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    \textbf{NUM001-2.ax} Number theory div axioms






$\mbox{divides}(a,b) \ \Rightarrow \ ( \mbox{less}(a,b)\mbox{ or }a{=}b )\qquad\mbox{cnf}(\mbox{divides\_only\_less\_or\_equal},\mbox{axiom})$

$\mbox{less}(a,b) \ \Rightarrow \ \mbox{divides}(a,b)\qquad\mbox{cnf}(\mbox{divides\_if\_less},\mbox{axiom})$

$a{=}b \ \Rightarrow \ \mbox{divides}(a,b)\qquad\mbox{cnf}(\mbox{divides\_if\_equal},\mbox{axiom})$

\medskip\textbf{NUM006$\wedge$0.ax} Church Numerals in Simple Type Theory






$0{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{zero},\mbox{type})$

$1{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{one},\mbox{type})$

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

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

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

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

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

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

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

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

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

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

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

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

$0 = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;y )\qquad\mbox{thf}(\mbox{zero\_ax},\mbox{definition})$

$1 = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ y ) )\qquad\mbox{thf}(\mbox{one\_ax},\mbox{definition})$

$\mbox{two} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ y ) ) )\qquad\mbox{thf}(\mbox{two\_ax},\mbox{definition})$

$\mbox{three} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ y ) ) ) )\qquad\mbox{thf}(\mbox{three\_ax},\mbox{definition})$

$\mbox{four} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{four\_ax},\mbox{definition})$

$\mbox{five} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{five\_ax},\mbox{definition})$

$\mbox{six} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{six\_ax},\mbox{definition})$

$\mbox{seven} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{seven\_ax},\mbox{definition})$

$\mbox{eight} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{eight\_ax},\mbox{definition})$

$\mbox{nine} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{nine\_ax},\mbox{definition})$

$\mbox{ten} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ ( x@ y ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ten\_ax},\mbox{definition})$

$\mbox{succ} = ( \lambda n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x@ ( n@ x@ y ) ) )\qquad\mbox{thf}(\mbox{succ\_ax},\mbox{definition})$

$ +  = ( \lambda m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( m@ x@ ( n@ x@ y ) ) )\qquad\mbox{thf}(\mbox{plus\_ax},\mbox{definition})$

$\cdot  = ( \lambda m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( m@ ( n@ x )@ y ) )\qquad\mbox{thf}(\mbox{mult\_ax},\mbox{definition})$

\medskip\textbf{NUM001-0.ax} Number theory axioms






$a + n_0{=}a\qquad\mbox{cnf}(\mbox{adding\_zero},\mbox{axiom})$

$a + \mbox{successor}(b){=}\mbox{successor}(a + b)\qquad\mbox{cnf}(\mbox{addition},\mbox{axiom})$

$a\cdot n_0{=}n_0\qquad\mbox{cnf}(\mbox{times\_zero},\mbox{axiom})$

$a\cdot \mbox{successor}(b){=}a\cdot b + a\qquad\mbox{cnf}(\mbox{times},\mbox{axiom})$

$\mbox{successor}(a){=}\mbox{successor}(b) \ \Rightarrow \ a{=}b\qquad\mbox{cnf}(\mbox{successor\_equality}_1,\mbox{axiom})$

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

\medskip\textbf{NUM001-1.ax} Number theory less axioms






$( \mbox{less}(a,b)\mbox{ and }\mbox{less}(c,a) ) \ \Rightarrow \ \mbox{less}(c,b)\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$\mbox{successor}(a) + b{=}c \ \Rightarrow \ \mbox{less}(b,c)\qquad\mbox{cnf}(\mbox{smaller\_number},\mbox{axiom})$

$\mbox{less}(a,b) \ \Rightarrow \ \mbox{successor}(\mbox{predecessor\_of\_1st\_minus\_2nd}(b,a)) + a{=}b\qquad\mbox{cnf}(\mbox{less\_lemma},\mbox{axiom})$

\medskip\textbf{NUM005+1.ax} Less in RDN format

Impements a "human style" less using RDN format.




$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_1))\qquad\mbox{fof}(\mbox{rdn\_digit}_1,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_2))\qquad\mbox{fof}(\mbox{rdn\_digit}_2,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_3))\qquad\mbox{fof}(\mbox{rdn\_digit}_3,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_4))\qquad\mbox{fof}(\mbox{rdn\_digit}_4,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_5))\qquad\mbox{fof}(\mbox{rdn\_digit}_5,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_6))\qquad\mbox{fof}(\mbox{rdn\_digit}_6,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_7))\qquad\mbox{fof}(\mbox{rdn\_digit}_7,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_8))\qquad\mbox{fof}(\mbox{rdn\_digit}_8,\mbox{axiom})$

$\mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(n_9))\qquad\mbox{fof}(\mbox{rdn\_digit}_9,\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_0),\mbox{rdnn}(n_1))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{01},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_1),\mbox{rdnn}(n_2))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{12},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_2),\mbox{rdnn}(n_3))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{23},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_3),\mbox{rdnn}(n_4))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{34},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_4),\mbox{rdnn}(n_5))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{45},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_5),\mbox{rdnn}(n_6))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{56},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_6),\mbox{rdnn}(n_7))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{67},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_7),\mbox{rdnn}(n_8))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{78},\mbox{axiom})$

$\mbox{rdn\_positive\_less}(\mbox{rdnn}(n_8),\mbox{rdnn}(n_9))\qquad\mbox{fof}(\mbox{rdn\_positive\_less}_{89},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rdn\_positive\_less}(\mbox{rdnn}(x),\mbox{rdnn}(y))\mbox{ and }\mbox{rdn\_positive\_less}(\mbox{rdnn}(y),\mbox{rdnn}(z)) ) \ \Rightarrow \ \mbox{rdn\_positive\_less}(\mbox{rdnn}(x),\mbox{rdnn}(z)) )\qquad\mbox{fof}(\mbox{rdn\_positive\_less\_transitivity},\mbox{axiom})$

$\forall \mbox{ds},\mbox{os},\mbox{db},\mbox{ob}{:}\;( \mbox{rdn\_positive\_less}(\mbox{os},\mbox{ob}) \ \Rightarrow \ \mbox{rdn\_positive\_less}(\mbox{rdn}(\mbox{rdnn}(\mbox{ds}),\mbox{os}),\mbox{rdn}(\mbox{rdnn}(\mbox{db}),\mbox{ob})) )\qquad\mbox{fof}(\mbox{rdn\_positive\_less\_multi\_digit\_high},\mbox{axiom})$

$\forall \mbox{ds},o,\mbox{db}{:}\;( ( \mbox{rdn\_positive\_less}(\mbox{rdnn}(\mbox{ds}),\mbox{rdnn}(\mbox{db}))\mbox{ and }\mbox{rdn\_non\_zero}(o) ) \ \Rightarrow \ \mbox{rdn\_positive\_less}(\mbox{rdn}(\mbox{rdnn}(\mbox{ds}),o),\mbox{rdn}(\mbox{rdnn}(\mbox{db}),o)) )\qquad\mbox{fof}(\mbox{rdn\_positive\_less\_multi\_digit\_low},\mbox{axiom})$

$\forall d,\mbox{db},\mbox{ob}{:}\;( \mbox{rdn\_non\_zero}(\mbox{ob}) \ \Rightarrow \ \mbox{rdn\_positive\_less}(\mbox{rdnn}(d),\mbox{rdn}(\mbox{rdnn}(\mbox{db}),\mbox{ob})) )\qquad\mbox{fof}(\mbox{rdn\_extra\_digits\_positive\_less},\mbox{axiom})$

$\forall x{:}\;( \mbox{rdn\_non\_zero\_digit}(\mbox{rdnn}(x)) \ \Rightarrow \ \mbox{rdn\_non\_zero}(\mbox{rdnn}(x)) )\qquad\mbox{fof}(\mbox{rdn\_non\_zero\_by\_digit},\mbox{axiom})$

$\forall d,o{:}\;( \mbox{rdn\_non\_zero}(o) \ \Rightarrow \ \mbox{rdn\_non\_zero}(\mbox{rdn}(\mbox{rdnn}(d),o)) )\qquad\mbox{fof}(\mbox{rdn\_non\_zero\_by\_structure},\mbox{axiom})$

$\forall x,y,\mbox{rDN\_X},\mbox{rDN\_Y}{:}\;( ( \mbox{rdn\_translate}(x,\mbox{rdn\_pos}(\mbox{rDN\_X}))\mbox{ and }\mbox{rdn\_translate}(y,\mbox{rdn\_pos}(\mbox{rDN\_Y}))\mbox{ and }\mbox{rdn\_positive\_less}(\mbox{rDN\_X},\mbox{rDN\_Y}) ) \ \Rightarrow \ \mbox{less}(x,y) )\qquad\mbox{fof}(\mbox{less\_entry\_point\_pos\_pos},\mbox{axiom})$

$\forall x,y,\mbox{rDN\_X},\mbox{rDN\_Y}{:}\;( ( \mbox{rdn\_translate}(x,\mbox{rdn\_neg}(\mbox{rDN\_X}))\mbox{ and }\mbox{rdn\_translate}(y,\mbox{rdn\_pos}(\mbox{rDN\_Y})) ) \ \Rightarrow \ \mbox{less}(x,y) )\qquad\mbox{fof}(\mbox{less\_entry\_point\_neg\_pos},\mbox{axiom})$

$\forall x,y,\mbox{rDN\_X},\mbox{rDN\_Y}{:}\;( ( \mbox{rdn\_translate}(x,\mbox{rdn\_neg}(\mbox{rDN\_X}))\mbox{ and }\mbox{rdn\_translate}(y,\mbox{rdn\_neg}(\mbox{rDN\_Y}))\mbox{ and }\mbox{rdn\_positive\_less}(\mbox{rDN\_Y},\mbox{rDN\_X}) ) \ \Rightarrow \ \mbox{less}(x,y) )\qquad\mbox{fof}(\mbox{less\_entry\_point\_neg\_neg},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{less}(x,y)\iff ( \neg\,\mbox{less}(y,x)\mbox{ and }y\ne x ) )\qquad\mbox{fof}(\mbox{less\_property},\mbox{axiom})$

$\forall x,y{:}\;( x \leq y\iff ( \mbox{less}(x,y)\mbox{ or }x = y ) )\qquad\mbox{fof}(\mbox{less\_or\_equal},\mbox{axiom})$

$\forall x,y,z{:}\;( ( x + n_1{=}y\mbox{ and }\mbox{less}(z,y) ) \ \Rightarrow \ z \leq x )\qquad\mbox{fof}(\mbox{less\_successor},\mbox{axiom})$

\medskip\textbf{NUM002-0.ax} Number theory (equality) axioms






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

$( a{=}b\mbox{ and }b{=}c ) \ \Rightarrow \ a{=}c\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$a + b{=}b + a\qquad\mbox{cnf}(\mbox{commutativity\_of\_addition},\mbox{axiom})$

$a + ( b + c ){=}( a + b ) + c\qquad\mbox{cnf}(\mbox{associativity\_of\_addition},\mbox{axiom})$

$\mbox{subtract}(a + b,b){=}a\qquad\mbox{cnf}(\mbox{addition\_inverts\_subtraction}_1,\mbox{axiom})$

$a{=}\mbox{subtract}(a + b,b)\qquad\mbox{cnf}(\mbox{addition\_inverts\_subtraction}_2,\mbox{axiom})$

$\mbox{subtract}(a,b) + c{=}\mbox{subtract}(a + c,b)\qquad\mbox{cnf}(\mbox{commutativity}_1,\mbox{axiom})$

$\mbox{subtract}(a + b,c){=}\mbox{subtract}(a,c) + b\qquad\mbox{cnf}(\mbox{commutativity}_2,\mbox{axiom})$

$( a{=}b\mbox{ and }c{=}a + d ) \ \Rightarrow \ c{=}b + d\qquad\mbox{cnf}(\mbox{add\_substitution}_1,\mbox{axiom})$

$( a{=}b\mbox{ and }c{=}d + a ) \ \Rightarrow \ c{=}d + b\qquad\mbox{cnf}(\mbox{add\_substitution}_2,\mbox{axiom})$

$( a{=}b\mbox{ and }c{=}\mbox{subtract}(a,d) ) \ \Rightarrow \ c{=}\mbox{subtract}(b,d)\qquad\mbox{cnf}(\mbox{subtract\_substitution}_1,\mbox{axiom})$

$( a{=}b\mbox{ and }c{=}\mbox{subtract}(d,a) ) \ \Rightarrow \ c{=}\mbox{subtract}(d,b)\qquad\mbox{cnf}(\mbox{subtract\_substitution}_2,\mbox{axiom})$

\medskip\textbf{NUM268-1.p} Ordinal addition property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{image}(\mbox{image}(\mbox{add\_relation},\mbox{singleton}(x)),y))\ne \mbox{ordinal\_add}(x,y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM140-1.p} The successor of a set is a set, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_set\_is\_set1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_set\_is\_set1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM077-1.p} Corollary 1 to well ordering property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_well\_ordering\_property7}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_well\_ordering\_property7}_2,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{xr},y,u) = y\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_well\_ordering\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM133-1.p} Corollary to union of successor ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_union\_of\_successor\_ordinal}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_union\_of\_successor\_ordinal}_2,\mbox{negated\_conjecture})$

$\mbox{successor}(x) = \mbox{successor}(y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_union\_of\_successor\_ordinal}_3,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_union\_of\_successor\_ordinal}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM304+1.p} ? != 12






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x\ne n_{12}\qquad\mbox{fof}(\mbox{something\_not\_n}_{12},\mbox{conjecture})$

\medskip\textbf{NUM254-1.p} Transfinite recursion property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property10}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{domain\_of}(\mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x))\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property10}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM353+1.p} XYZ ((X+Y)+Z) != (Z+X)+Y)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y,z,z_1,z_2,z_3,z_4{:}\;( x + y{=}z_1\mbox{ and }z_1 + z{=}z_2\mbox{ and }z + x{=}z_3\mbox{ and }z_3 + y{=}z_4\mbox{ and }z_2\ne z_4 )\qquad\mbox{fof}(\mbox{anti\_associativity},\mbox{conjecture})$

\medskip\textbf{NUM920=1.p} No such positive number






$\neg\,\exists u{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(0,u)\mbox{ and }\forall v{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(v,u) \ \Rightarrow \ \mbox{\$less}(\mbox{\$sum}(v,1),u) ) )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM293+1.p} ? $<$ 13






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{less}(x,n_{13})\qquad\mbox{fof}(\mbox{something\_less\_n}_{13},\mbox{conjecture})$

\medskip\textbf{NUM137-1.p} Condition 1 for complete induction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),z),x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction1}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction1}_2,\mbox{negated\_conjecture})$

$y\in z\qquad\mbox{cnf}(\mbox{prove\_complete\_induction1}_3,\mbox{negated\_conjecture})$

$\neg\,y\in x\qquad\mbox{cnf}(\mbox{prove\_complete\_induction1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM652$\wedge$1.p} Landau theorem 10c

~(less x y)




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\neg\,\mbox{less}@ x@ y\qquad\mbox{thf}(\mbox{satz10c},\mbox{conjecture})$

\medskip\textbf{NUM871=1.p} Sum is not associativity






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$sum}(z_1,z) = z_2\mbox{ and }\mbox{\$sum}(z,x) = z_3\mbox{ and }\mbox{\$sum}(z_3,y) = z_4\mbox{ and }z_2\ne z_4 )\qquad\mbox{tff}(\mbox{anti\_associativity\_sum},\mbox{conjecture})$

\medskip\textbf{NUM658$\wedge$1.p} Landau theorem 10j

less x y




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{less}@ x@ y\qquad\mbox{thf}(\mbox{satz10j},\mbox{conjecture})$

\medskip\textbf{NUM110-1.p} Corollary to ordinal property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{connected}(\mbox{element\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_property10}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM883=1.p} Product of something and 1 is not that something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,1)\ne x\qquad\mbox{tff}(\mbox{anti\_product\_identity}_1,\mbox{conjecture})$

\medskip\textbf{NUM242-1.p} Transfinite recursion lemma 9.3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_1,\mbox{negated\_conjecture})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_3,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(y) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_4,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_5,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',y))),\mbox{universal\_class})\ne \mbox{restrict}(y,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',y))),\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_3}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM651$\wedge$1.p} Landau theorem 9b

~((x = y $\to$ ~(~(forall x\_0:nat.~(x = pl y x\_0)))) $\to$
 ~(~((~(forall x\_0:nat.~(x = pl y x\_0)) $\to$
 ~(~(forall x\_0:nat.~(y = pl x x\_0)))) $\to$
 ~(~(forall x\_0:nat.~(y = pl x x\_0)) $\to$ ~(x = y)))))




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\mbox{xy}\ne ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz}_7,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\neg\,( x = y \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,\neg\,( \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) \ \Rightarrow \ x\ne y\qquad\mbox{thf}(\mbox{satz9b},\mbox{conjecture})$

\medskip\textbf{NUM188-1.p} Omega is transitive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{omega}),\mbox{omega})\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_omega}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM310+1.p} 2 + 3 != 6






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\neg\,n_2 + n_3{=}n_6\qquad\mbox{fof}(\mbox{sum\_n2\_n3\_not\_n}_6,\mbox{conjecture})$

\medskip\textbf{NUM645$\wedge$1.p} Landau theorem 7

~(y = pl x y)




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} )\ne n_1\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \mbox{suc}@ \mbox{xx} )\ne ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz}_1,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4b},\mbox{axiom})$

$y\ne ( \mbox{pl}@ x@ y )\qquad\mbox{thf}(\mbox{satz}_7,\mbox{conjecture})$

\medskip\textbf{NUM036-1.p} Symmetrization property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{union}(\mbox{symmetrization\_of}(x),\mbox{symmetrization\_of}(y))\ne \mbox{symmetrization\_of}(\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM373+1.p} ?XYZ, (X+Y) = (Y+X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z_1,z_2{:}\;( ( x + y{=}z_1\mbox{ and }y + x{=}z_2 ) \ \Rightarrow \ z_1 = z_2 )\qquad\mbox{fof}(\mbox{communative},\mbox{conjecture})$

\medskip\textbf{NUM277-2.p} Ordinal addition property 7.1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_3,\mbox{axiom})$

$\mbox{null\_class}\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{lemma\_1\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers}))\qquad\mbox{cnf}(\mbox{lemma\_2\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{ordinals\_with\_null\_class\_as\_identity})\qquad\mbox{cnf}(\mbox{lemma\_3\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{domain\_of}(\mbox{recursion}(\mbox{null\_class},\mbox{successor\_relation},\mbox{union\_of\_range\_map})))\qquad\mbox{cnf}(\mbox{lemma\_4\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{domain\_of}(\mbox{intersection}(\mbox{recursion}(\mbox{null\_class},\mbox{successor\_relation},\mbox{union\_of\_range\_map}),\mbox{identity\_relation})))\qquad\mbox{cnf}(\mbox{lemma\_5\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{limit\_ordinals}),\mbox{ordinals\_with\_null\_class\_as\_identity})\qquad\mbox{cnf}(\mbox{lemma\_6\_for\_ordinal\_addition\_property}_7,\mbox{axiom})$

$\mbox{ordinals\_with\_null\_class\_as\_identity}\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property7\_1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM291+1.p} 3 !$<$ 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\neg\,\mbox{less}(n_3,n_2)\qquad\mbox{fof}(\mbox{n3\_not\_less\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM882=1.p} Product is not associative






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,y) = z_1\mbox{ and }\mbox{\$product}(z_1,z) = z_2\mbox{ and }\mbox{\$product}(z,x) = z_3\mbox{ and }\mbox{\$product}(z_3,y) = z_4\mbox{ and }z_2\ne z_4 )\qquad\mbox{tff}(\mbox{anti\_associativity\_product},\mbox{conjecture})$

\medskip\textbf{NUM682$\wedge$1.p} Landau theorem 20c

less x y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{less}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ z )\qquad\mbox{thf}(l,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz19b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\mbox{less}@ x@ y\qquad\mbox{thf}(\mbox{satz20c},\mbox{conjecture})$

\medskip\textbf{NUM422+1.p} Fuerstenberg's infinitude of primes 02, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\qquad\mbox{fof}(\mbox{m\_}_{446},\mbox{hypothesis})$

$\mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),\mbox{xa}) = \mbox{smndt}_0(\mbox{xa})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM361+1.p} !XY (X+Y = 8) =$>$ X - Y = 1,






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;( x + y{=}n_8 \ \Rightarrow \ x\setminus y{=}n_1 )\qquad\mbox{fof}(\mbox{all\_sum\_consecutive\_n}_8,\mbox{conjecture})$

\medskip\textbf{NUM862=1.p} Upper bound replace maximum embedded in a context (1)+(2)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{summation}{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{summation\_type},\mbox{type})$

$\mbox{ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{ub\_type},\mbox{type})$

$\mbox{model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_max\_type},\mbox{type})$

$\mbox{model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_ub\_type},\mbox{type})$

$\mbox{minsol\_model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_max\_type},\mbox{type})$

$\mbox{minsol\_model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_ub\_type},\mbox{type})$

$\mbox{max}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{max\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff \mbox{\$lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{tff}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{\$lesseq}(y,x) )\qquad\mbox{tff}(\mbox{max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{\$lesseq}(x,y) )\qquad\mbox{tff}(\mbox{max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{\$lesseq}(x,z)\mbox{ and }\mbox{\$lesseq}(y,z) ) )\qquad\mbox{tff}(\mbox{ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{\$lesseq}(\mbox{\$sum}(c,\mbox{summation}(\mbox{max}(x,y))),n) )\qquad\mbox{tff}(\mbox{model\_max}_5,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(c,\mbox{summation}(z)),n) ) )\qquad\mbox{tff}(\mbox{model\_ub}_5,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{tff}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM112-1.p} Ordinal property 12






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{well\_ordering}(\mbox{element\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property12}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM031-1.p} Symmetrization property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x',x)\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{symmetrization\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM103-1.p} Corollary to ordinal property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_\_ordinal\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_\_ordinal\_property4}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_\_ordinal\_property4}_3,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_\_ordinal\_property4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM180-1.p} Limit ordinals are ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{limit\_ordinals},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_limit\_ordinals\_are\_ordinals}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM042-1.p} Irreflexive class property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{identity\_relation}')\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{irreflexive}(x,\mbox{domain\_of}(\mbox{symmetrization\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM229-1.p} Transfinite recursion lemma 0






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma0}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma0}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{intersection}(y',x)),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma0}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM294+1.p} 12 $<$ ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{less}(n_{12},x)\qquad\mbox{fof}(\mbox{n12\_less\_something},\mbox{conjecture})$

\medskip\textbf{NUM135-1.p} The null class is the smallest ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{least}(\mbox{element\_relation},\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_least\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM808$\wedge$5.p} TPS problem THM130A






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }r@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( r@ \mbox{xx}@ \mbox{xx} ) \ \Rightarrow \ ( r@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( r@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM130A},\mbox{conjecture})$

\medskip\textbf{NUM233-1.p} Transfinite recursion lemma 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma4}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma4}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x)\in \mbox{domain\_of}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma4}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma4}_4,\mbox{negated\_conjecture})$

$\mbox{apply}(y,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x))))\ne \mbox{apply}(x,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x))))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma4}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM196-1.p} The union of a set of ordinals is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_is\_ordinal}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_is\_ordinal}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_is\_ordinal}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM287-1.p} Number theory less axioms






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

\medskip\textbf{NUM234-1.p} Transfinite recursion lemma 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma5}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma5}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x)\in \mbox{domain\_of}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma5}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma5}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x))),\mbox{apply}(y,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x)))))\in y\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma5}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM848+2.p} holds(286,441,4)






$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}),\mbox{vd}_{436})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{439})))\qquad\mbox{fof}(\mbox{'holds(286, 441, 4)'},\mbox{conjecture})$

$\mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439})),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}),\mbox{vd}_{436}))\qquad\mbox{fof}(\mbox{'holds(286, 441, 3)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 2)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vsucc}(\mbox{vd}_{439}))) = \mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})))\qquad\mbox{fof}(\mbox{'holds(286, 441, 0)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}))\qquad\mbox{fof}(\mbox{'holds(285, 440, 0)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},v_1))\qquad\mbox{fof}(\mbox{'holds(284, 438, 2)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{437})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(284, 438, 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{386},\mbox{vd}_{387}{:}\;( \mbox{less}(\mbox{vd}_{386},\mbox{vplus}(\mbox{vd}_{387},v_1)) \ \Rightarrow \ \mbox{leq}(\mbox{vd}_{386},\mbox{vd}_{387}) )\qquad\mbox{fof}(\mbox{'ass(cond(241, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

\medskip\textbf{NUM020-1.p} a + 1 = successor(a)






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

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

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

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

$n_1{=}\mbox{successor}(n_0)\qquad\mbox{cnf}(\mbox{one\_succeeds\_zero},\mbox{axiom})$

$\neg\,a + \mbox{successor}(n_0){=}\mbox{successor}(a)\qquad\mbox{cnf}(\mbox{deny\_addition\_lemma},\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{prove\_a\_contradiction},\mbox{negated\_conjecture})$

\medskip\textbf{NUM800$\wedge$1.p} Some function of two and three is six, and of one and two is two






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists h{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i} )\to ( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( ( h@ \mbox{two}@ \mbox{three} ) = \mbox{six}\mbox{ and }( h@ 1@ \mbox{two} ) = \mbox{two} )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM288-1.p} Number theory div axioms






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-2.ax'})$

\medskip\textbf{NUM220-1.p} Corollary to the domain of rest\_of(X) being the domain of X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{rest\_of}(\mbox{union}(u,\mbox{singleton}(\mbox{ordered\_pair}(x,y)))))\ne \mbox{union}(\mbox{domain\_of}(u),\mbox{singleton}(x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_of\_rest\_of}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM162-1.p} Successor property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_successor\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM069-1.p} Corollary to well ordering property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property3}_2,\mbox{negated\_conjecture})$

$u\in v\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property3}_3,\mbox{negated\_conjecture})$

$v\in u\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM864=1.p} Sum idempotent element






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,x) = x\qquad\mbox{tff}(\mbox{sum\_idempotent\_element},\mbox{conjecture})$

\medskip\textbf{NUM342+1.p} XY (X + Y = 8) =$>$ X - Y = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;( x + y{=}n_8\mbox{ and }x\setminus y{=}n_0 )\qquad\mbox{fof}(\mbox{sum\_and\_difference},\mbox{conjecture})$

\medskip\textbf{NUM017-1.p} Square root of this prime is irrational

If a is prime, and a is not b$\wedge$2/c$\wedge$2, then the square root
 of a is irrational.




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

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

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

$( d{=}b\mbox{ and }c\cdot a{=}d ) \ \Rightarrow \ c\cdot a{=}b\qquad\mbox{cnf}(\mbox{product\_substitution}_1,\mbox{axiom})$

$( d{=}b\mbox{ and }c\cdot d{=}a ) \ \Rightarrow \ c\cdot b{=}a\qquad\mbox{cnf}(\mbox{product\_substitution}_2,\mbox{axiom})$

$( c{=}b\mbox{ and }c\cdot d{=}a ) \ \Rightarrow \ b\cdot d{=}a\qquad\mbox{cnf}(\mbox{product\_substitution}_3,\mbox{axiom})$

$( b{=}a\mbox{ and }\mbox{divides}(c,b) ) \ \Rightarrow \ \mbox{divides}(c,a)\qquad\mbox{cnf}(\mbox{divides\_substitution}_1,\mbox{axiom})$

$( a{=}b\mbox{ and }\mbox{divides}(a,c) ) \ \Rightarrow \ \mbox{divides}(b,c)\qquad\mbox{cnf}(\mbox{divides\_substitution}_2,\mbox{axiom})$

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

$a\cdot b{=}a\cdot b\qquad\mbox{cnf}(\mbox{closure\_of\_product},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot e{=}b\mbox{ and }a\cdot d{=}f ) \ \Rightarrow \ f\cdot e{=}c\qquad\mbox{cnf}(\mbox{product\_associativity}_1,\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot b{=}e\mbox{ and }f\cdot d{=}a ) \ \Rightarrow \ f\cdot e{=}c\qquad\mbox{cnf}(\mbox{product\_associativity}_2,\mbox{axiom})$

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

$( a\cdot b{=}c\mbox{ and }a\cdot d{=}c ) \ \Rightarrow \ b{=}d\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( \mbox{divides}(a,b)\mbox{ and }\mbox{divides}(c,a) ) \ \Rightarrow \ \mbox{divides}(c,b)\qquad\mbox{cnf}(\mbox{transitivity\_of\_divides},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }a\cdot b{=}d ) \ \Rightarrow \ d{=}c\qquad\mbox{cnf}(\mbox{well\_defined\_product},\mbox{axiom})$

$\mbox{divides}(a,b) \ \Rightarrow \ a\cdot \mbox{second\_divided\_by\_1st}(a,b){=}b\qquad\mbox{cnf}(\mbox{divides\_implies\_product},\mbox{axiom})$

$a\cdot b{=}c \ \Rightarrow \ \mbox{divides}(a,c)\qquad\mbox{cnf}(\mbox{product\_divisible\_by\_operand},\mbox{axiom})$

$( \mbox{divides}(a,b)\mbox{ and }c\cdot c{=}b\mbox{ and }\mbox{prime}(a) ) \ \Rightarrow \ \mbox{divides}(a,c)\qquad\mbox{cnf}(\mbox{primes\_lemma}_1,\mbox{axiom})$

$\mbox{prime}(a)\qquad\mbox{cnf}(\mbox{a\_is\_prime},\mbox{hypothesis})$

$b\cdot b{=}d\qquad\mbox{cnf}(\mbox{b\_squared},\mbox{hypothesis})$

$c\cdot c{=}e\qquad\mbox{cnf}(\mbox{c\_squared},\mbox{hypothesis})$

$\neg\,a\cdot e{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_squared\_is\_not\_b\_squared},\mbox{hypothesis})$

$\mbox{divides}(a,c) \ \Rightarrow \ \neg\,\mbox{divides}(a,b)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_no\_common\_divisor},\mbox{negated\_conjecture})$

\medskip\textbf{NUM798$\wedge$1.p} Something times one is one






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( \cdot @ n@ 1 ) = 1\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM908=1.p} Difference everything and iteself is 0.0






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$difference}(x,x) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_difference\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{NUM141-1.p} The successor of a set is a set, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_set\_is\_set2}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_set\_is\_set2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM793$\wedge$1.p} Landau theorem 87d

more x0 z0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$z_0{:}\;\mbox{rat}\qquad\mbox{thf}(z_0,\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ x_0@ y_0\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{moreis}@ y_0@ z_0\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{83},\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat},\mbox{xz}_0{:}\;\mbox{rat}{:}\;( ( \mbox{lessis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( ( \mbox{less}@ \mbox{xy}_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xz}_0 ) ) )\qquad\mbox{thf}(\mbox{satz87a},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{moreis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{84},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{82},\mbox{axiom})$

$\mbox{more}@ x_0@ z_0\qquad\mbox{thf}(\mbox{satz87d},\mbox{conjecture})$

\medskip\textbf{NUM311+1.p} 2 + 3 = only 5






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_2 + n_3{=}x \ \Rightarrow \ x = n_5 )\qquad\mbox{fof}(\mbox{sum\_n2\_n3\_only\_n}_5,\mbox{conjecture})$

\medskip\textbf{NUM383+1.p} Ordinal numbers, theorem 7






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }b\subseteq a\qquad\mbox{fof}(\mbox{t7\_ordinal}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM258-1.p} Transfinite recursion property 14






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property14}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property14}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM834$\wedge$5.p} TPS problem from PETER-THMS






