{cnf}(\mbox{element\_of\_set},\mbox{hypothesis})$

$\neg\,\mbox{an\_element}(\mbox{the\_element}')\qquad\mbox{cnf}(\mbox{prove\_b\_inverse\_is\_in\_set},\mbox{negated\_conjecture})$

\medskip\textbf{GRP146-1.p} Prove least upper-bound axiom using the LUB transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,c) = c\qquad\mbox{cnf}(\mbox{ax\_lub1a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,c) = c\qquad\mbox{cnf}(\mbox{ax\_lub1a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{least\_upper\_bound}(a,b),c)\ne c\qquad\mbox{cnf}(\mbox{prove\_ax\_lub1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP038-3.p} In subgroups, if a and b are members, then a.b$\wedge$-1 is a member






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a) \ \Rightarrow \ \mbox{subgroup\_member}(a')\qquad\mbox{cnf}(\mbox{closure\_of\_inverse},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_is\_in\_subgroup},\mbox{axiom})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{hypothesis})$

$a\cdot b'{=}c\qquad\mbox{cnf}(\mbox{a\_times\_inverse\_b\_is\_c},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{prove\_c\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP614-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b')',c)',\mbox{double\_divide}(a,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP762+1.p} Linearly ordered group






$\forall a{:}\;a\cdot i(a) = e\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot e = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall a,b,c{:}\;a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a,b{:}\;a\cdot b = b\cdot a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;o(a,a)\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a,b{:}\;( ( a\ne b\mbox{ and }o(a,b) ) \ \Rightarrow \ \neg\,o(b,a) )\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall a,b,c{:}\;( ( o(a,b)\mbox{ and }o(b,c) ) \ \Rightarrow \ o(a,c) )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( o(a,b)\mbox{ and }o(c,d) ) \ \Rightarrow \ o(a\cdot c,b\cdot d) )\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a,b{:}\;( o(a,b)\mbox{ or }o(b,a) )\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$a\ne e\qquad\mbox{fof}(f_{10},\mbox{axiom})$

\medskip\textbf{GRP065-1.p} Single axiom for group theory, in division

This is a single axiom for group theory, in terms of division




$\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),\mbox{divide}(x,\mbox{divide}(y,\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),x),z)))),z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP408-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b'\cdot ( b'\cdot b ) ) )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP109-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(y',\mbox{double\_divide}(x,z))')',z) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP665+1.p} Conjugacy closed implies commutant in the nucleus






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a{:}\;\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall x_0,x_1{:}\;( \mbox{op\_c}\cdot ( x_0\cdot x_1 ) = ( \mbox{op\_c}\cdot x_0 )\cdot x_1\mbox{ and }( x_0\cdot \mbox{op\_c} )\cdot x_1 = x_0\cdot ( \mbox{op\_c}\cdot x_1 )\mbox{ and }( x_0\cdot x_1 )\cdot \mbox{op\_c} = x_0\cdot ( x_1\cdot \mbox{op\_c} ) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP458-1.p} Axiom for group theory, in division and identity, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{identity},a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-1.004.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP514-1.p} Axiom for Abelian group theory, in product and inverse, part 2






$a\cdot ( ( b\cdot c )\cdot ( a\cdot c )' ) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP677-1.p} In CC-loops, associators are in the center of the nucleus - 1c






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{asoc}(c,d,e) )\ne ( a\cdot b )\cdot \mbox{asoc}(c,d,e)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP179-3.p} For converting between GLB and LUB






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p18}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p18}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p18}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a',\mbox{identity})\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\qquad\mbox{cnf}(\mbox{prove\_p}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{GRP186-2.p} Application of distributivity and group theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p23}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p23}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p23}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\ne a\cdot \mbox{greatest\_lower\_bound}(a,b')'\qquad\mbox{cnf}(\mbox{prove\_p}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{GRP419-1.p} Axiom for group theory, in product \& inverse, part 2






$( ( a\cdot ( b'\cdot ( c\cdot ( c'\cdot c )' )' )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP711+1.p} A magma with 2-sided inverses satisfying the C-law is a loop - 1b






$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a{:}\;a\cdot i(a) = 1\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;i(a)\cdot a = 1\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_6,x_7,x_8{:}\;( ( x_6\cdot x_7 = x_6\cdot x_8 \ \Rightarrow \ x_7 = x_8 )\mbox{ and }( x_7\cdot x_6 = x_8\cdot x_6 \ \Rightarrow \ x_7 = x_8 ) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP124-1.004.p} (3,1,2) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*a=b iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP579-1.p} Axiom for Abelian group theory, in double div and id, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(b,a),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP348-1.p} An identity generated by HR, number 19633






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\mbox{ and }x_2' = \mbox{sk\_c}_6\mbox{ and }x_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_4' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_1\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP602-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{double\_divide}(a,c))')',c)' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP705-1.p} Property of commutative C-loop

In a commutative C-loop, if a has order 4 and b has order 9, then
 a.bx = ab.x




$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{op\_a}\cdot ( \mbox{op\_a}\cdot ( \mbox{op\_a}\cdot \mbox{op\_a} ) ) = 1\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot ( \mbox{op\_b}\cdot \mbox{op\_b} ) ) ) ) ) ) ) = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_a}\cdot ( \mbox{op\_b}\cdot a )\ne ( \mbox{op\_a}\cdot \mbox{op\_b} )\cdot a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP002-4.p} Commutator equals identity in groups of order 3

In a group, if (for all x) the cube of x is the identity
 (i.e. a group of order 3), then the equation [[x,y],y]=
 identity holds, where [x,y] is the product of x, y, the
 inverse of x and the inverse of y (i.e. the commutator
 of x and y).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{commutator}(x,y) = x\cdot ( y\cdot ( x'\cdot y' ) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$x\cdot ( x\cdot x ) = \mbox{identity}\qquad\mbox{cnf}(\mbox{x\_cubed\_is\_identity},\mbox{hypothesis})$

$\mbox{commutator}(\mbox{commutator}(a,b),b)\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_commutator},\mbox{negated\_conjecture})$

\medskip\textbf{GRP188-2.p} Consequence of lattice theory






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p38a}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p38a}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p38a}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,\mbox{least\_upper\_bound}(a,b))\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_p38a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP477-1.p} Axiom for group theory, in division and inverse, part 3






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),\mbox{divide}(d,c))',\mbox{divide}(b,a)) = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP672+1.p} Extra loop commutation property 1

In an extra loop, z commutes with [x;y; t] if and only if t
 commutes with [x;y; z] if and only if [x;y; z][x;y; t] =
 [x;y; zt].




$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot a ) ) = ( ( a\cdot b )\cdot c )\cdot a\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\mbox{op\_z}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t})\cdot \mbox{op\_z}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\mbox{op\_t}\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{op\_t}\mbox{ and }\mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z})\cdot \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_t}) = \mbox{asoc}(\mbox{op\_x},\mbox{op\_y},\mbox{op\_z}\cdot \mbox{op\_t})\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP581-1.p} Axiom for Abelian group theory, in double div and id, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},b),\mbox{double\_divide}(c,\mbox{double\_divide}(b,a)))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP099-1.p} Single axiom for Abelian group theory, in double div and id

This is a single axiom for Abelian group theory, in terms
 of double division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(y,\mbox{double\_divide}(x,z)),\mbox{double\_divide}(\mbox{identity},z))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP608-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b',\mbox{double\_divide}(a,c))')',c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP666+1.p} Inverse property A-loops are Moufang






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall b,a{:}\;i(a)\cdot ( a\cdot b ) = b\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;( a\cdot b )\cdot i(b) = a\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( x_0\cdot ( x_1\cdot ( x_2\cdot x_1 ) ) = ( ( x_0\cdot x_1 )\cdot x_2 )\cdot x_1 \ \Rightarrow \ x_1\cdot ( x_0\cdot ( x_1\cdot x_2 ) ) = ( ( x_1\cdot x_0 )\cdot x_1 )\cdot x_2 )\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$\forall x_3,x_4,x_5{:}\;( ( x_3\cdot x_4 )\cdot ( x_5\cdot x_3 ) = ( x_3\cdot ( x_4\cdot x_5 ) )\cdot x_3 \ \Rightarrow \ x_3\cdot ( x_4\cdot ( x_3\cdot x_5 ) ) = ( ( x_3\cdot x_4 )\cdot x_3 )\cdot x_5 )\qquad\mbox{fof}(f_{13},\mbox{axiom})$

$\forall x_6,x_7,x_8{:}\;( ( x_6\cdot x_7 )\cdot ( x_8\cdot x_6 ) = x_6\cdot ( ( x_7\cdot x_8 )\cdot x_6 ) \ \Rightarrow \ x_6\cdot ( x_7\cdot ( x_6\cdot x_8 ) ) = ( ( x_6\cdot x_7 )\cdot x_6 )\cdot x_8 )\qquad\mbox{fof}(f_{14},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP127-1.004.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP173-1.p} Each subgroup of negative elements is trivial






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p05a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},a') = \mbox{identity}\qquad\mbox{cnf}(\mbox{p05a}_2,\mbox{hypothesis})$

$\mbox{identity}\ne a\qquad\mbox{cnf}(\mbox{prove\_p05a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP167-1.p} Product of positive and negative parts

Each element in a lattice ordered group can be stated as a
 product of it's positive and it's negative part.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{positive\_part}(x) = \mbox{least\_upper\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_1,\mbox{axiom})$

$\mbox{negative\_part}(x) = \mbox{greatest\_lower\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_2,\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,\mbox{greatest\_lower\_bound}(y,z)) = \mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(x,y),\mbox{least\_upper\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_3,\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,\mbox{least\_upper\_bound}(y,z)) = \mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(x,y),\mbox{greatest\_lower\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_4,\mbox{axiom})$

$a\ne \mbox{positive\_part}(a)\cdot \mbox{negative\_part}(a)\qquad\mbox{cnf}(\mbox{prove\_lat}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP182-4.p} Positive part of the negative part is identity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p17b}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p17b}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p17b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},\mbox{least\_upper\_bound}(a,\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p17b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP059-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( ( ( ( ( x\cdot y )\cdot z )'\cdot x )\cdot y )\cdot ( u\cdot u' ) )' = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP133-2.003.p} (a.b).(b.a) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP131-1.002.p} (3,2,1) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-4.003.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_1\cdot z_2{=}x\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ x\cdot y{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_1\cdot z_2{=}x\mbox{ and }x\cdot y{=}z_1 ) \ \Rightarrow \ y\cdot x{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP132-1.002.p} (3,1,2) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP585-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c')',b)') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP031-1.p} In semigroups, left inverse and id =$>$ right inverse exists

If there are left inverses and left identity, then every
 element has a right inverse.




$\mbox{include}(\mbox{'Axioms/GRP002-0.ax'})$

$\mbox{identity}\cdot a{=}a\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{hypothesis})$

$a'\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{hypothesis})$

$\neg\,a\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_a\_has\_an\_inverse},\mbox{negated\_conjecture})$

\medskip\textbf{GRP510-1.p} Axiom for Abelian group theory, in product and inverse, part 2






$( ( a\cdot b )\cdot c )\cdot ( a\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP482-1.p} Axiom for group theory, in double division and identity, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity})),\mbox{double\_divide}(\mbox{double\_divide}(c,\mbox{double\_divide}(d,\mbox{double\_divide}(d,\mbox{identity}))),\mbox{double\_divide}(a,\mbox{identity}))),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP009-1.p} The left inverse of an element is unique






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot b{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{a\_is\_an\_inverse\_of\_b},\mbox{hypothesis})$

$c\cdot b{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{c\_is\_an\_inverse\_of\_b},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{prove\_a\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP025-1.p} All groups of order 2 are isomorphic

If G1 has exactly two elements and G2 has exactly two
 elements, then there exists an isomorphism [a one-to-one and
 onto homomorphism] between them.




