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    \textbf{ALG002+0.ax} Median algebra axioms






$\forall x,y{:}\;f(x,x,y) = x\qquad\mbox{fof}(\mbox{majority},\mbox{axiom})$

$\forall x,y,z{:}\;f(x,y,z) = f(z,x,y)\qquad\mbox{fof}(\mbox{permute}_1,\mbox{axiom})$

$\forall x,y,z{:}\;f(x,y,z) = f(x,z,y)\qquad\mbox{fof}(\mbox{permute}_2,\mbox{axiom})$

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

\medskip\textbf{ALG001-0.ax} Abstract algebra axioms, based on Godel set theory






$( \mbox{associative}(\mbox{xs},\mbox{xf})\mbox{ and }x\in \mbox{xs}\mbox{ and }y\in \mbox{xs}\mbox{ and }z\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{apply\_to\_two\_arguments}(\mbox{xf},x,y),z) = \mbox{apply\_to\_two\_arguments}(\mbox{xf},x,\mbox{apply\_to\_two\_arguments}(\mbox{xf},y,z))\qquad\mbox{cnf}(\mbox{associative\_system}_1,\mbox{axiom})$

$\mbox{associative}(\mbox{xs},\mbox{xf})\mbox{ or }f_{34}(\mbox{xs},\mbox{xf})\in \mbox{xs}\qquad\mbox{cnf}(\mbox{associative\_system}_2,\mbox{axiom})$

$\mbox{associative}(\mbox{xs},\mbox{xf})\mbox{ or }f_{35}(\mbox{xs},\mbox{xf})\in \mbox{xs}\qquad\mbox{cnf}(\mbox{associative\_system}_3,\mbox{axiom})$

$\mbox{associative}(\mbox{xs},\mbox{xf})\mbox{ or }f_{36}(\mbox{xs},\mbox{xf})\in \mbox{xs}\qquad\mbox{cnf}(\mbox{associative\_system}_4,\mbox{axiom})$

$\mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{34}(\mbox{xs},\mbox{xf}),f_{35}(\mbox{xs},\mbox{xf})),f_{36}(\mbox{xs},\mbox{xf})) = \mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{34}(\mbox{xs},\mbox{xf}),\mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{35}(\mbox{xs},\mbox{xf}),f_{36}(\mbox{xs},\mbox{xf}))) \ \Rightarrow \ \mbox{associative}(\mbox{xs},\mbox{xf})\qquad\mbox{cnf}(\mbox{associative\_system}_5,\mbox{axiom})$

$\mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe}) \ \Rightarrow \ \mbox{xe}\in \mbox{xs}\qquad\mbox{cnf}(\mbox{identity}_1,\mbox{axiom})$

$( \mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe})\mbox{ and }x\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{xe},x) = x\qquad\mbox{cnf}(\mbox{identity}_2,\mbox{axiom})$

$( \mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe})\mbox{ and }x\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},x,\mbox{xe}) = x\qquad\mbox{cnf}(\mbox{identity}_3,\mbox{axiom})$

$\mbox{xe}\in \mbox{xs} \ \Rightarrow \ ( \mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe})\mbox{ or }f_{37}(\mbox{xs},\mbox{xf},\mbox{xe})\in \mbox{xs} )\qquad\mbox{cnf}(\mbox{identity}_4,\mbox{axiom})$

$( \mbox{xe}\in \mbox{xs}\mbox{ and }\mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{xe},f_{37}(\mbox{xs},\mbox{xf},\mbox{xe})) = f_{37}(\mbox{xs},\mbox{xf},\mbox{xe})\mbox{ and }\mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{37}(\mbox{xs},\mbox{xf},\mbox{xe}),\mbox{xe}) = f_{37}(\mbox{xs},\mbox{xf},\mbox{xe}) ) \ \Rightarrow \ \mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe})\qquad\mbox{cnf}(\mbox{identity}_5,\mbox{axiom})$

$\mbox{xs}' \ \Rightarrow \ \mbox{maps}(\mbox{xg},\mbox{xs},\mbox{xs})\qquad\mbox{cnf}(\mbox{inverse}_1,\mbox{axiom})$

$( \mbox{xs}'\mbox{ and }x\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{apply}(\mbox{xg},x),x) = \mbox{xe}\qquad\mbox{cnf}(\mbox{inverse}_2,\mbox{axiom})$

$( \mbox{xs}'\mbox{ and }x\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},x,\mbox{apply}(\mbox{xg},x)) = \mbox{xe}\qquad\mbox{cnf}(\mbox{inverse}_3,\mbox{axiom})$

$\mbox{maps}(\mbox{xg},\mbox{xs},\mbox{xs}) \ \Rightarrow \ ( \mbox{xs}'\mbox{ or }f_{38}(\mbox{xs},\mbox{xf},\mbox{xe},\mbox{xg})\in \mbox{xs} )\qquad\mbox{cnf}(\mbox{inverse}_4,\mbox{axiom})$

$( \mbox{maps}(\mbox{xg},\mbox{xs},\mbox{xs})\mbox{ and }\mbox{apply\_to\_two\_arguments}(\mbox{xf},\mbox{apply}(\mbox{xg},f_{38}(\mbox{xs},\mbox{xf},\mbox{xe},\mbox{xg})),f_{38}(\mbox{xs},\mbox{xf},\mbox{xe},\mbox{xg})) = \mbox{xe}\mbox{ and }\mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{38}(\mbox{xs},\mbox{xf},\mbox{xe},\mbox{xg}),\mbox{apply}(\mbox{xg},f_{38}(\mbox{xs},\mbox{xf},\mbox{xe},\mbox{xg}))) = \mbox{xe} ) \ \Rightarrow \ \mbox{xs}'\qquad\mbox{cnf}(\mbox{inverse}_5,\mbox{axiom})$

$\mbox{group}(\mbox{xs},\mbox{xf}) \ \Rightarrow \ \mbox{closed}(\mbox{xs},\mbox{xf})\qquad\mbox{cnf}(\mbox{group}_1,\mbox{axiom})$

$\mbox{group}(\mbox{xs},\mbox{xf}) \ \Rightarrow \ \mbox{associative}(\mbox{xs},\mbox{xf})\qquad\mbox{cnf}(\mbox{group}_2,\mbox{axiom})$

$\mbox{group}(\mbox{xs},\mbox{xf}) \ \Rightarrow \ \mbox{identity}(\mbox{xs},\mbox{xf},f_{39}(\mbox{xs},\mbox{xf}))\qquad\mbox{cnf}(\mbox{group}_3,\mbox{axiom})$

$\mbox{group}(\mbox{xs},\mbox{xf}) \ \Rightarrow \ \mbox{xs}'\qquad\mbox{cnf}(\mbox{group}_4,\mbox{axiom})$

$( \mbox{closed}(\mbox{xs},\mbox{xf})\mbox{ and }\mbox{associative}(\mbox{xs},\mbox{xf})\mbox{ and }\mbox{identity}(\mbox{xs},\mbox{xf},\mbox{xe})\mbox{ and }\mbox{xs}' ) \ \Rightarrow \ \mbox{group}(\mbox{xs},\mbox{xf})\qquad\mbox{cnf}(\mbox{group}_5,\mbox{axiom})$

$( \mbox{commutes}(\mbox{xs},\mbox{xf})\mbox{ and }x\in \mbox{xs}\mbox{ and }y\in \mbox{xs} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}(\mbox{xf},x,y) = \mbox{apply\_to\_two\_arguments}(\mbox{xf},y,x)\qquad\mbox{cnf}(\mbox{commutes}_1,\mbox{axiom})$

$\mbox{commutes}(\mbox{xs},\mbox{xf})\mbox{ or }f_{41}(\mbox{xs},\mbox{xf})\in \mbox{xs}\qquad\mbox{cnf}(\mbox{commutes}_2,\mbox{axiom})$

$\mbox{commutes}(\mbox{xs},\mbox{xf})\mbox{ or }f_{42}(\mbox{xs},\mbox{xf})\in \mbox{xs}\qquad\mbox{cnf}(\mbox{commutes}_3,\mbox{axiom})$

$\mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{41}(\mbox{xs},\mbox{xf}),f_{42}(\mbox{xs},\mbox{xf})) = \mbox{apply\_to\_two\_arguments}(\mbox{xf},f_{42}(\mbox{xs},\mbox{xf}),f_{41}(\mbox{xs},\mbox{xf})) \ \Rightarrow \ \mbox{commutes}(\mbox{xs},\mbox{xf})\qquad\mbox{cnf}(\mbox{commutes}_4,\mbox{axiom})$

\medskip\textbf{ALG301-1.p} Identity with no nontrivial finite model

One of the two shortest identities with no nontrivial finite 
 models (in a single binary operation).




$a\cdot ( a\cdot ( a\cdot ( b\cdot ( c\cdot a ) ) ) ) = b\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$\mbox{tptp}_1\ne \mbox{tptp}_0\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

\medskip\textbf{ALG282$\wedge$5.p} TPS problem from GRP-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cE}{:}\;a\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cJ}{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ \forall x{:}\;a,y{:}\;a{:}\;\exists w{:}\;a{:}\;( \mbox{cP}@ w@ x ) = y\qquad\mbox{thf}(\mbox{cTHM21\_pme},\mbox{conjecture})$

\medskip\textbf{ALG003-1.p} Cancellative medial algebras

We prove a property of cancellative medial algebras.




$( x\cdot y = z\mbox{ and }u\cdot y = z ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( x\cdot y = z\mbox{ and }x\cdot u = z ) \ \Rightarrow \ y = u\qquad\mbox{cnf}(\mbox{right\_cancellation},\mbox{axiom})$

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

$\mbox{an\_element}\cdot \mbox{an\_element} = \mbox{an\_element}\qquad\mbox{cnf}(\mbox{idempotent\_element},\mbox{hypothesis})$

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

\medskip\textbf{ALG073+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-5






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{op}_1(u,u) = v\mbox{ and }\mbox{op}_1(v,v) = u\mbox{ and }\mbox{op}_1(u,v)\ne u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{op}_2(u,u) = v\mbox{ and }\mbox{op}_2(v,v) = u\mbox{ and }\mbox{op}_2(u,v)\ne u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG305-1.p} Random graph 3, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG010-1.p} Prove J follows from HBCK

Axioms for the quasivariety HBCK are given below. Show that
 equation J follows.