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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$\exists \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ c_0@ \mbox{xn}@ ( \mbox{cS}@ \mbox{xn} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xm}@ ( \mbox{cS}@ c_0 )@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ c_0@ \mbox{xk} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i},\mbox{xl}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ \mbox{xn}@ \mbox{xl} ) \ \Rightarrow \ ( ( \mbox{xr}@ \mbox{xm}@ \mbox{xl}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ ( \mbox{cS}@ \mbox{xn} )@ \mbox{xk} ) ) )\mbox{ and }\forall t{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( t@ c_0@ \mbox{xn}@ ( \mbox{cS}@ \mbox{xn} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( t@ \mbox{xm}@ ( \mbox{cS}@ c_0 )@ \mbox{xk} ) \ \Rightarrow \ ( t@ ( \mbox{cS}@ \mbox{xm} )@ c_0@ \mbox{xk} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i},\mbox{xl}{:}\;\mbox{\$i}{:}\;( ( t@ ( \mbox{cS}@ \mbox{xm} )@ \mbox{xn}@ \mbox{xl} ) \ \Rightarrow \ ( ( t@ \mbox{xm}@ \mbox{xl}@ \mbox{xk} ) \ \Rightarrow \ ( t@ ( \mbox{cS}@ \mbox{xm} )@ ( \mbox{cS}@ \mbox{xn} )@ \mbox{xk} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xx}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( t@ \mbox{xx}@ \mbox{xy}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{603},\mbox{conjecture})$

\medskip\textbf{NUM881=1.p} Product is not a function






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,y) = z_1\mbox{ and }\mbox{\$product}(x,y) = z_2\mbox{ and }z_1\ne z_2 )\qquad\mbox{tff}(\mbox{anti\_unique\_product},\mbox{conjecture})$

\medskip\textbf{NUM080-1.p} Well ordering property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(y,\mbox{intersection}(\mbox{ordinal\_numbers},\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y,\mbox{ordinal\_numbers}))))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property9}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM901=1.p} Difference something and itself is 0/1






$\forall x{:}\;\mbox{\$rat}{:}\;\mbox{\$difference}(x,x) = 0/1\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_{12},\mbox{conjecture})$

\medskip\textbf{NUM650$\wedge$1.p} Landau theorem 9a

~(x = y) $\to$ ~(~(forall x\_0:nat.~(x = pl y x\_0))) $\to$
 ~(forall x\_0:nat.~(y = pl x x\_0))




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne n_1 \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{suc}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ n_1@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4f},\mbox{axiom})$

$x\ne y \ \Rightarrow \ ( \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz9a},\mbox{conjecture})$

\medskip\textbf{NUM531+1.p} Ramsey's Infinite Theorem 01, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM235-1.p} Transfinite recursion lemma 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma6}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma6}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x)\in \mbox{domain\_of}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma6}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma6}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM272-1.p} Lemma 2 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers}))\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM091-1.p} Sections property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{transitive}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property2}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_sections\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{section}(\mbox{xr},\mbox{segment}(\mbox{xr},y,u),y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM767$\wedge$1.p} Landau theorem 67b

eq $\vee$ w




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

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

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

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

$w{:}\;\mbox{frac}\qquad\mbox{thf}(w,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{eq}@ ( \mbox{pf}@ y@ v )@ x\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{eq}@ ( \mbox{pf}@ y@ w )@ x\qquad\mbox{thf}(f,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{pf}@ \mbox{xz}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz63e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\mbox{eq}@ v@ w\qquad\mbox{thf}(\mbox{satz67b},\mbox{conjecture})$

\medskip\textbf{NUM760$\wedge$1.p} Landau theorem 64

moref (pf x z) (pf y u)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moref}@ z@ u\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{43},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{lessf}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{50},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{42},\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{61},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u )\qquad\mbox{thf}(\mbox{satz}_{64},\mbox{conjecture})$

\medskip\textbf{NUM185-1.p} Limit ordinals are not members






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{limit\_ordinals}\in x\qquad\mbox{cnf}(\mbox{prove\_limit\_ordinals\_are\_not\_members}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM124-1.p} Condensed statement of transfinite induction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),\mbox{ordinal\_numbers}),x)\qquad\mbox{cnf}(\mbox{prove\_condensed\_statement\_of\_transfinite\_induction}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},x)\qquad\mbox{cnf}(\mbox{prove\_condensed\_statement\_of\_transfinite\_induction}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM093-1.p} Corollary 2 to sections property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_sections\_property2}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_sections\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{section}(\mbox{element\_relation},\mbox{intersection}(y,u),y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_sections\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM259-1.p} The uniqueness of the function defined by transfinite recursion






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_function\_unique}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_function\_unique}_2,\mbox{negated\_conjecture})$

$z\circ \mbox{rest\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_function\_unique}_3,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))\ne x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_function\_unique}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM654$\wedge$1.p} Landau theorem 10e

~(less x y) $\to$ x = y




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x@ y\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(\mbox{satz10e},\mbox{conjecture})$

\medskip\textbf{NUM761$\wedge$1.p} Landau theorem 65a

moref (pf x z) (pf y u)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{moref}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moref}@ z@ u\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz62g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{64},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u )\qquad\mbox{thf}(\mbox{satz65a},\mbox{conjecture})$

\medskip\textbf{NUM789$\wedge$1.p} Landau theorem 81h

~(~(more x0 y0) $\to$ is x0 y0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ x_0@ y_0\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;\neg\,( ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0\qquad\mbox{thf}(\mbox{satz81b},\mbox{axiom})$

$\neg\,\neg\,\mbox{more}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(\mbox{satz81h},\mbox{conjecture})$

\medskip\textbf{NUM334+1.p} 7 - 5 = 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_7\setminus n_5{=}n_2\qquad\mbox{fof}(\mbox{diff\_n7\_n5\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM656$\wedge$1.p} Landau theorem 10g

~(~(less x y) $\to$ x = y)




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\neg\,\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(\mbox{satz10g},\mbox{conjecture})$

\medskip\textbf{NUM336+1.p} only 5 - 2 = 3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( x\setminus n_2{=}n_3 \ \Rightarrow \ x = n_5 )\qquad\mbox{fof}(\mbox{diff\_only\_n5\_n2\_n}_3,\mbox{conjecture})$

\medskip\textbf{NUM278-1.p} Ordinal addition property 7\_2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property7\_2}_1,\mbox{negated\_conjecture})$

$\mbox{ordinal\_add}(\mbox{null\_class},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property7\_2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM347+1.p} 2 + 2 = 5






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_2 + n_2{=}n_5\qquad\mbox{fof}(\mbox{anti\_sum\_n2\_n2\_n}_5,\mbox{conjecture})$

\medskip\textbf{NUM699$\wedge$1.p} Landau theorem 25c

lessis (suc y) x




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ y@ x\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{lessis}@ ( \mbox{pl}@ \mbox{xy}@ n_1 )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz25b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\mbox{lessis}@ ( \mbox{suc}@ y )@ x\qquad\mbox{thf}(\mbox{satz25c},\mbox{conjecture})$

\medskip\textbf{NUM062-1.p} Segments property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{element\_relation},y,z)\ne \mbox{intersection}(y,z)\qquad\mbox{cnf}(\mbox{prove\_segments\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM169-1.p} Successor property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(x)\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_successor\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_successor\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM040-1.p} Irreflexive class property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{irreflexive}(x,y)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{irreflexive}(x,z)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM536+1.p} Ramsey's Infinite Theorem 05, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{sdtpldt}_0(\mbox{sdtmndt}_0(w_0,w_1),w_1) = w_0 ) )\qquad\mbox{fof}(\mbox{mConsDiff},\mbox{axiom})$

$\mbox{aElement}_0(\mbox{xx})\mbox{ and }\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{679},\mbox{hypothesis})$

$\neg\,\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_679}_{02},\mbox{hypothesis})$

$\mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}) = \mbox{xS}\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM816$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cEVEN}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN1\_type},\mbox{type})$

$\mbox{cODD}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD1\_type},\mbox{type})$

$\mbox{cEVEN}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{cEVEN1\_def},\mbox{definition})$

$\mbox{cODD}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cEVEN}_1@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cODD1\_def},\mbox{definition})$

$( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xu} )\ne c_0\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xv} ) = ( \mbox{cS}@ \mbox{xw} ) \ \Rightarrow \ \mbox{xv} = \mbox{xw} ) ) \ \Rightarrow \ ( \mbox{cODD}_1@ ( \mbox{cS}@ c_0 ) )\qquad\mbox{thf}(\mbox{cTHM}_{406},\mbox{conjecture})$

\medskip\textbf{NUM016-1.p} There exist infinitely many primes






$\neg\,\mbox{less}(x,x)\qquad\mbox{cnf}(\mbox{nothing\_is\_less\_than\_itself},\mbox{axiom})$

$\mbox{less}(x,y) \ \Rightarrow \ \neg\,\mbox{less}(y,x)\qquad\mbox{cnf}(\mbox{numbers\_are\_different},\mbox{axiom})$

$\mbox{divides}(x,x)\qquad\mbox{cnf}(\mbox{everything\_divides\_itself},\mbox{axiom})$

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

$\mbox{divides}(x,y) \ \Rightarrow \ \neg\,\mbox{less}(y,x)\qquad\mbox{cnf}(\mbox{small\_divides\_large},\mbox{axiom})$

$\mbox{less}(x,\mbox{factorial\_plus\_one}(x))\qquad\mbox{cnf}(\mbox{a\_prime\_is\_less\_than\_the\_next\_one},\mbox{axiom})$

$\mbox{divides}(x,\mbox{factorial\_plus\_one}(y)) \ \Rightarrow \ \mbox{less}(y,x)\qquad\mbox{cnf}(\mbox{divisor\_is\_smaller},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{divides}(\mbox{prime\_divisor}(x),x)\qquad\mbox{cnf}(\mbox{division\_by\_prime\_divisor},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{prime}(\mbox{prime\_divisor}(x))\qquad\mbox{cnf}(\mbox{prime\_divsiors},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{less}(\mbox{prime\_divisor}(x),x)\qquad\mbox{cnf}(\mbox{smaller\_prime\_divisors},\mbox{axiom})$

$\mbox{prime}(a)\qquad\mbox{cnf}(\mbox{a\_is\_prime},\mbox{hypothesis})$

$( \mbox{prime}(x)\mbox{ and }\mbox{less}(a,x) ) \ \Rightarrow \ \mbox{less}(\mbox{factorial\_plus\_one}(a),x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_another\_prime},\mbox{negated\_conjecture})$

\medskip\textbf{NUM102-1.p} Ordinal property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property4}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property4}_3,\mbox{negated\_conjecture})$

$y\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM241-1.p} Transfinite recursion lemma 9.2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{image}(\mbox{comp}(z),\mbox{image}(\mbox{rest},\mbox{recursion\_equation\_functions}(z)))\ne \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM025-1.p} If a$<$b then not b$<$a






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

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

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

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

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_the\_first\_number},\mbox{axiom})$

$\neg\,\mbox{less}(a,a)\qquad\mbox{cnf}(\mbox{no\_number\_less\_than\_itself},\mbox{axiom})$

$\mbox{less}(a,b)\qquad\mbox{cnf}(\mbox{a\_less\_than\_b},\mbox{hypothesis})$

$\mbox{less}(b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_not\_less\_than\_a},\mbox{negated\_conjecture})$

\medskip\textbf{NUM055-1.p} Asymmetric class property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{asymmetric}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{asymmetric}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM868=1.p} Sum X and X is Y






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,x) = y\qquad\mbox{tff}(\mbox{anti\_sum\_x\_x\_y},\mbox{conjecture})$

\medskip\textbf{NUM350+1.p} !XY, (X + X) = Y






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;x + x{=}y\qquad\mbox{fof}(\mbox{anti\_sum\_x\_x\_y},\mbox{conjecture})$

\medskip\textbf{NUM251-1.p} Transfinite recursion property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM710$\wedge$1.p} Landau theorem 29

ts x y = ts y x




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

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

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ n_1 ) = \mbox{xx}\qquad\mbox{thf}(\mbox{satz28a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ n_1@ \mbox{xx} ) = \mbox{xx}\qquad\mbox{thf}(\mbox{satz28c},\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz28f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xy} ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz28d},\mbox{axiom})$

$( \mbox{ts}@ x@ y ) = ( \mbox{ts}@ y@ x )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{conjecture})$

\medskip\textbf{NUM312+1.p} only 2 + 3 = 5






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( x + n_3{=}n_5 \ \Rightarrow \ x = n_2 )\qquad\mbox{fof}(\mbox{sum\_only\_n2\_n3\_n}_5,\mbox{conjecture})$

\medskip\textbf{NUM671$\wedge$1.p} Landau theorem 19b

pl x z = pl y z




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$x = y\qquad\mbox{thf}(i,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$( \mbox{pl}@ x@ z ) = ( \mbox{pl}@ y@ z )\qquad\mbox{thf}(\mbox{satz19b},\mbox{conjecture})$

\medskip\textbf{NUM721$\wedge$1.p} Landau theorem 34a

some (lambda v.diffprop (ts y u) (ts x z) v)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ y@ x@ \mbox{xv} )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ u@ z@ \mbox{xv} )\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xu}@ \mbox{xu}_0 ) ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xu} )@ \mbox{xu}_0 ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{34},\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{ts}@ y@ u )@ ( \mbox{ts}@ x@ z )@ \mbox{xv} )\qquad\mbox{thf}(\mbox{satz34a},\mbox{conjecture})$

\medskip\textbf{NUM100-1.p} Ordinal property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{singleton}(\mbox{null\_class})\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM316+1.p} 2 + -5 = -3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_2 + \mbox{nn}_5{=}\mbox{nn}_3\qquad\mbox{fof}(\mbox{sum\_n2\_nn5\_nn}_3,\mbox{conjecture})$

\medskip\textbf{NUM184-1.p} Corollary to ordinals are either kind 1 or limit






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{union}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers}),\mbox{limit\_ordinals}))\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinals\_are\_kind\_1\_or\_limit}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM690$\wedge$1.p} Landau theorem 22d

some (lambda v.diffprop (pl y u) (pl x z) v)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ y@ x@ \mbox{xv} )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{lessis}@ z@ u\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{moreis}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} )@ \mbox{xu}_0 ) ) ) )\qquad\mbox{thf}(\mbox{satz22b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{14},\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ y@ u )@ ( \mbox{pl}@ x@ z )@ \mbox{xv} )\qquad\mbox{thf}(\mbox{satz22d},\mbox{conjecture})$

\medskip\textbf{NUM211-1.p} Lemma 2 for least upper bound property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},x),x)\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_for\_least\_upper\_bound\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{element\_relation},\mbox{intersection}(\mbox{intersection}(\mbox{power\_class}(x),x)',\mbox{ordinal\_numbers}))\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_for\_least\_upper\_bound\_property8}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},\mbox{intersection}(\mbox{power\_class}(x),x))\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_for\_least\_upper\_bound\_property8}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM139-1.p} Condition 3 for complete induction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{not\_subclass\_element}(\mbox{intersection}(\mbox{power\_class}(x),z),x)\in z\qquad\mbox{cnf}(\mbox{prove\_complete\_induction3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),z),x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM696$\wedge$1.p} Landau theorem 25

~(~(forall x\_0:nat.~(y = pl (pl x n\_1) x\_0))) $\to$ y = pl x n\_1




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 )\qquad\mbox{thf}(m,\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{pl}@ n_1@ \mbox{xx}_0 ) \ \Rightarrow \ \mbox{xx} = n_1 )\qquad\mbox{thf}(\mbox{satz}_{24},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{pl}@ \mbox{xy}@ \mbox{xx}_0 ) \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )\ne ( \mbox{pl}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} )@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ ( \mbox{pl}@ x@ n_1 )@ \mbox{xx}_0 ) \ \Rightarrow \ y = ( \mbox{pl}@ x@ n_1 )\qquad\mbox{thf}(\mbox{satz}_{25},\mbox{conjecture})$

\medskip\textbf{NUM228-1.p} Corollary to recursion equation functions definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary}_1,\mbox{negated\_conjecture})$

$\mbox{recursion\_equation\_functions}(z)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM365+1.p} !XY (X - Y $>$ X + Y)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z_1,z_2{:}\;( x + y{=}z_1\mbox{ and }x\setminus y{=}z_2\mbox{ and }\mbox{less}(z_2,z_1) )\qquad\mbox{fof}(\mbox{anti\_sum\_diff\_less}_2,\mbox{conjecture})$

\medskip\textbf{NUM328+1.p} ?1 + ? = ?1






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;x + y{=}x\qquad\mbox{fof}(\mbox{sum\_something\_anotherthing\_firstthing},\mbox{conjecture})$

\medskip\textbf{NUM822$\wedge$5.p} TPS problem from IND-THMS






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

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

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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cEVEN\_ODD}_2,\mbox{conjecture})$

\medskip\textbf{NUM885=1.p} Product idempotence






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,x) = x\qquad\mbox{tff}(\mbox{anti\_product\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM890=1.p} Sum of X and - X is 0






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,\mbox{\$uminus}(x)) = 0\qquad\mbox{tff}(\mbox{sum\_uminus\_to}_0,\mbox{conjecture})$

\medskip\textbf{NUM204-1.p} Least upper bound property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{successor}(\mbox{sum\_class}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM335+1.p} 5 - 3 = only 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_5\setminus n_3{=}x \ \Rightarrow \ x = n_2 )\qquad\mbox{fof}(\mbox{diff\_n5\_n3\_only\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM202-1.p} If the lub of a set of ordinals is a successor, it's in the set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property4}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property4}_2,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x) = \mbox{successor}(y)\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property4}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM232-1.p} Transfinite recursion lemma 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma3}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma3}_3,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x))),\mbox{universal\_class})\ne \mbox{restrict}(y,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x))),\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM244-1.p} Transfinite recursion lemma 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma11}_1,\mbox{negated\_conjecture})$

$z\circ \mbox{rest}(x) = x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma11}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma11}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))))),\mbox{apply}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)),\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))))))\in \mbox{intersection}(x',\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma11}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)),x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma11}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM755$\wedge$1.p} Landau theorem 63a

moref x y




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ z )\qquad\mbox{thf}(m,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;\neg\,( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy}\qquad\mbox{thf}(\mbox{satz41b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessf}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz41a},\mbox{axiom})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(\mbox{satz63a},\mbox{conjecture})$

\medskip\textbf{NUM086-1.p} Transitive class property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{union}(x,y)),\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM338+1.p} 5 - 3 = only 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_5\setminus n_3{=}x \ \Rightarrow \ x = n_2 )\qquad\mbox{fof}(\mbox{diff\_n5\_n3\_only\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM065-1.p} Well ordering property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(u,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property1}_2,\mbox{negated\_conjecture})$

$v\in u\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property1}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{least}(\mbox{xr},u),v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property1}_4,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{xr},u)\ne v\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property1}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM261-1.p} Alternate 4 for transfinite induction, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{recursion}(x,y,z))\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM786$\wedge$1.p} Landau theorem 81e

~(less x0 y0) $\to$ is x0 y0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x_0@ y_0\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) )\qquad\mbox{thf}(\mbox{satz81a},\mbox{axiom})$

$\neg\,\mbox{less}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(\mbox{satz81e},\mbox{conjecture})$

\medskip\textbf{NUM791$\wedge$1.p} Landau theorem 81k

more x0 y0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\neg\,\neg\,\mbox{less}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) )\qquad\mbox{thf}(\mbox{satz81a},\mbox{axiom})$

$\mbox{more}@ x_0@ y_0\qquad\mbox{thf}(\mbox{satz81k},\mbox{conjecture})$

\medskip\textbf{NUM172-1.p} The successor relation of a set is different from the set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_is\_different}_1,\mbox{negated\_conjecture})$

$\mbox{successor}(x) = x\qquad\mbox{cnf}(\mbox{prove\_successor\_is\_different}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM083-1.p} Transitive class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property1}_1,\mbox{negated\_conjecture})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(u,x)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM771$\wedge$1.p} Landau theorem 69

eq (fr (ts (num x) (num y)) (ts (den x) (den y))) (fr (ts (num y)
 (num x)) (ts (den y) (den x)))




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

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

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

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

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{num}@ y ) )@ ( \mbox{ts}@ ( \mbox{den}@ x )@ ( \mbox{den}@ y ) ) )@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{num}@ x ) )@ ( \mbox{ts}@ ( \mbox{den}@ y )@ ( \mbox{den}@ x ) ) )\qquad\mbox{thf}(\mbox{satz}_{69},\mbox{conjecture})$

\medskip\textbf{NUM329+1.p} ? + ? = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + x{=}x\qquad\mbox{fof}(\mbox{idempotent\_element},\mbox{conjecture})$

\medskip\textbf{NUM008-1.p} Peano axiom 0






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$\neg\,\mbox{little\_set}(\mbox{natural\_numbers})\qquad\mbox{cnf}(\mbox{prove\_naturals\_are\_a\_set},\mbox{negated\_conjecture})$

\medskip\textbf{NUM092-1.p} Corollary 1 to sections property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_sections\_property2}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_sections\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{section}(\mbox{xr},\mbox{segment}(\mbox{xr},y,u),y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_sections\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM160-1.p} Kind 1 ordinals is a class of ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{kind\_1\_ordinals},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_kind\_1\_ordinals\_are\_ordinals}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM231-1.p} Transfinite recursion lemma 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in y\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_3,\mbox{negated\_conjecture})$

$u\in \mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x)))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,v)\in x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma2}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM013-1.p} Peano axiom 5






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$\mbox{empty\_set}\in f_{78}\qquad\mbox{cnf}(\mbox{zero\_in\_set},\mbox{hypothesis})$

$\mbox{xk}\in f_{78} \ \Rightarrow \ \mbox{successor}(\mbox{xk})\in f_{78}\qquad\mbox{cnf}(\mbox{successor\_in\_set},\mbox{hypothesis})$

$\neg\,\mbox{natural\_numbers}\subseteq f_{78}\qquad\mbox{cnf}(\mbox{prove\_set\_is\_in\_naturals},\mbox{negated\_conjecture})$

\medskip\textbf{NUM025-2.p} If a$<$b then not b$<$a






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

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

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

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

$\mbox{greater\_or\_equalish}(c,b) \ \Rightarrow \ ( \mbox{greater\_or\_equalish}(a,b)\mbox{ or }\mbox{greater\_or\_equalish}(c,a) )\qquad\mbox{cnf}(\mbox{transitivity\_of\_less},\mbox{axiom})$

$\mbox{successor}(a) + b{=}c \ \Rightarrow \ \neg\,\mbox{greater\_or\_equalish}(b,c)\qquad\mbox{cnf}(\mbox{smaller\_number},\mbox{axiom})$

$\mbox{greater\_or\_equalish}(a,b)\mbox{ or }\mbox{successor}(\mbox{predecessor\_of\_1st\_minus\_2nd}(b,a)) + a{=}b\qquad\mbox{cnf}(\mbox{less\_lemma},\mbox{axiom})$

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_the\_first\_number},\mbox{axiom})$

$\mbox{greater\_or\_equalish}(a,a)\qquad\mbox{cnf}(\mbox{no\_number\_less\_than\_itself},\mbox{axiom})$

$\neg\,\mbox{greater\_or\_equalish}(a,b)\qquad\mbox{cnf}(\mbox{a\_less\_than\_b},\mbox{hypothesis})$

$\neg\,\mbox{greater\_or\_equalish}(b,a)\qquad\mbox{cnf}(\mbox{prove\_b\_not\_less\_than\_a},\mbox{negated\_conjecture})$

\medskip\textbf{NUM114-1.p} Corollary to ordinal property 13






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_property13}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_property13}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM157-1.p} Membership condition 2 for kind 1 ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_successor\_is\_kind\_1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{kind\_1\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_successor\_is\_kind\_1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM457+1.p} Square root of a prime is irrational 01, 00 expansion






$\forall w_0{:}\;( \mbox{aNaturalNumber}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mNatSort},\mbox{axiom})$

$\mbox{aNaturalNumber}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mSortsC},\mbox{axiom})$

$\mbox{aNaturalNumber}_0(\mbox{sz}_{10})\mbox{ and }\mbox{sz}_{10}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{mSortsC}_{01},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1) ) \ \Rightarrow \ \mbox{aNaturalNumber}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mSortsB},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1) ) \ \Rightarrow \ \mbox{aNaturalNumber}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mSortsB}_{02},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1)\mbox{ and }\mbox{aNaturalNumber}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aNaturalNumber}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{m\_AddZero},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1)\mbox{ and }\mbox{aNaturalNumber}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) = \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aNaturalNumber}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{m\_MulUnit},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aNaturalNumber}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{m\_MulZero},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1)\mbox{ and }\mbox{aNaturalNumber}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_1,w_2),w_0) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_1,w_0),\mbox{sdtasdt}_0(w_2,w_0)) ) )\qquad\mbox{fof}(\mbox{mAMDistr},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1)\mbox{ and }\mbox{aNaturalNumber}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_0,w_2)\mbox{ or }\mbox{sdtpldt}_0(w_1,w_0) = \mbox{sdtpldt}_0(w_2,w_0) ) \ \Rightarrow \ w_1 = w_2 ) )\qquad\mbox{fof}(\mbox{mAddCanc},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aNaturalNumber}_0(w_0) \ \Rightarrow \ ( w_0\ne \mbox{sz}_{00} \ \Rightarrow \ \forall w_1,w_2{:}\;( ( \mbox{aNaturalNumber}_0(w_1)\mbox{ and }\mbox{aNaturalNumber}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_0,w_2)\mbox{ or }\mbox{sdtasdt}_0(w_1,w_0) = \mbox{sdtasdt}_0(w_2,w_0) ) \ \Rightarrow \ w_1 = w_2 ) ) ) )\qquad\mbox{fof}(\mbox{mMulCanc},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aNaturalNumber}_0(w_0)\mbox{ and }\mbox{aNaturalNumber}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ and }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroAdd},\mbox{axiom})$

$\mbox{aNaturalNumber}_0(\mbox{xm})\mbox{ and }\mbox{aNaturalNumber}_0(\mbox{xn})\qquad\mbox{fof}(\mbox{m\_}_{624},\mbox{hypothesis})$

$\mbox{sdtasdt}_0(\mbox{xm},\mbox{xn}) = \mbox{sz}_{00} \ \Rightarrow \ ( \mbox{xm} = \mbox{sz}_{00}\mbox{ or }\mbox{xn} = \mbox{sz}_{00} )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM269-1.p} Ordinal addition property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property5}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property5}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordinal\_add}(x,y)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property5}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM047-1.p} Connected class property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{connected}(x,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{connected}(x,z)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM644$\wedge$1.p} Landau theorem 6

pl x y = pl y x




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ n_1@ \mbox{xx} ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xy} ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4d},\mbox{axiom})$

$( \mbox{pl}@ x@ y ) = ( \mbox{pl}@ y@ x )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{conjecture})$

\medskip\textbf{NUM426+1.p} Fuerstenberg's infinitude of primes 04\_02, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{704},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_}_{724},\mbox{hypothesis})$

$\mbox{aInteger}_0(\mbox{xn})\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},\mbox{xn}) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb}))\qquad\mbox{fof}(\mbox{m\_}_{747},\mbox{hypothesis})$

$\mbox{sdtasdt}_0(\mbox{xq},\mbox{smndt}_0(\mbox{xn})) = \mbox{sdtpldt}_0(\mbox{xb},\mbox{smndt}_0(\mbox{xa}))\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM270-1.p} Ordinal addition property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_add}(x,\mbox{successor}(\mbox{null\_class}))\ne \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property6}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM889=1.p} - - X is X






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$uminus}(\mbox{\$uminus}(x)) = x\qquad\mbox{tff}(\mbox{uminus\_uminus},\mbox{conjecture})$

\medskip\textbf{NUM753$\wedge$1.p} Landau theorem 62g

moref (pf x z) (pf y u)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ z@ u\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{pf}@ \mbox{xz}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz62d},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u )\qquad\mbox{thf}(\mbox{satz62g},\mbox{conjecture})$

\medskip\textbf{NUM693$\wedge$1.p} Landau theorem 24

~(more x n\_1) $\to$ x = n\_1




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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne n_1 \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{suc}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{18},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ n_1@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4g},\mbox{axiom})$

$\neg\,\mbox{more}@ x@ n_1 \ \Rightarrow \ x = n_1\qquad\mbox{thf}(\mbox{satz}_{24},\mbox{conjecture})$

\medskip\textbf{NUM156-1.p} Membership condition 1 for kind 1 ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{kind\_1\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_kind\_1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM635$\wedge$2.p} Landau theorem 1






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne 1\qquad\mbox{thf}(\mbox{one\_is\_first},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{succ}@ x ) = ( \mbox{succ}@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{succ\_injective},\mbox{axiom})$

$\forall m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( m@ 1\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( m@ y ) )\qquad\mbox{thf}(\mbox{induction},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x\ne y \ \Rightarrow \ ( \mbox{succ}@ x )\ne ( \mbox{succ}@ y ) )\qquad\mbox{thf}(\mbox{satz}_1,\mbox{conjecture})$

\medskip\textbf{NUM811$\wedge$5.p} TPS problem THM129

Induction theorem for addition.