$\mbox{include}(\mbox{'Axioms/GRP006-0.ax'})$

$\mbox{group\_member}(a,g_1)\qquad\mbox{cnf}(\mbox{a\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(b,g_1)\qquad\mbox{cnf}(\mbox{b\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(c,g_2)\qquad\mbox{cnf}(\mbox{c\_member\_of\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(d,g_2)\qquad\mbox{cnf}(\mbox{d\_member\_of\_group}_2,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_1) \ \Rightarrow \ ( x = a\mbox{ or }x = b )\qquad\mbox{cnf}(\mbox{a\_and\_b\_only\_members\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(x,g_2) \ \Rightarrow \ ( x = c\mbox{ or }x = d )\qquad\mbox{cnf}(\mbox{c\_and\_d\_only\_members\_of\_group}_2,\mbox{hypothesis})$

$g_1\cdot a{=}a\qquad\mbox{cnf}(\mbox{a\_times\_a\_is\_a},\mbox{hypothesis})$

$g_1\cdot a{=}b\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_b},\mbox{hypothesis})$

$g_1\cdot b{=}a\qquad\mbox{cnf}(\mbox{b\_times\_a\_is\_b},\mbox{hypothesis})$

$g_1\cdot b{=}b\qquad\mbox{cnf}(\mbox{b\_times\_b\_is\_a},\mbox{hypothesis})$

$g_2\cdot c{=}c\qquad\mbox{cnf}(\mbox{c\_times\_c\_is\_c},\mbox{hypothesis})$

$g_2\cdot c{=}d\qquad\mbox{cnf}(\mbox{c\_times\_d\_is\_d},\mbox{hypothesis})$

$g_2\cdot d{=}c\qquad\mbox{cnf}(\mbox{d\_times\_c\_is\_d},\mbox{hypothesis})$

$g_2\cdot d{=}d\qquad\mbox{cnf}(\mbox{d\_times\_d\_is\_c},\mbox{hypothesis})$

$\mbox{an\_isomorphism}(a) = c\qquad\mbox{cnf}(\mbox{a\_maps\_to\_c},\mbox{hypothesis})$

$\mbox{an\_isomorphism}(b) = d\qquad\mbox{cnf}(\mbox{b\_maps\_to\_d},\mbox{hypothesis})$

$\mbox{group\_member}(d_1,g_1)\qquad\mbox{cnf}(\mbox{d1\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(d_2,g_1)\qquad\mbox{cnf}(\mbox{d2\_member\_of\_group}_1,\mbox{hypothesis})$

$\mbox{group\_member}(d_3,g_1)\qquad\mbox{cnf}(\mbox{d3\_member\_of\_group}_1,\mbox{hypothesis})$

$g_1\cdot d_1{=}d_2\qquad\mbox{cnf}(\mbox{d1\_times\_d2\_is\_d}_3,\mbox{hypothesis})$

$\neg\,g_2\cdot \mbox{an\_isomorphism}(d_1){=}\mbox{an\_isomorphism}(d_2)\qquad\mbox{cnf}(\mbox{prove\_product\_holds\_in\_group}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP610-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b)',c)',\mbox{double\_divide}(a,c))' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP456-1.p} Axiom for group theory, in division and identity, part 3






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP121-1.p} Derive right identity from a single axiom for groups order 4






$y\cdot ( ( y\cdot ( ( y\cdot y )\cdot ( x\cdot z ) ) )\cdot ( z\cdot ( z\cdot z ) ) ) = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$\mbox{identity}\cdot \mbox{identity} = \mbox{identity}\qquad\mbox{cnf}(\mbox{single\_axiom}_2,\mbox{axiom})$

$a\cdot \mbox{identity}\ne a\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP007-1.p} The identity element is unique






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$c\cdot a{=}a\qquad\mbox{cnf}(\mbox{another\_left\_identity},\mbox{hypothesis})$

$a\cdot c{=}a\qquad\mbox{cnf}(\mbox{another\_right\_identity},\mbox{hypothesis})$

$\mbox{identity}\ne c\qquad\mbox{cnf}(\mbox{prove\_identity\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP518-1.p} Axiom for Abelian group theory, in product and inverse, part 2






$a\cdot ( ( ( a\cdot b )'\cdot c )\cdot b ) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP135-1.002.p} ((b.a).b).b) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 2 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP180-1.p} Consequence of converting between GLB and LUB






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$a\cdot ( \mbox{greatest\_lower\_bound}(a,b)'\cdot b )\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_p}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{GRP728-1.p} Bruck loops that are centrally nilpotent - 2nd easy part a

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - 2nd easy part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot b = ( b\cdot a )\cdot \mbox{op\_k}(a,b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e) = 1\qquad\mbox{cnf}(c_{21},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{asoc}(c,d,e)) = 1\qquad\mbox{cnf}(c_{22},\mbox{axiom})$

$\mbox{op\_k}(\mbox{op\_k}(a,b),c)\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP666-4.p} Inverse property A-loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP185-4.p} Application of monotonicity and distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p22b}_1,\mbox{hypothesis})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p22b}_2,\mbox{hypothesis})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p22b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a\cdot b,\mbox{identity}),\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{least\_upper\_bound}(b,\mbox{identity}))\ne \mbox{least\_upper\_bound}(a\cdot b,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p22b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP216-1.p} An identity generated by HR, number 00407






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_7' = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_2 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = x_2 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP211-1.p} An identity generated by HR, number 00349






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( x_4\cdot x_5 = \mbox{sk\_c}_8\mbox{ and }x_4' = x_5\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_6' = \mbox{sk\_c}_8\mbox{ and }x_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_2 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_8 = x_2 ) \ \Rightarrow \ x_3'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP262-1.p} An identity generated by HR, number 00951






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP410-1.p} Axiom for group theory, in product \& inverse, part 2






$( ( a\cdot ( b\cdot c )' )'\cdot ( a\cdot c' ) )\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP457-1.p} Axiom for group theory, in division and identity, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{identity},a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP486-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c),\mbox{double\_divide}(b,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP265-1.p} An identity generated by HR, number 01100






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_9\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_9\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{27},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{28},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ and }x_3' = \mbox{sk\_c}_9\mbox{ and }x_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_4' = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_9 = \mbox{sk\_c}_8\mbox{ and }x_1' = \mbox{sk\_c}_9\mbox{ and }x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_5 = \mbox{sk\_c}_8\mbox{ and }x_6\cdot \mbox{sk\_c}_7 = x_5 ) \ \Rightarrow \ x_6'\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{GRP572-1.p} Axiom for Abelian group theory, in double div and id, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP438-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( b\cdot ( c\cdot ( ( c'\cdot ( d\cdot b )' )\cdot a ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP183-3.p} Orthogonal elements form a subgroup with orthogonal parts






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,\mbox{identity}),\mbox{least\_upper\_bound}(a',\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_20x},\mbox{negated\_conjecture})$

\medskip\textbf{GRP655+2.p} A quasigroup satisfying Moufang 2 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{rd}(x_1,x_1) = x_0\mbox{ and }\mbox{rd}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP464-1.p} Axiom for group theory, in division and identity, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),b),c),\mbox{divide}(\mbox{divide}(\mbox{identity},a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$\mbox{identity}\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP034-3.p} In subgroups, inverse is closed






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(a')\qquad\mbox{cnf}(\mbox{prove\_a\_inverse\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP182-3.p} Positive part of the negative part is identity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},\mbox{least\_upper\_bound}(a,\mbox{identity}))\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_p17b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP725-1.p} Loops with abelian inner mapping group - commutativity

Uniquely 2-divisible loops with abelian inner mapping group of
 exponent 2 are commutative.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$s(a)\cdot s(a) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$s(a\cdot a) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = \mbox{ld}(c\cdot b,c\cdot ( b\cdot a ))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_t}(a,b) = \mbox{ld}(b,a\cdot b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),b) = a\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),b,c) = a\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),b,c) = a\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP057-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$x\cdot ( ( ( y'\cdot ( x'\cdot z ) )'\cdot u )\cdot ( y\cdot u )' )' = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP014-1.p} Product is associative in this group theory

The group theory specified by the axiom given implies the
 associativity of multiply.




$x\cdot ( ( ( y'\cdot ( x'\cdot w ) )'\cdot z )\cdot ( y\cdot z )' )' = w\qquad\mbox{cnf}(\mbox{group\_axiom},\mbox{axiom})$

$a\cdot ( b\cdot c )\ne ( a\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{prove\_associativity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP064-1.p} Single axiom for group theory, in division

This is a single axiom for group theory, in terms of division




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(y,y),y),z),\mbox{divide}(\mbox{divide}(\mbox{divide}(y,y),x),z))) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP127-2.005.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-3.004.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP702+1.p} In C-loops the nucleus is normal - a






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( b\cdot c ) ) = ( ( a\cdot b )\cdot b )\cdot c\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_c}\cdot ( a\cdot b ) = ( \mbox{op\_c}\cdot a )\cdot b\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( b\cdot \mbox{op\_c} ) = ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( \mbox{op\_c}\cdot b ) = ( a\cdot \mbox{op\_c} )\cdot b\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall a{:}\;\mbox{op\_d} = \mbox{ld}(a,\mbox{op\_c}\cdot a)\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_e} = ( \mbox{rd}(\mbox{op\_c},a\cdot b)\cdot b )\cdot a\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$\forall b,a{:}\;\mbox{op\_f} = a\cdot ( b\cdot \mbox{ld}(a\cdot b,\mbox{op\_c}) )\qquad\mbox{fof}(f_{13},\mbox{axiom})$

$\forall x_0,x_1{:}\;( \mbox{op\_d}\cdot ( x_0\cdot x_1 ) = ( \mbox{op\_d}\cdot x_0 )\cdot x_1\mbox{ and }x_0\cdot ( x_1\cdot \mbox{op\_d} ) = ( x_0\cdot x_1 )\cdot \mbox{op\_d}\mbox{ and }x_0\cdot ( \mbox{op\_d}\cdot x_1 ) = ( x_0\cdot \mbox{op\_d} )\cdot x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP177-1.p} A consequence of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p08a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p08a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},c) = c\qquad\mbox{cnf}(\mbox{p08a}_3,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b\cdot c),\mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c))\ne \mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c)\qquad\mbox{cnf}(\mbox{prove\_p08a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP690-1.p} Bruck loop elements of order 2$\wedge$4 commute with elems of order 3$\wedge$2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) ) ) ) ) ) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot ( \mbox{op\_d}\cdot \mbox{op\_d} ) ) ) ) ) ) ) = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d}\ne \mbox{op\_d}\cdot \mbox{op\_c}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP731-1.p} Bruck loops that are centrally nilpotent - 2nd easy part d

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - 2nd easy part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot b = ( b\cdot a )\cdot \mbox{op\_k}(a,b)\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e) = 1\qquad\mbox{cnf}(c_{21},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{asoc}(c,d,e)) = 1\qquad\mbox{cnf}(c_{22},\mbox{axiom})$

$\mbox{op\_k}(\mbox{asoc}(a,b,c),d)\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP523-1.p} Axiom for Abelian group theory, in division, part 3






$\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,\mbox{divide}(a,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP090-1.p} Single axiom for Abelian group theory, in division

This is a single axiom for Abelian group theory, in terms
 of division




$\mbox{divide}(\mbox{divide}(x,\mbox{divide}(y,z)),\mbox{divide}(x,y)) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{divide}(z,z),y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{divide}(z,z),x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP671-1.p} RIF loops are ARIF - b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot ( a\cdot b ) ) = ( ( a\cdot ( b\cdot c ) )\cdot a )\cdot b\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot a\ne a\cdot ( b\cdot a )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP771-1.p} Buchsteiner loop lemma 8






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$\mbox{quotient}(j(a),a) = j(a)\cdot i(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$( ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot b )\cdot c = ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{20},\mbox{axiom})$

$i(a\cdot b)\cdot i(i(a)) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{21},\mbox{axiom})$

$j(j(a))\cdot j(b\cdot a) = j(b)\qquad\mbox{cnf}(\mbox{sos}_{22},\mbox{axiom})$

$( a\cdot ( b\cdot c ) )\cdot a(a,b,c) = ( a\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{23},\mbox{axiom})$

$( a\cdot b )\cdot c(b,a) = b\cdot a\qquad\mbox{cnf}(\mbox{sos}_{24},\mbox{axiom})$

$( x_0\cdot x_1 )\cdot c(x_2,x_3)\ne x_0\cdot ( x_1\cdot c(x_2,x_3) )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP507-1.p} Axiom for Abelian group theory, in product and inverse, part 3






$( ( ( a\cdot b )'\cdot ( b\cdot a ) )'\cdot ( ( c\cdot d )'\cdot ( c\cdot ( ( e\cdot f' )\cdot d' )' ) ) )'\cdot f = e\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP720+1.p} In commutative A-loops, squares form a subloop






$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall b,a{:}\;a\cdot b = b\cdot a\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall d,c,b,a{:}\;\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall c,b,a{:}\;\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall x_0,x_1{:}\;\exists x_2{:}\;( x_0\cdot x_0 )\cdot ( x_1\cdot x_1 ) = x_2\cdot x_2\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP681-1.p} Commutants in Bol loops 3

If Q is a Bol loop, and if a,b in C(Q), then so are a$\wedge$2, b$\wedge$-1,
 and a$\wedge$2b.