$a\cdot 1 = 1\qquad\mbox{cnf}(m_3,\mbox{axiom})$

$1\cdot a = a\qquad\mbox{cnf}(m_4,\mbox{axiom})$

$( a\cdot b )\cdot ( ( c\cdot a )\cdot ( c\cdot b ) ) = 1\qquad\mbox{cnf}(m_5,\mbox{axiom})$

$( a\cdot b = 1\mbox{ and }b\cdot a = 1 ) \ \Rightarrow \ a = b\qquad\mbox{cnf}(m_7,\mbox{axiom})$

$a\cdot a = 1\qquad\mbox{cnf}(m_8,\mbox{axiom})$

$a\cdot ( b\cdot c ) = b\cdot ( a\cdot c )\qquad\mbox{cnf}(m_9,\mbox{axiom})$

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

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

\medskip\textbf{ALG029+1.p} Groups 6: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(u,v) = \mbox{op}_1(v,u) ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(u,v) = \mbox{op}_2(v,u) ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG288$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x@ \mbox{xu} ) \ \Rightarrow \ ( x@ ( \mbox{cL}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cZ} ) ) ) \ \Rightarrow \ \forall x{:}\;a\to \mbox{\$o},\mbox{xz}{:}\;a{:}\;( \exists \mbox{xy}{:}\;a{:}\;( x@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\iff \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{29}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{29} ) ) \ \Rightarrow \ ( x@ \mbox{xx}_{29} ) )\mbox{ and }\exists \mbox{xy}{:}\;a{:}\;\forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_SETR\_CTS\_pme},\mbox{conjecture})$

\medskip\textbf{ALG258$\wedge$2.p} HOASlam is injective






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslaminj\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG001-1.p} The composition of homomorphisms is a homomorphism






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\mbox{ and }x\in \mbox{xd} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_3,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\subseteq \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_1,\mbox{axiom})$

$\mbox{function}(\mbox{xf}) \ \Rightarrow \ \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\subseteq \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_2,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x)) = \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_3,\mbox{axiom})$

$( x\in \mbox{xs}_1\mbox{ and }y\in \mbox{xs}_2 ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{xs}_1,\mbox{xs}_2)\qquad\mbox{cnf}(\mbox{ordered\_pair\_in\_cross\_product},\mbox{axiom})$

$\mbox{homomorphism}(f_{60},f_{62},f_{63},f_{64},f_{65})\qquad\mbox{cnf}(\mbox{one\_homomorphism},\mbox{hypothesis})$

$\mbox{homomorphism}(f_{61},f_{64},f_{65},f_{66},f_{67})\qquad\mbox{cnf}(\mbox{another\_homomorphism},\mbox{hypothesis})$

$\neg\,\mbox{homomorphism}(f_{60}\circ f_{61},f_{62},f_{63},f_{66},f_{67})\qquad\mbox{cnf}(\mbox{prove\_composition\_is\_a\_homomorphism},\mbox{negated\_conjecture})$

\medskip\textbf{ALG291$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall z{:}\;a,p{:}\;a\to a\to a,l{:}\;a\to a,r{:}\;a\to a,x{:}\;a\to \mbox{\$o}{:}\;( ( ( l@ z ) = z\mbox{ and }( r@ z ) = z\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( l@ ( p@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( r@ ( p@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne z\iff \mbox{xt} = ( p@ ( l@ \mbox{xt} )@ ( r@ \mbox{xt} ) ) )\mbox{ and }\forall x_0{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x_0@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x_0@ \mbox{xu} ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x_0@ z ) ) ) \ \Rightarrow \ \forall x_0{:}\;a\to \mbox{\$o},\mbox{xz}{:}\;a{:}\;( \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_9{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_9 ) ) \ \Rightarrow \ ( x@ \mbox{xx}_9 ) )\mbox{ and }x_0@ ( p@ \mbox{xx}@ \mbox{xz} ) )\iff \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{10}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_{10} ) ) \ \Rightarrow \ ( x_0@ \mbox{xx}_{10} ) )\mbox{ and }\exists \mbox{xx}_{12}{:}\;a{:}\;( \forall \mbox{xx}_{11}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}_{12}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_{11} ) ) \ \Rightarrow \ ( x@ \mbox{xx}_{11} ) )\mbox{ and }\forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = ( p@ \mbox{xx}_{12}@ \mbox{xz} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_X238B\_pme},\mbox{conjecture})$

\medskip\textbf{ALG253$\wedge$2.p} Induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{induction2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG202+1.p} Quasigroups 7 QG5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{op}_1(u,u) = u )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \mbox{op}_2(u,u) = u )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG289$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x@ \mbox{xu} ) \ \Rightarrow \ ( x@ ( \mbox{cL}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cZ} ) ) ) \ \Rightarrow \ \forall x{:}\;a\to \mbox{\$o},\mbox{xz}{:}\;a{:}\;( \exists \mbox{xz}_0{:}\;a{:}\;( x@ ( \mbox{cP}@ \mbox{xz}@ \mbox{xz}_0 ) )\iff \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{13}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{13} ) ) \ \Rightarrow \ ( x@ \mbox{xx}_{13} ) )\mbox{ and }\exists \mbox{xz}_1{:}\;a{:}\;\forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = ( \mbox{cP}@ \mbox{xz}@ \mbox{xz}_1 ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_SETL\_CTS\_pme},\mbox{conjecture})$

\medskip\textbf{ALG011-1.p} Partition a monoid into 2 partitions

If C,D is a partition of a monoid M, we cannot have
 C * C subset D and D * D subset C.




$f(x,f(y,z)) = f(f(x,y),z)\qquad\mbox{cnf}(\mbox{f\_is\_associative},\mbox{axiom})$

$c(x)\mbox{ or }d(x)\qquad\mbox{cnf}(\mbox{partitions\_union},\mbox{axiom})$

$c(x) \ \Rightarrow \ \neg\,d(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive},\mbox{hypothesis})$

$c(a_1)\qquad\mbox{cnf}(\mbox{partition\_c\_not\_empty},\mbox{hypothesis})$

$d(a_2)\qquad\mbox{cnf}(\mbox{partition\_d\_not\_empty},\mbox{hypothesis})$

$( c(x)\mbox{ and }c(y) ) \ \Rightarrow \ d(f(x,y))\qquad\mbox{cnf}(\mbox{conjecture}_1,\mbox{negated\_conjecture})$

$( d(x)\mbox{ and }d(y) ) \ \Rightarrow \ c(f(x,y))\qquad\mbox{cnf}(\mbox{conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{ALG009-1.p} Abstract algebra axioms, based on Godel set theory






$\mbox{include}(\mbox{'Axioms/ALG001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

\medskip\textbf{ALG249$\wedge$3.p} Push property lemma 2






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem2v2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG008-1.p} TC + right identity does not give RC.

An algebra with a right identity satisfying the Thomsen
 Closure (RC) condition does not necessarily satisfy the
 Reidemeister Closure (RC) condition.




$( x\cdot y = z\mbox{ and }u\cdot v = z\mbox{ and }x\cdot w = v_6\mbox{ and }v_7\cdot v = v_6 ) \ \Rightarrow \ u\cdot w = v_7\cdot y\qquad\mbox{cnf}(\mbox{thomsen\_closure},\mbox{axiom})$

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

$c_4\cdot a = c_3\cdot b\qquad\mbox{cnf}(\mbox{prove\_reidimeister}_1,\mbox{negated\_conjecture})$

$c_2\cdot a = c_1\cdot b\qquad\mbox{cnf}(\mbox{prove\_reidimeister}_2,\mbox{negated\_conjecture})$

$c_4\cdot f = c_3\cdot \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_reidimeister}_3,\mbox{negated\_conjecture})$

$c_2\cdot f\ne c_1\cdot \mbox{identity}\qquad\mbox{cnf}(\mbox{prove\_reidimeister}_4,\mbox{negated\_conjecture})$

\medskip\textbf{ALG315-1.p} Random graph 13, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG263$\wedge$1.p} HOAS induction lemma 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem1\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG257$\wedge$2.p} HOASap is injective 2






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapinj2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG286$\wedge$5.p} TPS problem from PAIRING-UNPAIRING-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$u{:}\;a\qquad\mbox{thf}(u,\mbox{type})$

$y{:}\;a\qquad\mbox{thf}(y,\mbox{type})$

$x{:}\;a\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}_0{:}\;a,\mbox{xy}_0{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) = \mbox{xx}_0\mbox{ and }\forall \mbox{xx}_0{:}\;a,\mbox{xy}_0{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) = \mbox{xy}_0\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) ) ) \ \Rightarrow \ ( u = ( \mbox{cP}@ x@ y ) \ \Rightarrow \ u\ne \mbox{cZ} )\qquad\mbox{thf}(\mbox{cPU\_PAIR\_NOT\_ZERO\_pme},\mbox{conjecture})$

\medskip\textbf{ALG298$\wedge$5.p} TPS problem THM270






$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{c\_starc}{:}\;c\to c\to c\qquad\mbox{thf}(\mbox{c\_starc},\mbox{type})$

$\mbox{c\_starb}{:}\;b\to b\to b\qquad\mbox{thf}(\mbox{c\_starb},\mbox{type})$

$\mbox{c\_stara}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{c\_stara},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to b,\mbox{xg}{:}\;a\to c,\mbox{xh}{:}\;b\to c{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( \mbox{xh}@ ( \mbox{xf}@ \mbox{xx} ) ) = ( \mbox{xg}@ \mbox{xx} )\mbox{ and }\forall \mbox{xy}{:}\;b{:}\;\exists \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{c\_stara}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{c\_starb}@ ( \mbox{xf}@ \mbox{xx} )@ ( \mbox{xf}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{xg}@ ( \mbox{c\_stara}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{c\_starc}@ ( \mbox{xg}@ \mbox{xx} )@ ( \mbox{xg}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( \mbox{xh}@ ( \mbox{c\_starb}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{c\_starc}@ ( \mbox{xh}@ \mbox{xx} )@ ( \mbox{xh}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cTHM270\_pme},\mbox{conjecture})$

\medskip\textbf{ALG013-1.p} Partition a monoid into 4 partitions

If C1,C2,D1,D2 is a partition of a monoid M, we cannot have
 Ci * Ci subset D1 u D2 and Dj * Dj subset C1 u C2.