$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{c\_plus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_plus\_type},\mbox{type})$

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{c\_plus}@ c_0@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{c\_plus}@ \mbox{xy}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{c\_plus}@ ( \mbox{cS}@ \mbox{xy} )@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xz} ) ) ) ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{c\_plus}@ \mbox{xy}@ \mbox{xx}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cTHM}_{129},\mbox{conjecture})$

\medskip\textbf{NUM738$\wedge$1.p} Landau theorem 45

lessf z u




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{lessf}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{eq}@ x@ z\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{eq}@ y@ u\qquad\mbox{thf}(f,\mbox{axiom})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{42},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{43},\mbox{axiom})$

$\mbox{lessf}@ z@ u\qquad\mbox{thf}(\mbox{satz}_{45},\mbox{conjecture})$

\medskip\textbf{NUM799$\wedge$1.p} Something times four equal five plus seven






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( \cdot @ n@ \mbox{four} ) = (  + @ \mbox{five}@ \mbox{seven} )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM697$\wedge$1.p} Landau theorem 25a

moreis y (suc x)




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ y@ x\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ ( \mbox{pl}@ \mbox{xx}@ n_1 ) ) )\qquad\mbox{thf}(\mbox{satz}_{25},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\mbox{moreis}@ y@ ( \mbox{suc}@ x )\qquad\mbox{thf}(\mbox{satz25a},\mbox{conjecture})$

\medskip\textbf{NUM821$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cEVEN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN\_type},\mbox{type})$

$\mbox{cNUMBER}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cNUMBER\_type},\mbox{type})$

$\mbox{cODD}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cNUMBER}@ \mbox{xn} )\iff ( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cNUMBER}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM303A},\mbox{conjecture})$

\medskip\textbf{NUM874=1.p} Sum idempotence






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,x) = x\qquad\mbox{tff}(\mbox{anti\_sum\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM427+1.p} Fuerstenberg's infinitude of primes 04\_03, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{704},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_}_{724},\mbox{hypothesis})$

$\mbox{aInteger}_0(\mbox{xn})\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},\mbox{xn}) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb}))\qquad\mbox{fof}(\mbox{m\_}_{747},\mbox{hypothesis})$

$\mbox{sdtasdt}_0(\mbox{xq},\mbox{smndt}_0(\mbox{xn})) = \mbox{sdtpldt}_0(\mbox{xb},\mbox{smndt}_0(\mbox{xa}))\qquad\mbox{fof}(\mbox{m\_}_{767},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xb},\mbox{xa},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM098-1.p} Ordinal property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM273-1.p} Lemma 3 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{ordinals\_with\_null\_class\_as\_identity})\qquad\mbox{cnf}(\mbox{prove\_lemma\_3\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM286-3.p} Number theory axioms, based on Godel set theory






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

\medskip\textbf{NUM359+1.p} !X (X + X != X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;\neg\,x + x{=}x\qquad\mbox{fof}(\mbox{anti\_not\_sum\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM782$\wedge$1.p} Landau theorem 80

x0 = z0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$z_0{:}\;\mbox{rat}\qquad\mbox{thf}(z_0,\mbox{type})$

$x_0 = y_0\qquad\mbox{thf}(i,\mbox{axiom})$

$y_0 = z_0\qquad\mbox{thf}(j,\mbox{axiom})$

$x_0 = z_0\qquad\mbox{thf}(\mbox{satz}_{80},\mbox{conjecture})$

\medskip\textbf{NUM706$\wedge$1.p} Landau theorem 28e

x = ts x n\_1




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

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ n_1 ) = \mbox{xx}\qquad\mbox{thf}(\mbox{satz28a},\mbox{axiom})$

$x = ( \mbox{ts}@ x@ n_1 )\qquad\mbox{thf}(\mbox{satz28e},\mbox{conjecture})$

\medskip\textbf{NUM421+1.p} Fuerstenberg's infinitude of primes 01, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\qquad\mbox{fof}(\mbox{m\_}_{419},\mbox{hypothesis})$

$\mbox{sdtasdt}_0(\mbox{xa},\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},\mbox{xa})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM066-1.p} Corollary to well ordering property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(u,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property1}_2,\mbox{negated\_conjecture})$

$v\in u\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property1}_3,\mbox{negated\_conjecture})$

$v\in \mbox{least}(\mbox{element\_relation},u)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM844+2.p} holds(266,415,3)






$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{413},\mbox{vsucc}(\mbox{vd}_{411}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}))\qquad\mbox{fof}(\mbox{'holds(266, 415, 3)'},\mbox{conjecture})$

$\mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{413},\mbox{vsucc}(\mbox{vd}_{411})))\qquad\mbox{fof}(\mbox{'holds(266, 415, 2)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411}))\qquad\mbox{fof}(\mbox{'holds(266, 415, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vsucc}(\mbox{vd}_{413})) = \mbox{vplus}(\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411}))\qquad\mbox{fof}(\mbox{'holds(266, 415, 0)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413})\qquad\mbox{fof}(\mbox{'holds(265, 414, 0)'},\mbox{axiom})$

$\mbox{vsucc}(\mbox{vmul}(\mbox{vd}_{411},v_1)) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},v_1),v_1)\qquad\mbox{fof}(\mbox{'holds(264, 412, 2)'},\mbox{axiom})$

$\mbox{vsucc}(\mbox{vd}_{411}) = \mbox{vsucc}(\mbox{vmul}(\mbox{vd}_{411},v_1))\qquad\mbox{fof}(\mbox{'holds(264, 412, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),v_1) = \mbox{vsucc}(\mbox{vd}_{411})\qquad\mbox{fof}(\mbox{'holds(264, 412, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{400}{:}\;\mbox{vmul}(v_1,\mbox{vd}_{400}) = \mbox{vd}_{400}\qquad\mbox{fof}(\mbox{'ass(cond(253, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{386},\mbox{vd}_{387}{:}\;( \mbox{less}(\mbox{vd}_{386},\mbox{vplus}(\mbox{vd}_{387},v_1)) \ \Rightarrow \ \mbox{leq}(\mbox{vd}_{386},\mbox{vd}_{387}) )\qquad\mbox{fof}(\mbox{'ass(cond(241, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{375},\mbox{vd}_{376}{:}\;( \mbox{greater}(\mbox{vd}_{375},\mbox{vd}_{376}) \ \Rightarrow \ \mbox{geq}(\mbox{vd}_{375},\mbox{vplus}(\mbox{vd}_{376},v_1)) )\qquad\mbox{fof}(\mbox{'ass(cond(234, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{16}{:}\;\mbox{vsucc}(\mbox{vd}_{16})\ne \mbox{vd}_{16}\qquad\mbox{fof}(\mbox{'ass(cond(12, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_7,\mbox{vd}_8{:}\;( \mbox{vd}_7\ne \mbox{vd}_8 \ \Rightarrow \ \mbox{vsucc}(\mbox{vd}_7)\ne \mbox{vsucc}(\mbox{vd}_8) )\qquad\mbox{fof}(\mbox{'ass(cond(6, 0), 0)'},\mbox{axiom})$

\medskip\textbf{NUM906=1.p} Definition of lesseq in terms of less and equality






$\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$lesseq}(x,y)\iff ( \mbox{\$less}(x,y)\mbox{ or }x = y ) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_1,\mbox{conjecture})$

\medskip\textbf{NUM101-1.p} Ordinal property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property3}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property3}_3,\mbox{negated\_conjecture})$

$\neg\,y\in x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property3}_4,\mbox{negated\_conjecture})$

$y\ne x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property3}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM749$\wedge$1.p} Landau theorem 60a

less x (pf x y)




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

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

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

$\mbox{less}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{more}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{42},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{more}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{60},\mbox{axiom})$

$\mbox{less}@ x@ ( \mbox{pf}@ x@ y )\qquad\mbox{thf}(\mbox{satz60a},\mbox{conjecture})$

\medskip\textbf{NUM071-1.p} Well ordering is irreflexive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_is\_irreflexive}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,u)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_is\_irreflexive}_2,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_is\_irreflexive}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM236-1.p} Corollary 1 to transfinite recursion lemma 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_transfinite\_recursion\_lemma6}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_transfinite\_recursion\_lemma6}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{union}(x,y)\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_transfinite\_recursion\_lemma6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM296+1.p} -2 $<$ 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\mbox{less}(\mbox{nn}_2,n_2)\qquad\mbox{fof}(\mbox{nn2\_less\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM041-1.p} Irreflexive class property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{irreflexive}(x,\mbox{domain\_of}(\mbox{symmetrization\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{identity\_relation}')\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM026-1.p} Less preserved over multiplication by a number






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

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

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

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

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_the\_first\_number},\mbox{axiom})$

$\neg\,\mbox{less}(a,a)\qquad\mbox{cnf}(\mbox{no\_number\_less\_than\_itself},\mbox{axiom})$

$\neg\,c{=}n_0\qquad\mbox{cnf}(\mbox{c\_not}_0,\mbox{hypothesis})$

$\mbox{less}(a,b)\qquad\mbox{cnf}(\mbox{a\_less\_than\_b},\mbox{hypothesis})$

$\neg\,\mbox{less}(a\cdot c,b\cdot c)\qquad\mbox{cnf}(\mbox{prove\_a\_times\_c\_less\_b\_times\_c},\mbox{negated\_conjecture})$

\medskip\textbf{NUM163-1.p} Inductive is closed under union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},x)),x)\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_union}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},x)),y)\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_union}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},\mbox{union}(x,y))),\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_union}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM323+1.p} ? + -5 = -7






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + \mbox{nn}_5{=}\mbox{nn}_7\qquad\mbox{fof}(\mbox{sum\_what\_nn5\_nn}_7,\mbox{conjecture})$

\medskip\textbf{NUM195-1.p} The union of a class of ordinals is transitive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_union\_of\_ordinal\_class}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{sum\_class}(x)),\mbox{sum\_class}(x))\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_union\_of\_ordinal\_class}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM372+1.p} if (X - Y) = 0, then X = Y






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;( x\setminus y{=}n_0 \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{identity},\mbox{conjecture})$

\medskip\textbf{NUM695$\wedge$1.p} Landau theorem 24b

more (suc x) n\_1




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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne n_1 \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{suc}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} )\ne n_1\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{18},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ n_1@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4g},\mbox{axiom})$

$\mbox{more}@ ( \mbox{suc}@ x )@ n_1\qquad\mbox{thf}(\mbox{satz24b},\mbox{conjecture})$

\medskip\textbf{NUM060-1.p} Segments property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordered\_pair}(u,z)\in \mbox{intersection}(\mbox{xr},\mbox{cross\_product}(y,\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_segments\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{segment}(\mbox{xr},y,z)\qquad\mbox{cnf}(\mbox{prove\_segments\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM688$\wedge$1.p} Landau theorem 22b

more (pl x z) (pl y u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\neg\,\mbox{more}@ z@ u \ \Rightarrow \ z = u\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{pl}@ \mbox{xu}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{satz19h},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{21},\mbox{axiom})$

$\mbox{more}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ u )\qquad\mbox{thf}(\mbox{satz22b},\mbox{conjecture})$

\medskip\textbf{NUM765$\wedge$1.p} Landau theorem 66

~(moref (pf x z) (pf y u)) $\to$ eq (pf x z) (pf y u)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{moref}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\neg\,\mbox{moref}@ z@ u \ \Rightarrow \ ( \mbox{eq}@ z@ u )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \neg\,\mbox{moref}@ \mbox{xz}@ \mbox{xu} \ \Rightarrow \ ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz65b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz65a},\mbox{axiom})$

$\neg\,\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u ) )\qquad\mbox{thf}(\mbox{satz}_{66},\mbox{conjecture})$

\medskip\textbf{NUM726$\wedge$1.p} Landau theorem 38

ts (num y) (den x) = ts (num x) (den y)




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

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

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

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) ) = ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(e,\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) ) = ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{conjecture})$

\medskip\textbf{NUM919=1.p} No number inbetween






$\forall u{:}\;\mbox{\$int}{:}\;\exists v{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(v,u)\mbox{ and }\neg\,\exists w{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(v,w)\mbox{ and }\mbox{\$less}(w,u) ) )\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM284-1.014.p} Calculation of fibonacci numbers

Compute the 14th Fibonacci number.




$\mbox{fibonacci}(n_0,s(n_0))\qquad\mbox{cnf}(\mbox{fibonacci}_0,\mbox{axiom})$

$\mbox{fibonacci}(s(n_0),s(n_0))\qquad\mbox{cnf}(\mbox{fibonacci}_1,\mbox{axiom})$

$( n_1 + s(n_0){=}n\mbox{ and }n_2 + s(s(n_0)){=}n\mbox{ and }\mbox{fibonacci}(n_1,f_1)\mbox{ and }\mbox{fibonacci}(n_2,f_2)\mbox{ and }f_1 + f_2{=}\mbox{fN} ) \ \Rightarrow \ \mbox{fibonacci}(n,\mbox{fN})\qquad\mbox{cnf}(\mbox{fibonacci\_N},\mbox{axiom})$

$x + n_0{=}x\qquad\mbox{cnf}(\mbox{add}_0,\mbox{axiom})$

$x + y{=}z \ \Rightarrow \ x + s(y){=}s(z)\qquad\mbox{cnf}(\mbox{add},\mbox{axiom})$

$\neg\,\mbox{fibonacci}(s(s(s(s(s(s(s(s(s(s(s(s(s(s(n_0)))))))))))))),\mbox{result})\qquad\mbox{cnf}(\mbox{prove\_fibonacci},\mbox{negated\_conjecture})$

\medskip\textbf{NUM636$\wedge$2.p} Landau theorem 2






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne 1\qquad\mbox{thf}(\mbox{one\_is\_first},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{succ}@ x ) = ( \mbox{succ}@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{succ\_injective},\mbox{axiom})$

$\forall m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( m@ 1\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( m@ y ) )\qquad\mbox{thf}(\mbox{induction},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne x\qquad\mbox{thf}(\mbox{satz}_2,\mbox{conjecture})$

\medskip\textbf{NUM914=1.p} Sum and difference






$\exists x{:}\;\mbox{\$real},y{:}\;\mbox{\$real},z{:}\;\mbox{\$real}{:}\;( \mbox{\$sum}(x,y) = z\iff ( \mbox{\$difference}(z,y) = x\mbox{ and }\mbox{\$difference}(z,x) = y ) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_2,\mbox{conjecture})$

\medskip\textbf{NUM125-1.p} Complete induction upto omega






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),\mbox{omega}),x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction\_upto\_omega}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{omega},x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction\_upto\_omega}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM266-1.p} Ordinal addition property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{ordinal\_add}(x,\mbox{successor}(y))\ne \mbox{successor}(\mbox{ordinal\_add}(x,y))\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM105-1.p} Ordinal property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property6}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM225-1.p} Rest relation property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{rest\_relation})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM534+2.p} Ramsey's Infinite Theorem 04, 01 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{617},\mbox{hypothesis})$

$\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_617}_{02},\mbox{hypothesis})$

$( \mbox{aSet}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}))\mbox{ and }\forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}))\iff ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElementOf}_0(w_0,\mbox{xS})\mbox{ and }w_0\ne \mbox{xx} ) ) ) \ \Rightarrow \ ( ( \mbox{aSet}_0(\mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\mbox{ and }\forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\iff ( \mbox{aElement}_0(w_0)\mbox{ and }( \mbox{aElementOf}_0(w_0,\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}))\mbox{ or }w_0 = \mbox{xx} ) ) ) ) \ \Rightarrow \ \mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}) = \mbox{xS} )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM215-1.p} Corollary to induction upto y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{null\_class}\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_induction\_upto\_y}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(x,\mbox{omega})),x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_induction\_upto\_y}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{omega},x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_induction\_upto\_y}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM369+1.p} ! + 0 = !






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;x + n_0{=}x\qquad\mbox{fof}(\mbox{n0\_identity},\mbox{conjecture})$

\medskip\textbf{NUM805$\wedge$5.p} TPS problem from NATS






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

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

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{cS}@ ( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cPA}_2,\mbox{conjecture})$

\medskip\textbf{NUM907=1.p} Sum and difference






$\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat},z{:}\;\mbox{\$rat}{:}\;( \mbox{\$sum}(x,y) = z\iff ( \mbox{\$difference}(z,y) = x\mbox{ and }\mbox{\$difference}(z,x) = y ) )\qquad\mbox{tff}(\mbox{rat\_combined\_problem}_2,\mbox{conjecture})$

\medskip\textbf{NUM747$\wedge$1.p} Landau theorem 57a

eq (fr (pl x1 x2) n) (pf (fr x1 n) (fr x2 n))




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

$x_1{:}\;\mbox{nat}\qquad\mbox{thf}(x_1,\mbox{type})$

$x_2{:}\;\mbox{nat}\qquad\mbox{thf}(x_2,\mbox{type})$

$n{:}\;\mbox{nat}\qquad\mbox{thf}(n,\mbox{type})$

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}_1{:}\;\mbox{nat},\mbox{xx}_2{:}\;\mbox{nat},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{pf}@ ( \mbox{fr}@ \mbox{xx}_1@ \mbox{xn} )@ ( \mbox{fr}@ \mbox{xx}_2@ \mbox{xn} ) )@ ( \mbox{fr}@ ( \mbox{pl}@ \mbox{xx}_1@ \mbox{xx}_2 )@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{satz}_{57},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{fr}@ ( \mbox{pl}@ x_1@ x_2 )@ n )@ ( \mbox{pf}@ ( \mbox{fr}@ x_1@ n )@ ( \mbox{fr}@ x_2@ n ) )\qquad\mbox{thf}(\mbox{satz57a},\mbox{conjecture})$

\medskip\textbf{NUM904=1.p} Sum something and - something is 0/1






$\forall x{:}\;\mbox{\$rat}{:}\;\mbox{\$sum}(x,\mbox{\$uminus}(x)) = 0/1\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_8,\mbox{conjecture})$

\medskip\textbf{NUM819$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cEVEN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN\_type},\mbox{type})$

$\mbox{cODD}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cIND} ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cEVEN\_ODD}_1,\mbox{conjecture})$

\medskip\textbf{NUM203-1.p} Corollary to least upper bound being a successor relation






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property4}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property4}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property4}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property4}_4,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property4}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM705$\wedge$1.p} Landau theorem 27a

~((forall x:nat.forall y:nat.~((forall x\_0:nat.p x\_0 $\to$
 lessis x x\_0) $\to$ ~(p x)) $\to$ ~((forall x\_0:nat.p x\_0 $\to$
 lessis y x\_0) $\to$ ~(p y)) $\to$ x = y) $\to$ ~(some (lambda x.
 ~((forall x\_0:nat.p x\_0 $\to$ lessis x x\_0) $\to$ ~(p x)))))




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

$p{:}\;\mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{some}@ p\qquad\mbox{thf}(s,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xy}@ \mbox{xx} \ \Rightarrow \ \mbox{xy} = \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{14},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz10d},\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o}{:}\;( ( \mbox{some}@ \mbox{xp} ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xx}{:}\;\mbox{nat}{:}\;\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xx}@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,\mbox{xp}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{27},\mbox{axiom})$

$\neg\,\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( ( p@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xx}@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,p@ \mbox{xx} \ \Rightarrow \ ( \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( ( p@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,p@ \mbox{xy} \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ \neg\,\mbox{some}@ \lambda \mbox{xx}{:}\;\mbox{nat}{:}\;\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( ( p@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xx}@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,p@ \mbox{xx}\qquad\mbox{thf}(\mbox{satz27a},\mbox{conjecture})$

\medskip\textbf{NUM703$\wedge$1.p} Landau theorem 26c

moreis y x




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{more}@ ( \mbox{suc}@ y )@ x\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ ( \mbox{pl}@ \mbox{xy}@ n_1 )@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz26b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{axiom})$

$\mbox{moreis}@ y@ x\qquad\mbox{thf}(\mbox{satz26c},\mbox{conjecture})$

\medskip\textbf{NUM636$\wedge$3.p} Landau theorem 2






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne 1\qquad\mbox{thf}(\mbox{one\_is\_first},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{succ}@ x ) = ( \mbox{succ}@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{succ\_injective},\mbox{axiom})$

$\forall m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( m@ 1\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( m@ y ) )\qquad\mbox{thf}(\mbox{induction},\mbox{axiom})$

$m{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{m\_type},\mbox{type})$

$m = ( \lambda e{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ e )\ne e )\qquad\mbox{thf}(\mbox{m\_defn},\mbox{definition})$

$m@ 1\qquad\mbox{thf}(\mbox{m\_is\_one},\mbox{lemma})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) )\qquad\mbox{thf}(\mbox{m\_is\_next},\mbox{lemma})$

$\forall x{:}\;\mbox{\$i}{:}\;( m@ x )\qquad\mbox{thf}(\mbox{m\_is\_all},\mbox{conjecture})$

\medskip\textbf{NUM209-1.p} Corollary to least upper bound property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property7}_2,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_least\_upper\_bound\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM828$\wedge$5.p} TPS problem from PA-THMS






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

$c_0{:}\;n\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS\_type},\mbox{type})$

$\mbox{c\_plus}{:}\;n\to n\to n\qquad\mbox{thf}(\mbox{c\_plus\_type},\mbox{type})$

$\mbox{cPA}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_1\_type},\mbox{type})$

$\mbox{cPA}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_2\_type},\mbox{type})$

$\mbox{cPA\_IND\_EQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_type},\mbox{type})$

$\mbox{cPA}_1 = ( \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_1\_def},\mbox{definition})$

$\mbox{cPA}_2 = ( \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{cS}@ ( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cPA\_2\_def},\mbox{definition})$

$\mbox{cPA\_IND\_EQ} = ( \forall \mbox{xp}{:}\;n\to n,\mbox{xq}{:}\;n\to n{:}\;( ( ( \mbox{xp}@ c_0 ) = ( \mbox{xq}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;n{:}\;( ( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) = ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_def},\mbox{definition})$

$( \mbox{cPA}_1\mbox{ and }\mbox{cPA}_2\mbox{ and }\mbox{cPA\_IND\_EQ} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{c\_plus}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_THM}_4,\mbox{conjecture})$

\medskip\textbf{NUM281-1.p} Ordinal multiplication property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_multiplication\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{ordinal\_multiply}(x,\mbox{successor}(y))\ne \mbox{ordinal\_add}(\mbox{ordinal\_multiply}(x,y),y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_multiplication\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM866=1.p} Prove sum with 0 is the identity






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,0) = x\qquad\mbox{tff}(\mbox{prove\_sum\_0\_identity},\mbox{conjecture})$

\medskip\textbf{NUM290+1.p} 2 $<$ 3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\mbox{less}(n_2,n_3)\qquad\mbox{fof}(\mbox{n2\_less\_n}_3,\mbox{conjecture})$

\medskip\textbf{NUM762$\wedge$1.p} Landau theorem 65b

moref (pf x z) (pf y u)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{moref}@ z@ u \ \Rightarrow \ ( \mbox{eq}@ z@ u )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{61},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{64},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ u )\qquad\mbox{thf}(\mbox{satz65b},\mbox{conjecture})$

\medskip\textbf{NUM665$\wedge$1.p} Landau theorem 16c

some (lambda u.diffprop x z u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{moreis}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ y@ z@ \mbox{xu} )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xy}@ \mbox{xx}@ \mbox{xv} ) ) \ \Rightarrow \ ( ( \mbox{lessis}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xx}@ \mbox{xv} ) ) ) )\qquad\mbox{thf}(\mbox{satz16b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ x@ z@ \mbox{xu} )\qquad\mbox{thf}(\mbox{satz16c},\mbox{conjecture})$

\medskip\textbf{NUM712$\wedge$1.p} Landau theorem 31

ts (ts x y) z = ts x (ts y z)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;\mbox{xx} = ( \mbox{ts}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz28e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ n_1 ) = \mbox{xx}\qquad\mbox{thf}(\mbox{satz28a},\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz28f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_{30},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz28b},\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{ts}@ x@ y )@ z ) = ( \mbox{ts}@ x@ ( \mbox{ts}@ y@ z ) )\qquad\mbox{thf}(\mbox{satz}_{31},\mbox{conjecture})$

\medskip\textbf{NUM117-1.p} Corollary to ordinal class and numbers






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_class}(x) \ \Rightarrow \ \mbox{well\_ordering}(\mbox{element\_relation},x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_1,\mbox{axiom})$

$\mbox{ordinal\_class}(x) \ \Rightarrow \ \mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_2,\mbox{axiom})$

$( \mbox{well\_ordering}(\mbox{element\_relation},x)\mbox{ and }\mbox{subclass}(\mbox{sum\_class}(x),x) ) \ \Rightarrow \ \mbox{ordinal\_class}(x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_3,\mbox{axiom})$

$\mbox{ordinal\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_class\_and\_numbers}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_class\_and\_numbers}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM194-1.p} The union of a class of ordinals is a class of ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_class\_is\_ordinal\_class}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(x),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_class\_is\_ordinal\_class}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM176-1.p} Successor property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_property10}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{successor\_relation},x)\ne \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_successor\_property10}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM533+1.p} Ramsey's Infinite Theorem 03, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\mbox{aSet}_0(\mbox{xA})\mbox{ and }\mbox{aSet}_0(\mbox{xB})\mbox{ and }\mbox{aSet}_0(\mbox{xC})\qquad\mbox{fof}(\mbox{m\_}_{522},\mbox{hypothesis})$

$( \mbox{aSubsetOf}_0(\mbox{xA},\mbox{xB})\mbox{ and }\mbox{aSubsetOf}_0(\mbox{xB},\mbox{xC}) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(\mbox{xA},\mbox{xC})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM198-1.p} The union of a set of ordinals is $>$= each ordinal in the set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_exceeds\_members}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_exceeds\_members}_2,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_exceeds\_members}_3,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{sum\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_exceeds\_members}_4,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x)\ne y\qquad\mbox{cnf}(\mbox{prove\_union\_of\_ordinal\_set\_exceeds\_members}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM313+1.p} 2 + only 3 = 5






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_2 + x{=}n_5 \ \Rightarrow \ x = n_3 )\qquad\mbox{fof}(\mbox{sumn2\_only\_n3\_n}_5,\mbox{conjecture})$

\medskip\textbf{NUM275-1.p} Lemma 5 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{domain\_of}(\mbox{intersection}(\mbox{recursion}(\mbox{null\_class},\mbox{successor\_relation},\mbox{union\_of\_range\_map}),\mbox{identity\_relation})))\qquad\mbox{cnf}(\mbox{prove\_lemma\_5\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM637$\wedge$1.p} Landau theorem 3

~(forall x\_0:nat.~(x = suc x\_0))




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

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

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$x\ne n_1\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{suc}@ \mbox{xx}_0 )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{conjecture})$

\medskip\textbf{NUM037-1.p} Symmetrization property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class}),\mbox{cross\_product}(\mbox{domain\_of}(\mbox{symmetrization\_of}(x)),\mbox{domain\_of}(\mbox{symmetrization\_of}(x))))\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property8}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM887=1.p} Product with 0 is identity






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,0) = x\qquad\mbox{tff}(\mbox{anti\_product\_0\_identity},\mbox{conjecture})$

\medskip\textbf{NUM143-1.p} Corollary to the successor of a set being a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(\mbox{successor}(x),x)\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_successor\_of\_set\_is\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM016$\wedge$5.p} TPS problem NUM016-1

There exist infinitely many primes.




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

$\mbox{factorial\_plus\_one}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{factorial\_plus\_one},\mbox{type})$

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

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

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

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

$\neg\,\forall x{:}\;\mbox{\$i}{:}\;\neg\,\mbox{less}@ x@ x\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{less}@ x@ y\mbox{ or }\neg\,\mbox{less}@ y@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{divides}@ x@ x )\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{divides}@ x@ y\mbox{ or }\neg\,\mbox{divides}@ y@ z\mbox{ or }\mbox{divides}@ x@ z )\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{divides}@ x@ y\mbox{ or }\neg\,\mbox{less}@ y@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{less}@ x@ ( \mbox{factorial\_plus\_one}@ x ) )\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{divides}@ x@ ( \mbox{factorial\_plus\_one}@ y )\mbox{ or }\mbox{less}@ y@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{prime}@ x\mbox{ or }\mbox{divides}@ ( \mbox{prime\_divisor}@ x )@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{prime}@ x\mbox{ or }\mbox{prime}@ ( \mbox{prime\_divisor}@ x ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{prime}@ x\mbox{ or }\mbox{less}@ ( \mbox{prime\_divisor}@ x )@ x )\mbox{ and }\mbox{prime}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{prime}@ x\mbox{ or }\neg\,\mbox{less}@ a@ x\mbox{ or }\mbox{less}@ ( \mbox{factorial\_plus\_one}@ a )@ x )\qquad\mbox{thf}(\mbox{cNUM016}_1,\mbox{conjecture})$

\medskip\textbf{NUM068-1.p} Well ordering property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property3}_1,\mbox{negated\_conjecture})$

$u\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property3}_2,\mbox{negated\_conjecture})$

$u\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property3}_3,\mbox{negated\_conjecture})$

$u\in \mbox{xr}'\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM641$\wedge$1.p} Landau theorem 4g

suc x = pl n\_1 x




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

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

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ n_1@ \mbox{xx} ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4c},\mbox{axiom})$

$( \mbox{suc}@ x ) = ( \mbox{pl}@ n_1@ x )\qquad\mbox{thf}(\mbox{satz4g},\mbox{conjecture})$

\medskip\textbf{NUM322+1.p} 5 + -5 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_5 + \mbox{nn}_5{=}x\qquad\mbox{fof}(\mbox{sum\_n5\_nn5\_what},\mbox{conjecture})$

\medskip\textbf{NUM201-1.p} Least upper bound property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property3}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property3}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{sum\_class}(x))\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM360+1.p} ?XY (X + Y = 8) =$>$ X - Y = 1






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;( x + y{=}n_8\mbox{ and }x\setminus y{=}n_1 )\qquad\mbox{fof}(\mbox{exists\_sum\_consecutive\_n}_8,\mbox{conjecture})$

\medskip\textbf{NUM640$\wedge$1.p} Landau theorem 4f

suc (pl x y) = pl x (suc y)




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

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

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

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4b},\mbox{axiom})$

$( \mbox{suc}@ ( \mbox{pl}@ x@ y ) ) = ( \mbox{pl}@ x@ ( \mbox{suc}@ y ) )\qquad\mbox{thf}(\mbox{satz4f},\mbox{conjecture})$

\medskip\textbf{NUM423+1.p} Fuerstenberg's infinitude of primes 03, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{671},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xa},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM345+1.p} 2 + 3 $<$ 6






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_2 + n_3{=}x \ \Rightarrow \ \mbox{less}(x,n_6) )\qquad\mbox{fof}(\mbox{sum\_n2\_n3\_less\_n}_6,\mbox{conjecture})$

\medskip\textbf{NUM085-1.p} Alternate transitive class definition, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{power\_class}(w))\qquad\mbox{cnf}(\mbox{prove\_alternate\_trasitive\_class\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(x),w)\qquad\mbox{cnf}(\mbox{prove\_alternate\_trasitive\_class\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM913=1.p} Definition of lesseq in terms of less and equality






$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$lesseq}(x,y)\iff ( \mbox{\$less}(x,y)\mbox{ or }x = y ) )\qquad\mbox{tff}(\mbox{real\_combined\_problem}_1,\mbox{conjecture})$

\medskip\textbf{NUM814$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cEVEN}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN1\_type},\mbox{type})$

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cODD}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD1\_type},\mbox{type})$

$\mbox{cPEANO}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPEANO1\_type},\mbox{type})$

$\mbox{cEVEN}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{cEVEN1\_def},\mbox{definition})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$\mbox{cODD}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cEVEN}_1@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cODD1\_def},\mbox{definition})$

$\mbox{cPEANO}_1 = ( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xu} )\ne c_0\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xv} ) = ( \mbox{cS}@ \mbox{xw} ) \ \Rightarrow \ \mbox{xv} = \mbox{xw} )\mbox{ and }\mbox{cIND} )\qquad\mbox{thf}(\mbox{cPEANO1\_def},\mbox{definition})$

$\mbox{cPEANO}_1 \ \Rightarrow \ \exists \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cODD}_1@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM}_{404},\mbox{conjecture})$

\medskip\textbf{NUM792$\wedge$1.p} Landau theorem 87c 

more x0 z0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$z_0{:}\;\mbox{rat}\qquad\mbox{thf}(z_0,\mbox{type})$

$\mbox{moreis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{moreis}@ x_0@ y_0\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ y_0@ z_0\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{83},\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat},\mbox{xz}_0{:}\;\mbox{rat}{:}\;( ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( ( \mbox{lessis}@ \mbox{xy}_0@ \mbox{xz}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xz}_0 ) ) )\qquad\mbox{thf}(\mbox{satz87b},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{82},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{moreis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{84},\mbox{axiom})$

$\mbox{more}@ x_0@ z_0\qquad\mbox{thf}(\mbox{satz87c},\mbox{conjecture})$

\medskip\textbf{NUM787$\wedge$1.p} Landau theorem 81f

~(more x0 y0) $\to$ is x0 y0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\mbox{less}@ x_0@ y_0\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) )\qquad\mbox{thf}(\mbox{satz81a},\mbox{axiom})$

$\neg\,\mbox{more}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(\mbox{satz81f},\mbox{conjecture})$

\medskip\textbf{NUM827$\wedge$5.p} TPS problem PA-THM2






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

$c_0{:}\;n\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS\_type},\mbox{type})$

$\mbox{c\_plus}{:}\;n\to n\to n\qquad\mbox{thf}(\mbox{c\_plus\_type},\mbox{type})$

$\mbox{cPA}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_1\_type},\mbox{type})$

$\mbox{cPA}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_2\_type},\mbox{type})$

$\mbox{cPA\_IND\_EQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_type},\mbox{type})$

$\mbox{cPA}_1 = ( \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_1\_def},\mbox{definition})$

$\mbox{cPA}_2 = ( \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{cS}@ ( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cPA\_2\_def},\mbox{definition})$

$\mbox{cPA\_IND\_EQ} = ( \forall \mbox{xp}{:}\;n\to n,\mbox{xq}{:}\;n\to n{:}\;( ( ( \mbox{xp}@ c_0 ) = ( \mbox{xq}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;n{:}\;( ( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) = ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_def},\mbox{definition})$

$( \mbox{cPA}_1\mbox{ and }\mbox{cPA}_2\mbox{ and }\mbox{cPA\_IND\_EQ} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = ( \mbox{c\_plus}@ c_0@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_THM}_2,\mbox{conjecture})$

\medskip\textbf{NUM657$\wedge$1.p} Landau theorem 10h

~(~(more x y) $\to$ x = y)




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\neg\,\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(\mbox{satz10h},\mbox{conjecture})$

\medskip\textbf{NUM096-1.p} Sections property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{section}(\mbox{xr},w,y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property5}_2,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{xr},y,\mbox{least}(\mbox{xr},\mbox{intersection}(w',y)))\ne w\qquad\mbox{cnf}(\mbox{prove\_sections\_property5}_3,\mbox{negated\_conjecture})$

$y\ne w\qquad\mbox{cnf}(\mbox{prove\_sections\_property5}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM374+1.p} Disprove Wilkie identity from Tarski's identities






$\forall x,y{:}\;x + y = y + x\qquad\mbox{fof}(\mbox{sum\_symmetry},\mbox{axiom})$

$\forall x,y,z{:}\;x + ( y + z ) = ( x + y ) + z\qquad\mbox{fof}(\mbox{sum\_associativity},\mbox{axiom})$

$\forall x{:}\;x\cdot n_1 = x\qquad\mbox{fof}(\mbox{product\_identity},\mbox{axiom})$