$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{op\_d}\cdot a = a\cdot \mbox{op\_d}\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( ( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot \mbox{op\_d} )\cdot a\ne a\cdot ( ( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot \mbox{op\_d} )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP654+1.p} A quasigroup satisfying Moufang 1 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\exists x_0{:}\;\forall x_1{:}\;( x_1\cdot x_0 = x_1\mbox{ and }x_0\cdot x_1 = x_1 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP053-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( y\cdot ( ( ( z\cdot y )'\cdot ( z\cdot x' ) )'\cdot ( y'\cdot y )' )' )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP012-2.p} Inverse of products = Product of inverses

The inverse of products equals the product of the inverse,
 in opposite order.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_multiply\_b\_is\_c},\mbox{hypothesis})$

$b'\cdot a'{=}d\qquad\mbox{cnf}(\mbox{inverse\_b\_multiply\_inverse\_a\_is\_d},\mbox{hypothesis})$

$c'\ne d\qquad\mbox{cnf}(\mbox{prove\_c\_inverse\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP309-1.p} An identity generated by HR, number 17391






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_3 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = x_3\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP414-1.p} Axiom for group theory, in product \& inverse, part 3






$a\cdot ( ( ( b'\cdot b )\cdot ( ( a\cdot b' )'\cdot c )' )\cdot b )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP287-1.p} An identity generated by HR, number 03165






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_7\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP490-1.p} Axiom for group theory, in double division and identity, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},a),\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{identity}),\mbox{double\_divide}(c,b)))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP169-1.p} Inverses reverse inequalities






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a',b') = b'\qquad\mbox{cnf}(\mbox{p02a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,b)\ne a\qquad\mbox{cnf}(\mbox{prove\_p02a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP012+5.p} Inverse of products = Product of inverses

The inverse of products equals the product of the inverse,
 in opposite order




$\forall e{:}\;( ( \forall x,y{:}\;\exists z{:}\;x\cdot y{=}z\mbox{ and }\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w )\mbox{ and }\forall x,y,z,u,v,w{:}\;( ( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w )\mbox{ and }\forall x{:}\;x\cdot e{=}x\mbox{ and }\forall x{:}\;e\cdot x{=}x\mbox{ and }\forall x{:}\;x\cdot x'{=}e\mbox{ and }\forall x{:}\;x'\cdot x{=}e ) \ \Rightarrow \ \forall u,v,w,x{:}\;( ( u'\cdot v'{=}w\mbox{ and }v\cdot u{=}x ) \ \Rightarrow \ w'\cdot x'{=}e ) )\qquad\mbox{fof}(\mbox{prove\_distribution},\mbox{conjecture})$

\medskip\textbf{GRP450-1.p} Axiom for group theory, in division, part 3






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP168-2.p} Inner group automorphisms are order preserving






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{p01b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(c'\cdot ( a\cdot c ),c'\cdot ( b\cdot c ))\ne c'\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{prove\_p01b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP473-1.p} Axiom for group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b)',\mbox{divide}(\mbox{divide}(c,d),a)),\mbox{divide}(d,c)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP769-1.p} Buchsteiner loop lemma 6






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$\mbox{quotient}(j(a),a) = j(a)\cdot i(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$( ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot b )\cdot c = ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{20},\mbox{axiom})$

$i(a\cdot b)\cdot i(i(a)) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{21},\mbox{axiom})$

$j(j(a))\cdot j(b\cdot a) = j(b)\qquad\mbox{cnf}(\mbox{sos}_{22},\mbox{axiom})$

$a\cdot i(b\cdot a) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{23},\mbox{axiom})$

$j(a\cdot b)\cdot a = j(b)\qquad\mbox{cnf}(\mbox{sos}_{24},\mbox{axiom})$

$( x_0\cdot x_1 )\cdot x_2\ne ( x_0\cdot x_2 )\cdot ( x_2\setminus x_1\cdot x_2 )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP083-1.p} Single axiom for group theory, in double division and inverse

This is a single axiom for group theory, in terms of double
 division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(y,z)'),\mbox{double\_divide}(y',\mbox{double\_divide}(u,\mbox{double\_divide}(x,u))')) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP418-1.p} Axiom for group theory, in product \& inverse, part 1






$( ( a\cdot ( b'\cdot ( c\cdot ( c'\cdot c )' )' )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP001-5.p} X$\wedge$2 = identity =$>$ commutativity

If the square of every element is the identity, the system
 is commutative.




$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$x\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{square\_element},\mbox{hypothesis})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{hypothesis})$

$\neg\,b\cdot a{=}c\qquad\mbox{cnf}(\mbox{prove\_b\_times\_a\_is\_c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP028-1.p} In semigroups, left and right solutions =$>$ right id exists

If there are left and right solutions, then there is a right
 identity element.




$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity},\mbox{axiom})$

$\mbox{left\_solution}(x,y)\cdot x{=}y\qquad\mbox{cnf}(\mbox{left\_soln},\mbox{hypothesis})$

$x\cdot \mbox{right\_solution}(x,y){=}y\qquad\mbox{cnf}(\mbox{right\_soln},\mbox{hypothesis})$

$\neg\,\mbox{not\_identity}(x)\cdot x{=}\mbox{not\_identity}(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP106-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,y),\mbox{double\_divide}(x,\mbox{double\_divide}(z,y)')')' = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP470-1.p} Axiom for group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(c,d)))',\mbox{divide}(\mbox{divide}(d,c),a)) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP576-1.p} Axiom for Abelian group theory, in double div and id, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP394-2.p} Group theory axioms






$\mbox{include}(\mbox{'Axioms/GRP005-0.ax'})$

\medskip\textbf{GRP727-1.p} Bruck loops that are centrally nilpotent - 1st easy part

Bruck loops with abelian inner mapping group are centrally
 nilpotent of class two - 1st easy part.




$1\cdot a = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot ( b\cdot a ) )\cdot c = a\cdot ( b\cdot ( a\cdot c ) )\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot ( b\cdot c ) )\cdot \mbox{asoc}(a,b,c)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_l}(a,b,c) = i(c\cdot b)\cdot ( c\cdot ( b\cdot a ) )\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_r}(a,b,c) = \mbox{rd}(( a\cdot b )\cdot c,b\cdot c)\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_t}(a,b) = i(b)\cdot ( a\cdot b )\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_r}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_r}(a,d,e),b,c)\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_r}(a,b,c),d,e) = \mbox{op\_r}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{op\_l}(\mbox{op\_l}(a,b,c),d,e) = \mbox{op\_l}(\mbox{op\_l}(a,d,e),b,c)\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_r}(a,b,c),d) = \mbox{op\_r}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_l}(a,b,c),d) = \mbox{op\_l}(\mbox{op\_t}(a,d),b,c)\qquad\mbox{cnf}(c_{18},\mbox{axiom})$

$\mbox{op\_t}(\mbox{op\_t}(a,b),c) = \mbox{op\_t}(\mbox{op\_t}(a,c),b)\qquad\mbox{cnf}(c_{19},\mbox{axiom})$

$\mbox{asoc}(\mbox{asoc}(a,b,c),d,e) = 1\qquad\mbox{cnf}(c_{20},\mbox{axiom})$

$\mbox{asoc}(a,b,\mbox{asoc}(c,d,e))\ne 1\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP080-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{identity},\mbox{double\_divide}(x,\mbox{double\_divide}(y,\mbox{identity}))),\mbox{double\_divide}(\mbox{double\_divide}(y,\mbox{double\_divide}(z,x)),\mbox{identity})) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP573-1.p} Axiom for Abelian group theory, in double div and id, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(c,a)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP708-1.p} Bol loop commutant element squared in left and right nucleus - 1






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{op\_c}\cdot a = a\cdot \mbox{op\_c}\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot ( a\cdot b ) = ( ( \mbox{op\_c}\cdot \mbox{op\_c} )\cdot a )\cdot b\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{op\_c} )\ne ( a\cdot b )\cdot \mbox{op\_c}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP191-1.p} Something useful for estimations






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p39b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a',b')\ne a'\qquad\mbox{cnf}(\mbox{prove\_p39b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP439-1.p} Axiom for group theory, in product \& inverse, part 1






$( a\cdot ( b\cdot ( ( b'\cdot c )\cdot ( d\cdot ( a\cdot c ) )' ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP571-1.p} Axiom for Abelian group theory, in double div and id, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(b,\mbox{double\_divide}(a,c)),\mbox{double\_divide}(c,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP715+1.p} Strongly right alternative rings 1

If a has a 2-sided inverse, then R(a$\wedge$-1) = R(a)$\wedge$-1 and
 L(a)$\wedge$-1 = R(a)L(a$\wedge$-1)R(a$\wedge$-1).




$\forall c,b,a{:}\;( a + b ) + c = a + ( b + c )\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;a + b = b + a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall a{:}\;a + \mbox{op}_0 = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a{:}\;a + -a = \mbox{op}_0\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b + c ) = a\cdot b + a\cdot c\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( b\cdot b ) = ( a\cdot b )\cdot b\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\mbox{op\_a}\cdot \mbox{op\_b} = 1\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\mbox{op\_b}\cdot \mbox{op\_a} = 1\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall x_0{:}\;( ( x_0\cdot \mbox{op\_a} )\cdot \mbox{op\_b} = x_0\mbox{ and }( x_0\cdot \mbox{op\_b} )\cdot \mbox{op\_a} = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP128-1.003.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP114-1.p} Product of positive and negative parts of X equals X

Prove that for each element X in a group, X is equal to the
 product of its positive part (the union with the identity)
 and its negative part (the intersection with the identity).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{inverse\_of\_identity},\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{inverse\_involution},\mbox{axiom})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{inverse\_product\_lemma},\mbox{axiom})$

$\mbox{intersection}(x,x) = x\qquad\mbox{cnf}(\mbox{intersection\_idempotent},\mbox{axiom})$

$\mbox{union}(x,x) = x\qquad\mbox{cnf}(\mbox{union\_idempotent},\mbox{axiom})$

$\mbox{intersection}(x,y) = \mbox{intersection}(y,x)\qquad\mbox{cnf}(\mbox{intersection\_commutative},\mbox{axiom})$

$\mbox{union}(x,y) = \mbox{union}(y,x)\qquad\mbox{cnf}(\mbox{union\_commutative},\mbox{axiom})$

$\mbox{intersection}(x,\mbox{intersection}(y,z)) = \mbox{intersection}(\mbox{intersection}(x,y),z)\qquad\mbox{cnf}(\mbox{intersection\_associative},\mbox{axiom})$

$\mbox{union}(x,\mbox{union}(y,z)) = \mbox{union}(\mbox{union}(x,y),z)\qquad\mbox{cnf}(\mbox{union\_associative},\mbox{axiom})$

$\mbox{union}(\mbox{intersection}(x,y),y) = y\qquad\mbox{cnf}(\mbox{union\_intersection\_absorbtion},\mbox{axiom})$

$\mbox{intersection}(\mbox{union}(x,y),y) = y\qquad\mbox{cnf}(\mbox{intersection\_union\_absorbtion},\mbox{axiom})$

$x\cdot \mbox{union}(y,z) = \mbox{union}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{multiply\_union}_1,\mbox{axiom})$

$x\cdot \mbox{intersection}(y,z) = \mbox{intersection}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{multiply\_intersection}_1,\mbox{axiom})$

$\mbox{union}(y,z)\cdot x = \mbox{union}(y\cdot x,z\cdot x)\qquad\mbox{cnf}(\mbox{multiply\_union}_2,\mbox{axiom})$

$\mbox{intersection}(y,z)\cdot x = \mbox{intersection}(y\cdot x,z\cdot x)\qquad\mbox{cnf}(\mbox{multiply\_intersection}_2,\mbox{axiom})$

$\mbox{positive\_part}(x) = \mbox{union}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{positive\_part},\mbox{axiom})$

$\mbox{negative\_part}(x) = \mbox{intersection}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{negative\_part},\mbox{axiom})$

$\mbox{positive\_part}(a)\cdot \mbox{negative\_part}(a)\ne a\qquad\mbox{cnf}(\mbox{prove\_product},\mbox{negated\_conjecture})$

\medskip\textbf{GRP400-1.p} Prove associativity implies distribution in cancellative semigroup

Assume a cancellative semigroup admits a commutator operation.
 Then the following three properties are equivalent:
 (1) commutator is associative;
 (2) commutator distributes over product;
 (3) the semigroup is nilpotent of class 2.
 This is a generalization of the corresponding theorem for group
 theory. The problem here is to prove (1) implies (2).