$f(x,f(y,z)) = f(f(x,y),z)\qquad\mbox{cnf}(\mbox{f\_is\_associative},\mbox{axiom})$

$c_2(x)\mbox{ or }c_1(x)\mbox{ or }d_1(x)\mbox{ or }d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_union},\mbox{axiom})$

$c_1(x) \ \Rightarrow \ \neg\,c_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c1\_c}_2,\mbox{hypothesis})$

$c_1(x) \ \Rightarrow \ \neg\,d_1(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c1\_d}_1,\mbox{hypothesis})$

$c_1(x) \ \Rightarrow \ \neg\,d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c1\_d}_2,\mbox{hypothesis})$

$c_2(x) \ \Rightarrow \ \neg\,d_1(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c2\_d}_1,\mbox{hypothesis})$

$c_2(x) \ \Rightarrow \ \neg\,d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c2\_d}_2,\mbox{hypothesis})$

$d_1(x) \ \Rightarrow \ \neg\,d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_d1\_d}_2,\mbox{hypothesis})$

$c_1(a_1)\qquad\mbox{cnf}(\mbox{partition\_c1\_not\_empty},\mbox{hypothesis})$

$c_2(a_2)\qquad\mbox{cnf}(\mbox{partition\_c2\_not\_empty},\mbox{hypothesis})$

$d_1(a_3)\qquad\mbox{cnf}(\mbox{partition\_d1\_not\_empty},\mbox{hypothesis})$

$d_2(a_4)\qquad\mbox{cnf}(\mbox{partition\_d2\_not\_empty},\mbox{hypothesis})$

$( c_1(x)\mbox{ and }c_1(y) ) \ \Rightarrow \ ( d_2(f(x,y))\mbox{ or }d_1(f(x,y)) )\qquad\mbox{cnf}(\mbox{conjecture}_1,\mbox{negated\_conjecture})$

$( c_2(x)\mbox{ and }c_2(y) ) \ \Rightarrow \ ( d_2(f(x,y))\mbox{ or }d_1(f(x,y)) )\qquad\mbox{cnf}(\mbox{conjecture}_2,\mbox{negated\_conjecture})$

$( d_1(x)\mbox{ and }d_1(y) ) \ \Rightarrow \ ( c_2(f(x,y))\mbox{ or }c_1(f(x,y)) )\qquad\mbox{cnf}(\mbox{conjecture}_3,\mbox{negated\_conjecture})$

$( d_2(x)\mbox{ and }d_2(y) ) \ \Rightarrow \ ( c_2(f(x,y))\mbox{ or }c_1(f(x,y)) )\qquad\mbox{cnf}(\mbox{conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{ALG213+1.p} Distributivity, short version






$\mbox{include}(\mbox{'Axioms/ALG002+0.ax'})$

$\forall u,w,x,y,z{:}\;f(f(x,y,z),u,w) = f(x,f(y,u,w),f(z,u,w))\qquad\mbox{fof}(\mbox{dist\_long},\mbox{conjecture})$

\medskip\textbf{ALG268$\wedge$5.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3v2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG245-1.p} Idempotent selfdistributive groupoids are symmetric-by-medial - 4






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

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

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

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

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

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

\medskip\textbf{ALG072+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-4






$\mbox{sorti}_1(\mbox{unit}_1)\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ ( \mbox{op}_1(\mbox{unit}_1,u) = u\mbox{ and }\mbox{op}_1(u,\mbox{unit}_1) = u ) )\mbox{ and }\exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{unit}_1 = v )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\mbox{sorti}_2(\mbox{unit}_2)\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ ( \mbox{op}_2(\mbox{unit}_2,u) = u\mbox{ and }\mbox{op}_2(u,\mbox{unit}_2) = u ) )\mbox{ and }\exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{unit}_2 = v )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_5,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_6,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ ( \mbox{op}_1(v,w)\ne u\mbox{ or }( \mbox{op}_1(v,u) = w\mbox{ and }u\ne \mbox{unit}_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{ax}_7,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \forall w{:}\;( \mbox{sorti}_2(w) \ \Rightarrow \ ( \mbox{op}_2(v,w)\ne u\mbox{ or }( \mbox{op}_2(v,u) = w\mbox{ and }u\ne \mbox{unit}_2 ) ) ) ) )\qquad\mbox{fof}(\mbox{ax}_8,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG239-1.p} Selfdistributive groupoids are symmetric-by-medial - part 3






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

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

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

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

\medskip\textbf{ALG268$\wedge$2.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG237-1.p} Selfdistributive groupoids are symmetric-by-medial - part 1






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

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

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

\medskip\textbf{ALG312-1.p} Random graph 10, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG210+1.p} Star-algebras are closed under multiplication






$\forall a,b,c{:}\;\mbox{times}(\mbox{times}(a,b),c) = \mbox{times}(b,\mbox{times}(c,a))\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall b{:}\;( \mbox{element}(b)\iff \exists c{:}\;( \mbox{times}(b,c) = b\mbox{ and }\mbox{times}(b,b) = c ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{element}(a)\mbox{ and }\mbox{element}(b) ) \ \Rightarrow \ \mbox{element}(\mbox{times}(a,b)) )\qquad\mbox{fof}(\mbox{conjecture}_1,\mbox{conjecture})$

\medskip\textbf{ALG211+1.p} Vector spaces and bases






$\forall b,v{:}\;( \mbox{basis\_of}(b,v) \ \Rightarrow \ ( \mbox{lin\_ind\_subset}(b,v)\mbox{ and }\mbox{a\_subset\_of}(b,\mbox{vec\_to\_class}(v)) ) )\qquad\mbox{fof}(\mbox{basis\_of},\mbox{axiom})$

$\forall s,t,v{:}\;( ( \mbox{lin\_ind\_subset}(s,v)\mbox{ and }\mbox{basis\_of}(t,v) ) \ \Rightarrow \ \exists u{:}\;( \mbox{a\_subset\_of}(u,t)\mbox{ and }\mbox{basis\_of}(\mbox{union}(s,u),v) ) )\qquad\mbox{fof}(\mbox{bg\_2\_2}_5,\mbox{axiom})$

$\forall a{:}\;( \mbox{a\_vector\_space}(a) \ \Rightarrow \ \exists b{:}\;\mbox{basis\_of}(b,a) )\qquad\mbox{fof}(\mbox{bg\_remark\_63\_a},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{a\_vector\_subspace\_of}(a,b) \ \Rightarrow \ \mbox{a\_vector\_space}(a) )\qquad\mbox{fof}(\mbox{bg\_2\_4\_a},\mbox{axiom})$

$\forall w,v,e{:}\;( ( \mbox{a\_vector\_subspace\_of}(w,v)\mbox{ and }\mbox{a\_subset\_of}(e,\mbox{vec\_to\_class}(w)) ) \ \Rightarrow \ ( \mbox{lin\_ind\_subset}(e,w)\iff \mbox{lin\_ind\_subset}(e,v) ) )\qquad\mbox{fof}(\mbox{bg\_2\_4}_2,\mbox{axiom})$

$\forall w,v{:}\;( ( \mbox{a\_vector\_subspace\_of}(w,v)\mbox{ and }\mbox{a\_vector\_space}(v) ) \ \Rightarrow \ \exists e,f{:}\;( \mbox{basis\_of}(\mbox{union}(e,f),v)\mbox{ and }\mbox{basis\_of}(e,w) ) )\qquad\mbox{fof}(\mbox{bg\_2\_4}_3,\mbox{conjecture})$

\medskip\textbf{ALG292$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall z{:}\;a,p{:}\;a\to a\to a,l{:}\;a\to a,r{:}\;a\to a,f{:}\;a\to \mbox{\$o}{:}\;( ( ( l@ z ) = z\mbox{ and }( r@ z ) = z\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( l@ ( p@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( r@ ( p@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne z\iff \mbox{xt} = ( p@ ( l@ \mbox{xt} )@ ( r@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x@ \mbox{xu} ) \ \Rightarrow \ ( x@ ( l@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x@ z ) ) ) \ \Rightarrow \ \forall x{:}\;a\to \mbox{\$o},\mbox{xz}{:}\;a{:}\;( \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_5{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_5 ) ) \ \Rightarrow \ ( x@ \mbox{xx}_5 ) )\mbox{ and }f@ ( p@ \mbox{xx}@ \mbox{xz} ) )\iff \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_6{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_6 ) ) \ \Rightarrow \ ( x@ \mbox{xx}_6 ) )\mbox{ and }\exists \mbox{xx}_8{:}\;a{:}\;( \forall \mbox{xx}_7{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}_8\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_7 ) ) \ \Rightarrow \ \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( l@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( r@ \mbox{xv} ) = \mbox{xx}_7 ) ) )\mbox{ and }f@ ( p@ \mbox{xx}_8@ \mbox{xz} ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_X238A\_pme},\mbox{conjecture})$

\medskip\textbf{ALG248$\wedge$1.p} Push property lemma 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem1\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG243-1.p} Idempotent selfdistributive groupoids are symmetric-by-medial - 2






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

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

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

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

\medskip\textbf{ALG299-1.p} An equational theory with no nontrivial finite models

A classical example of an equational theory with no nontrivial 
 finite models (found idependently by Tarski, Jonsson, Skornyakov 
 and others).




$f(a\cdot b) = a\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$g(a\cdot b) = b\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

$\mbox{tptp}_1\ne \mbox{tptp}_0\qquad\mbox{cnf}(\mbox{sos}_{03},\mbox{axiom})$

\medskip\textbf{ALG071+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-3






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{op}_1(v,\mbox{op}_1(v,u))\ne u\mbox{ and }\mbox{op}_1(u,\mbox{op}_1(v,u)) = v ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{op}_2(v,\mbox{op}_2(v,u))\ne u\mbox{ and }\mbox{op}_2(u,\mbox{op}_2(v,u)) = v ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG268$\wedge$6.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3v2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG300-1.p} Identity with no nontrivial finite model

One of the two shortest identities with no nontrivial finite 
 models (in a single binary operation).