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

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

$\forall x,y,z{:}\;x\cdot ( y + z ) = x\cdot y + x\cdot z\qquad\mbox{fof}(\mbox{sum\_product\_distribution},\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(n_1,x) = n_1\qquad\mbox{fof}(\mbox{exponent\_n}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(x,n_1) = x\qquad\mbox{fof}(\mbox{exponent\_identity},\mbox{axiom})$

$\forall x,y,z{:}\;\mbox{exponent}(x,y + z) = \mbox{exponent}(x,y)\cdot \mbox{exponent}(x,z)\qquad\mbox{fof}(\mbox{exponent\_sum\_product},\mbox{axiom})$

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

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

$\forall c,p,q,r,s,a,b{:}\;( ( c = a\cdot a\mbox{ and }p = n_1 + a\mbox{ and }q = p + c\mbox{ and }r = n_1 + a\cdot c\mbox{ and }s = ( n_1 + c ) + c\cdot c ) \ \Rightarrow \ \mbox{exponent}(\mbox{exponent}(p,a) + \mbox{exponent}(q,a),b)\cdot \mbox{exponent}(\mbox{exponent}(r,b) + \mbox{exponent}(s,b),a) = \mbox{exponent}(\mbox{exponent}(p,b) + \mbox{exponent}(q,b),a)\cdot \mbox{exponent}(\mbox{exponent}(r,a) + \mbox{exponent}(s,a),b) )\qquad\mbox{fof}(\mbox{wilkie},\mbox{conjecture})$

\medskip\textbf{NUM356+1.p} ? + 0 != ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\neg\,x + n_0{=}x\qquad\mbox{fof}(\mbox{anti\_sum\_identity},\mbox{conjecture})$

\medskip\textbf{NUM221-1.p} Rest\_of property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(x)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_rest\_of\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{rest\_of}(\mbox{union}(x,\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),y))))\ne \mbox{union}(\mbox{rest\_of}(x),\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),x)))\qquad\mbox{cnf}(\mbox{prove\_rest\_of\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM858+1.p} Basic upper bound replace maximum

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\forall x{:}\;\mbox{lesseq}(x,x)\qquad\mbox{fof}(\mbox{lesseq\_ref},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,z) ) \ \Rightarrow \ \mbox{lesseq}(x,z) )\qquad\mbox{fof}(\mbox{lesseq\_trans},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,x) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{lesseq\_antisymmetric},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\mbox{ or }\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{lesseq\_total},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(z + x,z + y) )\qquad\mbox{fof}(\mbox{sum\_monotone}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{fof}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{max}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{lesseq}(x,y) )\qquad\mbox{fof}(\mbox{max}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{lesseq}(x,z)\mbox{ and }\mbox{lesseq}(y,z) ) )\qquad\mbox{fof}(\mbox{ub},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_max}(x,y,n)\iff n = \mbox{max}(x,y) )\qquad\mbox{fof}(\mbox{model\_max}_1,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_ub}(x,y,n)\iff \mbox{ub}(x,y,n) )\qquad\mbox{fof}(\mbox{model\_ub}_1,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{fof}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM646$\wedge$1.p} Landau theorem 8

~(pl x y = pl x z)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$y\ne z\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \mbox{suc}@ \mbox{xx} )\ne ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz}_1,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ n_1@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{pl}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz4h},\mbox{axiom})$

$( \mbox{pl}@ x@ y )\ne ( \mbox{pl}@ x@ z )\qquad\mbox{thf}(\mbox{satz}_8,\mbox{conjecture})$

\medskip\textbf{NUM217-1.p} Corollary 2 to rest definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{rest\_of}(\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_rest\_defn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM700$\wedge$1.p} Landau theorem 26

lessis y x




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{less}@ y@ ( \mbox{pl}@ x@ n_1 )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz10e},\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{moreis}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz10h},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ ( \mbox{pl}@ \mbox{xx}@ n_1 ) ) )\qquad\mbox{thf}(\mbox{satz}_{25},\mbox{axiom})$

$\mbox{lessis}@ y@ x\qquad\mbox{thf}(\mbox{satz}_{26},\mbox{conjecture})$

\medskip\textbf{NUM249-1.p} Transfinite recursion property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$z\circ \mbox{rest\_of}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\ne \mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property5}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM728$\wedge$1.p} Landau theorem 40a

eq (fr (ts (1x x) n) (ts (2x x) n)) x




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

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

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

$n{:}\;\mbox{nat}\qquad\mbox{thf}(n,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{c1x}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c1x},\mbox{type})$

$\mbox{c2x}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c2x},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ \mbox{xx}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{c1x}@ \mbox{xx} )@ \mbox{xn} )@ ( \mbox{ts}@ ( \mbox{c2x}@ \mbox{xx} )@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{satz}_{40},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{c1x}@ x )@ n )@ ( \mbox{ts}@ ( \mbox{c2x}@ x )@ n ) )@ x\qquad\mbox{thf}(\mbox{satz40a},\mbox{conjecture})$

\medskip\textbf{NUM888=1.p} Product with 0 is identity






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(0,x) = x\qquad\mbox{tff}(\mbox{anti\_product\_0\_identity\_rev},\mbox{conjecture})$

\medskip\textbf{NUM206-1.p} Corollary 2 to least upper bound property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_least\_upper\_bound\_property1}_1,\mbox{negated\_conjecture})$

$1_{\text{dom}(\mbox{successor\_relation})}\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_least\_upper\_bound\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM836+2.p} dis(ex(cond(conseq(131),0),1))






$\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204})\qquad\mbox{fof}(\mbox{'dis(ex(cond(conseq(131), 0), 1))'},\mbox{conjecture})$

$\mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\qquad\mbox{fof}(\mbox{'dis(ex(cond(conseq(131), 0), 0))'},\mbox{axiom})$

$\forall \mbox{vd}_{198},\mbox{vd}_{199}{:}\;( \mbox{less}(\mbox{vd}_{199},\mbox{vd}_{198})\iff \exists \mbox{vd}_{201}{:}\;\mbox{vd}_{198} = \mbox{vplus}(\mbox{vd}_{199},\mbox{vd}_{201}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 12), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{193},\mbox{vd}_{194}{:}\;( \mbox{greater}(\mbox{vd}_{194},\mbox{vd}_{193})\iff \exists \mbox{vd}_{196}{:}\;\mbox{vd}_{194} = \mbox{vplus}(\mbox{vd}_{193},\mbox{vd}_{196}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 11), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120} = \mbox{vd}_{121}\mbox{ or }\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{104},\mbox{vd}_{105}{:}\;( \mbox{vd}_{104}\ne \mbox{vd}_{105} \ \Rightarrow \ \forall \mbox{vd}_{107}{:}\;\mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{104})\ne \mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{105}) )\qquad\mbox{fof}(\mbox{'ass(cond(81, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

\medskip\textbf{NUM638$\wedge$1.p} Landau theorem 3a

~((forall x\_0:nat.forall y:nat.x = suc x\_0 $\to$ x = suc y $\to$
 x\_0 = y) $\to$ ~(some (lambda u.x = suc u)))




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

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

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$x\ne n_1\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{suc}@ \mbox{xx} ) = ( \mbox{suc}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne n_1 \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;\mbox{xx} = ( \mbox{suc}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( x = ( \mbox{suc}@ \mbox{xx}_0 ) \ \Rightarrow \ ( x = ( \mbox{suc}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}_0 = \mbox{xy} ) ) \ \Rightarrow \ \neg\,\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;x = ( \mbox{suc}@ \mbox{xu} )\qquad\mbox{thf}(\mbox{satz3a},\mbox{conjecture})$

\medskip\textbf{NUM119-1.p} Corollary to transitive class property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$u\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_transitive\_class\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{element\_relation},\mbox{ordinal\_numbers},u)\ne u\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_transitive\_class\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM327+1.p} ? + 0 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + n_0{=}x\qquad\mbox{fof}(\mbox{sum\_zero\_identity},\mbox{conjecture})$

\medskip\textbf{NUM331+1.p} 6 + 7 = 7 + 6






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall z_1,z_2{:}\;( ( n_6 + n_7{=}z_1\mbox{ and }n_7 + n_6{=}z_2 ) \ \Rightarrow \ z_1 = z_2 )\qquad\mbox{fof}(\mbox{communative\_sum\_n6\_n}_7,\mbox{conjecture})$

\medskip\textbf{NUM043-1.p} Irreflexive class property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{irreflexive}(x,\mbox{domain\_of}(\mbox{symmetrization\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,u)\in x\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM252-1.p} Transfinite recursion property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{domain\_of}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM341+1.p} x + y = z $\leq$$>$ z - y = x \& z - x = y






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y,z{:}\;( x + y{=}z\iff ( z\setminus y{=}x\mbox{ and }z\setminus x{=}y ) )\qquad\mbox{fof}(\mbox{add\_same\_as\_subtract},\mbox{conjecture})$

\medskip\textbf{NUM770$\wedge$1.p} Landau theorem 67e

eq $\vee$ (mf x y)




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

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

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

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{eq}@ ( \mbox{pf}@ y@ v )@ x\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{mf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{mf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xv}{:}\;\mbox{frac},\mbox{xw}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xv} )@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xw} )@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xv}@ \mbox{xw} ) ) )\qquad\mbox{thf}(\mbox{satz67b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xy}@ ( \mbox{mf}@ \mbox{xx}@ \mbox{xy} ) )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz67c},\mbox{axiom})$

$\mbox{eq}@ v@ ( \mbox{mf}@ x@ y )\qquad\mbox{thf}(\mbox{satz67e},\mbox{conjecture})$

\medskip\textbf{NUM894=1.p} If Z is less than X + 1 then Z is less than or equal to X






$\forall x{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{\$less}(z,\mbox{\$sum}(x,1)) \ \Rightarrow \ \mbox{\$lesseq}(z,x) )\qquad\mbox{tff}(\mbox{less\_successor},\mbox{conjecture})$

\medskip\textbf{NUM302+1.p} -2 $<$ ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{less}(\mbox{nn}_2,x)\qquad\mbox{fof}(\mbox{nn2\_less\_something},\mbox{conjecture})$

\medskip\textbf{NUM325+1.p} ? + -2 = 3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + \mbox{nn}_2{=}n_3\qquad\mbox{fof}(\mbox{sum\_what\_nn2\_n}_3,\mbox{conjecture})$

\medskip\textbf{NUM301+1.p} ? $<$ -2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{less}(x,\mbox{nn}_2)\qquad\mbox{fof}(\mbox{something\_less\_nn}_2,\mbox{conjecture})$

\medskip\textbf{NUM424+1.p} Fuerstenberg's infinitude of primes 04, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{704},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq}) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xb},\mbox{xa},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM860=1.p} Upper bound replace maximum embedded in a context (1)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\mbox{summation}{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{summation\_type},\mbox{type})$

$\mbox{ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{ub\_type},\mbox{type})$

$\mbox{model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_max\_type},\mbox{type})$

$\mbox{model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_ub\_type},\mbox{type})$

$\mbox{minsol\_model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_max\_type},\mbox{type})$

$\mbox{minsol\_model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_ub\_type},\mbox{type})$

$\mbox{max}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{max\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff \mbox{\$lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{tff}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{\$lesseq}(y,x) )\qquad\mbox{tff}(\mbox{max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{\$lesseq}(x,y) )\qquad\mbox{tff}(\mbox{max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{\$lesseq}(x,z)\mbox{ and }\mbox{\$lesseq}(y,z) ) )\qquad\mbox{tff}(\mbox{ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{\$lesseq}(\mbox{summation}(\mbox{max}(x,y)),n) )\qquad\mbox{tff}(\mbox{model\_max}_3,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{\$lesseq}(\mbox{summation}(z),n) ) )\qquad\mbox{tff}(\mbox{model\_ub}_3,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{tff}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM766$\wedge$1.p} Landau theorem 67a

~(forall x\_0:frac.~(eq (pf y x\_0) x))




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

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

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

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )\ne ( \mbox{pl}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )@ \mbox{xx}_0 )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{ind}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{nat}\qquad\mbox{thf}(\mbox{ind},\mbox{type})$

$\mbox{amone}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{amone},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{amone}@ \lambda \mbox{xz}{:}\;\mbox{nat}{:}\;\mbox{xx} = ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz8b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ \mbox{xx}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ \mbox{xn} )@ ( \mbox{ts}@ ( \mbox{den}@ \mbox{xx} )@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{satz}_{40},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}_1{:}\;\mbox{nat},\mbox{xx}_2{:}\;\mbox{nat},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{pf}@ ( \mbox{fr}@ \mbox{xx}_1@ \mbox{xn} )@ ( \mbox{fr}@ \mbox{xx}_2@ \mbox{xn} ) )@ ( \mbox{fr}@ ( \mbox{pl}@ \mbox{xx}_1@ \mbox{xx}_2 )@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{satz}_{57},\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o}{:}\;( \neg\,( \mbox{amone}@ \mbox{xp} ) \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}{:}\;\mbox{nat}{:}\;\neg\,\mbox{xp}@ \mbox{xx} \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{ind}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{oneax},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ \mbox{xn} )@ ( \mbox{ts}@ ( \mbox{den}@ \mbox{xx} )@ \mbox{xn} ) )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz40a},\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{frac}{:}\;\neg\,\mbox{eq}@ ( \mbox{pf}@ y@ \mbox{xx}_0 )@ x\qquad\mbox{thf}(\mbox{satz67a},\mbox{conjecture})$

\medskip\textbf{NUM022-1.p} Numerator divisble by smaller denominators

If a numerator is divisible by a denominator, then the numerator
 is divisible by numbers smaller than the denominator.




$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-2.ax'})$

$( a{=}b\mbox{ and }c{=}a ) \ \Rightarrow \ c{=}b\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$\mbox{less}(a,b)\qquad\mbox{cnf}(\mbox{a\_less\_than\_b},\mbox{hypothesis})$

$\mbox{divides}(b,d)\qquad\mbox{cnf}(\mbox{b\_divides\_d},\mbox{hypothesis})$

$\neg\,\mbox{less}(a,d)\qquad\mbox{cnf}(\mbox{prove\_a\_less\_than\_d},\mbox{negated\_conjecture})$

\medskip\textbf{NUM170-1.p} Successor property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(x) = y\qquad\mbox{cnf}(\mbox{prove\_successor\_property5}_1,\mbox{negated\_conjecture})$

$y\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_successor\_property5}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_successor\_property5}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM089-1.p} Sections property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{section}(\mbox{xr},y,z)\qquad\mbox{cnf}(\mbox{prove\_sections\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_sections\_property1}_2,\mbox{negated\_conjecture})$

$u\in z\qquad\mbox{cnf}(\mbox{prove\_sections\_property1}_3,\mbox{negated\_conjecture})$

$v\in y\qquad\mbox{cnf}(\mbox{prove\_sections\_property1}_4,\mbox{negated\_conjecture})$

$\neg\,u\in y\qquad\mbox{cnf}(\mbox{prove\_sections\_property1}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM667$\wedge$1.p} Landau theorem 17

~(less x z) $\to$ x = z




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(l,\mbox{axiom})$

$\neg\,\mbox{less}@ y@ z \ \Rightarrow \ y = z\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{less}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz16a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \neg\,\mbox{less}@ \mbox{xy}@ \mbox{xz} \ \Rightarrow \ \mbox{xy} = \mbox{xz} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz16b},\mbox{axiom})$

$\neg\,\mbox{less}@ x@ z \ \Rightarrow \ x = z\qquad\mbox{thf}(\mbox{satz}_{17},\mbox{conjecture})$

\medskip\textbf{NUM314+1.p} Show upper boundary






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_{126} + n_1{=}x\qquad\mbox{fof}(\mbox{show\_upper\_boundary},\mbox{conjecture})$

\medskip\textbf{NUM915=1.p} Every sum right exists






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\exists w{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(u,w) = v\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM018-1.p} There is an infinite number of twin prime numbers






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$\mbox{finite}(\mbox{twin\_prime\_numbers})\qquad\mbox{cnf}(\mbox{prove\_infinite\_number\_of\_twin\_primes},\mbox{negated\_conjecture})$

\medskip\textbf{NUM126-1.p} Alternate 1 for transfinite induction, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(z,\mbox{ordinal\_numbers}))\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction1}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(z,\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM740$\wedge$1.p} Landau theorem 48

~(lessf y x) $\to$ eq y x




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

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

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{moref}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{42},\mbox{axiom})$

$\neg\,\mbox{lessf}@ y@ x \ \Rightarrow \ ( \mbox{eq}@ y@ x )\qquad\mbox{thf}(\mbox{satz}_{48},\mbox{conjecture})$

\medskip\textbf{NUM807$\wedge$5.p} TPS problem from NATS






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

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$c_0{:}\;n\qquad\mbox{thf}(c_0,\mbox{type})$

$\forall \mbox{xp}{:}\;n\to n,\mbox{xq}{:}\;n\to n{:}\;( ( ( \mbox{xp}@ c_0 ) = ( \mbox{xq}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;n{:}\;( ( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) = ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ},\mbox{conjecture})$

\medskip\textbf{NUM295+1.p} ? $<$ ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;\mbox{less}(x,y)\qquad\mbox{fof}(\mbox{something\_less\_something},\mbox{conjecture})$

\medskip\textbf{NUM004-1.p} (A + B) - C = A + (B - C)






$\mbox{include}(\mbox{'Axioms/NUM002-0.ax'})$

$\neg\,\mbox{subtract}(a + b,c){=}a + \mbox{subtract}(b,c)\qquad\mbox{cnf}(\mbox{prove\_equation},\mbox{negated\_conjecture})$

\medskip\textbf{NUM867=1.p} Prove sum with 0 is the identity






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(0,x) = x\qquad\mbox{tff}(\mbox{prove\_sum\_0\_identity\_rev},\mbox{conjecture})$

\medskip\textbf{NUM349+1.p} X (-128 - 1 = X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{nn}_{128}\setminus n_1{=}x\qquad\mbox{fof}(\mbox{anti\_lower\_boundary},\mbox{conjecture})$

\medskip\textbf{NUM698$\wedge$1.p} Landau theorem 25b

lessis (pl y n\_1) x




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

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

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ x@ y@ \mbox{xv} )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xy}@ \mbox{xx}@ \mbox{xu} ) ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ ( \mbox{pl}@ \mbox{xx}@ n_1 ) ) )\qquad\mbox{thf}(\mbox{satz}_{25},\mbox{axiom})$

$\mbox{lessis}@ ( \mbox{pl}@ y@ n_1 )@ x\qquad\mbox{thf}(\mbox{satz25b},\mbox{conjecture})$

\medskip\textbf{NUM897=1.p} Sum less than difference






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$difference}(x,y) = z_2\mbox{ and }\mbox{\$less}(z_1,z_2) )\qquad\mbox{tff}(\mbox{anti\_sum\_diff\_less}_1,\mbox{conjecture})$

\medskip\textbf{NUM736$\wedge$1.p} Landau theorem 42

less (ts (num y) (den x)) (ts (num x) (den y))




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

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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\mbox{more}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{11},\mbox{axiom})$

$\mbox{less}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )\qquad\mbox{thf}(\mbox{satz}_{42},\mbox{conjecture})$

\medskip\textbf{NUM178-1.p} Condition 2 for a class to be inductive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,1_{\text{dom}(\mbox{successor\_relation})}\in x\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition22}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},x),x)\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition22}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM321+1.p} 5 + -2 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_5 + \mbox{nn}_2{=}x\qquad\mbox{fof}(\mbox{sum\_n5\_nn2\_what},\mbox{conjecture})$

\medskip\textbf{NUM127-1.p} Alternate 1 for transfinite induction, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{least}(\mbox{element\_relation},\mbox{intersection}(z,\mbox{ordinal\_numbers})),z')\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction2}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(z,\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM884=1.p} Not product identity






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$product}(x,1) = y\mbox{ and }y\ne x )\qquad\mbox{tff}(\mbox{anti\_product\_identity}_2,\mbox{conjecture})$

\medskip\textbf{NUM175-1.p} Successor property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{successor\_relation})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_property9}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM146-1.p} The successor of a transitive set is transitive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{prove\_successor\_or\_transitive\_set\_is\_set}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{successor}(x)),\mbox{successor}(x))\qquad\mbox{cnf}(\mbox{prove\_successor\_or\_transitive\_set\_is\_set}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM171-1.p} Successor property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation}',\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})),\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals}))\qquad\mbox{cnf}(\mbox{prove\_successor\_property6}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM131-1.p} Alternate 2 for transfinite induction, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{least}(\mbox{element\_relation},y)\in y\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction3}_2,\mbox{negated\_conjecture})$

$y\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM174-1.p} Successor property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{successor\_relation})\qquad\mbox{cnf}(\mbox{prove\_successor\_property8}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM687$\wedge$1.p} Landau theorem 22a

more (pl x z) (pl y u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{more}@ z@ u\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz19g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{21},\mbox{axiom})$

$\mbox{more}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ u )\qquad\mbox{thf}(\mbox{satz22a},\mbox{conjecture})$

\medskip\textbf{NUM729$\wedge$1.p} Landau theorem 40c

eq (fr (ts x1 n) (ts x2 n)) (fr x1 x2)




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

$x_1{:}\;\mbox{nat}\qquad\mbox{thf}(x_1,\mbox{type})$

$x_2{:}\;\mbox{nat}\qquad\mbox{thf}(x_2,\mbox{type})$

$n{:}\;\mbox{nat}\qquad\mbox{thf}(n,\mbox{type})$

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}_1{:}\;\mbox{nat},\mbox{xx}_2{:}\;\mbox{nat},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{fr}@ \mbox{xx}_1@ \mbox{xx}_2 )@ ( \mbox{fr}@ ( \mbox{ts}@ \mbox{xx}_1@ \mbox{xn} )@ ( \mbox{ts}@ \mbox{xx}_2@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{satz40b},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{fr}@ ( \mbox{ts}@ x_1@ n )@ ( \mbox{ts}@ x_2@ n ) )@ ( \mbox{fr}@ x_1@ x_2 )\qquad\mbox{thf}(\mbox{satz40c},\mbox{conjecture})$

\medskip\textbf{NUM078-1.p} Corollary 2 to well ordering property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_well\_ordering\_property7}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_well\_ordering\_property7}_2,\mbox{negated\_conjecture})$

$u\in \mbox{intersection}(u,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_well\_ordering\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM286-1.p} Number theory axioms






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

\medskip\textbf{NUM015-1.p} Any number greater than 1 has a prime divisor






$\mbox{divides}(x,x)\qquad\mbox{cnf}(\mbox{divide\_self},\mbox{axiom})$

$( \mbox{divides}(x,y)\mbox{ and }\mbox{divides}(y,z) ) \ \Rightarrow \ \mbox{divides}(x,z)\qquad\mbox{cnf}(\mbox{transitive\_divide},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{divides}(\mbox{divisor}(x),x)\qquad\mbox{cnf}(\mbox{prime},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{less}(n_1,\mbox{divisor}(x))\qquad\mbox{cnf}(\mbox{divisor}_1,\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{less}(\mbox{divisor}(x),x)\qquad\mbox{cnf}(\mbox{divisor}_2,\mbox{axiom})$

$( \mbox{less}(n_1,x)\mbox{ and }\mbox{less}(x,a) ) \ \Rightarrow \ \mbox{prime}(\mbox{factor\_of}(x))\qquad\mbox{cnf}(\mbox{factor}_1,\mbox{axiom})$

$( \mbox{less}(n_1,x)\mbox{ and }\mbox{less}(x,a) ) \ \Rightarrow \ \mbox{divides}(\mbox{factor\_of}(x),x)\qquad\mbox{cnf}(\mbox{factor}_2,\mbox{axiom})$

$\mbox{less}(n_1,a)\qquad\mbox{cnf}(\mbox{a\_is\_greater\_than}_1,\mbox{hypothesis})$

$\mbox{prime}(x) \ \Rightarrow \ \neg\,\mbox{divides}(x,a)\qquad\mbox{cnf}(\mbox{prove\_a\_has\_prime\_divisor},\mbox{negated\_conjecture})$

\medskip\textbf{NUM023-1.p} Zero is less than all successor numbers






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

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

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

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

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_first\_number},\mbox{axiom})$

$\mbox{less}(a,\mbox{successor}(a))\qquad\mbox{cnf}(\mbox{numbers\_less\_than\_its\_successor},\mbox{axiom})$

$\neg\,\mbox{less}(n_0,\mbox{successor}(a))\qquad\mbox{cnf}(\mbox{prove\_zero\_is\_less\_than\_all\_successors},\mbox{negated\_conjecture})$

\medskip\textbf{NUM151-1.p} Corollary 2 to successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_successor\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{successor}(1_{\text{dom}(\mbox{successor\_relation})})\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_successor\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM781$\wedge$3.p} Landau theorem 79

Symmetry of equality.




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

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

$\mbox{cY} = \mbox{cX} \ \Rightarrow \ \mbox{cX} = \mbox{cY}\qquad\mbox{thf}(\mbox{cTHM}_{76},\mbox{conjecture})$

\medskip\textbf{NUM222-1.p} Rest\_of is monotonic.






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_rest\_of}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{rest\_of}(x),\mbox{rest\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_rest\_of}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM768$\wedge$1.p} Landau theorem 67c

eq (pf y (fr (ind (lambda t.ts (num x) (den y) = pl (ts (num y)
 (den x)) t)) (ts (den x) (den y)))) x




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

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

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

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) ) = ( \mbox{pl}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )@ \mbox{xu} )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{ind}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{nat}\qquad\mbox{thf}(\mbox{ind},\mbox{type})$

$\mbox{amone}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{amone},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{amone}@ \lambda \mbox{xz}{:}\;\mbox{nat}{:}\;\mbox{xx} = ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz8b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ \mbox{xx}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ \mbox{xn} )@ ( \mbox{ts}@ ( \mbox{den}@ \mbox{xx} )@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{satz}_{40},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}_1{:}\;\mbox{nat},\mbox{xx}_2{:}\;\mbox{nat},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{pf}@ ( \mbox{fr}@ \mbox{xx}_1@ \mbox{xn} )@ ( \mbox{fr}@ \mbox{xx}_2@ \mbox{xn} ) )@ ( \mbox{fr}@ ( \mbox{pl}@ \mbox{xx}_1@ \mbox{xx}_2 )@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{satz}_{57},\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o}{:}\;( \neg\,( \mbox{amone}@ \mbox{xp} ) \ \Rightarrow \ \neg\,\mbox{some}@ \mbox{xp} \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{ind}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{oneax},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xn}{:}\;\mbox{nat}{:}\;( \mbox{eq}@ ( \mbox{fr}@ ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ \mbox{xn} )@ ( \mbox{ts}@ ( \mbox{den}@ \mbox{xx} )@ \mbox{xn} ) )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz40a},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{pf}@ y@ ( \mbox{fr}@ ( \mbox{ind}@ \lambda \mbox{xt}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) ) = ( \mbox{pl}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )@ \mbox{xt} ) )@ ( \mbox{ts}@ ( \mbox{den}@ x )@ ( \mbox{den}@ y ) ) ) )@ x\qquad\mbox{thf}(\mbox{satz67c},\mbox{conjecture})$

\medskip\textbf{NUM344+1.p} x+1 = z =$>$ z $>$ x






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;( x + n_1{=}y\mbox{ and }\mbox{less}(x,y) )\qquad\mbox{fof}(\mbox{exist\_bigger\_plus\_one},\mbox{conjecture})$

\medskip\textbf{NUM381+1.p} Ordinal numbers, theorem 5






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( f = \mbox{unordered\_quintuple}(a,b,c,d,e)\iff \forall g{:}\;( \mbox{in}(g,f)\iff \neg\,g\ne a\mbox{ and }g\ne b\mbox{ and }g\ne c\mbox{ and }g\ne d\mbox{ and }g\ne e ) )\qquad\mbox{fof}(\mbox{d3\_enumset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c,d,e{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{in}(b,c)\mbox{ and }\mbox{in}(c,d)\mbox{ and }\mbox{in}(d,e)\mbox{ and }\mbox{in}(e,a)\qquad\mbox{fof}(\mbox{t5\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\mbox{in}(d,c)\qquad\mbox{fof}(\mbox{t7\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM088-1.p} Transitive class property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property4}_1,\mbox{negated\_conjecture})$

$z\in y\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property4}_2,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{element\_relation},y,z)\ne z\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM164-1.p} Inductive is closed under intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},x)),x)\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_intersection}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},x)),y)\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_intersection}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{union}(\mbox{singleton}(\mbox{null\_class}),\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(x,y))),\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_inductive\_closed\_under\_intersection}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM264-1.p} Alternate 4 for transfinite induction, part 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$u\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction5}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(z,\mbox{restrict}(\mbox{recursion}(x,y,z),u,\mbox{universal\_class}))\ne \mbox{apply}(\mbox{recursion}(x,y,z),u)\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM669$\wedge$1.p} Landau theorem 18b

more (suc x) x




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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{18},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\mbox{more}@ ( \mbox{suc}@ x )@ x\qquad\mbox{thf}(\mbox{satz18b},\mbox{conjecture})$

\medskip\textbf{NUM727$\wedge$1.p} Landau theorem 39

ts (num x) (den z) = ts (num z) (den x)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) ) = ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(e,\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ z ) ) = ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ y ) )\qquad\mbox{thf}(f,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{satz33b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_{31},\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ z ) ) = ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{conjecture})$

\medskip\textbf{NUM182-1.p} Only limit ordinals equal their successors






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_only\_limit\_ordinals\_equal\_successor}_1,\mbox{negated\_conjecture})$

$\mbox{successor}(y) = x\qquad\mbox{cnf}(\mbox{prove\_only\_limit\_ordinals\_equal\_successor}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM927=2.p} Related to the Collatz Conjecture

There are two sequences of different length that lead to the
 same value.




$f{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$\mbox{iterate\_f}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{iterate\_f\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$remainder\_t}(x,2) = 1 \ \Rightarrow \ f(x) = \mbox{\$sum}(\mbox{\$product}(3,x),1) )\qquad\mbox{tff}(\mbox{f\_odd},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$remainder\_t}(x,2) = 0 \ \Rightarrow \ f(x) = \mbox{\$quotient\_t}(x,2) )\qquad\mbox{tff}(\mbox{f\_even},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int},x{:}\;\mbox{\$int}{:}\;( i = 1 \ \Rightarrow \ \mbox{iterate\_f}(i,x) = f(x) )\qquad\mbox{tff}(\mbox{iterate\_f\_base},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int},x{:}\;\mbox{\$int}{:}\;( \mbox{\$greater}(i,1) \ \Rightarrow \ \mbox{iterate\_f}(i,x) = \mbox{iterate\_f}(\mbox{\$difference}(i,1),f(x)) )\qquad\mbox{tff}(\mbox{iterate\_f},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$greatereq}(x,1) \ \Rightarrow \ \exists i_1{:}\;\mbox{\$int},i_2{:}\;\mbox{\$int}{:}\;( \mbox{\$greatereq}(i_1,1)\mbox{ and }\mbox{\$greater}(i_2,i_1)\mbox{ and }\mbox{iterate\_f}(i_1,x) = \mbox{iterate\_f}(i_2,x) ) )\qquad\mbox{tff}(\mbox{iterates},\mbox{conjecture})$

\medskip\textbf{NUM340+1.p} ? - 0 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x\setminus n_0{=}x\qquad\mbox{fof}(\mbox{diff\_zero\_identity},\mbox{conjecture})$

\medskip\textbf{NUM674$\wedge$1.p} Landau theorem 19e

pl z x = pl z y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$x = y\qquad\mbox{thf}(i,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$( \mbox{pl}@ z@ x ) = ( \mbox{pl}@ z@ y )\qquad\mbox{thf}(\mbox{satz19e},\mbox{conjecture})$

\medskip\textbf{NUM012-1.p} Peano axiom 4






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$f_{76}\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{a\_natural\_number},\mbox{hypothesis})$

$f_{77}\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{another\_natural\_number},\mbox{hypothesis})$

$\mbox{successor}(f_{76}) = \mbox{successor}(f_{77})\qquad\mbox{cnf}(\mbox{successors\_are\_equal},\mbox{hypothesis})$

$f_{76}\ne f_{77}\qquad\mbox{cnf}(\mbox{prove\_well\_definedness\_of\_successor},\mbox{negated\_conjecture})$

\medskip\textbf{NUM689$\wedge$1.p} Landau theorem 22c

some (lambda v.diffprop (pl y u) (pl x z) v)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{lessis}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ u@ z@ \mbox{xv} )\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xu}@ \mbox{xu}_0 ) ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} )@ \mbox{xu}_0 ) ) ) )\qquad\mbox{thf}(\mbox{satz22a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{14},\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ y@ u )@ ( \mbox{pl}@ x@ z )@ \mbox{xv} )\qquad\mbox{thf}(\mbox{satz22c},\mbox{conjecture})$

\medskip\textbf{NUM809$\wedge$5.p} TPS problem THM130

Induction theorem in which the conclusion is weaker than the 
 statement which must be proved by induction.