$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$a\cdot b = b\cdot ( a\cdot \mbox{commutator}(a,b) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$\mbox{commutator}(\mbox{commutator}(a,b),c) = \mbox{commutator}(a,\mbox{commutator}(b,c))\qquad\mbox{cnf}(\mbox{associativity\_of\_commutator},\mbox{axiom})$

$\mbox{commutator}(a\cdot b,c)\ne \mbox{commutator}(a,c)\cdot \mbox{commutator}(b,c)\qquad\mbox{cnf}(\mbox{prove\_commutator\_distributes\_over\_product},\mbox{negated\_conjecture})$

\medskip\textbf{GRP695-1.p} In power associative conjugacy closed loop c$\wedge$12 is in nucleus - c






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) = \mbox{op\_d}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_d}\cdot \mbox{op\_d} = \mbox{op\_e}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_e}\cdot \mbox{op\_e} = \mbox{op\_f}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{op\_f} )\ne ( a\cdot b )\cdot \mbox{op\_f}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP734+1.p} Non-commutative pure DTS loop.






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(c_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(c_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(c_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(c_{04},\mbox{axiom})$

$\forall a{:}\;a\cdot 1 = a\qquad\mbox{fof}(c_{05},\mbox{axiom})$

$\forall a{:}\;1\cdot a = a\qquad\mbox{fof}(c_{06},\mbox{axiom})$

$\forall a{:}\;a\cdot a = 1\qquad\mbox{fof}(c_{07},\mbox{axiom})$

$\mbox{op\_a}\cdot \mbox{op\_b}\ne \mbox{op\_b}\cdot \mbox{op\_a}\qquad\mbox{fof}(c_{08},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( x_0\cdot x_1 = x_2 \ \Rightarrow \ ( ( x_0\cdot x_2 = x_1\mbox{ and }x_1\cdot x_2 = x_0 )\mbox{ or }( x_0\cdot x_2 = x_1\mbox{ and }x_2\cdot x_1 = x_0 )\mbox{ or }( x_2\cdot x_0 = x_1\mbox{ and }x_2\cdot x_1 = x_0 ) ) )\qquad\mbox{fof}(c_{09},\mbox{axiom})$

$\forall x_3,x_4{:}\;( x_3\cdot x_4 = x_4\cdot x_3 \ \Rightarrow \ ( x_3 = 1\mbox{ or }x_4 = 1\mbox{ or }x_3\cdot x_4 = 1 ) )\qquad\mbox{fof}(c_{10},\mbox{axiom})$

\medskip\textbf{GRP184-1.p} Orthogonal elements commute and form a subgroup

For each X {Y: X orth Y} is a subgroup. X orthogonal to Y
 implies that X and Y commute. Moreover, pp(a) orthogonal to
 np(a).




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\ne \mbox{greatest\_lower\_bound}(a,\mbox{identity})'\cdot \mbox{least\_upper\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP130-3.003.p} (a.(a.b)).b = a

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_3,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP010-1.p} Inverse is a symmetric relationship

If a is an inverse of b then b is an inverse of a.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot b{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{a\_multiply\_b\_is\_identity},\mbox{hypothesis})$

$\neg\,b\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_b\_multiply\_a\_is\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP167-4.p} Product of positive and negative parts






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{p19}_1,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{p19}_2,\mbox{axiom})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p19}_3,\mbox{axiom})$

$a\ne \mbox{least\_upper\_bound}(a,\mbox{identity})\cdot \mbox{greatest\_lower\_bound}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_p}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{GRP536-1.p} Axiom for Abelian group theory, in division, part 4






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(\mbox{divide}(a,b),c)),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP031-2.p} In semigroups, left inverse and id =$>$ right inverse exists

If there are right inverses and right identity, then every
 element has a left inverse.




$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}z\mbox{ and }x\cdot y{=}w ) \ \Rightarrow \ z{=}w\qquad\mbox{cnf}(\mbox{total\_function}_2,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$a\cdot a'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{hypothesis})$

$a\cdot \mbox{identity}{=}a\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{hypothesis})$

$\neg\,a\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_a\_has\_a\_left\_inverse},\mbox{negated\_conjecture})$

\medskip\textbf{GRP028-3.p} In semigroups, left and right solutions =$>$ right id exists

If there are left and right solutions, then there is a right
 identity element.




$x\cdot y{=}x\cdot y\qquad\mbox{cnf}(\mbox{total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\mbox{left\_solution}(x,y)\cdot x{=}y\qquad\mbox{cnf}(\mbox{left\_soln},\mbox{hypothesis})$

$x\cdot \mbox{right\_solution}(x,y){=}y\qquad\mbox{cnf}(\mbox{right\_soln},\mbox{hypothesis})$

$\neg\,\mbox{not\_identity}(x)\cdot x{=}\mbox{not\_identity}(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP222-1.p} An identity generated by HR, number 00453






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( x_3' = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_4\cdot x_5 = \mbox{sk\_c}_6\mbox{ and }x_4' = x_5\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_6\ne \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP039-5.p} Subgroups of index 2 are normal

If O is a subgroup of G and there are exactly 2 cosets
 in G/O, then O is normal [that is, for all x in G and
 y in O, x*y*inverse(x) is back in O].




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-1.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{subgroup\_member}(\mbox{identity})\qquad\mbox{cnf}(\mbox{identity\_in\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }\mbox{subgroup\_member}(\mbox{element\_in\_O}_2(x,y))\qquad\mbox{cnf}(\mbox{an\_element\_in\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(x)\mbox{ or }\mbox{subgroup\_member}(y)\mbox{ or }x\cdot \mbox{element\_in\_O}_2(x,y) = y\qquad\mbox{cnf}(\mbox{property\_of\_O}_2,\mbox{axiom})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_in\_O}_2,\mbox{negated\_conjecture})$

$b\cdot a' = c\qquad\mbox{cnf}(\mbox{b\_times\_a\_inverse\_is\_c},\mbox{negated\_conjecture})$

$a\cdot c = d\qquad\mbox{cnf}(\mbox{a\_times\_c\_is\_d},\mbox{negated\_conjecture})$

$\neg\,\mbox{subgroup\_member}(d)\qquad\mbox{cnf}(\mbox{prove\_d\_in\_O}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP547-1.p} Axiom for Abelian group theory, in division and identity, part 3






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,b)),\mbox{divide}(\mbox{divide}(b,c),a)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP590-1.p} Axiom for Abelian group theory, in double div and inv, part 2






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,c')')',b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP108-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(y,\mbox{double\_divide}(x,z))')',z)' = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP198-1.p} In cancellative semigroups, xyyyxy=yyyyxx $\to$ babbba=aabbbb.






$\mbox{include}(\mbox{'Axioms/GRP008-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP008-1.ax'})$

$x\cdot ( y\cdot ( y\cdot ( y\cdot ( x\cdot y ) ) ) ) = y\cdot ( y\cdot ( y\cdot ( y\cdot ( x\cdot x ) ) ) )\qquad\mbox{cnf}(\mbox{condition},\mbox{hypothesis})$

$b\cdot ( a\cdot ( b\cdot ( b\cdot ( b\cdot a ) ) ) )\ne a\cdot ( a\cdot ( b\cdot ( b\cdot ( b\cdot b ) ) ) )\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-6.005.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c
 Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4)\mbox{ or }\mbox{product}_1(x,y,e_5) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4)\mbox{ or }\mbox{product}_2(x,y,e_5) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP003-1.p} The left identity is also a right identity






$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\neg\,a\cdot \mbox{identity}{=}a\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_right\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP688-1.p} Bruck loop elements of order 2$\wedge$2 commute with elements of order 3






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( a\cdot ( b\cdot a ) )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a\cdot b) = i(a)\cdot i(b)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) ) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_d}\cdot ( \mbox{op\_d}\cdot \mbox{op\_d} ) = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_c}\cdot \mbox{op\_d}\ne \mbox{op\_d}\cdot \mbox{op\_c}\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP160-1.p} Prove reflexivity axiom using the LUB transformation

This problem proves the original reflexivity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,a)\ne a\qquad\mbox{cnf}(\mbox{prove\_ax\_refla},\mbox{negated\_conjecture})$

\medskip\textbf{GRP132-1.005.p} (3,1,2) conjugate orthogonality, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg2}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot x_1{=}y_1\mbox{ and }z_2\cdot x_2{=}y_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP051-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( ( z\cdot ( x\cdot y )' )'\cdot ( z\cdot y' ) )\cdot ( y'\cdot y )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP750-1.p} Simplifying a basis for trimedial quasigroups: part 2






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$( a\cdot ( a\cdot a ) )\cdot ( b\cdot c ) = ( a\cdot b )\cdot ( ( a\cdot a )\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot c ) = ( a\cdot c )\cdot ( b\cdot c )\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot c )\ne ( a\cdot b )\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP134-1.005.p} (a.b).(b.a) = b, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP148-1.p} Prove least upper-bound axiom using a transformation

This problem proves the original least upper-bound axiom from
 the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,c) = c\qquad\mbox{cnf}(\mbox{ax\_lub1c}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(b,c) = c\qquad\mbox{cnf}(\mbox{ax\_lub1c}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(a,b),c)\ne \mbox{least\_upper\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_lub1c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP175-4.p} Positivity is preserved under inner automorphisms






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p06d}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},a'\cdot ( b\cdot a ))\ne a'\cdot ( b\cdot a )\qquad\mbox{cnf}(\mbox{prove\_p06d},\mbox{negated\_conjecture})$

\medskip\textbf{GRP226-1.p} An identity generated by HR, number 00460






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_3 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_3\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$( x_3' = \mbox{sk\_c}_7\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_5\cdot \mbox{sk\_c}_7 = x_4\mbox{ and }x_5' = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{GRP738-1.p} Proper Buchsteiner loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( a\cdot b )\cdot c) = \mbox{rd}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot c )\ne \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP605-1.p} Axiom for Abelian group theory, in double div and inv, part 1






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b',\mbox{double\_divide}(a,c))')',c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP437-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( b\cdot ( c\cdot ( ( c'\cdot ( d\cdot b )' )\cdot a ) ) )' = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP483-1.p} Axiom for group theory, in double division and identity, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{identity})),\mbox{double\_divide}(\mbox{double\_divide}(c,\mbox{double\_divide}(d,\mbox{double\_divide}(d,\mbox{identity}))),\mbox{double\_divide}(a,\mbox{identity}))),b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(\mbox{double\_divide}(b,a),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{double\_divide}(a,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(a,a')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP774+1.p} Green's relation D is a congruence






$\forall c,b,a{:}\;( a\cdot b )\cdot c = a\cdot ( b\cdot c )\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall x_0,x_1{:}\;( d(x_0,x_1)\iff ( x_0\cdot ( x_1\cdot x_0 ) = x_0\mbox{ and }x_1\cdot ( x_0\cdot x_1 ) = x_1 ) )\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall x_2,x_3,x_4,x_5{:}\;( ( d(x_2,x_3)\mbox{ and }d(x_4,x_5) ) \ \Rightarrow \ d(x_2\cdot x_4,x_3\cdot x_5) )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP426-1.p} Axiom for group theory, in product \& inverse, part 3






$( ( a\cdot ( b'\cdot c )' )'\cdot ( a\cdot c' ) )'\cdot ( c'\cdot c )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP005-1.p} Identity is in this subset of a group

If S is a non-empty subset of a group such that
 if X, Y belong to S, the XY$\wedge$-1 belongs to S, then the
 identity e belongs to S.