$( ( ( a\cdot a )\cdot a )\cdot b )\cdot ( a\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$\mbox{tptp}_1\ne \mbox{tptp}_0\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

\medskip\textbf{ALG238-1.p} Selfdistributive groupoids are symmetric-by-medial - part 2






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

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

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

\medskip\textbf{ALG002-1.p} In an ordered field, if X $>$ 0 then X$\wedge$-1 $>$ 0






$x\cdot 1{=}x\qquad\mbox{cnf}(\mbox{right\_identity},\mbox{axiom})$

$\neg\,1\cdot 1{=}0\qquad\mbox{cnf}(\mbox{not\_abelian},\mbox{axiom})$

$x\cdot y{=}z \ \Rightarrow \ ( -x )\cdot ( -y ){=}z\qquad\mbox{cnf}(\mbox{product\_of\_inverses}_1,\mbox{axiom})$

$( -x )\cdot ( -y ){=}z \ \Rightarrow \ x\cdot y{=}z\qquad\mbox{cnf}(\mbox{product\_of\_inverses}_2,\mbox{axiom})$

$x\cdot y{=}z \ \Rightarrow \ x\cdot ( -y ){=}-z\qquad\mbox{cnf}(\mbox{product\_to\_inverse},\mbox{axiom})$

$x\cdot x^{-1}{=}1\mbox{ or }x\cdot x{=}0\qquad\mbox{cnf}(\mbox{inverse\_and\_identity},\mbox{axiom})$

$\mbox{greater\_than}_0(x) \ \Rightarrow \ \neg\,\mbox{greater\_than}_0(-x)\qquad\mbox{cnf}(\mbox{inverse\_greater\_than}_0,\mbox{axiom})$

$\mbox{greater\_than}_0(x) \ \Rightarrow \ \neg\,x\cdot x{=}0\qquad\mbox{cnf}(\mbox{greater\_than\_0\_square},\mbox{axiom})$

$x\cdot y{=}z \ \Rightarrow \ y\cdot x{=}z\qquad\mbox{cnf}(\mbox{commutativity\_of\_product},\mbox{axiom})$

$\mbox{greater\_than}_0(x)\mbox{ or }x\cdot x{=}0\mbox{ or }\mbox{greater\_than}_0(-x)\qquad\mbox{cnf}(\mbox{product\_and\_inverse},\mbox{axiom})$

$( y\cdot z{=}x\mbox{ and }y\cdot y{=}0 ) \ \Rightarrow \ x\cdot x{=}0\qquad\mbox{cnf}(\mbox{square\_to}_0,\mbox{axiom})$

$( y\cdot z{=}x\mbox{ and }\mbox{greater\_than}_0(y)\mbox{ and }\mbox{greater\_than}_0(z) ) \ \Rightarrow \ \mbox{greater\_than}_0(x)\qquad\mbox{cnf}(\mbox{product\_and\_greater\_than}_0,\mbox{axiom})$

$\mbox{greater\_than}_0(a)\qquad\mbox{cnf}(\mbox{a\_greater\_than}_0,\mbox{hypothesis})$

$\neg\,\mbox{greater\_than}_0(a^{-1})\qquad\mbox{cnf}(\mbox{prove\_a\_inverse\_greater\_than}_0,\mbox{negated\_conjecture})$

\medskip\textbf{ALG268$\wedge$1.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG284$\wedge$5.p} TPS problem from GRP-THMS






$\mbox{cJ}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cE}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{cJ}@ ( \mbox{cP}@ x@ y ) ) = ( \mbox{cP}@ ( \mbox{cJ}@ y )@ ( \mbox{cJ}@ x ) )\qquad\mbox{thf}(\mbox{cTHM18\_pme},\mbox{conjecture})$

\medskip\textbf{ALG297$\wedge$5.p} TPS problem from S-SEQ-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$t{:}\;a\qquad\mbox{thf}(t,\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}_0{:}\;a{:}\;( \mbox{xt}_0\ne \mbox{cZ}\iff \mbox{xt}_0 = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt}_0 )@ ( \mbox{cR}@ \mbox{xt}_0 ) ) ) ) \ \Rightarrow \ \forall \mbox{xs}{:}\;a{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }x@ \mbox{xy} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xb}{:}\;a{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xb} ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;a{:}\;( \exists \mbox{xb}_9{:}\;a,\mbox{xu}_{13}{:}\;a{:}\;( ( \mbox{cP}@ \mbox{xb}@ \mbox{xu} ) = ( \mbox{cP}@ \mbox{xb}_9@ \mbox{xu}_{13} )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ ( \mbox{cP}@ \mbox{cZ}@ t )\mbox{ and }\forall \mbox{xc}{:}\;a,\mbox{xv}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xc}@ \mbox{xv} ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ \mbox{cZ} )@ ( \mbox{cL}@ \mbox{xv} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) )@ ( \mbox{cR}@ \mbox{xv} ) ) ) ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xb}_9@ \mbox{xu}_{13} ) ) ) ) \ \Rightarrow \ \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }x@ \mbox{xy} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xu} ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM7\_pme},\mbox{conjecture})$

\medskip\textbf{ALG316-1.p} Random graph 14, siggers polymorphism






$t(x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polysiggers}_{01},\mbox{axiom})$

$t(x,y,x,z) = t(y,x,z,y)\qquad\mbox{cnf}(\mbox{polysiggers}_{02},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6),t(x_1,x_3,x_5,x_7))\qquad\mbox{cnf}(\mbox{preserves}_{03},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG302-1.p} Austin's identity






$( ( ( a\cdot a )\cdot a )\cdot b )\cdot ( ( ( a\cdot a )\cdot ( ( a\cdot a )\cdot a ) )\cdot c ) = b\qquad\mbox{cnf}(\mbox{sos}_{01},\mbox{axiom})$

$\mbox{tptp}_1\ne \mbox{tptp}_0\qquad\mbox{cnf}(\mbox{sos}_{02},\mbox{axiom})$

\medskip\textbf{ALG268$\wedge$4.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3v2\_f\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG314-1.p} Random graph 12, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG235-1.p} Short equational base for two varieties of groupoids - part 1a






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

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

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

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

\medskip\textbf{ALG276$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_1{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP1\_type},\mbox{type})$

$\mbox{cGROUP}_4{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP4\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_DIVISORS}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_type},\mbox{type})$

$\mbox{cGRP\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_DIVISORS} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g{:}\;( \exists \mbox{xx}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xx} ) = \mbox{xb}\mbox{ and }\exists \mbox{xy}{:}\;g{:}\;( \mbox{xf}@ \mbox{xy}@ \mbox{xa} ) = \mbox{xb} ) )\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_def},\mbox{definition})$

$\mbox{cGRP\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe}\mbox{ and }( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa}\mbox{ and }( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} ) )\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_1 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP1\_def},\mbox{definition})$

$\mbox{cGROUP}_4 = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_DIVISORS}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cGROUP4\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g{:}\;( \exists \mbox{xe}{:}\;g{:}\;( \mbox{cGROUP}_1@ \mbox{xf}@ \mbox{xe} )\iff ( \mbox{cGROUP}_4@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cEQUIV\_01}_{04},\mbox{conjecture})$

\medskip\textbf{ALG274$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_3{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP3\_type},\mbox{type})$

$\mbox{cGROUP}_4{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP4\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_DIVISORS}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_DIVISORS} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g{:}\;( \exists \mbox{xx}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xx} ) = \mbox{xb}\mbox{ and }\exists \mbox{xy}{:}\;g{:}\;( \mbox{xf}@ \mbox{xy}@ \mbox{xa} ) = \mbox{xb} ) )\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_3 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_RIGHT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_RIGHT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP3\_def},\mbox{definition})$

$\mbox{cGROUP}_4 = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_DIVISORS}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cGROUP4\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g{:}\;( \exists \mbox{xe}{:}\;g{:}\;( \mbox{cGROUP}_3@ \mbox{xf}@ \mbox{xe} )\iff ( \mbox{cGROUP}_4@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cEQUIV\_03}_{04},\mbox{conjecture})$

\medskip\textbf{ALG290$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cG}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cX}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cX},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cF}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x_0{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x_0@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x_0@ \mbox{xu} ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x_0@ \mbox{cZ} ) ) ) \ \Rightarrow \ ( \lambda \mbox{xy}{:}\;a{:}\;\exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{17}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;a{:}\;( ( x_0@ \mbox{xz} ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz} ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{17} ) ) \ \Rightarrow \ ( \mbox{cX}@ \mbox{xx}_{17} ) )\mbox{ and }( \mbox{cF}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ or }\mbox{cG}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{18}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{18} ) ) \ \Rightarrow \ ( \mbox{cX}@ \mbox{xx}_{18} ) )\mbox{ and }\mbox{cF}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xz} ) )\mbox{ or }\exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{19}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{19} ) ) \ \Rightarrow \ ( \mbox{cX}@ \mbox{xx}_{19} ) )\mbox{ and }\mbox{cG}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_X2310A\_pme},\mbox{conjecture})$

\medskip\textbf{ALG272$\wedge$5.p} TPS problem EQUIV-01-02






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_1{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP1\_type},\mbox{type})$

$\mbox{cGROUP}_2{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP2\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe}\mbox{ and }( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_def},\mbox{definition})$

$\mbox{cGRP\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa}\mbox{ and }( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} ) )\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_1 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP1\_def},\mbox{definition})$

$\mbox{cGROUP}_2 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_LEFT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_LEFT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP2\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \mbox{cGROUP}_1@ \mbox{xf}@ \mbox{xe} )\iff ( \mbox{cGROUP}_2@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cEQUIV\_01}_{02},\mbox{conjecture})$

\medskip\textbf{ALG287$\wedge$5.p} TPS problem from PAIRING-UNPAIRING-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$v{:}\;a\qquad\mbox{thf}(v,\mbox{type})$

$u{:}\;a\qquad\mbox{thf}(u,\mbox{type})$

$y{:}\;a\qquad\mbox{thf}(y,\mbox{type})$

$x{:}\;a\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}_0{:}\;a,\mbox{xy}_0{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) = \mbox{xx}_0\mbox{ and }\forall \mbox{xx}_0{:}\;a,\mbox{xy}_0{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) = \mbox{xy}_0\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) ) ) \ \Rightarrow \ ( ( \mbox{cP}@ x@ u ) = ( \mbox{cP}@ y@ v ) \ \Rightarrow \ ( x = y\mbox{ and }u = v ) )\qquad\mbox{thf}(\mbox{cPU\_P\_INJ\_pme},\mbox{conjecture})$

\medskip\textbf{ALG005-1.p} Associativity of intersection in terms of set difference.