$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }r@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( r@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( r@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( r@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM}_{130},\mbox{conjecture})$

\medskip\textbf{NUM011-1.p} Peano axiom 3






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$f_{75}\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{a\_natural\_number},\mbox{hypothesis})$

$\mbox{empty\_set} = \mbox{successor}(f_{75})\qquad\mbox{cnf}(\mbox{prove\_zero\_is\_first},\mbox{negated\_conjecture})$

\medskip\textbf{NUM683$\wedge$1.p} Landau theorem 20d

more x y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{more}@ ( \mbox{pl}@ z@ x )@ ( \mbox{pl}@ z@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz20a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(\mbox{satz20d},\mbox{conjecture})$

\medskip\textbf{NUM909=1.p} Difference is 0.0 implies equality






$\forall x{:}\;\mbox{\$real},y{:}\;\mbox{\$real}{:}\;( \mbox{\$difference}(x,y) = \mbox{0.}_0 \ \Rightarrow \ x = y )\qquad\mbox{tff}(\mbox{real\_difference\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{NUM659$\wedge$1.p} Landau theorem 10k

more x y




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(\mbox{satz10k},\mbox{conjecture})$

\medskip\textbf{NUM318+1.p} 5 + -5 = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_5 + \mbox{nn}_5{=}n_0\qquad\mbox{fof}(\mbox{sum\_n5\_nn5\_n}_0,\mbox{conjecture})$

\medskip\textbf{NUM815$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cPETER\_INDEQS}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cPETER\_INDEQS\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$\mbox{cPETER\_INDEQS} = ( \lambda x_0{:}\;\mbox{\$i},s{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{pSI}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{pSI}@ x_0@ \mbox{xn} ) = ( s@ \mbox{xn} )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i}{:}\;( \mbox{pSI}@ ( s@ \mbox{xm} )@ x_0 ) = ( \mbox{pSI}@ \mbox{xm}@ ( s@ x_0 ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{pSI}@ ( s@ \mbox{xm} )@ ( s@ \mbox{xn} ) ) = ( \mbox{pSI}@ \mbox{xm}@ ( \mbox{pSI}@ ( s@ \mbox{xm} )@ \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cPETER\_INDEQS\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\mbox{cPETER\_INDEQS}@ c_0@ \mbox{cS}@ \mbox{cPSI} ) \ \Rightarrow \ \forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;\exists \mbox{xk}{:}\;\mbox{\$i}{:}\;( \mbox{cPSI}@ \mbox{xm}@ \mbox{xn} ) = ( \mbox{cS}@ \mbox{xk} )\qquad\mbox{thf}(\mbox{cTHM}_{585},\mbox{conjecture})$

\medskip\textbf{NUM020$\wedge$1.p} Find N such that N * 3 = 6






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( \cdot @ n@ \mbox{three} ) = \mbox{six}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM875=1.p} Sum not idempotence






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,x)\ne x\qquad\mbox{tff}(\mbox{anti\_not\_sum\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM737$\wedge$1.p} Landau theorem 44

more (ts (num z) (den u)) (ts (num u) (den z))




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\mbox{more}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(m,\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ z ) ) = ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(e,\mbox{axiom})$

$( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ u ) ) = ( \mbox{ts}@ ( \mbox{num}@ u )@ ( \mbox{den}@ y ) )\qquad\mbox{thf}(f,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz33a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{ts}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{ts}@ \mbox{xz}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz32d},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_{31},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ ( \mbox{den}@ \mbox{xy} ) ) = ( \mbox{ts}@ ( \mbox{num}@ \mbox{xy} )@ ( \mbox{den}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{ts}@ ( \mbox{num}@ \mbox{xy} )@ ( \mbox{den}@ \mbox{xx} ) ) = ( \mbox{ts}@ ( \mbox{num}@ \mbox{xx} )@ ( \mbox{den}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\mbox{more}@ ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ u ) )@ ( \mbox{ts}@ ( \mbox{num}@ u )@ ( \mbox{den}@ z ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{conjecture})$

\medskip\textbf{NUM653$\wedge$1.p} Landau theorem 10d

~(more x y)




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\neg\,\mbox{more}@ x@ y\qquad\mbox{thf}(\mbox{satz10d},\mbox{conjecture})$

\medskip\textbf{NUM879=1.p} Product X and X is not Y






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,x) = y\qquad\mbox{tff}(\mbox{anti\_product\_x\_x\_y},\mbox{conjecture})$

\medskip\textbf{NUM810$\wedge$5.p} TPS problem THM140

Existence of doubles of naturals.




$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM}_{140},\mbox{conjecture})$

\medskip\textbf{NUM292+1.p} 2 $<$ 13






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\mbox{less}(n_2,n_{13})\qquad\mbox{fof}(\mbox{n2\_less\_n}_{13},\mbox{conjecture})$

\medskip\textbf{NUM730$\wedge$1.p} Landau theorem 41

orec3 (ts (1x x) (2y y) = ts (1y y) (2x x)) (more (ts (1x x)
 (2y y)) (ts (1y y) (2x x))) (less (ts (1x x) (2y y)) (ts (1y y)
 (2x x)))




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

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

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

$\mbox{orec}_3{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{orec}_3,\mbox{type})$

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{c1x}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c1x},\mbox{type})$

$\mbox{c2y}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c2y},\mbox{type})$

$\mbox{c1y}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c1y},\mbox{type})$

$\mbox{c2x}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{c2x},\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{orec}_3@ \mbox{xx} = \mbox{xy}@ ( \mbox{more}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz}_{10},\mbox{axiom})$

$\mbox{orec}_3@ ( \mbox{ts}@ ( \mbox{c1x}@ x )@ ( \mbox{c2y}@ y ) ) = ( \mbox{ts}@ ( \mbox{c1y}@ y )@ ( \mbox{c2x}@ x ) )@ ( \mbox{more}@ ( \mbox{ts}@ ( \mbox{c1x}@ x )@ ( \mbox{c2y}@ y ) )@ ( \mbox{ts}@ ( \mbox{c1y}@ y )@ ( \mbox{c2x}@ x ) ) )@ ( \mbox{less}@ ( \mbox{ts}@ ( \mbox{c1x}@ x )@ ( \mbox{c2y}@ y ) )@ ( \mbox{ts}@ ( \mbox{c1y}@ y )@ ( \mbox{c2x}@ x ) ) )\qquad\mbox{thf}(\mbox{satz}_{41},\mbox{conjecture})$

\medskip\textbf{NUM197-1.p} The union of a proper class of ordinals is the class of ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_union\_of\_proper\_ordinal\_class\_is\_ordinal}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x)\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_proper\_ordinal\_class\_is\_ordinal}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_proper\_ordinal\_class\_is\_ordinal}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM836+1.p} dis(ex(cond(conseq(131),0),1))






$\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204})\qquad\mbox{fof}(\mbox{'dis(ex(cond(conseq(131), 0), 1))'},\mbox{conjecture})$

$\mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\qquad\mbox{fof}(\mbox{'dis(ex(cond(conseq(131), 0), 0))'},\mbox{axiom})$

$\forall \mbox{vd}_{198},\mbox{vd}_{199}{:}\;( \mbox{less}(\mbox{vd}_{199},\mbox{vd}_{198})\iff \exists \mbox{vd}_{201}{:}\;\mbox{vd}_{198} = \mbox{vplus}(\mbox{vd}_{199},\mbox{vd}_{201}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 12), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{193},\mbox{vd}_{194}{:}\;( \mbox{greater}(\mbox{vd}_{194},\mbox{vd}_{193})\iff \exists \mbox{vd}_{196}{:}\;\mbox{vd}_{194} = \mbox{vplus}(\mbox{vd}_{193},\mbox{vd}_{196}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 11), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120} = \mbox{vd}_{121}\mbox{ or }\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{104},\mbox{vd}_{105}{:}\;( \mbox{vd}_{104}\ne \mbox{vd}_{105} \ \Rightarrow \ \forall \mbox{vd}_{107}{:}\;\mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{104})\ne \mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{105}) )\qquad\mbox{fof}(\mbox{'ass(cond(81, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{92},\mbox{vd}_{93}{:}\;\mbox{vd}_{93}\ne \mbox{vplus}(\mbox{vd}_{92},\mbox{vd}_{93})\qquad\mbox{fof}(\mbox{'ass(cond(73, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{16}{:}\;\mbox{vsucc}(\mbox{vd}_{16})\ne \mbox{vd}_{16}\qquad\mbox{fof}(\mbox{'ass(cond(12, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_7,\mbox{vd}_8{:}\;( \mbox{vd}_7\ne \mbox{vd}_8 \ \Rightarrow \ \mbox{vsucc}(\mbox{vd}_7)\ne \mbox{vsucc}(\mbox{vd}_8) )\qquad\mbox{fof}(\mbox{'ass(cond(6, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_3,\mbox{vd}_4{:}\;( \mbox{vsucc}(\mbox{vd}_3) = \mbox{vsucc}(\mbox{vd}_4) \ \Rightarrow \ \mbox{vd}_3 = \mbox{vd}_4 )\qquad\mbox{fof}(\mbox{'qu(antec(axiom(3)), imp(antec(axiom(3))))'},\mbox{axiom})$

$\forall \mbox{vd}_1{:}\;\mbox{vsucc}(\mbox{vd}_1)\ne v_1\qquad\mbox{fof}(\mbox{'qu(restrictor(axiom(1)), holds(scope(axiom(1)), 2, 0))'},\mbox{axiom})$

\medskip\textbf{NUM326+1.p} ? + -5 = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + \mbox{nn}_5{=}n_0\qquad\mbox{fof}(\mbox{sum\_what\_nn5\_n}_0,\mbox{conjecture})$

\medskip\textbf{NUM243-1.p} Transfinite recursion lemma 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma10}_1,\mbox{negated\_conjecture})$

$z\circ \mbox{rest}(x) = x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma10}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma10}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)),x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma10}_4,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)),\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))))))\ne \mbox{apply}(x,\mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(x',\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z))))))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma10}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM649$\wedge$1.p} Landau theorem 9

~((~(x = y) $\to$ ~(~(forall x\_0:nat.~(x = pl y x\_0))) $\to$
 ~(forall x\_0:nat.~(y = pl x x\_0))) $\to$ ~(~((x = y $\to$
 ~(~(forall x\_0:nat.~(x = pl y x\_0)))) $\to$
 ~(~((~(forall x\_0:nat.~(x = pl y x\_0)) $\to$
 ~(~(forall x\_0:nat.~(y = pl x x\_0)))) $\to$
 ~(~(forall x\_0:nat.~(y = pl x x\_0)) $\to$ ~(x = y)))))))




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne n_1 \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;\mbox{xx}\ne ( \mbox{suc}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ n_1@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\mbox{xy}\ne ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz}_7,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\neg\,( x\ne y \ \Rightarrow \ ( \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) \ \Rightarrow \ \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ \neg\,\neg\,( x = y \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,\neg\,( \neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 ) \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) ) \ \Rightarrow \ \neg\,\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 ) \ \Rightarrow \ x\ne y\qquad\mbox{thf}(\mbox{satz}_9,\mbox{conjecture})$

\medskip\textbf{NUM643$\wedge$1.p} Landau theorem 5

pl (pl x y) z = pl x (pl y z)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4b},\mbox{axiom})$

$( \mbox{pl}@ ( \mbox{pl}@ x@ y )@ z ) = ( \mbox{pl}@ x@ ( \mbox{pl}@ y@ z ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{conjecture})$

\medskip\textbf{NUM248-1.p} Transfinite recursion property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM044-1.p} Irreflexive class property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{irreflexive}(x,y)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{restrict}(\mbox{intersection}(x,\mbox{identity\_relation}),y,y)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM387+1.p} Ordinal numbers, theorem 14






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{succ}(a) = \mbox{set\_union}_2(a,\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{in}(a,\mbox{succ}(a))\qquad\mbox{fof}(\mbox{t10\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;a\ne \mbox{succ}(a)\qquad\mbox{fof}(\mbox{t14\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM824$\wedge$5.p} TPS problem from IND-THMS






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

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

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

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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xu} ) ) )\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xq}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xv} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xx} ) ) ) )\mbox{ and }\mbox{cN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cN}@ ( \mbox{cS}@ \mbox{xn} ) ) ) ) \ \Rightarrow \ \forall \mbox{xm}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xm}\mbox{ or }\mbox{cODD}@ \mbox{xm} ) \ \Rightarrow \ ( \mbox{cN}@ \mbox{xm} ) )\qquad\mbox{thf}(\mbox{cTHM623\_pme},\mbox{conjecture})$

\medskip\textbf{NUM210-1.p} Lemma 1 for least upper bound property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{null\_class}\in x\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_for\_least\_upper\_bound\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{element\_relation},\mbox{intersection}(\mbox{intersection}(\mbox{power\_class}(x),x)',\mbox{ordinal\_numbers})) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_for\_least\_upper\_bound\_property8}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},\mbox{intersection}(\mbox{power\_class}(x),x))\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_for\_least\_upper\_bound\_property8}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM136-1.p} Transitivity of ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_ordinals}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_ordinals}_2,\mbox{negated\_conjecture})$

$z\in y\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_ordinals}_3,\mbox{negated\_conjecture})$

$\neg\,z\in x\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_ordinals}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM320+1.p} 2 + -5 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists y{:}\;n_2 + \mbox{nn}_5{=}y\qquad\mbox{fof}(\mbox{sum\_n2\_nn5\_what},\mbox{conjecture})$

\medskip\textbf{NUM104-1.p} Ordinal property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property5}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property5}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property5}_3,\mbox{negated\_conjecture})$

$\neg\,y\in x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property5}_4,\mbox{negated\_conjecture})$

$y\ne x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property5}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM227-1.p} Rest relation property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{sum\_class}(\mbox{image}(\mbox{rest\_relation},x))\ne \mbox{rest\_of}(\mbox{sum\_class}(x))\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM319+1.p} -2 + -5 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{nn}_2 + \mbox{nn}_5{=}x\qquad\mbox{fof}(\mbox{sum\_nn2\_nn5\_what},\mbox{conjecture})$

\medskip\textbf{NUM276-1.p} Lemma 6 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(\mbox{ordinals\_with\_null\_class\_as\_identity}),\mbox{limit\_ordinals}),\mbox{ordinals\_with\_null\_class\_as\_identity})\qquad\mbox{cnf}(\mbox{prove\_lemma\_6\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM639$\wedge$1.p} Landau theorem 4e

suc x = pl x n\_1




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

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

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$( \mbox{suc}@ x ) = ( \mbox{pl}@ x@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{conjecture})$

\medskip\textbf{NUM878=1.p} Product idempotent element






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,x) = x\qquad\mbox{tff}(\mbox{product\_idempotent\_element},\mbox{conjecture})$

\medskip\textbf{NUM743$\wedge$1.p} Landau theorem 51b

lessf x z




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{lessf}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{lessf}@ y@ z \ \Rightarrow \ ( \mbox{eq}@ y@ z )\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{45},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{lessf}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{50},\mbox{axiom})$

$\mbox{lessf}@ x@ z\qquad\mbox{thf}(\mbox{satz51b},\mbox{conjecture})$

\medskip\textbf{NUM159-1.p} Membership condition 4 for kind 1 ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_kind\_1\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM535+1.p} Ramsey's Infinite Theorem 04\_01, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{617},\mbox{hypothesis})$

$\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_617}_{02},\mbox{hypothesis})$

$\mbox{aSubsetOf}_0(\mbox{xS},\mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\mbox{ and }\mbox{aSubsetOf}_0(\mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}),\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM684$\wedge$1.p} Landau theorem 20e

x = y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$( \mbox{pl}@ z@ x ) = ( \mbox{pl}@ z@ y )\qquad\mbox{thf}(i,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{satz20b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$x = y\qquad\mbox{thf}(\mbox{satz20e},\mbox{conjecture})$

\medskip\textbf{NUM118-1.p} Ordinal property 14






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property14}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property14}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property14}_3,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property14}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM367+1.p} ?XY (X + Y != Y + X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z_1,z_2{:}\;( ( x + y{=}z_1\mbox{ and }x\setminus y{=}z_2 ) \ \Rightarrow \ \mbox{less}(z_2,z_1) )\qquad\mbox{fof}(\mbox{x\_plus\_y\_greater\_x\_minus\_y},\mbox{conjecture})$

\medskip\textbf{NUM709$\wedge$1.p} Landau theorem 28h

pl (ts x y) y = ts (suc x) y




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xy} ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} )\qquad\mbox{thf}(\mbox{satz28d},\mbox{axiom})$

$( \mbox{pl}@ ( \mbox{ts}@ x@ y )@ y ) = ( \mbox{ts}@ ( \mbox{suc}@ x )@ y )\qquad\mbox{thf}(\mbox{satz28h},\mbox{conjecture})$

\medskip\textbf{NUM395+1.p} Ordinal numbers, theorem 27






$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\mbox{ordinal}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{t27\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{ordinal}(a)\iff ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{d4\_ordinal}_1,\mbox{axiom})$

$\mbox{epsilon\_transitive}(\mbox{empty\_set})\mbox{ and }\mbox{epsilon\_connected}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{l18\_ordinal}_1,\mbox{axiom})$

\medskip\textbf{NUM739$\wedge$1.p} Landau theorem 46

~(moref z u) $\to$ eq z u




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{moref}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}@ x@ z\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{eq}@ y@ u\qquad\mbox{thf}(f,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\neg\,\mbox{moref}@ z@ u \ \Rightarrow \ ( \mbox{eq}@ z@ u )\qquad\mbox{thf}(\mbox{satz}_{46},\mbox{conjecture})$

\medskip\textbf{NUM911=1.p} Sum something and - something is 0.0






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$sum}(x,\mbox{\$uminus}(x)) = \mbox{0.}_0\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_8,\mbox{conjecture})$

\medskip\textbf{NUM087-1.p} Transitive class property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{sum\_class}(y),y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{intersection}(x,y)),\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_transitive\_class\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM896=1.p} Sum implies both less






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z \ \Rightarrow \ ( \mbox{\$less}(x,z)\mbox{ and }\mbox{\$less}(y,z) ) )\qquad\mbox{tff}(\mbox{anti\_sum\_larger},\mbox{conjecture})$

\medskip\textbf{NUM801$\wedge$1.p} Something times four equals five plus something






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i},m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( \cdot @ n@ \mbox{four} ) = (  + @ \mbox{five}@ m )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM332+1.p} (2 + 3) + 6 = 2 + (3 + 6)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall z_1,z_2,z_3,z_4{:}\;( ( n_2 + n_3{=}z_1\mbox{ and }z_1 + n_6{=}z_2\mbox{ and }n_3 + n_6{=}z_3\mbox{ and }n_2 + z_3{=}z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{fof}(\mbox{associative\_sum},\mbox{conjecture})$

\medskip\textbf{NUM099-1.p} Corollary to ordinal property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{singleton}(\mbox{null\_class}),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_ordinal\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM303+1.p} 31 != 21






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_{31}\ne n_{21}\qquad\mbox{fof}(\mbox{n31\_not\_n}_{12},\mbox{conjecture})$

\medskip\textbf{NUM205-1.p} Corollary 1 to least upper bound property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_least\_upper\_bound\_property5}_1,\mbox{negated\_conjecture})$

$1_{\text{dom}(\mbox{successor\_relation})}\ne \mbox{sum\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_least\_upper\_bound\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM300+1.p} ? $<$ 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{less}(x,n_0)\qquad\mbox{fof}(\mbox{something\_less\_n}_0,\mbox{conjecture})$

\medskip\textbf{NUM917=1.p} Every difference right exists






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\exists w{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(u,w) = v\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM686$\wedge$1.p} Landau theorem 21

some (lambda u\_0.diffprop (pl x z) (pl y u) u\_0)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ x@ y@ \mbox{xu} )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ z@ u@ \mbox{xu}_0 )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xy}@ \mbox{xx}@ \mbox{xv} ) ) \ \Rightarrow \ ( ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xy}@ \mbox{xv} ) ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xx}@ \mbox{xv} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{15},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu} ) ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} )@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xu}_0{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ u )@ \mbox{xu}_0 )\qquad\mbox{thf}(\mbox{satz}_{21},\mbox{conjecture})$

\medskip\textbf{NUM417$\wedge$1.p} (10 * 10) * (10 * 10) = ((10 * 10) * 10) * 10






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$( \cdot @ ( \cdot @ \mbox{ten}@ \mbox{ten} )@ ( \cdot @ \mbox{ten}@ \mbox{ten} ) ) = ( \cdot @ ( \cdot @ ( \cdot @ \mbox{ten}@ \mbox{ten} )@ \mbox{ten} )@ \mbox{ten} )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM309+1.p} 23 + ? = 34






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_{23} + x{=}n_{34}\qquad\mbox{fof}(\mbox{sum\_n23\_something\_n}_{34},\mbox{conjecture})$

\medskip\textbf{NUM366+1.p} 2 + 3 $>$ 7






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_2 + n_3{=}x \ \Rightarrow \ \mbox{less}(n_7,x) )\qquad\mbox{fof}(\mbox{anti\_sum\_n2\_n3\_greater\_n}_7,\mbox{conjecture})$

\medskip\textbf{NUM895=1.p} Sum and difference






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z \ \Rightarrow \ \mbox{\$difference}(z,x) = y )\qquad\mbox{tff}(\mbox{sum\_difference},\mbox{conjecture})$

\medskip\textbf{NUM128-1.p} Alternate 1 for transfinite induction, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{least}(\mbox{element\_relation},\mbox{intersection}(z,\mbox{ordinal\_numbers}))\in z'\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction3}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(z,\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_1\_transfinite\_induction3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM021$\wedge$1.p} Find operator o such that 2 o 3 = 5 and 1 o 2 = 3






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists \mbox{op}{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i} )\to ( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{op}@ \mbox{two}@ \mbox{three} ) = \mbox{five}\mbox{ and }( \mbox{op}@ 1@ \mbox{two} ) = \mbox{three} )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM769$\wedge$1.p} Landau theorem 67d

eq x (pf y (mf x y))




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

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

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

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{mf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{mf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xy}@ ( \mbox{mf}@ \mbox{xx}@ \mbox{xy} ) )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz67c},\mbox{axiom})$

$\mbox{eq}@ x@ ( \mbox{pf}@ y@ ( \mbox{mf}@ x@ y ) )\qquad\mbox{thf}(\mbox{satz67d},\mbox{conjecture})$

\medskip\textbf{NUM317+1.p} 5 + -2 = 3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_5 + \mbox{nn}_2{=}n_3\qquad\mbox{fof}(\mbox{sum\_n5\_nn2\_n}_3,\mbox{conjecture})$

\medskip\textbf{NUM797$\wedge$1.p} Landau theorem 4






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne 1\qquad\mbox{thf}(\mbox{one\_is\_first},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{succ}@ x ) = ( \mbox{succ}@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{succ\_injective},\mbox{axiom})$

$\forall m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( m@ 1\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( m@ y ) )\qquad\mbox{thf}(\mbox{induction},\mbox{axiom})$

$\exists p{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;( \forall x{:}\;\mbox{\$i}{:}\;( p@ x@ 1 ) = ( \mbox{succ}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( p@ x@ ( \mbox{succ}@ y ) ) = ( \mbox{succ}@ ( p@ x@ y ) ) )\qquad\mbox{thf}(\mbox{satz}_4,\mbox{conjecture})$

\medskip\textbf{NUM354+1.p} ? != 0 such that ? + ? = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;( x\ne n_0\mbox{ and }x + x{=}n_0 )\qquad\mbox{fof}(\mbox{what\_what\_n}_0,\mbox{conjecture})$

\medskip\textbf{NUM812$\wedge$5.p} TPS problem THM578

Variant of THM6104, including induction in the hypothesis.




$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$\mbox{cIND} \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xn} = c_0\mbox{ or }\exists \mbox{xm}{:}\;\mbox{\$i}{:}\;\mbox{xn} = ( \mbox{cS}@ \mbox{xm} ) )\qquad\mbox{thf}(\mbox{cTHM}_{578},\mbox{conjecture})$

\medskip\textbf{NUM247-1.p} Transfinite recursion property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{recursion\_equation\_functions}(z)),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM045-1.p} Irreflexive class property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{restrict}(\mbox{intersection}(x,\mbox{identity\_relation}),y,y) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{irreflexive}(x,y)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM190-1.p} Omega is not the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{omega} = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_omega\_is\_not\_null}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM912=1.p} X is - X only for 0.0






$\forall x{:}\;\mbox{\$real}{:}\;( x = \mbox{\$uminus}(x)\iff x = \mbox{0.}_0 )\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_9,\mbox{conjecture})$

\medskip\textbf{NUM860+1.p} Upper bound replace maximum embedded in a context (1)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\forall x{:}\;\mbox{lesseq}(x,x)\qquad\mbox{fof}(\mbox{lesseq\_ref},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,z) ) \ \Rightarrow \ \mbox{lesseq}(x,z) )\qquad\mbox{fof}(\mbox{lesseq\_trans},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,x) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{lesseq\_antisymmetric},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\mbox{ or }\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{lesseq\_total},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(z + x,z + y) )\qquad\mbox{fof}(\mbox{sum\_monotone}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{fof}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{max}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{lesseq}(x,y) )\qquad\mbox{fof}(\mbox{max}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{lesseq}(x,z)\mbox{ and }\mbox{lesseq}(y,z) ) )\qquad\mbox{fof}(\mbox{ub},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{lesseq}(\mbox{summation}(\mbox{max}(x,y)),n) )\qquad\mbox{fof}(\mbox{model\_max}_3,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{lesseq}(\mbox{summation}(z),n) ) )\qquad\mbox{fof}(\mbox{model\_ub}_3,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{fof}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM642$\wedge$1.p} Landau theorem 4h

suc (pl x y) = pl (suc x) y




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

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

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

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xy} ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4d},\mbox{axiom})$

$( \mbox{suc}@ ( \mbox{pl}@ x@ y ) ) = ( \mbox{pl}@ ( \mbox{suc}@ x )@ y )\qquad\mbox{thf}(\mbox{satz4h},\mbox{conjecture})$

\medskip\textbf{NUM142-1.p} The successor of a set is a set, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(x,\mbox{successor}(x))\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_set\_is\_set3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM802$\wedge$5.p} TPS problem BLEDSOE-FENG-8

There is a set containing no nonnegative numbers and containing 
 -2.




$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(c_2,\mbox{type})$

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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$\mbox{c\_less\_}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_less\_},\mbox{type})$

$( \mbox{c\_less\_}@ c_2@ c_0 ) \ \Rightarrow \ ( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{c\_less\_}@ \mbox{xu}@ c_0 ) \ \Rightarrow \ \mbox{xu}\ne ( \mbox{absval}@ \mbox{xv} ) ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;\neg\,a@ ( \mbox{absval}@ \mbox{xy} )\mbox{ and }a@ c_2 ) )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG}_8,\mbox{conjecture})$

\medskip\textbf{NUM001-1.p} (A + B) + C = A + (B + C)






$\mbox{include}(\mbox{'Axioms/NUM002-0.ax'})$

$\neg\,( a + b ) + c{=}a + ( b + c )\qquad\mbox{cnf}(\mbox{prove\_equation},\mbox{negated\_conjecture})$

\medskip\textbf{NUM648$\wedge$1.p} Landau theorem 8b

(forall x\_0:nat.forall y\_0:nat.x = pl y x\_0 $\to$ x = pl y y\_0 $\to$
 x\_0 = y\_0)




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ \mbox{xy} = \mbox{xz} )\qquad\mbox{thf}(\mbox{satz8a},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{nat},\mbox{xy}_0{:}\;\mbox{nat}{:}\;( x = ( \mbox{pl}@ y@ \mbox{xx}_0 ) \ \Rightarrow \ ( x = ( \mbox{pl}@ y@ \mbox{xy}_0 ) \ \Rightarrow \ \mbox{xx}_0 = \mbox{xy}_0 ) )\qquad\mbox{thf}(\mbox{satz8b},\mbox{conjecture})$

\medskip\textbf{NUM245-1.p} Transfinite recursion property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property1}_1,\mbox{negated\_conjecture})$

$u\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(z,\mbox{restrict}(x,u,\mbox{universal\_class}))\ne \mbox{apply}(x,u)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM191-1.p} Omega is a limit ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{omega}\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_omega\_is\_a\_limit\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM262-1.p} Alternate 4 for transfinite induction, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{apply}(\mbox{recursion}(x,y,z),\mbox{null\_class})\ne x\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM655$\wedge$1.p} Landau theorem 10f

~(more x y) $\to$ x = y




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\mbox{less}@ x@ y\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(\mbox{satz10f},\mbox{conjecture})$

\medskip\textbf{NUM660$\wedge$1.p} Landau theorem 13

~(more y x) $\to$ y = x




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

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

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{11},\mbox{axiom})$

$\neg\,\mbox{less}@ y@ x \ \Rightarrow \ y = x\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{conjecture})$

\medskip\textbf{NUM677$\wedge$1.p} Landau theorem 19h

more (pl z x) (pl u y)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$x = y\qquad\mbox{thf}(i,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ z@ u\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz19g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\mbox{more}@ ( \mbox{pl}@ z@ x )@ ( \mbox{pl}@ u@ y )\qquad\mbox{thf}(\mbox{satz19h},\mbox{conjecture})$

\medskip\textbf{NUM279-1.p} Ordinal addition property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_add}(x,\mbox{successor}(\mbox{null\_class}))\ne \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_adition\_property8}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM759$\wedge$1.p} Landau theorem 63e

eq x y




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{eq}@ ( \mbox{pf}@ z@ x )@ ( \mbox{pf}@ z@ y )\qquad\mbox{thf}(e,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz63b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{axiom})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(\mbox{satz63e},\mbox{conjecture})$

\medskip\textbf{NUM419$\wedge$1.p} Find N such that N + 3 = 4






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;(  + @ n@ \mbox{three} ) = \mbox{four}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM662$\wedge$1.p} Landau theorem 15

~(forall x\_0:nat.~(z = pl x x\_0))




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;y\ne ( \mbox{pl}@ x@ \mbox{xx}_0 )\qquad\mbox{thf}(l,\mbox{axiom})$

$\neg\,\forall \mbox{xx}{:}\;\mbox{nat}{:}\;z\ne ( \mbox{pl}@ y@ \mbox{xx} )\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;z\ne ( \mbox{pl}@ x@ \mbox{xx}_0 )\qquad\mbox{thf}(\mbox{satz}_{15},\mbox{conjecture})$

\medskip\textbf{NUM208-1.p} Least upper bound property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property7}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM852+2.p} holds(conseq\_conjunct1(conseq\_conjunct2(conseq(304))),484,0)






$\mbox{greater}(\mbox{vmul}(\mbox{vd}_{481},\mbox{vd}_{469}),\mbox{vmul}(\mbox{vd}_{480},\mbox{vd}_{469}))\qquad\mbox{fof}(\mbox{'holds(conseq\_conjunct1(conseq\_conjunct2(conseq(304))), 484, 0)'},\mbox{conjecture})$

$\mbox{greater}(\mbox{vd}_{481},\mbox{vd}_{480})\qquad\mbox{fof}(\mbox{'holds(conseq\_conjunct1(conseq(304)), 483, 0)'},\mbox{axiom})$