$\mbox{identity}\cdot x{=}x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$x\cdot \mbox{identity}{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x'{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$x'\cdot x{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$\mbox{an\_element}(a)\qquad\mbox{cnf}(\mbox{element\_of\_set},\mbox{axiom})$

$( \mbox{an\_element}(x)\mbox{ and }\mbox{an\_element}(y)\mbox{ and }x\cdot y'{=}z ) \ \Rightarrow \ \mbox{an\_element}(z)\qquad\mbox{cnf}(\mbox{condition},\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }u\cdot z{=}w ) \ \Rightarrow \ x\cdot v{=}w\qquad\mbox{cnf}(\mbox{associativity}_1,\mbox{axiom})$

$( x\cdot y{=}u\mbox{ and }y\cdot z{=}v\mbox{ and }x\cdot v{=}w ) \ \Rightarrow \ u\cdot z{=}w\qquad\mbox{cnf}(\mbox{associativity}_2,\mbox{axiom})$

$\neg\,\mbox{an\_element}(\mbox{identity})\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_an\_element},\mbox{negated\_conjecture})$

\medskip\textbf{GRP693-1.p} In power associative conjugacy closed loop c$\wedge$12 is in nucleus - a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) = \mbox{op\_d}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_d}\cdot \mbox{op\_d} = \mbox{op\_e}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_e}\cdot \mbox{op\_e} = \mbox{op\_f}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{op\_f}\cdot ( a\cdot b )\ne ( \mbox{op\_f}\cdot a )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP190-1.p} Something useful for estimations






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = a\qquad\mbox{cnf}(\mbox{p39a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a',b')\ne b'\qquad\mbox{cnf}(\mbox{prove\_p39a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP277-1.p} An identity generated by HR, number 01755






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( x_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_2' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3\cdot x_4 = \mbox{sk\_c}_6\mbox{ and }x_3' = x_4 ) \ \Rightarrow \ x_4\cdot \mbox{sk\_c}_7\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP599-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c',b)')') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP171-1.p} Positive elements form a semigroup






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(\mbox{identity},a) = a\qquad\mbox{cnf}(\mbox{p04a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},b) = b\qquad\mbox{cnf}(\mbox{p04a}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{identity},a\cdot b)\ne a\cdot b\qquad\mbox{cnf}(\mbox{prove\_p04a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP519-1.p} Axiom for Abelian group theory, in product and inverse, part 3






$a\cdot ( ( ( a\cdot b )'\cdot c )\cdot b ) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP394-1.p} Group theory axioms






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

\medskip\textbf{GRP024-4.p} Associativity of commutator

The commutator operation is associative if and only if the
 commutator of any two elements lies in the center of the
 group, i.e. [[X,Y],Z]=[X,[Y,Z]] iff [U,V]*W=W*[U,V].




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$\mbox{commutator}(x,y) = x\cdot ( y\cdot ( x'\cdot y' ) )\qquad\mbox{cnf}(\mbox{commutator},\mbox{axiom})$

$\mbox{commutator}(\mbox{commutator}(a,b),c) = \mbox{commutator}(a,\mbox{commutator}(b,c))\mbox{ or }\mbox{commutator}(e,f)\cdot g = g\cdot \mbox{commutator}(e,f)\qquad\mbox{cnf}(\mbox{associativity\_or\_center},\mbox{negated\_conjecture})$

$\mbox{commutator}(\mbox{commutator}(a,b),c) = \mbox{commutator}(a,\mbox{commutator}(b,c)) \ \Rightarrow \ \mbox{commutator}(e,f)\cdot g\ne g\cdot \mbox{commutator}(e,f)\qquad\mbox{cnf}(\mbox{not\_both\_associativity\_and\_center},\mbox{negated\_conjecture})$

\medskip\textbf{GRP076-1.p} Single axiom for group theory, in double division and identity

This is a single axiom for group theory, in terms of double
 division and identity.




$\mbox{double\_divide}(\mbox{double\_divide}(x,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y),z),\mbox{double\_divide}(y,\mbox{identity}))),\mbox{double\_divide}(\mbox{identity},\mbox{identity})) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(\mbox{double\_divide}(y,x),\mbox{identity})\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{double\_divide}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{double\_divide}(x,x')\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP177-2.p} A consequence of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},a) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p08b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},b) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p08b}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{identity},c) = \mbox{identity}\qquad\mbox{cnf}(\mbox{p08b}_3,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(\mbox{greatest\_lower\_bound}(a,b\cdot c),\mbox{greatest\_lower\_bound}(a,b)\cdot \mbox{greatest\_lower\_bound}(a,c))\ne \mbox{greatest\_lower\_bound}(a,b\cdot c)\qquad\mbox{cnf}(\mbox{prove\_p08b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-2.004.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP545-1.p} Axiom for Abelian group theory, in division and identity, part 1






$\mbox{divide}(\mbox{divide}(\mbox{identity},\mbox{divide}(a,b)),\mbox{divide}(\mbox{divide}(b,c),a)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP654+3.p} A quasigroup satisfying Moufang 1 is a loop






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{ld}(x_1,x_1) = x_0\mbox{ and }\mbox{ld}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP129-1.003.p} a.(b.a) = (b.a).b

Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }x\cdot z_1{=}z_2 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP466-1.p} Axiom for group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(b,\mbox{divide}(\mbox{divide}(c,\mbox{divide}(d,b)),d'))) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP676-1.p} In CC-loops, associators are in the center of the nucleus - 1b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{asoc}(a,b,c) = \mbox{ld}(a\cdot ( b\cdot c ),( a\cdot b )\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot ( \mbox{asoc}(b,c,d)\cdot e )\ne ( a\cdot \mbox{asoc}(b,c,d) )\cdot e\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP591-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,c')')',b) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP475-1.p} Axiom for group theory, in division and inverse, part 1






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),\mbox{divide}(d,c))',\mbox{divide}(b,a)) = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a_1'\cdot a_1\ne b_1'\cdot b_1\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_1,\mbox{negated\_conjecture})$

\medskip\textbf{GRP413-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( ( ( b'\cdot b )\cdot ( ( a\cdot b' )'\cdot c )' )\cdot b )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP165-2.p} An application of monotonicity

Essentially a simple application of monotonicity, more
 difficult when proved from the equations replacing
 monotonicity.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,\mbox{identity}) = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat1b}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a,a\cdot a)\ne a\qquad\mbox{cnf}(\mbox{prove\_lat1b},\mbox{negated\_conjecture})$

\medskip\textbf{GRP587-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(a,\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(a,b),c')',b)') = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP105-1.p} Single axiom for Abelian group theory, in double div and inv

This is a single axiom for Abelian group theory, in terms
 of double division and inverse.




$\mbox{double\_divide}(\mbox{double\_divide}(\mbox{double\_divide}(x,y),\mbox{double\_divide}(x,z')')',y) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{double\_divide}(y,x)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-7.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP779+1.p} Napoleon's quasigroups: Gruenbaum's theorem 2






$\forall b,a{:}\;a\setminus a\cdot b = b\qquad\mbox{fof}(\mbox{sos}_{01},\mbox{axiom})$

$\forall b,a{:}\;a\cdot ( a\setminus b ) = b\qquad\mbox{fof}(\mbox{sos}_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall d,c,b,a{:}\;( a\cdot b )\cdot ( c\cdot d ) = ( a\cdot c )\cdot ( b\cdot d )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;a\cdot a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall b,a{:}\;( ( a\cdot b )\cdot b )\cdot ( b\cdot ( b\cdot a ) ) = b\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( d(x_0,x_1,x_2)\iff x_0\cdot x_1 = x_1\cdot x_2 )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_3,x_4,x_5{:}\;( m(x_3,x_4,x_5)\iff ( x_3\cdot x_4 )\cdot ( x_4\cdot x_5 ) = x_3\cdot x_5 )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$d(b,\mbox{bigA},c)\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$d(\mbox{bigB},a,c)\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$d(b,a,\mbox{bigC})\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$d(a,b_1,c_1)\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$d(a_1,b,c_1)\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$d(a_1,b_1,c)\qquad\mbox{fof}(\mbox{sos}_{15},\mbox{axiom})$

$m(\mbox{bigA},b_1,\mbox{bigC})\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP334-1.p} An identity generated by HR, number 19095






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_3' = \mbox{sk\_c}_6\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_7' = \mbox{sk\_c}_5\mbox{ and }\mbox{sk\_c}_6\cdot x_1 = \mbox{sk\_c}_5\mbox{ and }x_2\cdot \mbox{sk\_c}_6 = x_1 ) \ \Rightarrow \ x_2'\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP167-2.p} Product of positive and negative parts

Each element in a lattice ordered group can be stated as a
 product of it's positive and it's negative part.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{identity}' = \mbox{identity}\qquad\mbox{cnf}(\mbox{lat4}_1,\mbox{axiom})$

$( x' )' = x\qquad\mbox{cnf}(\mbox{lat4}_2,\mbox{axiom})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{lat4}_3,\mbox{axiom})$

$\mbox{positive\_part}(x) = \mbox{least\_upper\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_4,\mbox{axiom})$

$\mbox{negative\_part}(x) = \mbox{greatest\_lower\_bound}(x,\mbox{identity})\qquad\mbox{cnf}(\mbox{lat4}_5,\mbox{axiom})$

$\mbox{least\_upper\_bound}(x,\mbox{greatest\_lower\_bound}(y,z)) = \mbox{greatest\_lower\_bound}(\mbox{least\_upper\_bound}(x,y),\mbox{least\_upper\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_6,\mbox{axiom})$

$\mbox{greatest\_lower\_bound}(x,\mbox{least\_upper\_bound}(y,z)) = \mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(x,y),\mbox{greatest\_lower\_bound}(x,z))\qquad\mbox{cnf}(\mbox{lat4}_7,\mbox{axiom})$

$a\ne \mbox{positive\_part}(a)\cdot \mbox{negative\_part}(a)\qquad\mbox{cnf}(\mbox{prove\_lat}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP015-1.p} x,$<$$<$x x,X$>$ x,X$>$ is a group






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/ALG001-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\neg\,\mbox{group}(\mbox{singleton\_set}(a),\mbox{singleton\_set}(\mbox{ordered\_pair}(\mbox{ordered\_pair}(a,a),a)))\qquad\mbox{cnf}(\mbox{prove\_the\_group},\mbox{negated\_conjecture})$

\medskip\textbf{GRP770-1.p} Buchsteiner loop lemma 7






$a\cdot 1 = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$a\cdot ( a\setminus b ) = b\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

$a\setminus a\cdot b = b\qquad\mbox{cnf}(\mbox{sos}_{04},\mbox{axiom})$

$\mbox{quotient}(a\cdot b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{05},\mbox{axiom})$