Starting with Kalman's basis for families of sets closed under
 set difference, we define intersection and show it to be
 associative.




$x\setminus ( y\setminus x ) = x\qquad\mbox{cnf}(\mbox{set\_difference}_1,\mbox{axiom})$

$x\setminus ( x\setminus y ) = y\setminus ( y\setminus x )\qquad\mbox{cnf}(\mbox{set\_difference}_2,\mbox{axiom})$

$( x\setminus y )\setminus z = ( x\setminus z )\setminus ( y\setminus z )\qquad\mbox{cnf}(\mbox{set\_difference}_3,\mbox{axiom})$

$x\cdot y = x\setminus ( x\setminus y )\qquad\mbox{cnf}(\mbox{intersection},\mbox{axiom})$

$( a\cdot b )\cdot c\ne a\cdot ( b\cdot c )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_multiply},\mbox{negated\_conjecture})$

\medskip\textbf{ALG308-1.p} Random graph 6, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG320-1.p} Random graph 18, nu5 polymorphism 






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG265$\wedge$2.p} HOAS induction lemma 3aa






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3aa\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG442-1.p} Malcev, wnu3, wnu4 implies majority






$m(a,a,b) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$m(a,b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{001},\mbox{axiom})$

$u(a,a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{002},\mbox{axiom})$

$v(a,a,a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{003},\mbox{axiom})$

$u(a,a,b) = u(a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{004},\mbox{axiom})$

$u(a,a,b) = u(b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{005},\mbox{axiom})$

$v(a,a,a,b) = v(a,a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{006},\mbox{axiom})$

$v(a,a,b,a) = v(a,b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{007},\mbox{axiom})$

$v(a,b,a,a) = v(b,a,a,a)\qquad\mbox{cnf}(\mbox{sos}_{008},\mbox{axiom})$

$u(a,a,b) = v(a,a,a,b)\qquad\mbox{cnf}(\mbox{sos}_{009},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8) ) \ \Rightarrow \ r(m(x_0,x_3,x_6),m(x_1,x_4,x_7),m(x_2,x_5,x_8))\qquad\mbox{cnf}(\mbox{sos}_{010},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8) ) \ \Rightarrow \ r(u(x_0,x_3,x_6),u(x_1,x_4,x_7),u(x_2,x_5,x_8))\qquad\mbox{cnf}(\mbox{sos}_{011},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8)\mbox{ and }r(x_9,x_{10},x_{11}) ) \ \Rightarrow \ r(v(x_0,x_3,x_6,x_9),v(x_1,x_4,x_7,x_{10}),v(x_2,x_5,x_8,x_{11}))\qquad\mbox{cnf}(\mbox{sos}_{012},\mbox{axiom})$

$r(a,a,b)\qquad\mbox{cnf}(\mbox{sos}_{013},\mbox{axiom})$

$r(a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{014},\mbox{axiom})$

$r(b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{015},\mbox{axiom})$

$\neg\,r(a,a,a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{ALG273$\wedge$5.p} TPS problem EQUIV-02-03






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_2{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP2\_type},\mbox{type})$

$\mbox{cGROUP}_3{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP3\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_2 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_LEFT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_LEFT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP2\_def},\mbox{definition})$

$\mbox{cGROUP}_3 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_RIGHT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_RIGHT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP3\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \mbox{cGROUP}_2@ \mbox{xf}@ \mbox{xe} )\iff ( \mbox{cGROUP}_3@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cEQUIV\_02}_{03},\mbox{conjecture})$

\medskip\textbf{ALG263$\wedge$3.p} HOAS induction lemma 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem1v2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG179+1.p} Quasigroups 5 QG4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-4






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(\mbox{op}_1(u,v),v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(\mbox{op}_2(u,v),v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG177+1.p} Quasigroups 5 QG4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(u,v)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(u,v)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG268$\wedge$3.p} HOAS induction lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3v2a\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG267$\wedge$1.p} HOAS induction lemma 3b






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3b\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG260$\wedge$1.p} HOASlam not var






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslamnotvar\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG261$\wedge$1.p} HOASap not var






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapnotvar\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG256$\wedge$2.p} HOASap is injective 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapinj1\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG269$\wedge$3.p} HOAS induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lthm}_3\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG260$\wedge$2.p} HOASlam not var






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslamnotvar\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG269$\wedge$4.p} HOAS induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_no\_psi\_cond\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG212+1.p} Distributivity, long version






$\mbox{include}(\mbox{'Axioms/ALG002+0.ax'})$

$\forall u,w,x,y,z{:}\;f(f(x,y,z),u,w) = f(f(x,u,w),f(y,u,w),f(z,u,w))\qquad\mbox{fof}(\mbox{dist\_long},\mbox{conjecture})$

\medskip\textbf{ALG280$\wedge$5.p} TPS problem from GRP-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cE}{:}\;a\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cJ}{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ \forall x{:}\;a{:}\;( \mbox{cP}@ x@ \mbox{cE} ) = x\qquad\mbox{thf}(\mbox{cTHM17\_pme},\mbox{conjecture})$

\medskip\textbf{ALG313-1.p} Random graph 11, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG001$\wedge$5.p} TPS problem THM133

The composition of homomorphisms of binary operators is a
 homomorphisms. Boyer et al JAR 2 page 284.




$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xf}_1{:}\;g\to g\to g,\mbox{xf}_2{:}\;b\to b\to b,\mbox{xf}_3{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} )@ ( \mbox{xh}_1@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} )@ ( \mbox{xh}_2@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) )@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM133\_pme},\mbox{conjecture})$

\medskip\textbf{ALG267$\wedge$2.p} HOAS induction lemma 3b






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3b\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG281$\wedge$5.p} TPS problem from GRP-THMS






$\mbox{cE}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cJ}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ x@ ( \mbox{cJ}@ x ) ) = \mbox{cE}\qquad\mbox{thf}(\mbox{cTHM16\_pme},\mbox{conjecture})$

\medskip\textbf{ALG307-1.p} Random graph 5, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG256$\wedge$1.p} HOASap is injective 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapinj1\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG201+1.p} Quasigroups 7 QG5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{op}_1(u,u)\ne u )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \mbox{op}_2(u,u)\ne u )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG178+1.p} Quasigroups 5 QG4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-3






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(u,\mbox{op}_1(u,v)) = v ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(u,\mbox{op}_2(u,v)) = v ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG296$\wedge$5.p} TPS problem from SEQUENTIAL-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xb}{:}\;a{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xb} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;a{:}\;( \exists \mbox{xb}_0{:}\;a,\mbox{xu}_0{:}\;a{:}\;( ( \mbox{cP}@ \mbox{xb}@ \mbox{xu} ) = ( \mbox{cP}@ \mbox{xb}_0@ \mbox{xu}_0 )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{xt} )\mbox{ and }\forall \mbox{xc}{:}\;a,\mbox{xv}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xc}@ \mbox{xv} ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ \mbox{cZ} )@ ( \mbox{cL}@ \mbox{xv} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) )@ ( \mbox{cR}@ \mbox{xv} ) ) ) ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xb}_0@ \mbox{xu}_0 ) ) ) )\mbox{ and }\forall \mbox{xv}{:}\;a{:}\;( \exists \mbox{xb}_1{:}\;a,\mbox{xu}_0{:}\;a{:}\;( ( \mbox{cP}@ \mbox{xb}@ \mbox{xv} ) = ( \mbox{cP}@ \mbox{xb}_1@ \mbox{xu}_0 )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{xt} )\mbox{ and }\forall \mbox{xc}{:}\;a,\mbox{xv}_0{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xc}@ \mbox{xv}_0 ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ \mbox{cZ} )@ ( \mbox{cL}@ \mbox{xv}_0 ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) )@ ( \mbox{cR}@ \mbox{xv}_0 ) ) ) ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xb}_1@ \mbox{xu}_0 ) ) ) ) \ \Rightarrow \ \mbox{xu} = \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM6\_pme},\mbox{conjecture})$

\medskip\textbf{ALG258$\wedge$1.p} HOASlam is injective






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslaminj\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG074+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-6






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{op}_1(v,u)\ne \mbox{op}_1(u,v)\mbox{ and }\mbox{op}_1(\mbox{op}_1(u,v),v) = u\mbox{ and }\mbox{op}_1(\mbox{op}_1(u,v),u)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{op}_2(v,u)\ne \mbox{op}_2(u,v)\mbox{ and }\mbox{op}_2(\mbox{op}_2(u,v),v) = u\mbox{ and }\mbox{op}_2(\mbox{op}_2(u,v),u)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG262$\wedge$2.p} HOAS induction lemma 0






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem0\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG252$\wedge$1.p} Induction lemma






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{induction2lem\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG264$\wedge$1.p} HOAS induction lemma 2






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG004-1.p} Cancellative medial algebras satisfy the quotient condition.