$\mbox{less}(\mbox{vd}_{480},\mbox{vd}_{481})\qquad\mbox{fof}(\mbox{'holds(antec(304), 482, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{476},\mbox{vd}_{477}{:}\;( \mbox{vd}_{476} = \mbox{vd}_{477} \ \Rightarrow \ \mbox{vmul}(\mbox{vd}_{476},\mbox{vd}_{469}) = \mbox{vmul}(\mbox{vd}_{477},\mbox{vd}_{469}) )\qquad\mbox{fof}(\mbox{'ass(cond(303, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{470},\mbox{vd}_{471}{:}\;( \mbox{greater}(\mbox{vd}_{470},\mbox{vd}_{471}) \ \Rightarrow \ \mbox{greater}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{471},\mbox{vd}_{469}),\mbox{vmul}(\mbox{vskolem}_9(\mbox{vd}_{470},\mbox{vd}_{471}),\mbox{vd}_{469})),\mbox{vmul}(\mbox{vd}_{471},\mbox{vd}_{469})) )\qquad\mbox{fof}(\mbox{'ass(cond(302, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{470},\mbox{vd}_{471}{:}\;( \mbox{greater}(\mbox{vd}_{470},\mbox{vd}_{471}) \ \Rightarrow \ \mbox{vmul}(\mbox{vplus}(\mbox{vd}_{471},\mbox{vskolem}_9(\mbox{vd}_{470},\mbox{vd}_{471})),\mbox{vd}_{469}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{471},\mbox{vd}_{469}),\mbox{vmul}(\mbox{vskolem}_9(\mbox{vd}_{470},\mbox{vd}_{471}),\mbox{vd}_{469})) )\qquad\mbox{fof}(\mbox{'ass(cond(302, 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{470},\mbox{vd}_{471}{:}\;( \mbox{greater}(\mbox{vd}_{470},\mbox{vd}_{471}) \ \Rightarrow \ \mbox{vmul}(\mbox{vd}_{470},\mbox{vd}_{469}) = \mbox{vmul}(\mbox{vplus}(\mbox{vd}_{471},\mbox{vskolem}_9(\mbox{vd}_{470},\mbox{vd}_{471})),\mbox{vd}_{469}) )\qquad\mbox{fof}(\mbox{'ass(cond(302, 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{470},\mbox{vd}_{471}{:}\;( \mbox{greater}(\mbox{vd}_{470},\mbox{vd}_{471}) \ \Rightarrow \ \mbox{vd}_{470} = \mbox{vplus}(\mbox{vd}_{471},\mbox{vskolem}_9(\mbox{vd}_{470},\mbox{vd}_{471})) )\qquad\mbox{fof}(\mbox{'ass(cond(302, 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{444},\mbox{vd}_{445},\mbox{vd}_{446}{:}\;\mbox{vmul}(\mbox{vmul}(\mbox{vd}_{444},\mbox{vd}_{445}),\mbox{vd}_{446}) = \mbox{vmul}(\mbox{vd}_{444},\mbox{vmul}(\mbox{vd}_{445},\mbox{vd}_{446}))\qquad\mbox{fof}(\mbox{'ass(cond(290, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{432},\mbox{vd}_{433},\mbox{vd}_{434}{:}\;\mbox{vmul}(\mbox{vd}_{432},\mbox{vplus}(\mbox{vd}_{433},\mbox{vd}_{434})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{432},\mbox{vd}_{433}),\mbox{vmul}(\mbox{vd}_{432},\mbox{vd}_{434}))\qquad\mbox{fof}(\mbox{'ass(cond(281, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{418},\mbox{vd}_{419}{:}\;\mbox{vmul}(\mbox{vd}_{418},\mbox{vd}_{419}) = \mbox{vmul}(\mbox{vd}_{419},\mbox{vd}_{418})\qquad\mbox{fof}(\mbox{'ass(cond(270, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{408},\mbox{vd}_{409}{:}\;\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{408}),\mbox{vd}_{409}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{408},\mbox{vd}_{409}),\mbox{vd}_{409})\qquad\mbox{fof}(\mbox{'ass(cond(261, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{226},\mbox{vd}_{227}{:}\;( \mbox{less}(\mbox{vd}_{226},\mbox{vd}_{227}) \ \Rightarrow \ \mbox{greater}(\mbox{vd}_{227},\mbox{vd}_{226}) )\qquad\mbox{fof}(\mbox{'ass(cond(147, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{208},\mbox{vd}_{209}{:}\;( \mbox{greater}(\mbox{vd}_{208},\mbox{vd}_{209}) \ \Rightarrow \ \mbox{less}(\mbox{vd}_{209},\mbox{vd}_{208}) )\qquad\mbox{fof}(\mbox{'ass(cond(140, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203} = \mbox{vd}_{204}\mbox{ or }\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{198},\mbox{vd}_{199}{:}\;( \mbox{less}(\mbox{vd}_{199},\mbox{vd}_{198})\iff \exists \mbox{vd}_{201}{:}\;\mbox{vd}_{198} = \mbox{vplus}(\mbox{vd}_{199},\mbox{vd}_{201}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 12), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{193},\mbox{vd}_{194}{:}\;( \mbox{greater}(\mbox{vd}_{194},\mbox{vd}_{193})\iff \exists \mbox{vd}_{196}{:}\;\mbox{vd}_{194} = \mbox{vplus}(\mbox{vd}_{193},\mbox{vd}_{196}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 11), 1)'},\mbox{axiom})$

\medskip\textbf{NUM862+1.p} Upper bound replace maximum embedded in a context (1)+(2)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\forall x{:}\;\mbox{lesseq}(x,x)\qquad\mbox{fof}(\mbox{lesseq\_ref},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,z) ) \ \Rightarrow \ \mbox{lesseq}(x,z) )\qquad\mbox{fof}(\mbox{lesseq\_trans},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,x) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{lesseq\_antisymmetric},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\mbox{ or }\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{lesseq\_total},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(z + x,z + y) )\qquad\mbox{fof}(\mbox{sum\_monotone}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{fof}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{max}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{lesseq}(x,y) )\qquad\mbox{fof}(\mbox{max}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{lesseq}(x,z)\mbox{ and }\mbox{lesseq}(y,z) ) )\qquad\mbox{fof}(\mbox{ub},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{lesseq}(c + \mbox{summation}(\mbox{max}(x,y)),n) )\qquad\mbox{fof}(\mbox{model\_max}_5,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{lesseq}(c + \mbox{summation}(z),n) ) )\qquad\mbox{fof}(\mbox{model\_ub}_5,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{fof}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM283-1.005.p} Calculation of factorial

Compute 5 factorial.




$x + n_0{=}x\qquad\mbox{cnf}(\mbox{add}_0,\mbox{axiom})$

$x + y{=}z \ \Rightarrow \ x + s(y){=}s(z)\qquad\mbox{cnf}(\mbox{add},\mbox{axiom})$

$s(n_0)\cdot x{=}x\qquad\mbox{cnf}(\mbox{times}_1,\mbox{axiom})$

$( r + y{=}z\mbox{ and }x\cdot y{=}r ) \ \Rightarrow \ s(x)\cdot y{=}z\qquad\mbox{cnf}(\mbox{times},\mbox{axiom})$

$\mbox{factorial}(n_0,s(n_0))\qquad\mbox{cnf}(\mbox{factorial}_0,\mbox{axiom})$

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

$\neg\,\mbox{factorial}(s(s(s(s(s(n_0))))),\mbox{result})\qquad\mbox{cnf}(\mbox{prove\_factorial},\mbox{negated\_conjecture})$

\medskip\textbf{NUM825$\wedge$5.p} TPS problem from IND-THMS






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

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

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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xu} ) ) )\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xq}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xv} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ \exists \mbox{xm}{:}\;\mbox{\$i}{:}\;( \mbox{cEVEN}@ \mbox{xm}\mbox{ and }( \mbox{cS}@ \mbox{xm} ) = \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cTHM624\_pme},\mbox{conjecture})$

\medskip\textbf{NUM374+2.p} Disprove Wilkie identity from Tarski's identities






$\forall x,y{:}\;x + y = y + x\qquad\mbox{fof}(\mbox{sum\_symmetry},\mbox{axiom})$

$\forall x,y,z{:}\;x + ( y + z ) = ( x + y ) + z\qquad\mbox{fof}(\mbox{sum\_associativity},\mbox{axiom})$

$\forall x{:}\;x\cdot n_1 = x\qquad\mbox{fof}(\mbox{product\_identity},\mbox{axiom})$

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

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

$\forall x,y,z{:}\;x\cdot ( y + z ) = x\cdot y + x\cdot z\qquad\mbox{fof}(\mbox{sum\_product\_distribution},\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(n_1,x) = n_1\qquad\mbox{fof}(\mbox{exponent\_n}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(x,n_1) = x\qquad\mbox{fof}(\mbox{exponent\_identity},\mbox{axiom})$

$\forall x,y,z{:}\;\mbox{exponent}(x,y + z) = \mbox{exponent}(x,y)\cdot \mbox{exponent}(x,z)\qquad\mbox{fof}(\mbox{exponent\_sum\_product},\mbox{axiom})$

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

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

$\forall c,p,q,r,s,b{:}\;( \mbox{lemmas}(c,p,q,r,s,b)\iff ( n_2 = n_1 + n_1\mbox{ and }b\ne n_0\mbox{ and }b\ne n_1\mbox{ and }b\ne n_2\mbox{ and }\forall x{:}\;b\ne n_0\cdot x\mbox{ and }\forall x{:}\;p\ne q\cdot x\mbox{ and }\forall x{:}\;q\ne p\cdot x\mbox{ and }\forall x{:}\;r\ne s\cdot x\mbox{ and }\forall x{:}\;s\ne r\cdot x\mbox{ and }n_1 + n_0\ne n_1\mbox{ and }n_2 + n_0\ne n_1\mbox{ and }n_0 + n_0\ne n_1\mbox{ and }c\ne n_1\mbox{ and }n_1 + c\ne n_1\mbox{ and }c\cdot n_0\ne n_1\mbox{ and }n_1 + n_0\ne n_0\mbox{ and }n_2 + n_0\ne n_0\mbox{ and }n_0 + n_0\ne n_0\mbox{ and }c\ne n_0\mbox{ and }n_1 + c\ne n_0\mbox{ and }n_2 + n_0\ne n_1 + n_0\mbox{ and }c\ne n_1 + n_0\mbox{ and }c\cdot n_0\ne n_1 + n_0\mbox{ and }c\ne n_2 + n_0\mbox{ and }c\ne n_0 + n_0\mbox{ and }n_1 + c\ne c ) )\qquad\mbox{fof}(\mbox{lemmas},\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{fof}(\mbox{n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{fof}(\mbox{n0\_n}_2,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{fof}(\mbox{n1\_n}_2,\mbox{axiom})$

$\forall c,p,q,r,s,a,b{:}\;( ( c = a\cdot a\mbox{ and }p = n_1 + a\mbox{ and }q = p + c\mbox{ and }r = n_1 + a\cdot c\mbox{ and }s = ( n_1 + c ) + c\cdot c\mbox{ and }\mbox{lemmas}(c,p,q,r,s,b) ) \ \Rightarrow \ \mbox{exponent}(\mbox{exponent}(p,a) + \mbox{exponent}(q,a),b)\cdot \mbox{exponent}(\mbox{exponent}(r,b) + \mbox{exponent}(s,b),a) = \mbox{exponent}(\mbox{exponent}(p,b) + \mbox{exponent}(q,b),a)\cdot \mbox{exponent}(\mbox{exponent}(r,a) + \mbox{exponent}(s,a),b) )\qquad\mbox{fof}(\mbox{wilkie},\mbox{conjecture})$

\medskip\textbf{NUM216-1.p} Corollary 1 to rest definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$u\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_rest\_defn}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{rest\_of}(x),u)\ne \mbox{restrict}(x,u,\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_rest\_defn}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM788$\wedge$1.p} Landau theorem 81g

~(~(less x0 y0) $\to$ is x0 y0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ x_0@ y_0\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;\neg\,( ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0\qquad\mbox{thf}(\mbox{satz81b},\mbox{axiom})$

$\neg\,\neg\,\mbox{less}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(\mbox{satz81g},\mbox{conjecture})$

\medskip\textbf{NUM212-1.p} Lemma 3 for least upper bound property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),\mbox{limit\_ordinals}),x)\qquad\mbox{cnf}(\mbox{prove\_lemma\_3\_for\_least\_upper\_bound\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{element\_relation},\mbox{intersection}(\mbox{intersection}(\mbox{power\_class}(x),x)',\mbox{ordinal\_numbers}))\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_lemma\_3\_for\_least\_upper\_bound\_property8}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},\mbox{intersection}(\mbox{power\_class}(x),x))\qquad\mbox{cnf}(\mbox{prove\_lemma\_3\_for\_least\_upper\_bound\_property8}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM847+2.p} holds(286,441,3)






$\mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439})),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}),\mbox{vd}_{436}))\qquad\mbox{fof}(\mbox{'holds(286, 441, 3)'},\mbox{conjecture})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 2)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vsucc}(\mbox{vd}_{439}))) = \mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})))\qquad\mbox{fof}(\mbox{'holds(286, 441, 0)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}))\qquad\mbox{fof}(\mbox{'holds(285, 440, 0)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},v_1))\qquad\mbox{fof}(\mbox{'holds(284, 438, 2)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{437})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(284, 438, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},v_1)) = \mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{437}))\qquad\mbox{fof}(\mbox{'holds(284, 438, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{400}{:}\;\mbox{vmul}(v_1,\mbox{vd}_{400}) = \mbox{vd}_{400}\qquad\mbox{fof}(\mbox{'ass(cond(253, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{386},\mbox{vd}_{387}{:}\;( \mbox{less}(\mbox{vd}_{386},\mbox{vplus}(\mbox{vd}_{387},v_1)) \ \Rightarrow \ \mbox{leq}(\mbox{vd}_{386},\mbox{vd}_{387}) )\qquad\mbox{fof}(\mbox{'ass(cond(241, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{375},\mbox{vd}_{376}{:}\;( \mbox{greater}(\mbox{vd}_{375},\mbox{vd}_{376}) \ \Rightarrow \ \mbox{geq}(\mbox{vd}_{375},\mbox{vplus}(\mbox{vd}_{376},v_1)) )\qquad\mbox{fof}(\mbox{'ass(cond(234, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{16}{:}\;\mbox{vsucc}(\mbox{vd}_{16})\ne \mbox{vd}_{16}\qquad\mbox{fof}(\mbox{'ass(cond(12, 0), 0)'},\mbox{axiom})$

\medskip\textbf{NUM742$\wedge$1.p} Landau theorem 51a

lessf x z




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{lessf}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessf}@ y@ z\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{45},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{38},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{lessf}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{50},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\mbox{lessf}@ x@ z\qquad\mbox{thf}(\mbox{satz51a},\mbox{conjecture})$

\medskip\textbf{NUM672$\wedge$1.p} Landau theorem 19c

less (pl x z) (pl y z)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{11},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{12},\mbox{axiom})$

$\mbox{less}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ z )\qquad\mbox{thf}(\mbox{satz19c},\mbox{conjecture})$

\medskip\textbf{NUM533+2.p} Ramsey's Infinite Theorem 03, 01 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\mbox{aSet}_0(\mbox{xA})\mbox{ and }\mbox{aSet}_0(\mbox{xB})\mbox{ and }\mbox{aSet}_0(\mbox{xC})\qquad\mbox{fof}(\mbox{m\_}_{522},\mbox{hypothesis})$

$( \forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{xA}) \ \Rightarrow \ \mbox{aElementOf}_0(w_0,\mbox{xB}) )\mbox{ and }\mbox{aSubsetOf}_0(\mbox{xA},\mbox{xB})\mbox{ and }\forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{xB}) \ \Rightarrow \ \mbox{aElementOf}_0(w_0,\mbox{xC}) )\mbox{ and }\mbox{aSubsetOf}_0(\mbox{xB},\mbox{xC}) ) \ \Rightarrow \ ( \forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{xA}) \ \Rightarrow \ \mbox{aElementOf}_0(w_0,\mbox{xC}) )\mbox{ or }\mbox{aSubsetOf}_0(\mbox{xA},\mbox{xC}) )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM192-1.p} Omega is the smallest limit ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_omega\_is\_the\_smallest\_limit\_ordinal}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{omega}\in \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_omega\_is\_the\_smallest\_limit\_ordinal}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM532+1.p} Ramsey's Infinite Theorem 02, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\mbox{aSet}_0(\mbox{xA})\qquad\mbox{fof}(\mbox{m\_}_{467},\mbox{hypothesis})$

$\mbox{aSubsetOf}_0(\mbox{xA},\mbox{xA})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM691$\wedge$1.p} Landau theorem 23

~(more (pl x z) (pl y u)) $\to$ pl x z = pl y u




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\neg\,\mbox{more}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(m,\mbox{axiom})$

$\neg\,\mbox{more}@ z@ u \ \Rightarrow \ z = u\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz22a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \neg\,\mbox{more}@ \mbox{xz}@ \mbox{xu} \ \Rightarrow \ \mbox{xz} = \mbox{xu} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz22b},\mbox{axiom})$

$\neg\,\mbox{more}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ u ) \ \Rightarrow \ ( \mbox{pl}@ x@ z ) = ( \mbox{pl}@ y@ u )\qquad\mbox{thf}(\mbox{satz}_{23},\mbox{conjecture})$

\medskip\textbf{NUM230-1.p} Transfinite recursion lemma 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma1}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma1}_3,\mbox{negated\_conjecture})$

$u\in \mbox{least}(\mbox{element\_relation},\mbox{domain\_of}(\mbox{intersection}(y',x)))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma1}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,v)\in y\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma1}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM046-1.p} Connected class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{connected}(x,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,v)\in x\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property1}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(v,u)\in x\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property1}_4,\mbox{negated\_conjecture})$

$u\ne v\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property1}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM200-1.p} If every element of x is $\leq$ y, then sum class(x) $\leq$ y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{successor}(y))\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property2}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{sum\_class}(x)\in \mbox{successor}(y)\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM637$\wedge$2.p} Landau theorem 3






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one\_type},\mbox{type})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{succ}@ x )\ne 1\qquad\mbox{thf}(\mbox{one\_is\_first},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{succ}@ x ) = ( \mbox{succ}@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{succ\_injective},\mbox{axiom})$

$\forall m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( m@ 1\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( m@ x ) \ \Rightarrow \ ( m@ ( \mbox{succ}@ x ) ) ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( m@ y ) )\qquad\mbox{thf}(\mbox{induction},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( x\ne 1 \ \Rightarrow \ \exists u{:}\;\mbox{\$i}{:}\;x = ( \mbox{succ}@ u ) )\qquad\mbox{thf}(\mbox{satz}_3,\mbox{conjecture})$

\medskip\textbf{NUM680$\wedge$1.p} Landau theorem 20a

more x y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{more}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ z )\qquad\mbox{thf}(m,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz19b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(\mbox{satz20a},\mbox{conjecture})$

\medskip\textbf{NUM837+2.p} qe(171)






$\exists \mbox{vd}_{273}{:}\;\mbox{vd}_{269} = \mbox{vplus}(\mbox{vd}_{268},\mbox{vd}_{273})\qquad\mbox{fof}(\mbox{'qe(171)'},\mbox{conjecture})$

$\mbox{less}(\mbox{vd}_{269},\mbox{vd}_{271})\qquad\mbox{fof}(\mbox{'(conjunct2(170),272,0)'},\mbox{axiom})$

$\mbox{less}(\mbox{vd}_{268},\mbox{vd}_{269})\qquad\mbox{fof}(\mbox{'holds(conjunct1(170), 270, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{258},\mbox{vd}_{259}{:}\;( \mbox{leq}(\mbox{vd}_{258},\mbox{vd}_{259}) \ \Rightarrow \ \mbox{geq}(\mbox{vd}_{259},\mbox{vd}_{258}) )\qquad\mbox{fof}(\mbox{'ass(cond(163, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{254},\mbox{vd}_{255}{:}\;( \mbox{geq}(\mbox{vd}_{254},\mbox{vd}_{255}) \ \Rightarrow \ \mbox{leq}(\mbox{vd}_{255},\mbox{vd}_{254}) )\qquad\mbox{fof}(\mbox{'ass(cond(158, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{249},\mbox{vd}_{250}{:}\;( \mbox{leq}(\mbox{vd}_{250},\mbox{vd}_{249})\iff ( \mbox{less}(\mbox{vd}_{250},\mbox{vd}_{249})\mbox{ or }\mbox{vd}_{250} = \mbox{vd}_{249} ) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 17), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{244},\mbox{vd}_{245}{:}\;( \mbox{geq}(\mbox{vd}_{245},\mbox{vd}_{244})\iff ( \mbox{greater}(\mbox{vd}_{245},\mbox{vd}_{244})\mbox{ or }\mbox{vd}_{245} = \mbox{vd}_{244} ) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 16), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{226},\mbox{vd}_{227}{:}\;( \mbox{less}(\mbox{vd}_{226},\mbox{vd}_{227}) \ \Rightarrow \ \mbox{greater}(\mbox{vd}_{227},\mbox{vd}_{226}) )\qquad\mbox{fof}(\mbox{'ass(cond(147, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{208},\mbox{vd}_{209}{:}\;( \mbox{greater}(\mbox{vd}_{208},\mbox{vd}_{209}) \ \Rightarrow \ \mbox{less}(\mbox{vd}_{209},\mbox{vd}_{208}) )\qquad\mbox{fof}(\mbox{'ass(cond(140, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203} = \mbox{vd}_{204}\mbox{ or }\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203}\ne \mbox{vd}_{204}\mbox{ or }\neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{198},\mbox{vd}_{199}{:}\;( \mbox{less}(\mbox{vd}_{199},\mbox{vd}_{198})\iff \exists \mbox{vd}_{201}{:}\;\mbox{vd}_{198} = \mbox{vplus}(\mbox{vd}_{199},\mbox{vd}_{201}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 12), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{193},\mbox{vd}_{194}{:}\;( \mbox{greater}(\mbox{vd}_{194},\mbox{vd}_{193})\iff \exists \mbox{vd}_{196}{:}\;\mbox{vd}_{194} = \mbox{vplus}(\mbox{vd}_{193},\mbox{vd}_{196}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 11), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120} = \mbox{vd}_{121}\mbox{ or }\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123})\mbox{ or }\neg\,\exists \mbox{vd}_{125}{:}\;\mbox{vd}_{121} = \mbox{vplus}(\mbox{vd}_{120},\mbox{vd}_{125}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{104},\mbox{vd}_{105}{:}\;( \mbox{vd}_{104}\ne \mbox{vd}_{105} \ \Rightarrow \ \forall \mbox{vd}_{107}{:}\;\mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{104})\ne \mbox{vplus}(\mbox{vd}_{107},\mbox{vd}_{105}) )\qquad\mbox{fof}(\mbox{'ass(cond(81, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

\medskip\textbf{NUM905=1.p} X is - X only for 0






$\forall x{:}\;\mbox{\$rat}{:}\;( x = \mbox{\$uminus}(x)\iff x = 0/1 )\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_9,\mbox{conjecture})$

\medskip\textbf{NUM315+1.p} -2 + -5 = -7






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\mbox{nn}_2 + \mbox{nn}_5{=}\mbox{nn}_7\qquad\mbox{fof}(\mbox{sum\_nn2\_nn5\_nn}_7,\mbox{conjecture})$

\medskip\textbf{NUM385+1.p} Ordinal numbers, theorem 12






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{succ}(a) = \mbox{set\_union}_2(a,\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d1\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{in}(a,\mbox{succ}(a))\qquad\mbox{fof}(\mbox{t10\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{succ}(a) = \mbox{succ}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t12\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM831$\wedge$5.p} TPS problem from PETER-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} ) = ( \mbox{cS}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xn} )\ne c_0 ) \ \Rightarrow \ \exists \mbox{xd}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xd}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xd}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xd}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( \mbox{xd}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{xd}@ \mbox{xx}@ y ) \ \Rightarrow \ x = y ) ) )\qquad\mbox{thf}(\mbox{cTHM606\_pme},\mbox{conjecture})$

\medskip\textbf{NUM330+1.p} XY (X+Y = 8) \& X = 4 \& Y = 4






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;( x + y{=}n_8\mbox{ and }x = n_4\mbox{ and }y = n_4 )\qquad\mbox{fof}(\mbox{sum\_n4\_n4\_n}_8,\mbox{conjecture})$

\medskip\textbf{NUM803$\wedge$5.p} TPS problem from NATS






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

$c_0{:}\;n\qquad\mbox{thf}(c_0,\mbox{type})$

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

$\forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_star}@ \mbox{xx}@ c_0 ) = c_0\qquad\mbox{thf}(\mbox{cPA}_3,\mbox{conjecture})$

\medskip\textbf{NUM028-1.p} Symmetrization property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{symmetrization\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{symmetrization\_of}(x')\ne x'\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM299+1.p} -2 !$<$ -4






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\neg\,\mbox{less}(\mbox{nn}_2,\mbox{nn}_4)\qquad\mbox{fof}(\mbox{nn2\_not\_less\_nn}_4,\mbox{conjecture})$

\medskip\textbf{NUM061-1.p} Segments property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordered\_pair}(u,z)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_segments\_property5}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_segments\_property5}_2,\mbox{negated\_conjecture})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property5}_3,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{segment}(\mbox{xr},y,z)\qquad\mbox{cnf}(\mbox{prove\_segments\_property5}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM918=1.p} Every difference left exists






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\exists w{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(w,u) = v\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM308+1.p} ? + 23 = 34






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + n_{23}{=}n_{34}\qquad\mbox{fof}(\mbox{sum\_something\_n23\_n}_{34},\mbox{conjecture})$

\medskip\textbf{NUM910=1.p} - - something is something






$\forall x{:}\;\mbox{\$real}{:}\;\mbox{\$uminus}(\mbox{\$uminus}(x)) = x\qquad\mbox{tff}(\mbox{real\_uminus\_problem}_7,\mbox{conjecture})$

\medskip\textbf{NUM751$\wedge$1.p} Landau theorem 62d

moref (pf z x) (pf z y)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ z@ x )@ ( \mbox{pf}@ z@ y )\qquad\mbox{thf}(\mbox{satz62d},\mbox{conjecture})$

\medskip\textbf{NUM030-1.p} Symmetrization property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{symmetrization\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x',x)\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM081-1.p} Corollary to well ordering property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(y,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y,\mbox{ordinal\_numbers})))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property9}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(y,\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property9}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM138-1.p} Condition 2 for complete induction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{not\_subclass\_element}(\mbox{intersection}(\mbox{power\_class}(x),z),x),x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),z),x)\qquad\mbox{cnf}(\mbox{prove\_complete\_induction2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM240-1.p} Transfinite recursion lemma 9.1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_1}_1,\mbox{negated\_conjecture})$

$\mbox{image}(\mbox{image}(\mbox{composition\_function},\mbox{singleton}(z)),\mbox{image}(\mbox{rest},\mbox{recursion\_equation\_functions}(z)))\ne \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma9\_1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM305+1.p} 2 + 3 = 5






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_2 + n_3{=}n_5\qquad\mbox{fof}(\mbox{sum\_n2\_n3\_n}_5,\mbox{conjecture})$

\medskip\textbf{NUM271-1.p} Lemma 1 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{null\_class}\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM168-1.p} Corollary to successor property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{omega},\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals}))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_successor\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM796$\wedge$1.p} Landau theorem 99d

less (pl x0 z0) (pl y0 u0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$z_0{:}\;\mbox{rat}\qquad\mbox{thf}(z_0,\mbox{type})$

$u_0{:}\;\mbox{rat}\qquad\mbox{thf}(u_0,\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ x_0@ y_0\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{lessis}@ z_0@ u_0\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{rat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{82},\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat},\mbox{xz}_0{:}\;\mbox{rat},\mbox{xu}_0{:}\;\mbox{rat}{:}\;( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( ( \mbox{moreis}@ \mbox{xz}_0@ \mbox{xu}_0 ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}_0@ \mbox{xz}_0 )@ ( \mbox{pl}@ \mbox{xy}_0@ \mbox{xu}_0 ) ) ) )\qquad\mbox{thf}(\mbox{satz99b},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{83},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{lessis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{85},\mbox{axiom})$

$\mbox{less}@ ( \mbox{pl}@ x_0@ z_0 )@ ( \mbox{pl}@ y_0@ u_0 )\qquad\mbox{thf}(\mbox{satz99d},\mbox{conjecture})$

\medskip\textbf{NUM179-1.p} Condition 3 for a class to be inductive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(1_{\text{dom}(\mbox{successor\_relation})})\in x\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},x),x)\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM144-1.p} The successor of a proper class is a class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_proper\_class\_is\_class}_1,\mbox{negated\_conjecture})$

$\mbox{successor}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_proper\_class\_is\_class}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM757$\wedge$1.p} Landau theorem 63c

lessf x y




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{lessf}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ z )\qquad\mbox{thf}(l,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;\neg\,( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy}\qquad\mbox{thf}(\mbox{satz41b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz41a},\mbox{axiom})$

$\mbox{lessf}@ x@ y\qquad\mbox{thf}(\mbox{satz63c},\mbox{conjecture})$

\medskip\textbf{NUM014-1.p} If a is a prime and a = b$\wedge$2/c$\wedge$2 then a divides b






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

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

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

$( \mbox{prime}(x)\mbox{ and }y\cdot z{=}u\mbox{ and }\mbox{divides}(x,u) ) \ \Rightarrow \ ( \mbox{divides}(x,y)\mbox{ or }\mbox{divides}(x,z) )\qquad\mbox{cnf}(\mbox{remainder},\mbox{axiom})$

$\mbox{prime}(a)\qquad\mbox{cnf}(\mbox{a\_is\_prime},\mbox{hypothesis})$

$a\cdot \mbox{square}(c){=}\mbox{square}(b)\qquad\mbox{cnf}(\mbox{a\_equals\_b\_squared\_by\_c\_squared},\mbox{hypothesis})$

$\neg\,\mbox{divides}(a,b)\qquad\mbox{cnf}(\mbox{prove\_a\_divides\_b},\mbox{negated\_conjecture})$

\medskip\textbf{NUM711$\wedge$1.p} Landau theorem 30

ts x (pl y z) = pl (ts x y) (ts x z)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;\mbox{xx} = ( \mbox{ts}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz28e},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz28b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz28f},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{suc}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz4b},\mbox{axiom})$

$( \mbox{ts}@ x@ ( \mbox{pl}@ y@ z ) ) = ( \mbox{pl}@ ( \mbox{ts}@ x@ y )@ ( \mbox{ts}@ x@ z ) )\qquad\mbox{thf}(\mbox{satz}_{30},\mbox{conjecture})$

\medskip\textbf{NUM892=1.p} Definition of lesseq in terms of less and equality






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff ( \mbox{\$less}(x,y)\mbox{ or }x = y ) )\qquad\mbox{tff}(\mbox{less\_lesseq},\mbox{conjecture})$

\medskip\textbf{NUM063-1.p} Segments property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{segment}(\mbox{element\_relation},y,z)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM277-1.p} Ordinal addition property 7\_1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity}_3,\mbox{axiom})$

$\mbox{ordinals\_with\_null\_class\_as\_identity}\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property7\_1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM826$\wedge$5.p} TPS problem from IND-THMS






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

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

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

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

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

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

$( \mbox{cP}@ \mbox{cA}\mbox{ and }\mbox{cQ}@ \mbox{cB}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}\mbox{ and }\mbox{cQ}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cQ}@ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx}\mbox{ and }\mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{cA}\mbox{ and }\mbox{xq}@ \mbox{cB}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xq}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xq}@ \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xx} ) ) ) )\mbox{ and }( \mbox{cG}@ \mbox{cA} ) = \mbox{cB}\mbox{ and }( \mbox{cG}@ \mbox{cB} ) = \mbox{cA}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{cF}@ ( \mbox{cG}@ \mbox{xx} )@ ( \mbox{cG}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ ( \mbox{cG}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM622\_pme},\mbox{conjecture})$

\medskip\textbf{NUM059-1.p} Segments property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$u\in \mbox{segment}(\mbox{xr},y,z)\qquad\mbox{cnf}(\mbox{prove\_segments\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,z)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_segments\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM297+1.p} -4 $<$ -2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\mbox{less}(\mbox{nn}_4,\mbox{nn}_2)\qquad\mbox{fof}(\mbox{nn4\_less\_nn}_2,\mbox{conjecture})$