$\mbox{quotient}(a,b)\cdot b = a\qquad\mbox{cnf}(\mbox{sos}_{06},\mbox{axiom})$

$a\setminus ( a\cdot b )\cdot c = \mbox{quotient}(b\cdot ( c\cdot a ),a)\qquad\mbox{cnf}(\mbox{sos}_{07},\mbox{axiom})$

$a\cdot b\setminus a\cdot ( b\cdot c ) = \mbox{quotient}(\mbox{quotient}(c\cdot ( a\cdot b ),b),a)\qquad\mbox{cnf}(\mbox{sos}_{08},\mbox{axiom})$

$i(a) = a\setminus 1\qquad\mbox{cnf}(\mbox{sos}_{09},\mbox{axiom})$

$j(a) = \mbox{quotient}(1,a)\qquad\mbox{cnf}(\mbox{sos}_{10},\mbox{axiom})$

$i(a)\cdot a = a\cdot j(a)\qquad\mbox{cnf}(\mbox{sos}_{11},\mbox{axiom})$

$\mbox{eta}(a) = i(a)\cdot a\qquad\mbox{cnf}(\mbox{sos}_{12},\mbox{axiom})$

$i(i(a))\cdot b = \mbox{eta}(a)\cdot ( a\cdot b )\qquad\mbox{cnf}(\mbox{sos}_{13},\mbox{axiom})$

$a\cdot ( \mbox{eta}(a)\cdot b ) = j(j(a))\cdot b\qquad\mbox{cnf}(\mbox{sos}_{14},\mbox{axiom})$

$a\cdot ( b\cdot \mbox{eta}(a) ) = ( a\cdot b )\cdot \mbox{eta}(a)\qquad\mbox{cnf}(\mbox{sos}_{15},\mbox{axiom})$

$\mbox{eta}(a)\cdot ( b\cdot c ) = ( \mbox{eta}(a)\cdot b )\cdot c\qquad\mbox{cnf}(\mbox{sos}_{16},\mbox{axiom})$

$\mbox{quotient}(j(a),a) = j(a)\cdot i(a)\qquad\mbox{cnf}(\mbox{sos}_{17},\mbox{axiom})$

$( ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot b )\cdot c = ( \mbox{quotient}(j(a),a)\cdot ( a\cdot a ) )\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{sos}_{18},\mbox{axiom})$

$t(a,b) = \mbox{quotient}(a\cdot b,a)\qquad\mbox{cnf}(\mbox{sos}_{19},\mbox{axiom})$

$t(\mbox{eta}(a),b\cdot c) = t(\mbox{eta}(a),b)\cdot t(\mbox{eta}(a),c)\qquad\mbox{cnf}(\mbox{sos}_{20},\mbox{axiom})$

$i(a\cdot b)\cdot i(i(a)) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{21},\mbox{axiom})$

$j(j(a))\cdot j(b\cdot a) = j(b)\qquad\mbox{cnf}(\mbox{sos}_{22},\mbox{axiom})$

$a\cdot i(b\cdot a) = i(b)\qquad\mbox{cnf}(\mbox{sos}_{23},\mbox{axiom})$

$j(a\cdot b)\cdot a = j(b)\qquad\mbox{cnf}(\mbox{sos}_{24},\mbox{axiom})$

$x_0\cdot ( x_1\cdot x_2 )\ne \mbox{quotient}(x_0\cdot x_1,x_0)\cdot ( x_0\cdot x_2 )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP530-1.p} Axiom for Abelian group theory, in division, part 2






$\mbox{divide}(\mbox{divide}(a,\mbox{divide}(b,c)),\mbox{divide}(a,b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP165-1.p} An application of monotonicity

Essentially a simple application of monotonicity, more
 difficult when proved from the equations replacing
 monotonicity.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,\mbox{identity}) = a\qquad\mbox{cnf}(\mbox{lat1a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(a,a\cdot a)\ne a\cdot a\qquad\mbox{cnf}(\mbox{prove\_lat1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP128-3.004.p} (a.b).b = a.(a.b)

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ x\cdot z_1{=}z_2\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP417-1.p} Axiom for group theory, in product \& inverse, part 3






$( a\cdot ( ( ( b\cdot a )'\cdot ( b\cdot c' ) )'\cdot ( a'\cdot a )' )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP068-1.p} Single axiom for group theory, in division and identity

This is a single axiom for group theory, in terms of division
 and identity




$\mbox{divide}(x,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{identity},y),z),\mbox{divide}(\mbox{divide}(\mbox{divide}(x,x),x),z))) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = \mbox{identity}\mbox{ and }\mbox{identity}\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP452-1.p} Axiom for group theory, in division, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),\mbox{divide}(a,\mbox{divide}(b,\mbox{divide}(\mbox{divide}(\mbox{divide}(a,a),a),c)))),c) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP203-1.p} Left identity, left inverse, Moufang-3 =$>$ Moufang-2






$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( ( x\cdot y )\cdot x )\cdot z = x\cdot ( y\cdot ( x\cdot z ) )\qquad\mbox{cnf}(\mbox{moufang}_3,\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b\ne a\cdot ( b\cdot ( c\cdot b ) )\qquad\mbox{cnf}(\mbox{prove\_moufang}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP763+1.p} Lattice ordered group






$\forall a{:}\;a\cdot i(a) = e\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall a{:}\;a\cdot e = a\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall a,b,c{:}\;a\cdot ( b\cdot c ) = ( a\cdot b )\cdot c\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall a{:}\;m(a,a) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall a,b{:}\;m(a,b) = m(b,a)\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall a,b,c{:}\;m(a,m(b,c)) = m(m(a,b),c)\qquad\mbox{fof}(f_{06},\mbox{axiom})$

$\forall a{:}\;j(a,a) = a\qquad\mbox{fof}(f_{07},\mbox{axiom})$

$\forall a,b{:}\;j(a,b) = j(b,a)\qquad\mbox{fof}(f_{08},\mbox{axiom})$

$\forall a,b,c{:}\;j(a,j(b,c)) = j(j(a,b),c)\qquad\mbox{fof}(f_{09},\mbox{axiom})$

$\forall a,b{:}\;m(a,j(a,b)) = a\qquad\mbox{fof}(f_{10},\mbox{axiom})$

$\forall a,b{:}\;j(a,m(a,b)) = a\qquad\mbox{fof}(f_{11},\mbox{axiom})$

$\forall a,b,c{:}\;a\cdot j(b,c) = j(a\cdot b,a\cdot c)\qquad\mbox{fof}(f_{12},\mbox{axiom})$

$\forall a,b,c{:}\;j(b,c)\cdot a = j(b\cdot a,c\cdot a)\qquad\mbox{fof}(f_{13},\mbox{axiom})$

$a\ne e\qquad\mbox{fof}(f_{14},\mbox{axiom})$

\medskip\textbf{GRP123-9.004.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 4 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_1(x,y,e_1)\mbox{ or }\mbox{product}_1(x,y,e_2)\mbox{ or }\mbox{product}_1(x,y,e_3)\mbox{ or }\mbox{product}_1(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product1\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_1(x,y,w)\mbox{ and }\mbox{product}_1(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_1(x,w,y)\mbox{ and }\mbox{product}_1(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_1(w,y,x)\mbox{ and }\mbox{product}_1(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product1\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_1(x,x,x)\qquad\mbox{cnf}(\mbox{product1\_idempotence},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( \mbox{product}_2(x,y,e_1)\mbox{ or }\mbox{product}_2(x,y,e_2)\mbox{ or }\mbox{product}_2(x,y,e_3)\mbox{ or }\mbox{product}_2(x,y,e_4) )\qquad\mbox{cnf}(\mbox{product2\_total\_function}_1,\mbox{axiom})$

$( \mbox{product}_2(x,y,w)\mbox{ and }\mbox{product}_2(x,y,z) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_total\_function}_2,\mbox{axiom})$

$( \mbox{product}_2(x,w,y)\mbox{ and }\mbox{product}_2(x,z,y) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_right\_cancellation},\mbox{axiom})$

$( \mbox{product}_2(w,y,x)\mbox{ and }\mbox{product}_2(z,y,x) ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product2\_left\_cancellation},\mbox{axiom})$

$\mbox{product}_2(x,x,x)\qquad\mbox{cnf}(\mbox{product2\_idempotence},\mbox{axiom})$

$( \mbox{product}_1(x,y,z_1)\mbox{ and }\mbox{product}_1(z_1,y,z_2) ) \ \Rightarrow \ \mbox{product}_2(z_2,x,y)\qquad\mbox{cnf}(\mbox{qg1a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP157-1.p} Prove monotonicity axiom using the LUB transformation

This problem proves the original monotonicity axiom from the
 equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{ax\_mono2a}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(c\cdot a,c\cdot b)\ne c\cdot b\qquad\mbox{cnf}(\mbox{prove\_ax\_mono2a},\mbox{negated\_conjecture})$

\medskip\textbf{GRP115-1.p} Derive order 3 from a single axiom for groups order 3






$x\cdot ( ( x\cdot ( ( x\cdot y )\cdot z ) )\cdot ( \mbox{identity}\cdot ( z\cdot z ) ) ) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot ( a\cdot a )\ne \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_order}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP135-2.005.p} ((b.a).b).b) = a, no idempotence

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{next}(e_4,e_5)\qquad\mbox{cnf}(\mbox{e\_4\_then\_e}_5,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_5,e_1)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_5,e_2)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_3)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_5,e_4)\qquad\mbox{cnf}(\mbox{e\_5\_greater\_e}_4,\mbox{axiom})$

$( x\cdot e_1{=}y\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x_1)\qquad\mbox{cnf}(\mbox{no\_redundancy},\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP660+2.p} Bol-Moufang identity 3 implies a unit element






$\forall b,a{:}\;a\cdot \mbox{ld}(a,b) = b\qquad\mbox{fof}(f_{01},\mbox{axiom})$

$\forall b,a{:}\;\mbox{ld}(a,a\cdot b) = b\qquad\mbox{fof}(f_{02},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a,b)\cdot b = a\qquad\mbox{fof}(f_{03},\mbox{axiom})$

$\forall b,a{:}\;\mbox{rd}(a\cdot b,b) = a\qquad\mbox{fof}(f_{04},\mbox{axiom})$

$\forall c,b,a{:}\;( ( a\cdot b )\cdot c )\cdot a = a\cdot ( b\cdot ( c\cdot a ) )\qquad\mbox{fof}(f_{05},\mbox{axiom})$

$\forall x_0,x_1{:}\;( x_0\cdot \mbox{rd}(x_1,x_1) = x_0\mbox{ and }\mbox{rd}(x_1,x_1)\cdot x_0 = x_0 )\qquad\mbox{fof}(\mbox{goals},\mbox{conjecture})$

\medskip\textbf{GRP504-1.p} Axiom for group theory, in double division and inverse, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,c)'),\mbox{double\_divide}(b',\mbox{double\_divide}(d,\mbox{double\_divide}(a,d))')) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP476-1.p} Axiom for group theory, in division and inverse, part 2






$\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(a,b),c),\mbox{divide}(d,c))',\mbox{divide}(b,a)) = d\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,b')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP538-1.p} Axiom for Abelian group theory, in division, part 2






$\mbox{divide}(\mbox{divide}(a,b),\mbox{divide}(\mbox{divide}(a,c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP666-2.p} Inverse property A-loops are Moufang






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{rd}(( ( a\cdot b )\cdot c )\cdot d,c\cdot d) = \mbox{rd}(( a\cdot c )\cdot d,c\cdot d)\cdot \mbox{rd}(( b\cdot c )\cdot d,c\cdot d)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) )\ne ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP722-1.p} In commutative A-loops square-subloop operation is commutative






$a\cdot 1 = a\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{ld}(a\cdot b,a\cdot ( b\cdot ( c\cdot d ) )) = \mbox{ld}(a\cdot b,a\cdot ( b\cdot c ))\cdot \mbox{ld}(a\cdot b,a\cdot ( b\cdot d ))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{ld}(a,( b\cdot c )\cdot a) = \mbox{ld}(a,b\cdot a)\cdot \mbox{ld}(a,c\cdot a)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot a )\cdot ( b\cdot b ) = f(a,b)\cdot f(a,b)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$f(a,b)\ne f(b,a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP717-1.p} Strongly right alternative rings 2b

If a has a 2-sided inverse, then R(a$\wedge$-1) = R(a)$\wedge$-1 and
 L(a)$\wedge$-1 = R(a)L(a$\wedge$-1)R(a$\wedge$-1).