$( x\cdot y = z\mbox{ and }u\cdot y = z ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( x\cdot y = z\mbox{ and }x\cdot u = z ) \ \Rightarrow \ y = u\qquad\mbox{cnf}(\mbox{right\_cancelaation},\mbox{axiom})$

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

$b\cdot b_0 = a\cdot a_0\qquad\mbox{cnf}(\mbox{prove\_quotient\_condition}_1,\mbox{negated\_conjecture})$

$d\cdot b_0 = c\cdot a_0\qquad\mbox{cnf}(\mbox{prove\_quotient\_condition}_2,\mbox{negated\_conjecture})$

$b\cdot d_0 = a\cdot c_0\qquad\mbox{cnf}(\mbox{prove\_quotient\_condition}_3,\mbox{negated\_conjecture})$

$d\cdot d_0\ne c\cdot c_0\qquad\mbox{cnf}(\mbox{prove\_quotient\_condition}_4,\mbox{negated\_conjecture})$

\medskip\textbf{ALG241-1.p} Selfdistributive groupoids are symmetric-by-medial - part 5






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

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

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

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

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

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

\medskip\textbf{ALG248$\wedge$2.p} Push property lemma 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem1\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG210+2.p} Star-algebras are closed under multiplication






$\forall a,b,c{:}\;\mbox{times}(\mbox{times}(a,b),c) = \mbox{times}(b,\mbox{times}(c,a))\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall b{:}\;( \mbox{element}(b)\iff \exists c{:}\;( \mbox{times}(b,c) = b\mbox{ and }\mbox{times}(b,b) = c ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{element}(a)\mbox{ and }\mbox{element}(b)\mbox{ and }c = \mbox{times}(a,b) ) \ \Rightarrow \ \mbox{element}(c) )\qquad\mbox{fof}(\mbox{conjecture}_1,\mbox{conjecture})$

\medskip\textbf{ALG269$\wedge$1.p} HOAS induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG440-1.p} Malcev, wnu2, wnu3 implies majority






$m(a,a,b) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$m(a,b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{001},\mbox{axiom})$

$u(a,a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{002},\mbox{axiom})$

$v(a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{003},\mbox{axiom})$

$u(a,a,b) = u(a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{004},\mbox{axiom})$

$u(a,a,b) = u(b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{005},\mbox{axiom})$

$v(a,b) = v(b,a)\qquad\mbox{cnf}(\mbox{sos}_{006},\mbox{axiom})$

$u(a,a,b) = v(a,b)\qquad\mbox{cnf}(\mbox{sos}_{007},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8) ) \ \Rightarrow \ r(m(x_0,x_3,x_6),m(x_1,x_4,x_7),m(x_2,x_5,x_8))\qquad\mbox{cnf}(\mbox{sos}_{008},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8) ) \ \Rightarrow \ r(u(x_0,x_3,x_6),u(x_1,x_4,x_7),u(x_2,x_5,x_8))\qquad\mbox{cnf}(\mbox{sos}_{009},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5) ) \ \Rightarrow \ r(v(x_0,x_3),v(x_1,x_4),v(x_2,x_5))\qquad\mbox{cnf}(\mbox{sos}_{010},\mbox{axiom})$

$r(a,a,b)\qquad\mbox{cnf}(\mbox{sos}_{011},\mbox{axiom})$

$r(a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{012},\mbox{axiom})$

$r(b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{013},\mbox{axiom})$

$\neg\,r(a,a,a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{ALG254$\wedge$2.p} M is a monoid and T is an M-set






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{substmonoid\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG441-1.p} Malcev, wnu2, wnu4 implies majority






$m(a,a,b) = b\qquad\mbox{cnf}(\mbox{sos},\mbox{axiom})$

$m(a,b,b) = a\qquad\mbox{cnf}(\mbox{sos}_{001},\mbox{axiom})$

$u(a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{002},\mbox{axiom})$

$v(a,a,a,a) = a\qquad\mbox{cnf}(\mbox{sos}_{003},\mbox{axiom})$

$u(a,b) = u(b,a)\qquad\mbox{cnf}(\mbox{sos}_{004},\mbox{axiom})$

$v(a,a,a,b) = v(a,a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{005},\mbox{axiom})$

$v(a,a,b,a) = v(a,b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{006},\mbox{axiom})$

$v(a,b,a,a) = v(b,a,a,a)\qquad\mbox{cnf}(\mbox{sos}_{007},\mbox{axiom})$

$u(a,b) = v(a,a,a,b)\qquad\mbox{cnf}(\mbox{sos}_{008},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8) ) \ \Rightarrow \ r(m(x_0,x_3,x_6),m(x_1,x_4,x_7),m(x_2,x_5,x_8))\qquad\mbox{cnf}(\mbox{sos}_{009},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5) ) \ \Rightarrow \ r(u(x_0,x_3),u(x_1,x_4),u(x_2,x_5))\qquad\mbox{cnf}(\mbox{sos}_{010},\mbox{axiom})$

$( r(x_0,x_1,x_2)\mbox{ and }r(x_3,x_4,x_5)\mbox{ and }r(x_6,x_7,x_8)\mbox{ and }r(x_9,x_{10},x_{11}) ) \ \Rightarrow \ r(v(x_0,x_3,x_6,x_9),v(x_1,x_4,x_7,x_{10}),v(x_2,x_5,x_8,x_{11}))\qquad\mbox{cnf}(\mbox{sos}_{011},\mbox{axiom})$

$r(a,a,b)\qquad\mbox{cnf}(\mbox{sos}_{012},\mbox{axiom})$

$r(a,b,a)\qquad\mbox{cnf}(\mbox{sos}_{013},\mbox{axiom})$

$r(b,a,a)\qquad\mbox{cnf}(\mbox{sos}_{014},\mbox{axiom})$

$\neg\,r(a,a,a)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{ALG257$\wedge$1.p} HOASap is injective 2






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapinj2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG266$\wedge$2.p} HOAS induction lemma 3a






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3a\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG264$\wedge$3.p} HOAS induction lemma 2






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem2v2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG070+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ ( \mbox{op}_1(u,u)\ne v\mbox{ or }\mbox{op}_1(u,v) = u ) ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ ( \mbox{op}_2(u,u)\ne v\mbox{ or }\mbox{op}_2(u,v) = u ) ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG306-1.p} Random graph 4, edge5 polymorphism






$t(y,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polyedge5}_{01},\mbox{axiom})$

$t(y,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polyedge5}_{02},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polyedge5}_{03},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polyedge5}_{04},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG069+1.p} Loops 5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{op}_1(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{op}_2(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG444$\wedge$1.p} Axioms for Untyped lambda sigma calculus






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

\medskip\textbf{ALG248$\wedge$3.p} Push property lemma 1






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem1v2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG001-3.p} The composition of homomorphisms is a homomorphism






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{homomorphism}(\mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_homomorphisms}_1,\mbox{negated\_conjecture})$

$\mbox{homomorphism}(\mbox{xh}_2,\mbox{xf}_2,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_homomorphisms}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{homomorphism}(\mbox{xh}_2\circ \mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_homomorphisms}_3,\mbox{negated\_conjecture})$

\medskip\textbf{ALG254$\wedge$1.p} M is a monoid and T is an M-set






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{substmonoid\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG006-1.p} Simplification of Kalman's set difference basis (part 1)

This is part 1 of a proof that one of the axioms in Kalman's
 basis for set difference can be simplified.




$x\setminus ( y\setminus x ) = x\qquad\mbox{cnf}(\mbox{set\_difference}_1,\mbox{axiom})$

$x\setminus ( x\setminus y ) = y\setminus ( y\setminus x )\qquad\mbox{cnf}(\mbox{set\_difference}_2,\mbox{axiom})$

$( x\setminus y )\setminus z = ( x\setminus z )\setminus ( y\setminus z )\qquad\mbox{cnf}(\mbox{set\_difference}_3,\mbox{axiom})$

$( a\setminus c )\setminus b\ne ( a\setminus b )\setminus c\qquad\mbox{cnf}(\mbox{prove\_set\_difference\_3\_simplified},\mbox{negated\_conjecture})$

\medskip\textbf{ALG295$\wedge$5.p} TPS problem from SEQUENTIAL-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) ) ) \ \Rightarrow \ ( \forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x@ \mbox{xu} ) \ \Rightarrow \ ( x@ ( \mbox{cL}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cZ} ) )\iff \forall \mbox{xt}{:}\;a{:}\;\exists \mbox{xn}{:}\;a{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xn} ) )\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( \exists \mbox{xb}{:}\;a,\mbox{xu}_{11}{:}\;a{:}\;( ( \mbox{cP}@ \mbox{xn}@ \mbox{xu} ) = ( \mbox{cP}@ \mbox{xb}@ \mbox{xu}_{11} )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{xt} )\mbox{ and }\forall \mbox{xc}{:}\;a,\mbox{xv}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xc}@ \mbox{xv} ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ \mbox{cZ} )@ ( \mbox{cL}@ \mbox{xv} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xc}@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{cZ} ) )@ ( \mbox{cR}@ \mbox{xv} ) ) ) ) ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xb}@ \mbox{xu}_{11} ) ) ) ) \ \Rightarrow \ \mbox{xu} = \mbox{cZ} ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM8\_pme},\mbox{conjecture})$

\medskip\textbf{ALG319-1.p} Random graph 17, siggers polymorphism






$t(x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polysiggers}_{01},\mbox{axiom})$

$t(x,y,x,z) = t(y,x,z,y)\qquad\mbox{cnf}(\mbox{polysiggers}_{02},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6),t(x_1,x_3,x_5,x_7))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG246-1.p} Axioms of SBL algebras are not independent






$\mbox{tptp}_1(a,b) = \mbox{tptp}_1(b,a)\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{tptp}_1(\mbox{tptp}_5,a) = a\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{tptp}_1(a,\mbox{tptp}_0(a,b)) = \mbox{tptp}_1(b,\mbox{tptp}_0(b,a))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{tptp}_0(\mbox{tptp}_1(a,b),c) = \mbox{tptp}_0(a,\mbox{tptp}_0(b,c))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{tptp}_0(\mbox{tptp}_3,a) = \mbox{tptp}_5\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{tptp}_0(\mbox{tptp}_0(\mbox{tptp}_0(a,b),c),\mbox{tptp}_0(\mbox{tptp}_0(\mbox{tptp}_0(b,a),c),c)) = \mbox{tptp}_5\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{tptp}_4(a,b) = \mbox{tptp}_1(a,\mbox{tptp}_0(a,b))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$v(a,b) = \mbox{tptp}_4(\mbox{tptp}_0(\mbox{tptp}_0(a,b),b),\mbox{tptp}_0(\mbox{tptp}_0(b,a),a))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{tptp}_0(\mbox{tptp}_1(a,b),\mbox{tptp}_3) = v(\mbox{tptp}_0(a,\mbox{tptp}_3),\mbox{tptp}_0(b,\mbox{tptp}_3))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$m(a) = \mbox{tptp}_0(a,\mbox{tptp}_3)\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$n(a) = m(\mbox{tptp}_2(a))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{tptp}_2(\mbox{tptp}_2(a)) = a\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$n(\mbox{tptp}_0(a,b)) = n(\mbox{tptp}_0(\mbox{tptp}_2(b),\mbox{tptp}_2(a)))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$v(\mbox{tptp}_1(n(a),n(\mbox{tptp}_0(a,b))),n(b))\ne n(b)\qquad\mbox{cnf}(\mbox{goals},\mbox{negated\_conjecture})$