\medskip\textbf{NUM165-1.p} Corollary to omega definition, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{omega}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_omega\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{omega}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_omega\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM861+1.p} Upper bound replace maximum embedded in a context (2)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\forall x{:}\;\mbox{lesseq}(x,x)\qquad\mbox{fof}(\mbox{lesseq\_ref},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,z) ) \ \Rightarrow \ \mbox{lesseq}(x,z) )\qquad\mbox{fof}(\mbox{lesseq\_trans},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,x) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{lesseq\_antisymmetric},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\mbox{ or }\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{lesseq\_total},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(z + x,z + y) )\qquad\mbox{fof}(\mbox{sum\_monotone}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{fof}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{max}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{lesseq}(x,y) )\qquad\mbox{fof}(\mbox{max}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{lesseq}(x,z)\mbox{ and }\mbox{lesseq}(y,z) ) )\qquad\mbox{fof}(\mbox{ub},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{lesseq}(c + \mbox{max}(x,y),n) )\qquad\mbox{fof}(\mbox{model\_max}_4,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{lesseq}(c + z,n) ) )\qquad\mbox{fof}(\mbox{model\_ub}_4,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{fof}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM804$\wedge$5.p} TPS problem from NATS






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

$c_0{:}\;n\qquad\mbox{thf}(c_0,\mbox{type})$

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

$\forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = \mbox{xx}\qquad\mbox{thf}(\mbox{cPA}_1,\mbox{conjecture})$

\medskip\textbf{NUM238-1.p} Transfinite recursion lemma 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma7}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma7}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x)\in \mbox{domain\_of}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma7}_3,\mbox{negated\_conjecture})$

$u\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma7}_4,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,u,\mbox{universal\_class})\ne \mbox{restrict}(y,u,\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma7}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM537+1.p} Ramsey's Infinite Theorem 05\_01, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{sdtpldt}_0(\mbox{sdtmndt}_0(w_0,w_1),w_1) = w_0 ) )\qquad\mbox{fof}(\mbox{mConsDiff},\mbox{axiom})$

$\mbox{aElement}_0(\mbox{xx})\mbox{ and }\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{679},\mbox{hypothesis})$

$\neg\,\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_679}_{02},\mbox{hypothesis})$

$\mbox{aSubsetOf}_0(\mbox{xS},\mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\mbox{ and }\mbox{aSubsetOf}_0(\mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}),\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM052-1.p} Transitive ordering property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{transitive}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,w)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_4,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,w)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,w)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property1}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM074-1.p} Well ordering property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property5}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{transitive}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM076-1.p} Well ordering property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property7}_1,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property7}_2,\mbox{negated\_conjecture})$

$u\in \mbox{segment}(\mbox{xr},y,u)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM746$\wedge$1.p} Landau theorem 52

~(lessf x z) $\to$ eq x z




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\neg\,\mbox{lessf}@ x@ y \ \Rightarrow \ ( \mbox{eq}@ x@ y )\qquad\mbox{thf}(l,\mbox{axiom})$

$\neg\,\mbox{lessf}@ y@ z \ \Rightarrow \ ( \mbox{eq}@ y@ z )\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{39},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \neg\,\mbox{lessf}@ \mbox{xy}@ \mbox{xz} \ \Rightarrow \ ( \mbox{eq}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz51b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \neg\,\mbox{lessf}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( ( \mbox{lessf}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz51a},\mbox{axiom})$

$\neg\,\mbox{lessf}@ x@ z \ \Rightarrow \ ( \mbox{eq}@ x@ z )\qquad\mbox{thf}(\mbox{satz}_{52},\mbox{conjecture})$

\medskip\textbf{NUM224-1.p} Rest relation property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{compose\_class}(x)\circ \mbox{rest\_relation})\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM145-1.p} Corollary to the successor of a proper class being a class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM784$\wedge$1.p} Landau theorem 81c

~(less x0 y0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\neg\,\mbox{more}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;\neg\,( ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0\qquad\mbox{thf}(\mbox{satz81b},\mbox{axiom})$

$\neg\,\mbox{less}@ x_0@ y_0\qquad\mbox{thf}(\mbox{satz81c},\mbox{conjecture})$

\medskip\textbf{NUM274-1.p} Lemma 4 for ordinal addition property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_1,\mbox{axiom})$

$x\in \mbox{ordinals\_with\_null\_class\_as\_identity} \ \Rightarrow \ \mbox{ordinal\_add}(\mbox{null\_class},x) = x\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_2,\mbox{axiom})$

$( x\in \mbox{ordinal\_numbers}\mbox{ and }\mbox{ordinal\_add}(\mbox{null\_class},x) = x ) \ \Rightarrow \ x\in \mbox{ordinals\_with\_null\_class\_as\_identity}\qquad\mbox{cnf}(\mbox{ordinals\_with\_null\_class\_as\_identity\_def}_3,\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{ordinals\_with\_null\_class\_as\_identity},\mbox{domain\_of}(\mbox{recursion}(\mbox{null\_class},\mbox{successor\_relation},\mbox{union\_of\_range\_map})))\qquad\mbox{cnf}(\mbox{prove\_lemma\_4\_for\_ordinal\_addition\_property7}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM130-1.p} Alternate 2 for transfinite induction, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{least}(\mbox{element\_relation},y),y')\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction2}_2,\mbox{negated\_conjecture})$

$y\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM861=1.p} Upper bound replace maximum embedded in a context (2)

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$c{:}\;\mbox{\$int}\qquad\mbox{tff}(\mbox{c\_type},\mbox{type})$

$\mbox{summation}{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{summation\_type},\mbox{type})$

$\mbox{ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{ub\_type},\mbox{type})$

$\mbox{model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_max\_type},\mbox{type})$

$\mbox{model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_ub\_type},\mbox{type})$

$\mbox{minsol\_model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_max\_type},\mbox{type})$

$\mbox{minsol\_model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_ub\_type},\mbox{type})$

$\mbox{max}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{max\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff \mbox{\$lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{tff}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{\$lesseq}(y,x) )\qquad\mbox{tff}(\mbox{max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{\$lesseq}(x,y) )\qquad\mbox{tff}(\mbox{max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{\$lesseq}(x,z)\mbox{ and }\mbox{\$lesseq}(y,z) ) )\qquad\mbox{tff}(\mbox{ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{\$lesseq}(\mbox{\$sum}(c,\mbox{max}(x,y)),n) )\qquad\mbox{tff}(\mbox{model\_max}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(c,z),n) ) )\qquad\mbox{tff}(\mbox{model\_ub}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{tff}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM668$\wedge$1.p} Landau theorem 18

~(forall x\_0:nat.~(pl x y = pl x x\_0))




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( \mbox{pl}@ x@ y )\ne ( \mbox{pl}@ x@ \mbox{xx}_0 )\qquad\mbox{thf}(\mbox{satz}_{18},\mbox{conjecture})$

\medskip\textbf{NUM416$\wedge$1.p} 10 * (10 * 10) = (10 + 10) * (5 * 10)






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$( \cdot @ \mbox{ten}@ ( \cdot @ \mbox{ten}@ \mbox{ten} ) ) = ( \cdot @ (  + @ \mbox{ten}@ \mbox{ten} )@ ( \cdot @ \mbox{five}@ \mbox{ten} ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM107-1.p} Ordinal property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(\mbox{ordinal\_numbers},y)\ne y\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM189-1.p} Omega is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{omega}\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_omega\_is\_an\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM058-1.p} Segments property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{xr},y,z)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_segments\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM121-1.p} Transfinite induction, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y',\mbox{ordinal\_numbers})),y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM673$\wedge$1.p} Landau theorem 19d

more (pl z x) (pl z y)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ x@ y\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\mbox{more}@ ( \mbox{pl}@ z@ x )@ ( \mbox{pl}@ z@ y )\qquad\mbox{thf}(\mbox{satz19d},\mbox{conjecture})$

\medskip\textbf{NUM122-1.p} Transfinite induction, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y',\mbox{ordinal\_numbers}))\in y\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM051-1.p} Everything is connected to the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{connected}(x,\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_everything\_connected\_to\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM352+1.p} ?XY (X+Y) != (X+Y)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y,z_1,z_2{:}\;( x + y{=}z_1\mbox{ and }x + y{=}z_2\mbox{ and }z_1\ne z_2 )\qquad\mbox{fof}(\mbox{anti\_unique},\mbox{conjecture})$

\medskip\textbf{NUM823$\wedge$5.p} TPS problem from IND-THMS






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

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

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

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) )\mbox{ and }\mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM}_{302},\mbox{conjecture})$

\medskip\textbf{NUM108-1.p} Ordinal property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property9}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property9}_2,\mbox{negated\_conjecture})$

$\mbox{intersection}(x,y)\ne x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property9}_3,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property9}_4,\mbox{negated\_conjecture})$

$\mbox{intersection}(x,y)\ne y\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property9}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM708$\wedge$1.p} Landau theorem 28g

x = ts n\_1 x




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

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

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ n_1@ \mbox{xx} ) = \mbox{xx}\qquad\mbox{thf}(\mbox{satz28c},\mbox{axiom})$

$x = ( \mbox{ts}@ n_1@ x )\qquad\mbox{thf}(\mbox{satz28g},\mbox{conjecture})$

\medskip\textbf{NUM123-1.p} Alternate transfinite induction 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y',\mbox{ordinal\_numbers}))\in y'\qquad\mbox{cnf}(\mbox{prove\_alternate\_transfinite\_induction3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},y)\qquad\mbox{cnf}(\mbox{prove\_alternate\_transfinite\_induction3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM423+3.p} Fuerstenberg's infinitude of primes 03, 02 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{671},\mbox{hypothesis})$

$\exists w_0{:}\;( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},w_0) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xa})) )\mbox{ or }\mbox{aDivisorOf}_0(\mbox{xq},\mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xa})))\mbox{ or }\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xa},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM111-1.p} Ordinal property 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property11}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property11}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property11}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM161-1.p} Corollary to kind 1 ordinals being a class of ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(\mbox{ordinal\_numbers},\mbox{kind\_1\_ordinals})\ne \mbox{kind\_1\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_kind\_1\_ordinals\_are\_ordinals}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM386+1.p} Ordinal numbers, theorem 13






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{succ}(a) = \mbox{set\_union}_2(a,\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d1\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,\mbox{succ}(b))\iff ( \mbox{in}(a,b)\mbox{ or }a = b ) )\qquad\mbox{fof}(\mbox{t13\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM113-1.p} Ordinal property 13






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_class}(x) \ \Rightarrow \ \mbox{well\_ordering}(\mbox{element\_relation},x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_1,\mbox{axiom})$

$\mbox{ordinal\_class}(x) \ \Rightarrow \ \mbox{subclass}(\mbox{sum\_class}(x),x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_2,\mbox{axiom})$

$( \mbox{well\_ordering}(\mbox{element\_relation},x)\mbox{ and }\mbox{subclass}(\mbox{sum\_class}(x),x) ) \ \Rightarrow \ \mbox{ordinal\_class}(x)\qquad\mbox{cnf}(\mbox{ordinal\_class}_3,\mbox{axiom})$

$\neg\,\mbox{ordinal\_class}(\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property13}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM830$\wedge$5.p} TPS problem from PA-THMS






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

$c_0{:}\;n\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS\_type},\mbox{type})$

$\mbox{c\_plus}{:}\;n\to n\to n\qquad\mbox{thf}(\mbox{c\_plus\_type},\mbox{type})$

$\mbox{c\_star}{:}\;n\to n\to n\qquad\mbox{thf}(\mbox{c\_star\_type},\mbox{type})$

$\mbox{cPA}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_1\_type},\mbox{type})$

$\mbox{cPA}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_2\_type},\mbox{type})$

$\mbox{cPA}_3{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_3\_type},\mbox{type})$

$\mbox{cPA}_4{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_4\_type},\mbox{type})$

$\mbox{cPA}_1 = ( \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_1\_def},\mbox{definition})$

$\mbox{cPA}_2 = ( \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{cS}@ ( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cPA\_2\_def},\mbox{definition})$

$\mbox{cPA}_3 = ( \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_star}@ \mbox{xx}@ c_0 ) = c_0 )\qquad\mbox{thf}(\mbox{cPA\_3\_def},\mbox{definition})$

$\mbox{cPA}_4 = ( \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_star}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{c\_plus}@ ( \mbox{c\_star}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPA\_4\_def},\mbox{definition})$

$( \mbox{cPA}_1\mbox{ and }\mbox{cPA}_2\mbox{ and }\mbox{cPA}_3\mbox{ and }\mbox{cPA}_4 ) \ \Rightarrow \ ( \mbox{c\_star}@ ( \mbox{cS}@ ( \mbox{cS}@ c_0 ) )@ ( \mbox{cS}@ ( \mbox{cS}@ c_0 ) ) ) = ( \mbox{c\_plus}@ ( \mbox{cS}@ ( \mbox{cS}@ c_0 ) )@ ( \mbox{cS}@ ( \mbox{cS}@ c_0 ) ) )\qquad\mbox{thf}(\mbox{cPA\_THM}_1,\mbox{conjecture})$

\medskip\textbf{NUM095-1.p} Sections property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{section}(\mbox{xr},w,y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property4}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{least}(\mbox{xr},\mbox{intersection}(w',y))\in w'\qquad\mbox{cnf}(\mbox{prove\_sections\_property4}_3,\mbox{negated\_conjecture})$

$y\ne w\qquad\mbox{cnf}(\mbox{prove\_sections\_property4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM003-1.p} A + (B - C) = (A - C) + B






$\mbox{include}(\mbox{'Axioms/NUM002-0.ax'})$

$\neg\,a + \mbox{subtract}(b,c){=}\mbox{subtract}(a,c) + b\qquad\mbox{cnf}(\mbox{prove\_equation},\mbox{negated\_conjecture})$

\medskip\textbf{NUM670$\wedge$1.p} Landau theorem 19a

~(forall x\_0:nat.~(pl x z = pl (pl y z) x\_0))




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 )\qquad\mbox{thf}(m,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{pl}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_5,\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( \mbox{pl}@ x@ z )\ne ( \mbox{pl}@ ( \mbox{pl}@ y@ z )@ \mbox{xx}_0 )\qquad\mbox{thf}(\mbox{satz19a},\mbox{conjecture})$

\medskip\textbf{NUM415$\wedge$1.p} 2 * (3 + 7) = (2 * 5) * (1 + 1)






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$( \cdot @ \mbox{two}@ (  + @ \mbox{three}@ \mbox{seven} ) ) = ( \cdot @ ( \cdot @ \mbox{two}@ \mbox{five} )@ (  + @ 1@ 1 ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM374+3.p} Disprove Wilkie identity from Tarski's identities






$\forall x,y{:}\;x + y = y + x\qquad\mbox{fof}(\mbox{sum\_symmetry},\mbox{axiom})$

$\forall x,y,z{:}\;x + ( y + z ) = ( x + y ) + z\qquad\mbox{fof}(\mbox{sum\_associativity},\mbox{axiom})$

$\forall x{:}\;x\cdot n_1 = x\qquad\mbox{fof}(\mbox{product\_identity},\mbox{axiom})$

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

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

$\forall x,y,z{:}\;x\cdot ( y + z ) = x\cdot y + x\cdot z\qquad\mbox{fof}(\mbox{sum\_product\_distribution},\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(n_1,x) = n_1\qquad\mbox{fof}(\mbox{exponent\_n}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(x,n_1) = x\qquad\mbox{fof}(\mbox{exponent\_identity},\mbox{axiom})$

$\forall x,y,z{:}\;\mbox{exponent}(x,y + z) = \mbox{exponent}(x,y)\cdot \mbox{exponent}(x,z)\qquad\mbox{fof}(\mbox{exponent\_sum\_product},\mbox{axiom})$

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

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

$n_0\ne n_1\qquad\mbox{fof}(\mbox{n0\_n}_1,\mbox{axiom})$

$\forall c,p,q,r,s,b{:}\;( ( c = n_0\cdot n_0\mbox{ and }p = n_1 + n_0\mbox{ and }q = p + c\mbox{ and }r = n_1 + n_0\cdot c\mbox{ and }s = ( n_1 + c ) + c\cdot c ) \ \Rightarrow \ \mbox{exponent}(\mbox{exponent}(p,n_0) + \mbox{exponent}(q,n_0),b)\cdot \mbox{exponent}(\mbox{exponent}(r,b) + \mbox{exponent}(s,b),n_0) = \mbox{exponent}(\mbox{exponent}(p,b) + \mbox{exponent}(q,b),n_0)\cdot \mbox{exponent}(\mbox{exponent}(r,n_0) + \mbox{exponent}(s,n_0),b) )\qquad\mbox{fof}(\mbox{wilkie},\mbox{conjecture})$

\medskip\textbf{NUM754$\wedge$1.p} Landau theorem 62h

moref (pf z x) (pf u y)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{moref}@ z@ u\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz62g},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{axiom})$

$\mbox{moref}@ ( \mbox{pf}@ z@ x )@ ( \mbox{pf}@ u@ y )\qquad\mbox{thf}(\mbox{satz62h},\mbox{conjecture})$

\medskip\textbf{NUM876=1.p} X minus X equals 0






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(x,x) = 0\qquad\mbox{tff}(\mbox{x\_minus\_x\_equals}_0,\mbox{conjecture})$

\medskip\textbf{NUM333+1.p} ! XYZ, ((X+Y)+Z) = (X+(Y+Z))






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z,z_1,z_2,z_3,z_4{:}\;( ( x + y{=}z_1\mbox{ and }z_1 + z{=}z_2\mbox{ and }y + z{=}z_3\mbox{ and }x + z_3{=}z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{fof}(\mbox{associative},\mbox{conjecture})$

\medskip\textbf{NUM158-1.p} Membership condition 3 for kind 1 ordinals






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{kind\_1\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_kind\_1\_ordinal}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_kind\_1\_ordinal}_2,\mbox{negated\_conjecture})$

$\mbox{successor}(1_{\text{dom}(\mbox{successor\_relation})})\ne x\qquad\mbox{cnf}(\mbox{prove\_kind\_1\_ordinal}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM147-1.p} The successor of an ordinal is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_ordinal\_is\_ordinal}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_ordinal\_is\_ordinal}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM132-1.p} Union of successor relation ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_union\_of\_successor\_ordinal}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{successor}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_union\_of\_successor\_ordinal}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM027-1.p} If a $>$= b and b*c $\leq$ a*c, then c = 0






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

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

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

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

$a{=}b \ \Rightarrow \ a\cdot c{=}b\cdot c\qquad\mbox{cnf}(\mbox{equality\_preserved\_over\_times},\mbox{axiom})$

$\mbox{less}(a,b) \ \Rightarrow \ \neg\,a{=}b\qquad\mbox{cnf}(\mbox{not\_less\_and\_equal},\mbox{axiom})$

$\mbox{less}(a,b)\mbox{ or }b{=}a\mbox{ or }\mbox{less}(b,a)\qquad\mbox{cnf}(\mbox{numbers\_either\_less\_or\_equal},\mbox{axiom})$

$\neg\,\mbox{less}(a,a)\qquad\mbox{cnf}(\mbox{number\_not\_less\_than\_itself},\mbox{axiom})$

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_the\_first\_number},\mbox{axiom})$

$\mbox{less}(a,b) \ \Rightarrow \ ( c{=}n_0\mbox{ or }\mbox{less}(a\cdot c,b\cdot c) )\qquad\mbox{cnf}(\mbox{multiply\_lemma},\mbox{axiom})$

$\neg\,\mbox{less}(b,a)\qquad\mbox{cnf}(\mbox{b\_not\_less\_than\_a},\mbox{hypothesis})$

$\mbox{less}(b\cdot c,a\cdot c)\qquad\mbox{cnf}(\mbox{b\_times\_c\_less\_than\_a\_times\_c},\mbox{hypothesis})$

$\neg\,c{=}n_0\qquad\mbox{cnf}(\mbox{prove\_c\_is}_0,\mbox{negated\_conjecture})$

\medskip\textbf{NUM256-1.p} Transfinite recursion property 12






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property12}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property12}_2,\mbox{negated\_conjecture})$

$z\circ \mbox{rest\_of}(\mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x))\ne \mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property12}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM075-1.p} Well ordering property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property6}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{well\_ordering}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM382+1.p} Ordinal numbers, theorem 6






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d,e,f,g{:}\;( g = \mbox{unordered\_sextuple}(a,b,c,d,e,f)\iff \forall h{:}\;( \mbox{in}(h,g)\iff \neg\,h\ne a\mbox{ and }h\ne b\mbox{ and }h\ne c\mbox{ and }h\ne d\mbox{ and }h\ne e\mbox{ and }h\ne f ) )\qquad\mbox{fof}(\mbox{d4\_enumset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{in}(b,c)\mbox{ and }\mbox{in}(c,d)\mbox{ and }\mbox{in}(d,e)\mbox{ and }\mbox{in}(e,f)\mbox{ and }\mbox{in}(f,a)\qquad\mbox{fof}(\mbox{t6\_ordinal}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\mbox{in}(d,c)\qquad\mbox{fof}(\mbox{t7\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM183-1.p} Ordinals are either kind 1 or limit






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinals\_are\_kind\_1\_or\_limit}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{kind\_1\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_ordinals\_are\_kind\_1\_or\_limit}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_ordinals\_are\_kind\_1\_or\_limit}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM534+1.p} Ramsey's Infinite Theorem 04, 00 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{617},\mbox{hypothesis})$

$\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_617}_{02},\mbox{hypothesis})$

$\mbox{sdtpldt}_0(\mbox{sdtmndt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}) = \mbox{xS}\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM109-1.p} Ordinal property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property10}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property10}_2,\mbox{negated\_conjecture})$

$\neg\,x\in y\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property10}_3,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property10}_4,\mbox{negated\_conjecture})$

$\neg\,y\in x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property10}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM009-1.p} Peano axiom 1






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$\neg\,\mbox{empty\_set}\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{prove\_zero\_is\_a\_natural},\mbox{negated\_conjecture})$

\medskip\textbf{NUM702$\wedge$1.p} Landau theorem 26b

moreis y x




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

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

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

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ y@ n_1 )@ x@ \mbox{xu} )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{14},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ ( \mbox{pl}@ \mbox{xx}@ n_1 )@ \mbox{xy}@ \mbox{xv} ) ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{26},\mbox{axiom})$

$\mbox{moreis}@ y@ x\qquad\mbox{thf}(\mbox{satz26b},\mbox{conjecture})$

\medskip\textbf{NUM167-1.p} Successor property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})),\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals}))\qquad\mbox{cnf}(\mbox{prove\_successor\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM177-1.p} Condition 1 for a class to be inductive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},x),x)\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition1}_1,\mbox{negated\_conjecture})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(u)\in x\qquad\mbox{cnf}(\mbox{prove\_inductive\_class\_condition1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM006-1.p} Goldbach conjecture






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$f_{79}\in \mbox{even\_numbers}\qquad\mbox{cnf}(\mbox{an\_even\_number},\mbox{hypothesis})$

$\neg\,f_{79}\in \mbox{non\_ordered\_pair}(\mbox{empty\_set},\mbox{successor}(\mbox{successor}(\mbox{empty\_set})))\qquad\mbox{cnf}(\mbox{its\_not\_0\_or}_2,\mbox{hypothesis})$

$( x\in \mbox{prime\_numbers}\mbox{ and }y\in \mbox{prime\_numbers} ) \ \Rightarrow \ \mbox{apply\_to\_two\_arguments}( + ,x,y)\ne f_{79}\qquad\mbox{cnf}(\mbox{prove\_its\_not\_the\_sum\_of\_two\_primes},\mbox{negated\_conjecture})$

\medskip\textbf{NUM289-1.p} Number theory (ordinals) axioms, based on NBG set theory






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

\medskip\textbf{NUM024-1.p} No number is less than itself






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

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

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

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

$a + b{=}c + b \ \Rightarrow \ a{=}c\qquad\mbox{cnf}(\mbox{plus\_substitution},\mbox{axiom})$

$a + b{=}b + a\qquad\mbox{cnf}(\mbox{commutativity\_of\_plus},\mbox{axiom})$

$\mbox{less}(a,a)\qquad\mbox{cnf}(\mbox{impossible\_a\_is\_less\_than\_itself},\mbox{hypothesis})$

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{prove\_a\_contradiction},\mbox{negated\_conjecture})$

\medskip\textbf{NUM120-1.p} Transfinite induction, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y',\mbox{ordinal\_numbers}))\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_induction1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM282-1.p} Ordinal multiplication property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_multiplication\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{image}(\mbox{recursion}(\mbox{null\_class},\mbox{apply}(\mbox{add\_relation},x),\mbox{union\_of\_range\_map}),y))\ne \mbox{ordinal\_multiply}(x,y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_multiplication\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM016-2.p} There exist infinitely many primes






$\neg\,\mbox{less}(x,x)\qquad\mbox{cnf}(\mbox{nothing\_is\_less\_than\_itself},\mbox{axiom})$

$\mbox{less}(x,y) \ \Rightarrow \ \neg\,\mbox{less}(y,x)\qquad\mbox{cnf}(\mbox{numbers\_are\_different},\mbox{axiom})$

$\mbox{less}(x,\mbox{factorial\_plus\_one}(x))\qquad\mbox{cnf}(\mbox{a\_prime\_is\_less\_than\_the\_next\_one},\mbox{axiom})$

$\mbox{divides}(x,\mbox{factorial\_plus\_one}(y)) \ \Rightarrow \ \mbox{less}(y,x)\qquad\mbox{cnf}(\mbox{divisor\_is\_smaller},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{divides}(\mbox{prime\_divisor}(x),x)\qquad\mbox{cnf}(\mbox{division\_by\_prime\_divisor},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{prime}(\mbox{prime\_divisor}(x))\qquad\mbox{cnf}(\mbox{prime\_divsiors},\mbox{axiom})$

$\mbox{prime}(x)\mbox{ or }\mbox{less}(\mbox{prime\_divisor}(x),x)\qquad\mbox{cnf}(\mbox{smaller\_prime\_divisors},\mbox{axiom})$

$( \mbox{prime}(x)\mbox{ and }\mbox{less}(a,x) ) \ \Rightarrow \ \mbox{less}(\mbox{factorial\_plus\_one}(a),x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_another\_prime},\mbox{negated\_conjecture})$

\medskip\textbf{NUM380+1.p} Ordinal numbers, theorem 4






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d,e{:}\;( e = \mbox{unordered\_quadruple}(a,b,c,d)\iff \forall f{:}\;( \mbox{in}(f,e)\iff \neg\,f\ne a\mbox{ and }f\ne b\mbox{ and }f\ne c\mbox{ and }f\ne d ) )\qquad\mbox{fof}(\mbox{d2\_enumset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{in}(b,c)\mbox{ and }\mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a)\qquad\mbox{fof}(\mbox{t4\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\mbox{in}(d,c)\qquad\mbox{fof}(\mbox{t7\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM374+4.p} Disprove Wilkie identity from Tarski's identities






$\forall x,y{:}\;x + y = y + x\qquad\mbox{fof}(\mbox{sum\_symmetry},\mbox{axiom})$

$\forall x,y,z{:}\;x + ( y + z ) = ( x + y ) + z\qquad\mbox{fof}(\mbox{sum\_associativity},\mbox{axiom})$

$\forall x{:}\;x\cdot n_1 = x\qquad\mbox{fof}(\mbox{product\_identity},\mbox{axiom})$

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

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

$\forall x,y,z{:}\;x\cdot ( y + z ) = x\cdot y + x\cdot z\qquad\mbox{fof}(\mbox{sum\_product\_distribution},\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(n_1,x) = n_1\qquad\mbox{fof}(\mbox{exponent\_n}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{exponent}(x,n_1) = x\qquad\mbox{fof}(\mbox{exponent\_identity},\mbox{axiom})$

$\forall x,y,z{:}\;\mbox{exponent}(x,y + z) = \mbox{exponent}(x,y)\cdot \mbox{exponent}(x,z)\qquad\mbox{fof}(\mbox{exponent\_sum\_product},\mbox{axiom})$

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

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

$\forall c,p,q,r,s,b{:}\;( \mbox{lemmas}(c,p,q,r,s,b)\iff ( n_2 = n_1 + n_1\mbox{ and }b\ne n_0\mbox{ and }b\ne n_1\mbox{ and }b\ne n_2\mbox{ and }\forall x{:}\;b\ne n_0\cdot x\mbox{ and }\forall x{:}\;p\ne q\cdot x\mbox{ and }\forall x{:}\;q\ne p\cdot x\mbox{ and }\forall x{:}\;r\ne s\cdot x\mbox{ and }\forall x{:}\;s\ne r\cdot x\mbox{ and }n_1 + n_0\ne n_1\mbox{ and }n_2 + n_0\ne n_1\mbox{ and }n_0 + n_0\ne n_1\mbox{ and }c\ne n_1\mbox{ and }n_1 + c\ne n_1\mbox{ and }c\cdot n_0\ne n_1\mbox{ and }n_1 + n_0\ne n_0\mbox{ and }n_2 + n_0\ne n_0\mbox{ and }n_0 + n_0\ne n_0\mbox{ and }c\ne n_0\mbox{ and }n_1 + c\ne n_0\mbox{ and }n_2 + n_0\ne n_1 + n_0\mbox{ and }c\ne n_1 + n_0\mbox{ and }c\cdot n_0\ne n_1 + n_0\mbox{ and }c\ne n_2 + n_0\mbox{ and }c\ne n_0 + n_0\mbox{ and }n_1 + c\ne c ) )\qquad\mbox{fof}(\mbox{lemmas},\mbox{axiom})$

$n_0\ne n_1\qquad\mbox{fof}(\mbox{n0\_n}_1,\mbox{axiom})$

$n_0\ne n_2\qquad\mbox{fof}(\mbox{n0\_n}_2,\mbox{axiom})$

$n_1\ne n_2\qquad\mbox{fof}(\mbox{n1\_n}_2,\mbox{axiom})$

$\forall c,p,q,r,s,b{:}\;( ( c = n_0\cdot n_0\mbox{ and }p = n_1 + n_0\mbox{ and }q = p + c\mbox{ and }r = n_1 + n_0\cdot c\mbox{ and }s = ( n_1 + c ) + c\cdot c\mbox{ and }\mbox{lemmas}(c,p,q,r,s,b) ) \ \Rightarrow \ \mbox{exponent}(\mbox{exponent}(p,n_0) + \mbox{exponent}(q,n_0),b)\cdot \mbox{exponent}(\mbox{exponent}(r,b) + \mbox{exponent}(s,b),n_0) = \mbox{exponent}(\mbox{exponent}(p,b) + \mbox{exponent}(q,b),n_0)\cdot \mbox{exponent}(\mbox{exponent}(r,n_0) + \mbox{exponent}(s,n_0),b) )\qquad\mbox{fof}(\mbox{wilkie},\mbox{conjecture})$

\medskip\textbf{NUM246-1.p} Transfinite recursion property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{function}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x) = \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_3,\mbox{negated\_conjecture})$

$z\circ \mbox{rest\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_4,\mbox{negated\_conjecture})$

$u\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_5,\mbox{negated\_conjecture})$

$\mbox{apply}(z,\mbox{restrict}(x,u,\mbox{universal\_class}))\ne \mbox{apply}(x,u)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property2}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM148-1.p} The predecessor of an ordinal is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(x)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_predcessor\_of\_ordinal\_is\_ordinal}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_predcessor\_of\_ordinal\_is\_ordinal}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM880=1.p} Product of X and Y is not X






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,y) = x\qquad\mbox{tff}(\mbox{anti\_product\_x\_y\_x},\mbox{conjecture})$

\medskip\textbf{NUM872=1.p} Sum something and 0 is not something






$\exists x{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,0)\ne x\qquad\mbox{tff}(\mbox{anti\_sum\_identity}_1,\mbox{conjecture})$

\medskip\textbf{NUM368+1.p} ! - ! = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;x\setminus x{=}n_0\qquad\mbox{fof}(\mbox{x\_minus\_x\_equals}_0,\mbox{conjecture})$

\medskip\textbf{NUM813$\wedge$5.p} TPS problem THM303






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cEVEN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN\_type},\mbox{type})$

$\mbox{cNUMBER}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cNUMBER\_type},\mbox{type})$