$( a + b ) + c = a + ( b + c )\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$a + b = b + a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$a + \mbox{op}_0 = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$a + -a = \mbox{op}_0\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot ( b + c ) = a\cdot b + a\cdot c\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot b ) = ( a\cdot b )\cdot b\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{op\_a}\cdot \mbox{op\_b} = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_b}\cdot \mbox{op\_a} = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( \mbox{op\_b}\cdot ( ( \mbox{op\_a}\cdot a )\cdot \mbox{op\_b} ) )\cdot \mbox{op\_a}\ne a\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP126-3.004.p} (a.b).(b.a) = b

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{next}(e_3,e_4)\qquad\mbox{cnf}(\mbox{e\_3\_then\_e}_4,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_4,e_0)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_4,e_1)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_2)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_2,\mbox{axiom})$

$\mbox{greater}(e_4,e_3)\qquad\mbox{cnf}(\mbox{e\_4\_greater\_e}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2)\mbox{ or }\mbox{cycle}(x,e_3) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_4,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}y\qquad\mbox{cnf}(\mbox{qg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP420-1.p} Axiom for group theory, in product \& inverse, part 3






$( ( a\cdot ( b'\cdot ( c\cdot ( c'\cdot c )' )' )' )'\cdot ( a\cdot c ) )' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP306-1.p} An identity generated by HR, number 17237






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_2 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_2\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$( \mbox{sk\_c}_8\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_8' = \mbox{sk\_c}_6\mbox{ and }x_3\cdot x_4 = \mbox{sk\_c}_8\mbox{ and }x_3' = x_4\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_1\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_7\mbox{ and }x_1' = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_2 = \mbox{sk\_c}_7\mbox{ and }x_5\cdot \mbox{sk\_c}_8 = x_2 ) \ \Rightarrow \ x_5'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{GRP552-1.p} Axiom for Abelian group theory, in division and identity, part 4






$\mbox{divide}(\mbox{divide}(\mbox{identity},a),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,a),c),b)) = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{identity},b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{identity},a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(a,a)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP072-1.p} Single axiom for group theory, in division and inverse

This is a single axiom for group theory, in terms of division
 and inverse.




$\mbox{divide}(\mbox{divide}(\mbox{divide}(x,y)',\mbox{divide}(\mbox{divide}(z,u),x)),\mbox{divide}(u,z)) = y\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,y')\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP595-1.p} Axiom for Abelian group theory, in double div and inv, part 3






$\mbox{double\_divide}(\mbox{double\_divide}(a,b),\mbox{double\_divide}(a,\mbox{double\_divide}(c,b)')')' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP272-1.p} An identity generated by HR, number 01392






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4' = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_6\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

$( x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_6\mbox{ and }x_3' = \mbox{sk\_c}_7\mbox{ and }\mbox{sk\_c}_7\cdot \mbox{sk\_c}_5 = \mbox{sk\_c}_6\mbox{ and }x_4\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_5\mbox{ and }x_4' = \mbox{sk\_c}_6\mbox{ and }\mbox{sk\_c}_6\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_5\mbox{ and }x_1' = \mbox{sk\_c}_7\mbox{ and }x_1\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\mbox{ and }x_2' = \mbox{sk\_c}_6 ) \ \Rightarrow \ x_2\cdot \mbox{sk\_c}_5\ne \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{GRP127-4.005.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( e_1\cdot x{=}y\mbox{ or }e_2\cdot x{=}y\mbox{ or }e_3\cdot x{=}y\mbox{ or }e_4\cdot x{=}y\mbox{ or }e_5\cdot x{=}y )\qquad\mbox{cnf}(\mbox{row\_surjectivity},\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot e_1{=}y\mbox{ or }x\cdot e_2{=}y\mbox{ or }x\cdot e_3{=}y\mbox{ or }x\cdot e_4{=}y\mbox{ or }x\cdot e_5{=}y )\qquad\mbox{cnf}(\mbox{column\_surjectivity},\mbox{axiom})$

$( z_2\cdot y{=}x\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ y\cdot x{=}z_1\qquad\mbox{cnf}(\mbox{qg3}_1,\mbox{negated\_conjecture})$

$( z_2\cdot y{=}x\mbox{ and }y\cdot x{=}z_1 ) \ \Rightarrow \ z_1\cdot y{=}z_2\qquad\mbox{cnf}(\mbox{qg3}_2,\mbox{negated\_conjecture})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP449-1.p} Axiom for group theory, in division, part 2






$\mbox{divide}(a,\mbox{divide}(\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),b),c),\mbox{divide}(\mbox{divide}(\mbox{divide}(b,b),a),c))) = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{divide}(a,\mbox{divide}(\mbox{divide}(c,c),b))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a' = \mbox{divide}(\mbox{divide}(b,b),a)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP748-5.p} Right alternative loop rings: a lemma






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(f_{08},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(f_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(f_{10},\mbox{axiom})$

$a\cdot b = b\cdot a\mbox{ or }i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(f_{11},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a )\ne a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP013-1.p} Commutator equals identity in these conditions

If X.X=identity and if X$\wedge$-1.Y$\wedge$-1 = Z then X.Z = Y, then
 (X.Y).(X$\wedge$-1.Y$\wedge$-1) = identity.




$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$a\cdot a{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{squareness},\mbox{hypothesis})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{hypothesis})$

$a'\cdot b'{=}d\qquad\mbox{cnf}(\mbox{inverse\_a\_times\_inverse\_b\_is\_d},\mbox{hypothesis})$

$a'\cdot b'{=}c \ \Rightarrow \ a\cdot c{=}b\qquad\mbox{cnf}(\mbox{inverses\_have\_property},\mbox{hypothesis})$

$\neg\,c\cdot d{=}\mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_c\_times\_d\_is\_identity},\mbox{negated\_conjecture})$

\medskip\textbf{GRP393-1.p} Semigroup axioms






$\mbox{include}(\mbox{'Axioms/GRP002-0.ax'})$

\medskip\textbf{GRP095-1.p} Single axiom for Abelian group theory, in division and identity

This is a single axiom for Abelian group theory, in terms
 of division and identity




$\mbox{divide}(\mbox{divide}(\mbox{identity},x),\mbox{divide}(\mbox{divide}(\mbox{divide}(y,x),z),y)) = z\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$x\cdot y = \mbox{divide}(x,\mbox{divide}(\mbox{identity},y))\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$x' = \mbox{divide}(\mbox{identity},x)\qquad\mbox{cnf}(\mbox{inverse},\mbox{axiom})$

$\mbox{identity} = \mbox{divide}(x,x)\qquad\mbox{cnf}(\mbox{identity},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2\mbox{ and }( a_3\cdot b_3 )\cdot c_3 = a_3\cdot ( b_3\cdot c_3 ) ) \ \Rightarrow \ a_4\cdot b_4\ne b_4\cdot a_4\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP140-1.p} Prove greatest lower-bound axiom using a transformation

This problem proves the original greatest lower-bound axiom
 from the equational axiomatization.




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{greatest\_lower\_bound}(a,c) = c\qquad\mbox{cnf}(\mbox{ax\_glb1c}_1,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(b,c) = c\qquad\mbox{cnf}(\mbox{ax\_glb1c}_2,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(\mbox{greatest\_lower\_bound}(a,b),c)\ne \mbox{greatest\_lower\_bound}(a,b)\qquad\mbox{cnf}(\mbox{prove\_ax\_glb1c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP748-3.p} Right alternative loop rings: a lemma






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(f_{06},\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b = a\cdot ( ( b\cdot c )\cdot b )\qquad\mbox{cnf}(f_{07},\mbox{axiom})$

$( a\cdot b )\cdot i(b) = a\qquad\mbox{cnf}(f_{08},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(f_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(f_{10},\mbox{axiom})$

$a\cdot b = b\cdot a\mbox{ or }i(a)\cdot ( a\cdot b ) = b\qquad\mbox{cnf}(f_{11},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) )\ne ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP127-1.005.p} ((b.a).b).b) = a

Generate the multiplication table for the specified quasi-
 group with 5 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\mbox{group\_element}(e_5)\qquad\mbox{cnf}(\mbox{element}_5,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_1{=}e_5\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_5\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_5\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_4{=}e_5\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_5,\mbox{axiom})$

$\neg\,e_5{=}e_1\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_5{=}e_2\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_5{=}e_3\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_5{=}e_4\qquad\mbox{cnf}(\mbox{e\_5\_is\_not\_e}_4,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4\mbox{ or }x\cdot y{=}e_5 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( y\cdot x{=}z_1\mbox{ and }z_1\cdot y{=}z_2 ) \ \Rightarrow \ z_2\cdot y{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP012-4.p} Inverse of products = Product of inverses

The inverse of products equals the product of the inverse,
 in opposite order




$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$x\cdot \mbox{identity} = x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$x\cdot x' = \mbox{identity}\qquad\mbox{cnf}(\mbox{right\_inverse},\mbox{axiom})$

$( a\cdot b )'\ne b'\cdot a'\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_product\_is\_product\_of\_inverses},\mbox{negated\_conjecture})$

\medskip\textbf{GRP696-1.p} Variety of power associative, WIP conjugacy closed loops - 1a






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot i(b\cdot a) = i(b)\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( ( a\cdot b )\cdot a )\cdot ( a\cdot c )\ne a\cdot ( ( ( b\cdot a )\cdot a )\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP176-2.p} General form of distributivity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$( x\cdot y )' = y'\cdot x'\qquad\mbox{cnf}(\mbox{p07}_1,\mbox{hypothesis})$

$c\cdot ( \mbox{least\_upper\_bound}(a,b)\cdot d )\ne \mbox{least\_upper\_bound}(c\cdot ( a\cdot d ),c\cdot ( b\cdot d ))\qquad\mbox{cnf}(\mbox{prove\_p}_{07},\mbox{negated\_conjecture})$

\medskip\textbf{GRP170-3.p} General form of monotonicity






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP004-2.ax'})$

$\mbox{least\_upper\_bound}(a,b) = b\qquad\mbox{cnf}(\mbox{p03c}_1,\mbox{hypothesis})$

$\mbox{least\_upper\_bound}(c,d) = d\qquad\mbox{cnf}(\mbox{p03c}_2,\mbox{hypothesis})$

$\mbox{greatest\_lower\_bound}(a\cdot c,b\cdot d)\ne a\cdot c\qquad\mbox{cnf}(\mbox{prove\_p03c},\mbox{negated\_conjecture})$

\medskip\textbf{GRP740-1.p} Proper commutative Moufang loop






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot ( a\cdot c ) ) = ( ( a\cdot b )\cdot a )\cdot c\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$a\cdot ( b\cdot ( c\cdot b ) ) = ( ( a\cdot b )\cdot c )\cdot b\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = ( a\cdot ( b\cdot c ) )\cdot a\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$( a\cdot b )\cdot ( c\cdot a ) = a\cdot ( ( b\cdot c )\cdot a )\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$a\cdot b = b\cdot a\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP204-1.p} A non-basis for Moufang loops.

Left identity, left inverse, Moufang-1 do not imply Moufang-2;
 that is, is not a basis for Moufang loops.