\medskip\textbf{ALG275$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$f{:}\;g\to g\to g\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{cGROUP}_2{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP2\_type},\mbox{type})$

$\mbox{cGROUP}_4{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP4\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_DIVISORS}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_DIVISORS} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g{:}\;( \exists \mbox{xx}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xx} ) = \mbox{xb}\mbox{ and }\exists \mbox{xy}{:}\;g{:}\;( \mbox{xf}@ \mbox{xy}@ \mbox{xa} ) = \mbox{xb} ) )\qquad\mbox{thf}(\mbox{cGRP\_DIVISORS\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_2 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_LEFT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_LEFT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP2\_def},\mbox{definition})$

$\mbox{cGROUP}_4 = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_DIVISORS}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cGROUP4\_def},\mbox{definition})$

$\forall \mbox{xf}_0{:}\;g\to g\to g{:}\;\exists \mbox{xe}{:}\;g{:}\;( \mbox{cGROUP}_2@ \mbox{xf}_0@ \mbox{xe} )\iff ( \mbox{cGROUP}_4@ f )\qquad\mbox{thf}(\mbox{cEQUIV\_02}_{04},\mbox{conjecture})$

\medskip\textbf{ALG310-1.p} Random graph 8, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG266$\wedge$1.p} HOAS induction lemma 3a






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lem3a\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG293$\wedge$5.p} TPS problem from PU-LAMBDA-MODEL-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt}\mbox{ and }\forall \mbox{xu}{:}\;a{:}\;( ( x@ \mbox{xu} ) \ \Rightarrow \ ( x@ ( \mbox{cL}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cZ} ) ) ) \ \Rightarrow \ \forall x{:}\;a\to \mbox{\$o},\mbox{xz}{:}\;a{:}\;( \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{23}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{23} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;a{:}\;( x@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xx}_{23} ) ) )\mbox{ and }\exists \mbox{xz}_2{:}\;a{:}\;( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xz} )@ \mbox{xz}_2 ) ) )\iff \exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xx}_{24}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{24} ) ) \ \Rightarrow \ ( x@ \mbox{xx}_{24} ) )\mbox{ and }\exists \mbox{xx}_{26}{:}\;a{:}\;( \forall \mbox{xx}_{25}{:}\;a{:}\;( \forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}_{26}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = \mbox{xx}_{25} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;a{:}\;\forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = ( \mbox{cP}@ \mbox{xy}@ \mbox{xx}_{25} ) ) ) )\mbox{ and }\exists \mbox{xz}_3{:}\;a{:}\;\forall x_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}\mbox{ and }\forall \mbox{xz}_0{:}\;a{:}\;( ( x_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( x_0@ ( \mbox{cL}@ \mbox{xz}_0 ) ) ) ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( x_0@ \mbox{xv}\mbox{ and }( \mbox{cR}@ \mbox{xv} ) = ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}_{26}@ \mbox{xz} )@ \mbox{xz}_3 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_X239\_pme},\mbox{conjecture})$

\medskip\textbf{ALG236-1.p} Short equational base for two varieties of groupoids - part 1b






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

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

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

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

\medskip\textbf{ALG283$\wedge$5.p} TPS problem from GRP-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cE}{:}\;a\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cJ}{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ \forall x{:}\;a,y{:}\;a{:}\;\exists z{:}\;a{:}\;( \mbox{cP}@ x@ z ) = y\qquad\mbox{thf}(\mbox{cTHM20\_pme},\mbox{conjecture})$

\medskip\textbf{ALG279$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \forall \mbox{xb}{:}\;g,\mbox{xc}{:}\;g,\mbox{xa}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) )\mbox{ and }\mbox{cGRP\_RIGHT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_RIGHT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) \ \Rightarrow \ \forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cE13A2A},\mbox{conjecture})$

\medskip\textbf{ALG317-1.p} Random graph 15, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG285$\wedge$5.p} TPS problem from GRP-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cE}{:}\;a\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cJ}{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{cP}@ \mbox{cE}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cJ}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{cE} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{cP}@ \mbox{xx}@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall x{:}\;a,y{:}\;a{:}\;( \exists u{:}\;a{:}\;( \mbox{cP}@ x@ u ) = y\mbox{ and }\exists v{:}\;a{:}\;( \mbox{cP}@ v@ x ) = y ) )\qquad\mbox{thf}(\mbox{cTHM22\_pme},\mbox{conjecture})$

\medskip\textbf{ALG251$\wedge$2.p} Push property






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG270$\wedge$5.p} TPS problem THM23






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{c\_star}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{c\_star},\mbox{type})$

$\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{c\_star}@ ( \mbox{c\_star}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{c\_star}@ \mbox{xx}@ ( \mbox{c\_star}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ \forall w{:}\;a,x{:}\;a,y{:}\;a,z{:}\;a{:}\;( \mbox{c\_star}@ ( \mbox{c\_star}@ ( \mbox{c\_star}@ w@ x )@ y )@ z ) = ( \mbox{c\_star}@ w@ ( \mbox{c\_star}@ x@ ( \mbox{c\_star}@ y@ z ) ) )\qquad\mbox{thf}(\mbox{cTHM23\_pme},\mbox{conjecture})$

\medskip\textbf{ALG041+1.p} Loops 4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG247$\wedge$2.p} Push property lemma 0






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem0\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG007-1.p} Simplification of Kalman's set difference basis (part 2)

This is part 2 of a proof that one of the axioms in Kalman's
 basis for set difference can be simplified.




$x\setminus ( y\setminus x ) = x\qquad\mbox{cnf}(\mbox{set\_difference}_1,\mbox{axiom})$

$x\setminus ( x\setminus y ) = y\setminus ( y\setminus x )\qquad\mbox{cnf}(\mbox{set\_difference}_2,\mbox{axiom})$

$( x\setminus y )\setminus z = ( x\setminus z )\setminus y\qquad\mbox{cnf}(\mbox{set\_difference\_3\_simplified},\mbox{axiom})$

$( a\setminus b )\setminus c\ne ( a\setminus c )\setminus ( b\setminus c )\qquad\mbox{cnf}(\mbox{prove\_set\_difference}_3,\mbox{negated\_conjecture})$

\medskip\textbf{ALG311-1.p} Random graph 9, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG269$\wedge$2.p} HOAS induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasinduction\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG253$\wedge$1.p} Induction






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{induction2\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG176+1.p} Quasigroups 5 QG4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(v,u)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(v,u)\ne v ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG252$\wedge$2.p} Induction lemma






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{induction2lem\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG242-1.p} Idempotent selfdistributive groupoids are symmetric-by-medial - 1






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

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

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

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

\medskip\textbf{ALG259$\wedge$1.p} HOASlam not ap






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslamnotap\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG278$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_2{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP2\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_LEFT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_LEFT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_LEFT\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_2 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_LEFT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_LEFT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP2\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \mbox{cGROUP}_2@ \mbox{xf}@ \mbox{xe} ) \ \Rightarrow \ \forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cE12A}_1,\mbox{conjecture})$

\medskip\textbf{ALG250$\wedge$3.p} Push property lemma 3






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lem3v2\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG012-1.p} Partition a monoid into 3 partitions

If C,D1,D2 is a partition of a monoid M, we cannot have
 C * C subset D1 u D2 and Dj * Dj subset C.




$f(x,f(y,z)) = f(f(x,y),z)\qquad\mbox{cnf}(\mbox{f\_is\_associative},\mbox{axiom})$

$c(x)\mbox{ or }d_1(x)\mbox{ or }d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_union},\mbox{axiom})$

$c(x) \ \Rightarrow \ \neg\,d_1(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c\_d}_1,\mbox{hypothesis})$

$c(x) \ \Rightarrow \ \neg\,d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_c\_d}_2,\mbox{hypothesis})$

$d_1(x) \ \Rightarrow \ \neg\,d_2(x)\qquad\mbox{cnf}(\mbox{partitions\_exclusive\_d1\_d}_2,\mbox{hypothesis})$

$c(a_1)\qquad\mbox{cnf}(\mbox{partition\_c\_not\_empty},\mbox{hypothesis})$

$d_1(a_2)\qquad\mbox{cnf}(\mbox{partition\_d1\_not\_empty},\mbox{hypothesis})$

$d_2(a_3)\qquad\mbox{cnf}(\mbox{partition\_d2\_not\_empty},\mbox{hypothesis})$

$( c(x)\mbox{ and }c(y) ) \ \Rightarrow \ ( d_2(f(x,y))\mbox{ or }d_1(f(x,y)) )\qquad\mbox{cnf}(\mbox{conjecture}_1,\mbox{negated\_conjecture})$

$( d_1(x)\mbox{ and }d_1(y) ) \ \Rightarrow \ c(f(x,y))\qquad\mbox{cnf}(\mbox{conjecture}_2,\mbox{negated\_conjecture})$

$( d_2(x)\mbox{ and }d_2(y) ) \ \Rightarrow \ c(f(x,y))\qquad\mbox{cnf}(\mbox{conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{ALG271$\wedge$5.p} TPS problem EQUIV-01-03






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_1{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP1\_type},\mbox{type})$

$\mbox{cGROUP}_3{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP3\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe}\mbox{ and }( \mbox{xf}@ \mbox{xb}@ \mbox{xa} ) = \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGRP\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_def},\mbox{definition})$

$\mbox{cGRP\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( ( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa}\mbox{ and }( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} ) )\qquad\mbox{thf}(\mbox{cGRP\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_1 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP1\_def},\mbox{definition})$

$\mbox{cGROUP}_3 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_RIGHT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_RIGHT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP3\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \mbox{cGROUP}_1@ \mbox{xf}@ \mbox{xe} )\iff ( \mbox{cGROUP}_3@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cEQUIV\_01}_{03},\mbox{conjecture})$

\medskip\textbf{ALG040+1.p} Loops 4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG255$\wedge$1.p} T is an M-set






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{termmset\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG001-2.p} The composition of homomorphisms is a homomorphism






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{homomorphism}(f_{60},f_{62},f_{63},f_{64},f_{65})\qquad\mbox{cnf}(\mbox{one\_homomorphism},\mbox{hypothesis})$

$\mbox{homomorphism}(f_{61},f_{64},f_{65},f_{66},f_{67})\qquad\mbox{cnf}(\mbox{another\_homomorphism},\mbox{hypothesis})$

$\neg\,\mbox{homomorphism}(f_{60}\circ f_{61},f_{62},f_{63},f_{66},f_{67})\qquad\mbox{cnf}(\mbox{prove\_composition\_is\_a\_homomorphism},\mbox{negated\_conjecture})$

\medskip\textbf{ALG244-1.p} Idempotent selfdistributive groupoids are symmetric-by-medial - 3






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

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

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

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

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

\medskip\textbf{ALG251$\wedge$3.p} Push property






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_lthm\_orig}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG309-1.p} Random graph 7, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG259$\wedge$2.p} HOASlam not ap






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoaslamnotap\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG019+1.p} Groups 4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG277$\wedge$5.p} TPS problem from GRP-THMS






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{cGROUP}_3{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGROUP3\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC}{:}\;( g\to g\to g )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_INVERSE}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_type},\mbox{type})$

$\mbox{cGRP\_RIGHT\_UNIT}{:}\;( g\to g\to g )\to g\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_type},\mbox{type})$

$\mbox{cGRP\_ASSOC} = ( \lambda \mbox{xf}{:}\;g\to g\to g{:}\;\forall \mbox{xa}{:}\;g,\mbox{xb}{:}\;g,\mbox{xc}{:}\;g{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xa}@ \mbox{xb} )@ \mbox{xc} ) = ( \mbox{xf}@ \mbox{xa}@ ( \mbox{xf}@ \mbox{xb}@ \mbox{xc} ) ) )\qquad\mbox{thf}(\mbox{cGRP\_ASSOC\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_INVERSE} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;\exists \mbox{xb}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xe} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_INVERSE\_def},\mbox{definition})$

$\mbox{cGRP\_RIGHT\_UNIT} = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;\forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xa}@ \mbox{xe} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cGRP\_RIGHT\_UNIT\_def},\mbox{definition})$

$\mbox{cGROUP}_3 = ( \lambda \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( \mbox{cGRP\_ASSOC}@ \mbox{xf}\mbox{ and }\mbox{cGRP\_RIGHT\_UNIT}@ \mbox{xf}@ \mbox{xe}\mbox{ and }\mbox{cGRP\_RIGHT\_INVERSE}@ \mbox{xf}@ \mbox{xe} ) )\qquad\mbox{thf}(\mbox{cGROUP3\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;g\to g\to g,\mbox{xe}{:}\;g{:}\;( ( \mbox{cGROUP}_3@ \mbox{xf}@ \mbox{xe} ) \ \Rightarrow \ \forall \mbox{xa}{:}\;g{:}\;( \mbox{xf}@ \mbox{xe}@ \mbox{xa} ) = \mbox{xa} )\qquad\mbox{thf}(\mbox{cE13A}_2,\mbox{conjecture})$

\medskip\textbf{ALG321-1.p} Random graph 19, nu5 polymorphism






$t(y,x,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{01},\mbox{axiom})$

$t(x,y,x,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{02},\mbox{axiom})$

$t(x,x,y,x,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{03},\mbox{axiom})$

$t(x,x,x,y,x) = x\qquad\mbox{cnf}(\mbox{polynu5}_{04},\mbox{axiom})$

$t(x,x,x,x,y) = x\qquad\mbox{cnf}(\mbox{polynu5}_{05},\mbox{axiom})$

$( \mbox{gr}(x_0,x_1)\mbox{ and }\mbox{gr}(x_2,x_3)\mbox{ and }\mbox{gr}(x_4,x_5)\mbox{ and }\mbox{gr}(x_6,x_7)\mbox{ and }\mbox{gr}(x_8,x_9) ) \ \Rightarrow \ \mbox{gr}(t(x_0,x_2,x_4,x_6,x_8),t(x_1,x_3,x_5,x_7,x_9))\qquad\mbox{cnf}(\mbox{preserves},\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_0)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_0,\mbox{axiom})$

$\mbox{gr}(n_0,n_1)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_0,n_2)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_0,n_3)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_4)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_5)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_0,n_6)\qquad\mbox{cnf}(\mbox{graph\_n0\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_0)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_1)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_1,n_2)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_3)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_4)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_1,n_5)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_5,\mbox{axiom})$

$\mbox{gr}(n_1,n_6)\qquad\mbox{cnf}(\mbox{graph\_n1\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_0)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_1)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_2)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_3)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_4)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_4,\mbox{axiom})$

$\mbox{gr}(n_2,n_5)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_2,n_6)\qquad\mbox{cnf}(\mbox{graph\_n2\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_0)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_1)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_3,n_2)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_3)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_3,n_4)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_5)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_3,n_6)\qquad\mbox{cnf}(\mbox{graph\_n3\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_0)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_1)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_1,\mbox{axiom})$

$\mbox{gr}(n_4,n_2)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_2,\mbox{axiom})$

$\mbox{gr}(n_4,n_3)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_4,n_4)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_5)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_4,n_6)\qquad\mbox{cnf}(\mbox{graph\_n4\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_0)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_1)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_2)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_3)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_3,\mbox{axiom})$

$\mbox{gr}(n_5,n_4)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_5)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_5,n_6)\qquad\mbox{cnf}(\mbox{graph\_n5\_n}_6,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_0)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_0,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_1)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_1,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_2)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_2,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_3)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_3,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_4)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_4,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_5)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_5,\mbox{axiom})$

$\neg\,\mbox{gr}(n_6,n_6)\qquad\mbox{cnf}(\mbox{graph\_n6\_n}_6,\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_2,\mbox{axiom})$

$n_0\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_3,\mbox{axiom})$

$n_0\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_4,\mbox{axiom})$

$n_0\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_5,\mbox{axiom})$

$n_0\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n0\_n}_6,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_2,\mbox{axiom})$

$n_1\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_3,\mbox{axiom})$

$n_1\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_4,\mbox{axiom})$

$n_1\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_5,\mbox{axiom})$

$n_1\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n1\_n}_6,\mbox{axiom})$

$n_2\ne n_3\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_3,\mbox{axiom})$

$n_2\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_4,\mbox{axiom})$

$n_2\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_5,\mbox{axiom})$

$n_2\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n2\_n}_6,\mbox{axiom})$

$n_3\ne n_4\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_4,\mbox{axiom})$

$n_3\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_5,\mbox{axiom})$

$n_3\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n3\_n}_6,\mbox{axiom})$

$n_4\ne n_5\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_5,\mbox{axiom})$

$n_4\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n4\_n}_6,\mbox{axiom})$

$n_5\ne n_6\qquad\mbox{cnf}(\mbox{elems\_n5\_n}_6,\mbox{axiom})$

$x = n_0\mbox{ or }x = n_1\mbox{ or }x = n_2\mbox{ or }x = n_3\mbox{ or }x = n_4\mbox{ or }x = n_5\mbox{ or }x = n_6\qquad\mbox{cnf}(\mbox{elems},\mbox{axiom})$

\medskip\textbf{ALG443+1.p} Axioms for Median algebra






$\mbox{include}(\mbox{'Axioms/ALG002+0.ax'})$

\medskip\textbf{ALG030+1.p} Groups 6: CPROPS-SORTED-DISCRIMINANT-PROBLEM-2






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\neg\,\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(u,v) = \mbox{op}_1(v,u) ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\neg\,\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(u,v) = \mbox{op}_2(v,u) ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG203+1.p} Quasigroups 7 QG5: CPROPS-SORTED-DISCRIMINANT-PROBLEM-3






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\mbox{op}_1(u,u) = u )\mbox{ and }\exists v{:}\;( \mbox{sorti}_1(v)\mbox{ and }\mbox{op}_1(v,v)\ne v )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\mbox{op}_2(u,u) = u )\mbox{ and }\exists v{:}\;( \mbox{sorti}_2(v)\mbox{ and }\mbox{op}_2(v,v)\ne v )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG018+1.p} Groups 4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1






$\forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{sorti}_1(\mbox{op}_1(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_1,\mbox{axiom})$

$\forall u{:}\;( \mbox{sorti}_2(u) \ \Rightarrow \ \forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_2(\mbox{op}_2(u,v)) ) )\qquad\mbox{fof}(\mbox{ax}_2,\mbox{axiom})$

$\exists u{:}\;( \mbox{sorti}_1(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_1(v) \ \Rightarrow \ \mbox{op}_1(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_3,\mbox{axiom})$

$\neg\,\exists u{:}\;( \mbox{sorti}_2(u)\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{op}_2(v,v) = u ) )\qquad\mbox{fof}(\mbox{ax}_4,\mbox{axiom})$

$( \forall u{:}\;( \mbox{sorti}_1(u) \ \Rightarrow \ \mbox{sorti}_2(h(u)) )\mbox{ and }\forall v{:}\;( \mbox{sorti}_2(v) \ \Rightarrow \ \mbox{sorti}_1(j(v)) ) ) \ \Rightarrow \ \neg\,\forall w{:}\;( \mbox{sorti}_1(w) \ \Rightarrow \ \forall x{:}\;( \mbox{sorti}_1(x) \ \Rightarrow \ h(\mbox{op}_1(w,x)) = \mbox{op}_2(h(w),h(x)) ) )\mbox{ and }\forall y{:}\;( \mbox{sorti}_2(y) \ \Rightarrow \ \forall z{:}\;( \mbox{sorti}_2(z) \ \Rightarrow \ j(\mbox{op}_2(y,z)) = \mbox{op}_1(j(y),j(z)) ) )\mbox{ and }\forall x_1{:}\;( \mbox{sorti}_2(x_1) \ \Rightarrow \ h(j(x_1)) = x_1 )\mbox{ and }\forall x_2{:}\;( \mbox{sorti}_1(x_2) \ \Rightarrow \ j(h(x_2)) = x_2 )\qquad\mbox{fof}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{ALG261$\wedge$2.p} HOASap not var






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{hoasapnotvar\_lthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG251$\wedge$1.p} Push property






$\mbox{include}(\mbox{'Axioms/ALG003\string^0.ax'})$

$\mbox{pushprop\_gthm}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{ALG240-1.p} Selfdistributive groupoids are symmetric-by-medial - part 4






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

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

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

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

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

\medskip
\end{document}