$\mbox{identity}\cdot x = x\qquad\mbox{cnf}(\mbox{left\_identity},\mbox{axiom})$

$\mbox{left\_inverse}(x)\cdot x = \mbox{identity}\qquad\mbox{cnf}(\mbox{left\_inverse},\mbox{axiom})$

$( x\cdot ( y\cdot z ) )\cdot x = ( x\cdot y )\cdot ( z\cdot x )\qquad\mbox{cnf}(\mbox{moufang}_1,\mbox{axiom})$

$( ( a\cdot b )\cdot c )\cdot b\ne a\cdot ( b\cdot ( c\cdot b ) )\qquad\mbox{cnf}(\mbox{prove\_moufang}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP125-1.004.p} (a.b).(b.a) = a

Generate the multiplication table for the specified quasi-
 group with 4 elements.




$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\mbox{group\_element}(e_4)\qquad\mbox{cnf}(\mbox{element}_4,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_1{=}e_4\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_4\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_3{=}e_4\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_4,\mbox{axiom})$

$\neg\,e_4{=}e_1\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_4{=}e_2\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_4{=}e_3\qquad\mbox{cnf}(\mbox{e\_4\_is\_not\_e}_3,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3\mbox{ or }x\cdot y{=}e_4 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x\cdot y{=}z_1\mbox{ and }y\cdot x{=}z_2 ) \ \Rightarrow \ z_1\cdot z_2{=}x\qquad\mbox{cnf}(\mbox{qg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{GRP604-1.p} Axiom for Abelian group theory, in double div and inv, part 4






$\mbox{double\_divide}(\mbox{double\_divide}(a,\mbox{double\_divide}(b,\mbox{double\_divide}(a,c))')',c)' = b\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$a\cdot b = \mbox{double\_divide}(b,a)'\qquad\mbox{cnf}(\mbox{multiply},\mbox{axiom})$

$a\cdot b\ne b\cdot a\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_4,\mbox{negated\_conjecture})$

\medskip\textbf{GRP035-3.p} In subgroups, product is closed






$\mbox{include}(\mbox{'Axioms/GRP003-0.ax'})$

$\mbox{include}(\mbox{'Axioms/GRP003-2.ax'})$

$\mbox{subgroup\_member}(a)\qquad\mbox{cnf}(\mbox{a\_is\_in\_subgroup},\mbox{hypothesis})$

$\mbox{subgroup\_member}(b)\qquad\mbox{cnf}(\mbox{b\_is\_in\_subgroup},\mbox{hypothesis})$

$a\cdot b{=}c\qquad\mbox{cnf}(\mbox{a\_times\_b\_is\_c},\mbox{hypothesis})$

$\neg\,\mbox{subgroup\_member}(c)\qquad\mbox{cnf}(\mbox{prove\_c\_is\_in\_subgroup},\mbox{negated\_conjecture})$

\medskip\textbf{GRP756-1.p} Quasigroups satisfying certain Bol-Moufang identity are groups






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(f_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(f_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(f_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(f_{04},\mbox{axiom})$

$a\cdot ( ( b\cdot b )\cdot c ) = ( a\cdot b )\cdot ( b\cdot c )\qquad\mbox{cnf}(f_{05},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{GRP230-1.p} An identity generated by HR, number 00466






$\mbox{include}(\mbox{'Axioms/GRP004-0.ax'})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_1,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_2,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_3,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_4,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_5,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1' = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_6,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_7,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_8,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_9,\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{10},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{11},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{12},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{13},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{14},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{15},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{16},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{17},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{18},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3\cdot \mbox{sk\_c}_4 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{19},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_3' = \mbox{sk\_c}_4\qquad\mbox{cnf}(\mbox{prove\_this}_{20},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{21},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_8\cdot \mbox{sk\_c}_6 = \mbox{sk\_c}_7\qquad\mbox{cnf}(\mbox{prove\_this}_{22},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5\cdot \mbox{sk\_c}_8 = \mbox{sk\_c}_6\qquad\mbox{cnf}(\mbox{prove\_this}_{23},\mbox{negated\_conjecture})$

$\mbox{sk\_c}_2' = \mbox{sk\_c}_7\mbox{ or }\mbox{sk\_c}_5' = \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{24},\mbox{negated\_conjecture})$

$( x_3' = \mbox{sk\_c}_8\mbox{ and }x_3\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_4\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }x_4' = \mbox{sk\_c}_7\mbox{ and }x_2\cdot x_1 = \mbox{sk\_c}_8\mbox{ and }x_2' = x_1\mbox{ and }x_1\cdot \mbox{sk\_c}_7 = \mbox{sk\_c}_8\mbox{ and }\mbox{sk\_c}_8\cdot x_5 = \mbox{sk\_c}_7\mbox{ and }x_6\cdot \mbox{sk\_c}_8 = x_5 ) \ \Rightarrow \ x_6'\ne \mbox{sk\_c}_8\qquad\mbox{cnf}(\mbox{prove\_this}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{GRP056-1.p} Single axiom for group theory, in product \& inverse

This is a single axiom for group theory, in terms of product
 and inverse.




$( ( z\cdot ( x'\cdot y )' )'\cdot ( z\cdot y' ) )'\cdot ( y'\cdot y )' = x\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( a_1'\cdot a_1 = b_1'\cdot b_1\mbox{ and }( b_2'\cdot b_2 )\cdot a_2 = a_2 ) \ \Rightarrow \ ( a_3\cdot b_3 )\cdot c_3\ne a_3\cdot ( b_3\cdot c_3 )\qquad\mbox{cnf}(\mbox{prove\_these\_axioms},\mbox{negated\_conjecture})$

\medskip\textbf{GRP123-3.003.p} (3,2,1) conjugate orthogonality

If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
 Generate the multiplication table for the specified quasi-
 group with 3 elements.




$\mbox{next}(e_0,e_1)\qquad\mbox{cnf}(\mbox{e\_0\_then\_e}_1,\mbox{axiom})$

$\mbox{next}(e_1,e_2)\qquad\mbox{cnf}(\mbox{e\_1\_then\_e}_2,\mbox{axiom})$

$\mbox{next}(e_2,e_3)\qquad\mbox{cnf}(\mbox{e\_2\_then\_e}_3,\mbox{axiom})$

$\mbox{greater}(e_1,e_0)\qquad\mbox{cnf}(\mbox{e\_1\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_0)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_3,e_0)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_0,\mbox{axiom})$

$\mbox{greater}(e_2,e_1)\qquad\mbox{cnf}(\mbox{e\_2\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_1)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_1,\mbox{axiom})$

$\mbox{greater}(e_3,e_2)\qquad\mbox{cnf}(\mbox{e\_3\_greater\_e}_2,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(x,z) ) \ \Rightarrow \ y{=}z\qquad\mbox{cnf}(\mbox{cycle}_1,\mbox{axiom})$

$\mbox{group\_element}(x) \ \Rightarrow \ ( \mbox{cycle}(x,e_0)\mbox{ or }\mbox{cycle}(x,e_1)\mbox{ or }\mbox{cycle}(x,e_2) )\qquad\mbox{cnf}(\mbox{cycle}_2,\mbox{axiom})$

$\mbox{cycle}(e_3,e_0)\qquad\mbox{cnf}(\mbox{cycle}_3,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }\mbox{cycle}(w,z)\mbox{ and }\mbox{next}(x,w)\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(z,z_1) ) \ \Rightarrow \ y{=}z_1\qquad\mbox{cnf}(\mbox{cycle}_4,\mbox{axiom})$

$( \mbox{cycle}(x,z_1)\mbox{ and }\mbox{cycle}(y,e_0)\mbox{ and }\mbox{cycle}(w,z_2)\mbox{ and }\mbox{next}(y,w)\mbox{ and }\mbox{greater}(y,x) ) \ \Rightarrow \ \neg\,\mbox{greater}(z_1,z_2)\qquad\mbox{cnf}(\mbox{cycle}_5,\mbox{axiom})$

$( \mbox{cycle}(x,e_0)\mbox{ and }x\cdot e_1{=}y ) \ \Rightarrow \ \neg\,\mbox{greater}(y,x)\qquad\mbox{cnf}(\mbox{cycle}_6,\mbox{axiom})$

$( \mbox{cycle}(x,y)\mbox{ and }x\cdot e_1{=}z\mbox{ and }\mbox{greater}(y,e_0)\mbox{ and }\mbox{next}(x,x_1) ) \ \Rightarrow \ z{=}x_1\qquad\mbox{cnf}(\mbox{cycle}_7,\mbox{axiom})$

$\mbox{group\_element}(e_1)\qquad\mbox{cnf}(\mbox{element}_1,\mbox{axiom})$

$\mbox{group\_element}(e_2)\qquad\mbox{cnf}(\mbox{element}_2,\mbox{axiom})$

$\mbox{group\_element}(e_3)\qquad\mbox{cnf}(\mbox{element}_3,\mbox{axiom})$

$\neg\,e_1{=}e_2\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_2,\mbox{axiom})$

$\neg\,e_1{=}e_3\qquad\mbox{cnf}(\mbox{e\_1\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_2{=}e_1\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_2{=}e_3\qquad\mbox{cnf}(\mbox{e\_2\_is\_not\_e}_3,\mbox{axiom})$

$\neg\,e_3{=}e_1\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_1,\mbox{axiom})$

$\neg\,e_3{=}e_2\qquad\mbox{cnf}(\mbox{e\_3\_is\_not\_e}_2,\mbox{axiom})$

$( \mbox{group\_element}(x)\mbox{ and }\mbox{group\_element}(y) ) \ \Rightarrow \ ( x\cdot y{=}e_1\mbox{ or }x\cdot y{=}e_2\mbox{ or }x\cdot y{=}e_3 )\qquad\mbox{cnf}(\mbox{product\_total\_function}_1,\mbox{axiom})$

$( x\cdot y{=}w\mbox{ and }x\cdot y{=}z ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_total\_function}_2,\mbox{axiom})$

$( x\cdot w{=}y\mbox{ and }x\cdot z{=}y ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_right\_cancellation},\mbox{axiom})$

$( w\cdot y{=}x\mbox{ and }z\cdot y{=}x ) \ \Rightarrow \ w{=}z\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$x\cdot x{=}x\qquad\mbox{cnf}(\mbox{product\_idempotence},\mbox{axiom})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ x_1{=}x_2\qquad\mbox{cnf}(\mbox{qg1}_1,\mbox{negated\_conjecture})$

$( x_1\cdot y_1{=}z_1\mbox{ and }x_2\cdot y_2{=}z_1\mbox{ and }z_2\cdot y_1{=}x_1\mbox{ and }z_2\cdot y_2{=}x_2 ) \ \Rightarrow \ y_1{=}y_2\qquad\mbox{cnf}(\mbox{qg1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP404-1.p} Axiom for group theory, in product \& inverse, part 2






$a\cdot ( ( ( a\cdot b )'\cdot c )'\cdot ( b\cdot ( b'\cdot b ) )' )' = c\qquad\mbox{cnf}(\mbox{single\_axiom},\mbox{axiom})$

$( b_2'\cdot b_2 )\cdot a_2\ne a_2\qquad\mbox{cnf}(\mbox{prove\_these\_axioms}_2,\mbox{negated\_conjecture})$

\medskip\textbf{GRP694-1.p} In power associative conjugacy closed loop c$\wedge$12 is in nucleus - b






$a\cdot \mbox{ld}(a,b) = b\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{ld}(a,a\cdot b) = b\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{rd}(a,b)\cdot b = a\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{rd}(a\cdot b,b) = a\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$a\cdot 1 = a\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$a\cdot ( b\cdot c ) = \mbox{rd}(a\cdot b,a)\cdot ( a\cdot c )\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$( a\cdot b )\cdot c = ( a\cdot c )\cdot \mbox{ld}(c,b\cdot c)\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$i(a)\cdot a = 1\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$a\cdot i(a) = 1\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{op\_c}\cdot ( \mbox{op\_c}\cdot \mbox{op\_c} ) = \mbox{op\_d}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{op\_d}\cdot \mbox{op\_d} = \mbox{op\_e}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{op\_e}\cdot \mbox{op\_e} = \mbox{op\_f}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$a\cdot ( \mbox{op\_f}\cdot b )\ne ( a\cdot \mbox{op\_f} )\cdot b\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip
\end{document}