$\mbox{cODD}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cIND}\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cNUMBER}@ \mbox{xn} )\iff ( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cNUMBER}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM}_{303},\mbox{conjecture})$

\medskip\textbf{NUM181-1.p} The null class is not a limit ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{null\_class}\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_not\_a\_limit\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM790$\wedge$1.p} Landau theorem 81j

less x0 y0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\neg\,\neg\,\mbox{more}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) )\qquad\mbox{thf}(\mbox{satz81a},\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{less}@ x_0@ y_0\qquad\mbox{thf}(\mbox{satz81j},\mbox{conjecture})$

\medskip\textbf{NUM199-1.p} Least upper bound property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property1}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{successor}(\mbox{sum\_class}(x)))\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM298+1.p} 2 !$<$ -2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\neg\,\mbox{less}(n_2,\mbox{nn}_2)\qquad\mbox{fof}(\mbox{n2\_not\_less\_nn}_2,\mbox{conjecture})$

\medskip\textbf{NUM927=1.p} The Collatz Conjecture

f(X) = 3X + 1 if X is odd, X/2 if X is even. Prove this is cyclic.
 e.g., 3,10,5,16,8,4,2,1,4,2,1




$f{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$\mbox{iterate\_f}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{iterate\_f\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$remainder\_t}(x,2) = 1 \ \Rightarrow \ f(x) = \mbox{\$sum}(\mbox{\$product}(3,x),1) )\qquad\mbox{tff}(\mbox{f\_odd},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$remainder\_t}(x,2) = 0 \ \Rightarrow \ f(x) = \mbox{\$quotient\_t}(x,2) )\qquad\mbox{tff}(\mbox{f\_even},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int},x{:}\;\mbox{\$int}{:}\;( i = 1 \ \Rightarrow \ \mbox{iterate\_f}(i,x) = f(x) )\qquad\mbox{tff}(\mbox{iterate\_f\_base},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int},x{:}\;\mbox{\$int}{:}\;( \mbox{\$greater}(i,1) \ \Rightarrow \ \mbox{iterate\_f}(i,x) = \mbox{iterate\_f}(\mbox{\$difference}(i,1),f(x)) )\qquad\mbox{tff}(\mbox{iterate\_f},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int}{:}\;( \mbox{\$greatereq}(x,1) \ \Rightarrow \ \exists i{:}\;\mbox{\$int}{:}\;\mbox{iterate\_f}(i,x) = 1 )\qquad\mbox{tff}(\mbox{iterates\_to}_1,\mbox{conjecture})$

\medskip\textbf{NUM363+1.p} if (X+Y) = Z then Z $>$ X \& Z $>$ Y






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z{:}\;( x + y{=}z \ \Rightarrow \ ( \mbox{less}(x,z)\mbox{ and }\mbox{less}(y,z) ) )\qquad\mbox{fof}(\mbox{sum\_larger},\mbox{conjecture})$

\medskip\textbf{NUM750$\wedge$1.p} Landau theorem 62b

eq (pf x z) (pf y z)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ z )\qquad\mbox{thf}(\mbox{satz62b},\mbox{conjecture})$

\medskip\textbf{NUM647$\wedge$1.p} Landau theorem 8a

y = z




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$( \mbox{pl}@ x@ y ) = ( \mbox{pl}@ x@ z )\qquad\mbox{thf}(i,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{xy}\ne \mbox{xz} \ \Rightarrow \ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )\ne ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_8,\mbox{axiom})$

$y = z\qquad\mbox{thf}(\mbox{satz8a},\mbox{conjecture})$

\medskip\textbf{NUM886=1.p} Product non-idempotence






$\forall x{:}\;\mbox{\$int}{:}\;\mbox{\$product}(x,x)\ne x\qquad\mbox{tff}(\mbox{anti\_not\_product\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM193-1.p} The sum of ordinals is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{sum\_class}(\mbox{ordinal\_numbers})\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_ordinals\_is\_ordinal}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM351+1.p} XY (X + Y) = X






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;x + y{=}x\qquad\mbox{fof}(\mbox{anti\_sum\_x\_y\_x},\mbox{conjecture})$

\medskip\textbf{NUM029-1.p} Symmetrization property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{symmetrization\_of}(x') = x'\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{symmetrization\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM237-1.p} Corollary 2 to transfinite recursion lemma 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_transfinite\_recursion\_lemma6}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_transfinite\_recursion\_lemma6}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_transfinite\_recursion\_lemma6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM346+1.p} 2 + 3 $>$ 4






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_2 + n_3{=}x \ \Rightarrow \ \mbox{less}(n_4,x) )\qquad\mbox{fof}(\mbox{sum\_n2\_n3\_greater\_n}_4,\mbox{conjecture})$

\medskip\textbf{NUM267-1.p} Ordinal addition property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{limit\_ordinals}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{image}(\mbox{recursion}(x,\mbox{successor\_relation},\mbox{union\_of\_range\_map}),y))\ne \mbox{ordinal\_add}(x,y)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM056-1.p} Asymmetric class property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{asymmetric}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{irreflexive}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM010-1.p} Peano axiom 2






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

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

$\mbox{include}(\mbox{'Axioms/NUM003-0.ax'})$

$f_{74}\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{a\_natural\_number},\mbox{hypothesis})$

$\neg\,\mbox{successor}(f_{74})\in \mbox{natural\_numbers}\qquad\mbox{cnf}(\mbox{prove\_it\_has\_a\_successor},\mbox{negated\_conjecture})$

\medskip\textbf{NUM280-1.p} Ordinal multiplication property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_multiply}(x,\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_multiplication\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM428+1.p} Fuerstenberg's infinitude of primes 05, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_1,w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mEquModSym},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\mbox{ and }\mbox{aInteger}_0(\mbox{xc})\qquad\mbox{fof}(\mbox{m\_}_{818},\mbox{hypothesis})$

$( \mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq})\mbox{ and }\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xb},\mbox{xc},\mbox{xq}) ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xc},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM781$\wedge$1.p} Landau theorem 79

y0 = x0




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$x_0 = y_0\qquad\mbox{thf}(i,\mbox{axiom})$

$y_0 = x_0\qquad\mbox{thf}(\mbox{satz}_{79},\mbox{conjecture})$

\medskip\textbf{NUM070-1.p} A well-order is asymmetric






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_is\_asymmetric}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{asymmetric}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_is\_asymmetric}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM307+1.p} 23 + 34 = ?






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_{23} + n_{34}{=}x\qquad\mbox{fof}(\mbox{summ\_n23\_n34\_something},\mbox{conjecture})$

\medskip\textbf{NUM152-1.p} Corollary 3 to successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_successor\_property1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_successor\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM818$\wedge$5.p} TPS problem from IND-THMS






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

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

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

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

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

$\exists f{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to a{:}\;( ( f@ \mbox{cO} ) = g\mbox{ and }\forall \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall \mbox{xp}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{cO}\mbox{ and }\forall \mbox{xx}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) \ \Rightarrow \ ( f@ ( \mbox{cS}@ \mbox{xn} ) ) = ( h@ \mbox{xn}@ ( f@ \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cT6400A\_pme},\mbox{conjecture})$

\medskip\textbf{NUM050-1.p} Connected class property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{connected}(x,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property5}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{connected}(\mbox{restrict}(x,y,y),\mbox{domain\_of}(\mbox{symmetrization\_of}(\mbox{restrict}(x,y,y))))\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM039-1.p} Irreflexive class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{irreflexive}(x,y)\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,u)\in x\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property1}_2,\mbox{negated\_conjecture})$

$u\in y\qquad\mbox{cnf}(\mbox{prove\_irreflexive\_class\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM214-1.p} Induction up to y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_induction\_upto\_y}_1,\mbox{negated\_conjecture})$

$\mbox{null\_class}\in x\qquad\mbox{cnf}(\mbox{prove\_induction\_upto\_y}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(x,y)),x)\qquad\mbox{cnf}(\mbox{prove\_induction\_upto\_y}_3,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),\mbox{intersection}(\mbox{limit\_ordinals},y)),x)\qquad\mbox{cnf}(\mbox{prove\_induction\_upto\_y}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_induction\_upto\_y}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM306+1.p} 23 + 34 = 57






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$n_{23} + n_{34}{=}n_{57}\qquad\mbox{fof}(\mbox{sum\_n23\_n34\_n}_{57},\mbox{conjecture})$

\medskip\textbf{NUM666$\wedge$1.p} Landau theorem 16d

some (lambda u.diffprop x z u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{some}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{some},\mbox{type})$

$\mbox{diffprop}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{diffprop},\mbox{type})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ x@ y@ \mbox{xu} )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\mbox{moreis}@ y@ z\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xy}@ \mbox{xv} ) ) \ \Rightarrow \ ( \mbox{some}@ \lambda \mbox{xv}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ \mbox{xz}@ \mbox{xx}@ \mbox{xv} ) ) ) )\qquad\mbox{thf}(\mbox{satz16a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{axiom})$

$\mbox{some}@ \lambda \mbox{xu}{:}\;\mbox{nat}{:}\;( \mbox{diffprop}@ x@ z@ \mbox{xu} )\qquad\mbox{thf}(\mbox{satz16d},\mbox{conjecture})$

\medskip\textbf{NUM149-1.p} Successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers}),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_successor\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM053-1.p} Transitive ordering property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{transitive}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{transitive}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_transitive\_ordering\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM226-1.p} Rest relation property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{rest\_relation},x)\ne \mbox{rest\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM223-1.p} Rest relation is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{rest\_relation})\qquad\mbox{cnf}(\mbox{prove\_rest\_relation\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM859=1.p} Basic upper bound replace maximum with less-or-equal

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\mbox{summation}{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{summation\_type},\mbox{type})$

$\mbox{ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{ub\_type},\mbox{type})$

$\mbox{model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_max\_type},\mbox{type})$

$\mbox{model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_ub\_type},\mbox{type})$

$\mbox{minsol\_model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_max\_type},\mbox{type})$

$\mbox{minsol\_model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_ub\_type},\mbox{type})$

$\mbox{max}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{max\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff \mbox{\$lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{tff}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{\$lesseq}(y,x) )\qquad\mbox{tff}(\mbox{max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{\$lesseq}(x,y) )\qquad\mbox{tff}(\mbox{max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{\$lesseq}(x,z)\mbox{ and }\mbox{\$lesseq}(y,z) ) )\qquad\mbox{tff}(\mbox{ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{\$lesseq}(\mbox{max}(x,y),n) )\qquad\mbox{tff}(\mbox{model\_max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{\$lesseq}(z,n) ) )\qquad\mbox{tff}(\mbox{model\_ub}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{tff}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM355+1.p} XY (X+Y = 8) =$>$ X = 4, Y = 4






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;( x + y{=}n_8 \ \Rightarrow \ ( x = n_4\mbox{ and }y = n_4 ) )\qquad\mbox{fof}(\mbox{anti\_sum\_only\_n4\_only\_n4\_n}_8,\mbox{conjecture})$

\medskip\textbf{NUM849+2.p} qu(ind(296),imp(296))






$\forall \mbox{vd}_{454}{:}\;( \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{454}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{454})) \ \Rightarrow \ \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vsucc}(\mbox{vd}_{454})) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vsucc}(\mbox{vd}_{454}))) )\qquad\mbox{fof}(\mbox{'qu(ind(296), imp(296))'},\mbox{conjecture})$

$\forall \mbox{vd}_{451}{:}\;( \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})) \ \Rightarrow \ \mbox{vmul}(\mbox{vd}_{448},\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451}),\mbox{vd}_{449})) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vsucc}(\mbox{vd}_{451}))) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(292), 1), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{451}{:}\;( \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})),\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449})) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451}),\mbox{vd}_{449})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(292), 1), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{451}{:}\;( \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}),\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})),\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(292), 1), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{451}{:}\;( \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},\mbox{vd}_{451})) \ \Rightarrow \ \mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vsucc}(\mbox{vd}_{451})) = \mbox{vplus}(\mbox{vmul}(\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}),\mbox{vd}_{451}),\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(292), 1), 3)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{448},\mbox{vd}_{449}) = \mbox{vmul}(\mbox{vd}_{448},\mbox{vmul}(\mbox{vd}_{449},v_1))\qquad\mbox{fof}(\mbox{'holds(293, 450, 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{432},\mbox{vd}_{433},\mbox{vd}_{434}{:}\;\mbox{vmul}(\mbox{vd}_{432},\mbox{vplus}(\mbox{vd}_{433},\mbox{vd}_{434})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{432},\mbox{vd}_{433}),\mbox{vmul}(\mbox{vd}_{432},\mbox{vd}_{434}))\qquad\mbox{fof}(\mbox{'ass(cond(281, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{418},\mbox{vd}_{419}{:}\;\mbox{vmul}(\mbox{vd}_{418},\mbox{vd}_{419}) = \mbox{vmul}(\mbox{vd}_{419},\mbox{vd}_{418})\qquad\mbox{fof}(\mbox{'ass(cond(270, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{408},\mbox{vd}_{409}{:}\;\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{408}),\mbox{vd}_{409}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{408},\mbox{vd}_{409}),\mbox{vd}_{409})\qquad\mbox{fof}(\mbox{'ass(cond(261, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

\medskip\textbf{NUM048-1.p} Connected class property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(y,y),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{connected}(\mbox{restrict}(x,y,y),\mbox{domain\_of}(\mbox{symmetrization\_of}(\mbox{restrict}(x,y,y))))\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM748$\wedge$1.p} Landau theorem 58

eq (fr (pl (ts (num x) (den y)) (ts (num y) (den x))) (ts (den x)
 (den y))) (fr (pl (ts (num y) (den x)) (ts (num x) (den y)))
 (ts (den y) (den x)))




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

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

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

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

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

$\mbox{fr}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{frac}\qquad\mbox{thf}(\mbox{fr},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{pl}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_6,\mbox{axiom})$

$\mbox{eq}@ ( \mbox{fr}@ ( \mbox{pl}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) ) )@ ( \mbox{ts}@ ( \mbox{den}@ x )@ ( \mbox{den}@ y ) ) )@ ( \mbox{fr}@ ( \mbox{pl}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) ) )@ ( \mbox{ts}@ ( \mbox{den}@ y )@ ( \mbox{den}@ x ) ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{conjecture})$

\medskip\textbf{NUM255-1.p} Transfinite recursion property 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property11}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x))\ne \mbox{successor}(\mbox{domain\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property11}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM859+1.p} Basic upper bound replace maximum with less-or-equal

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\forall x{:}\;\mbox{lesseq}(x,x)\qquad\mbox{fof}(\mbox{lesseq\_ref},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,z) ) \ \Rightarrow \ \mbox{lesseq}(x,z) )\qquad\mbox{fof}(\mbox{lesseq\_trans},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{lesseq}(x,y)\mbox{ and }\mbox{lesseq}(y,x) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{lesseq\_antisymmetric},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\mbox{ or }\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{lesseq\_total},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(z + x,z + y) )\qquad\mbox{fof}(\mbox{sum\_monotone}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{lesseq}(x,y)\iff \mbox{lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{fof}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{lesseq}(y,x) )\qquad\mbox{fof}(\mbox{max}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{lesseq}(x,y) )\qquad\mbox{fof}(\mbox{max}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{lesseq}(x,z)\mbox{ and }\mbox{lesseq}(y,z) ) )\qquad\mbox{fof}(\mbox{ub},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_max}(x,y,n)\iff \mbox{lesseq}(\mbox{max}(x,y),n) )\qquad\mbox{fof}(\mbox{model\_max}_2,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{model\_ub}(x,y,n)\iff \exists z{:}\;( \mbox{ub}(x,y,z)\mbox{ and }\mbox{lesseq}(z,n) ) )\qquad\mbox{fof}(\mbox{model\_ub}_2,\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x,y,n{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{lesseq}(n,z) ) ) )\qquad\mbox{fof}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x,y,z{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{fof}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM425+1.p} Fuerstenberg's infinitude of primes 04\_01, 00 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{704},\mbox{hypothesis})$

$\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_}_{724},\mbox{hypothesis})$

$\exists w_0{:}\;( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},w_0) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb})) )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM260-1.p} Alternate 4 for transfinite induction, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{recursion}(x,y,z))\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM869=1.p} Sum X and Y is X






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(x,y) = x\qquad\mbox{tff}(\mbox{anti\_sum\_x\_y\_x},\mbox{conjecture})$

\medskip\textbf{NUM701$\wedge$1.p} Landau theorem 26a

lessis y x




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

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

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\mbox{less}@ y@ ( \mbox{suc}@ x )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xy}@ ( \mbox{pl}@ \mbox{xx}@ n_1 ) ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{26},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} ) = ( \mbox{pl}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz4e},\mbox{axiom})$

$\mbox{lessis}@ y@ x\qquad\mbox{thf}(\mbox{satz26a},\mbox{conjecture})$

\medskip\textbf{NUM795$\wedge$1.p} Landau theorem 99c

less (pl x0 z0) (pl y0 u0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$z_0{:}\;\mbox{rat}\qquad\mbox{thf}(z_0,\mbox{type})$

$u_0{:}\;\mbox{rat}\qquad\mbox{thf}(u_0,\mbox{type})$

$\mbox{lessis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{lessis}@ x_0@ y_0\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ z_0@ u_0\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{rat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{82},\mbox{axiom})$

$\mbox{moreis}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat},\mbox{xz}_0{:}\;\mbox{rat},\mbox{xu}_0{:}\;\mbox{rat}{:}\;( ( \mbox{moreis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( ( \mbox{more}@ \mbox{xz}_0@ \mbox{xu}_0 ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}_0@ \mbox{xz}_0 )@ ( \mbox{pl}@ \mbox{xy}_0@ \mbox{xu}_0 ) ) ) )\qquad\mbox{thf}(\mbox{satz99a},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{lessis}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{85},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;( ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ ( \mbox{more}@ \mbox{xy}_0@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{satz}_{83},\mbox{axiom})$

$\mbox{less}@ ( \mbox{pl}@ x_0@ z_0 )@ ( \mbox{pl}@ y_0@ u_0 )\qquad\mbox{thf}(\mbox{satz99c},\mbox{conjecture})$

\medskip\textbf{NUM741$\wedge$1.p} Landau theorem 50

less (ts (num x) (den z)) (ts (num z) (den x))




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{num}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{num},\mbox{type})$

$\mbox{den}{:}\;\mbox{frac}\to \mbox{nat}\qquad\mbox{thf}(\mbox{den},\mbox{type})$

$\mbox{less}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ y ) )@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{less}@ ( \mbox{ts}@ ( \mbox{num}@ y )@ ( \mbox{den}@ z ) )@ ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ y ) )\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz33c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{less}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{less}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz34a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{ts}@ \mbox{xx}@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_{31},\mbox{axiom})$

$\mbox{less}@ ( \mbox{ts}@ ( \mbox{num}@ x )@ ( \mbox{den}@ z ) )@ ( \mbox{ts}@ ( \mbox{num}@ z )@ ( \mbox{den}@ x ) )\qquad\mbox{thf}(\mbox{satz}_{50},\mbox{conjecture})$

\medskip\textbf{NUM870=1.p} Sum is not a function






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$sum}(x,y) = z_2\mbox{ and }z_1\ne z_2 )\qquad\mbox{tff}(\mbox{anti\_unique\_sum},\mbox{conjecture})$

\medskip\textbf{NUM054-1.p} Asymmetric class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{asymmetric}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property1}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,u)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_asymmetric\_class\_property1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM064-1.p} Least(xr,u) is unique






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(u,y)\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique}_2,\mbox{negated\_conjecture})$

$v\in u\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique}_3,\mbox{negated\_conjecture})$

$\mbox{restrict}(\mbox{xr},u,\mbox{singleton}(v)) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique}_4,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{xr},u)\ne v\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM832$\wedge$5.p} TPS problem from PETER-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ c_0@ \mbox{xn}@ ( \mbox{cS}@ \mbox{xn} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ \mbox{xm}@ ( \mbox{cS}@ c_0 )@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cR}@ ( \mbox{cS}@ \mbox{xm} )@ c_0@ \mbox{xk} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i},\mbox{xl}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ ( \mbox{cS}@ \mbox{xm} )@ \mbox{xn}@ \mbox{xl} ) \ \Rightarrow \ ( ( \mbox{cR}@ \mbox{xm}@ \mbox{xl}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cR}@ ( \mbox{cS}@ \mbox{xm} )@ ( \mbox{cS}@ \mbox{xn} )@ \mbox{xk} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx}@ \mbox{xy}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cTHM}_{604},\mbox{conjecture})$

\medskip\textbf{NUM173-1.p} Successor property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_successor\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{image}(\mbox{successor\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_successor\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM425+3.p} Fuerstenberg's infinitude of primes 04\_01, 02 expansion






$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mIntegers},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{00})\qquad\mbox{fof}(\mbox{mIntZero},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{sz}_{10})\qquad\mbox{fof}(\mbox{mIntOne},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{smndt}_0(w_0)) )\qquad\mbox{fof}(\mbox{mIntNeg},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtpldt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntPlus},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{aInteger}_0(\mbox{sdtasdt}_0(w_0,w_1)) )\qquad\mbox{fof}(\mbox{mIntMult},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mAddAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtpldt}_0(w_0,w_1) = \mbox{sdtpldt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mAddComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{sz}_{00}) = w_0\mbox{ and }w_0 = \mbox{sdtpldt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mAddZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_0)) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtpldt}_0(\mbox{smndt}_0(w_0),w_0) ) )\qquad\mbox{fof}(\mbox{mAddNeg},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,\mbox{sdtasdt}_0(w_1,w_2)) = \mbox{sdtasdt}_0(\mbox{sdtasdt}_0(w_0,w_1),w_2) )\qquad\mbox{fof}(\mbox{mMulAsso},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sdtasdt}_0(w_1,w_0) )\qquad\mbox{fof}(\mbox{mMulComm},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{10}) = w_0\mbox{ and }w_0 = \mbox{sdtasdt}_0(\mbox{sz}_{10},w_0) ) )\qquad\mbox{fof}(\mbox{mMulOne},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sdtpldt}_0(w_1,w_2)) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_1),\mbox{sdtasdt}_0(w_0,w_2))\mbox{ and }\mbox{sdtasdt}_0(\mbox{sdtpldt}_0(w_0,w_1),w_2) = \mbox{sdtpldt}_0(\mbox{sdtasdt}_0(w_0,w_2),\mbox{sdtasdt}_0(w_1,w_2)) ) )\qquad\mbox{fof}(\mbox{mDistrib},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,\mbox{sz}_{00}) = \mbox{sz}_{00}\mbox{ and }\mbox{sz}_{00} = \mbox{sdtasdt}_0(\mbox{sz}_{00},w_0) ) )\qquad\mbox{fof}(\mbox{mMulZero},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ ( \mbox{sdtasdt}_0(\mbox{smndt}_0(\mbox{sz}_{10}),w_0) = \mbox{smndt}_0(w_0)\mbox{ and }\mbox{smndt}_0(w_0) = \mbox{sdtasdt}_0(w_0,\mbox{smndt}_0(\mbox{sz}_{10})) ) )\qquad\mbox{fof}(\mbox{mMulMinOne},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1) ) \ \Rightarrow \ ( \mbox{sdtasdt}_0(w_0,w_1) = \mbox{sz}_{00} \ \Rightarrow \ ( w_0 = \mbox{sz}_{00}\mbox{ or }w_1 = \mbox{sz}_{00} ) ) )\qquad\mbox{fof}(\mbox{mZeroDiv},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aInteger}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aDivisorOf}_0(w_1,w_0)\iff ( \mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00}\mbox{ and }\exists w_2{:}\;( \mbox{aInteger}_0(w_2)\mbox{ and }\mbox{sdtasdt}_0(w_1,w_2) = w_0 ) ) ) )\qquad\mbox{fof}(\mbox{mDivisor},\mbox{definition})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }\mbox{aInteger}_0(w_2)\mbox{ and }w_2\ne \mbox{sz}_{00} ) \ \Rightarrow \ ( \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_1,w_2)\iff \mbox{aDivisorOf}_0(w_2,\mbox{sdtpldt}_0(w_0,\mbox{smndt}_0(w_1))) ) )\qquad\mbox{fof}(\mbox{mEquMod},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{aInteger}_0(w_1)\mbox{ and }w_1\ne \mbox{sz}_{00} ) \ \Rightarrow \ \mbox{sdteqdtlpzmzozddtrp}_0(w_0,w_0,w_1) )\qquad\mbox{fof}(\mbox{mEquModRef},\mbox{axiom})$

$\mbox{aInteger}_0(\mbox{xa})\mbox{ and }\mbox{aInteger}_0(\mbox{xb})\mbox{ and }\mbox{aInteger}_0(\mbox{xq})\mbox{ and }\mbox{xq}\ne \mbox{sz}_{00}\qquad\mbox{fof}(\mbox{m\_}_{704},\mbox{hypothesis})$

$\exists w_0{:}\;( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},w_0) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb})) )\mbox{ and }\mbox{aDivisorOf}_0(\mbox{xq},\mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb})))\mbox{ and }\mbox{sdteqdtlpzmzozddtrp}_0(\mbox{xa},\mbox{xb},\mbox{xq})\qquad\mbox{fof}(\mbox{m\_}_{724},\mbox{hypothesis})$

$\exists w_0{:}\;( \mbox{aInteger}_0(w_0)\mbox{ and }\mbox{sdtasdt}_0(\mbox{xq},w_0) = \mbox{sdtpldt}_0(\mbox{xa},\mbox{smndt}_0(\mbox{xb})) )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM900=1.p} Difference greater 0 implies less






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( ( \mbox{\$difference}(x,y) = z\mbox{ and }\mbox{\$less}(0,z) ) \ \Rightarrow \ \mbox{\$less}(y,x) )\qquad\mbox{tff}(\mbox{difference\_greater},\mbox{conjecture})$

\medskip\textbf{NUM154-1.p} Corollary 5 to successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_5\_to\_successor\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM257-1.p} Transfinite recursion property 13






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property13}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x)\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property13}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM676$\wedge$1.p} Landau theorem 19g

more (pl x z) (pl y u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$x = y\qquad\mbox{thf}(i,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{more}@ z@ u\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{pl}@ \mbox{xz}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz19d},\mbox{axiom})$

$\mbox{more}@ ( \mbox{pl}@ x@ z )@ ( \mbox{pl}@ y@ u )\qquad\mbox{thf}(\mbox{satz19g},\mbox{conjecture})$

\medskip\textbf{NUM681$\wedge$1.p} Landau theorem 20b

x = y




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$( \mbox{pl}@ x@ z ) = ( \mbox{pl}@ y@ z )\qquad\mbox{thf}(i,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;\neg\,( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy}\qquad\mbox{thf}(\mbox{satz10b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{less}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{more}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz10a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz19a},\mbox{axiom})$

$x = y\qquad\mbox{thf}(\mbox{satz20b},\mbox{conjecture})$

\medskip\textbf{NUM324+1.p} ? + -5 = -3






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;x + \mbox{nn}_5{=}\mbox{nn}_3\qquad\mbox{fof}(\mbox{sum\_what\_nn5\_nn}_3,\mbox{conjecture})$

\medskip\textbf{NUM357+1.p} ?X (X + 0 != X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x,y{:}\;( x + n_0{=}y\mbox{ and }y\ne x )\qquad\mbox{fof}(\mbox{anti\_sum\_identity},\mbox{conjecture})$

\medskip\textbf{NUM002-1.p} (X - Y) + Z = X + (Z - Y)






$\mbox{include}(\mbox{'Axioms/NUM002-0.ax'})$

$\neg\,\mbox{subtract}(a,b) + c{=}a + \mbox{subtract}(c,b)\qquad\mbox{cnf}(\mbox{prove\_equation},\mbox{negated\_conjecture})$

\medskip\textbf{NUM017-2.p} Square root of this prime is irrational

If a is prime, and a is not b$\wedge$2/c$\wedge$2, then the square root
 of a is irrational.




$a\cdot b{=}a\cdot b\qquad\mbox{cnf}(\mbox{closure\_of\_product},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot e{=}b\mbox{ and }a\cdot d{=}f ) \ \Rightarrow \ f\cdot e{=}c\qquad\mbox{cnf}(\mbox{product\_associativity}_1,\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }d\cdot b{=}e\mbox{ and }f\cdot d{=}a ) \ \Rightarrow \ f\cdot e{=}c\qquad\mbox{cnf}(\mbox{product\_associativity}_2,\mbox{axiom})$

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

$( a\cdot b{=}c\mbox{ and }a\cdot d{=}c ) \ \Rightarrow \ b = d\qquad\mbox{cnf}(\mbox{product\_left\_cancellation},\mbox{axiom})$

$( \mbox{divides}(a,b)\mbox{ and }\mbox{divides}(c,a) ) \ \Rightarrow \ \mbox{divides}(c,b)\qquad\mbox{cnf}(\mbox{transitivity\_of\_divides},\mbox{axiom})$

$( a\cdot b{=}c\mbox{ and }a\cdot b{=}d ) \ \Rightarrow \ d = c\qquad\mbox{cnf}(\mbox{well\_defined\_product},\mbox{axiom})$

$\mbox{divides}(a,b) \ \Rightarrow \ a\cdot \mbox{second\_divided\_by\_1st}(a,b){=}b\qquad\mbox{cnf}(\mbox{divides\_implies\_product},\mbox{axiom})$

$a\cdot b{=}c \ \Rightarrow \ \mbox{divides}(a,c)\qquad\mbox{cnf}(\mbox{product\_divisible\_by\_operand},\mbox{axiom})$

$( \mbox{divides}(a,b)\mbox{ and }c\cdot c{=}b\mbox{ and }\mbox{prime}(a) ) \ \Rightarrow \ \mbox{divides}(a,c)\qquad\mbox{cnf}(\mbox{primes\_lemma}_1,\mbox{axiom})$

$\mbox{prime}(a)\qquad\mbox{cnf}(\mbox{a\_is\_prime},\mbox{hypothesis})$

$b\cdot b{=}d\qquad\mbox{cnf}(\mbox{b\_squared},\mbox{hypothesis})$

$c\cdot c{=}e\qquad\mbox{cnf}(\mbox{c\_squared},\mbox{hypothesis})$

$\neg\,a\cdot e{=}d\qquad\mbox{cnf}(\mbox{a\_times\_c\_squared\_is\_not\_b\_squared},\mbox{hypothesis})$

$\mbox{divides}(a,c) \ \Rightarrow \ \neg\,\mbox{divides}(a,b)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_no\_common\_divisor},\mbox{negated\_conjecture})$

\medskip\textbf{NUM219-1.p} The domain of rest\_of(X) is the domain of X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{rest\_of}(u))\ne \mbox{domain\_of}(u)\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_rest\_of}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM902=1.p} Difference is 0/1 implies equal






$\forall x{:}\;\mbox{\$rat},y{:}\;\mbox{\$rat}{:}\;( \mbox{\$difference}(x,y) = 0/1 \ \Rightarrow \ x = y )\qquad\mbox{tff}(\mbox{rat\_difference\_problem}_{13},\mbox{conjecture})$

\medskip\textbf{NUM186-1.p} Omega property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\ne \mbox{omega}\qquad\mbox{cnf}(\mbox{prove\_omega\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM370+1.p} 0 + ! = !






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;n_0 + x{=}x\qquad\mbox{fof}(\mbox{n0\_identy\_rev},\mbox{conjecture})$

\medskip\textbf{NUM362+1.p} !XY (X+Y = 8) =$>$ X - Y = 0






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y{:}\;( x + y{=}n_8 \ \Rightarrow \ x\setminus y{=}n_0 )\qquad\mbox{fof}(\mbox{all\_sum\_same\_n}_8,\mbox{conjecture})$

\medskip\textbf{NUM371+1.p} if (X - Y) = Z and Z $>$ 0, then X $>$ Y






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z{:}\;( ( x\setminus y{=}z\mbox{ and }\mbox{less}(n_0,z) ) \ \Rightarrow \ \mbox{less}(y,x) )\qquad\mbox{fof}(\mbox{difference\_greater},\mbox{conjecture})$

\medskip\textbf{NUM756$\wedge$1.p} Landau theorem 63b

eq x y




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{eq}@ ( \mbox{pf}@ x@ z )@ ( \mbox{pf}@ y@ z )\qquad\mbox{thf}(e,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{lessf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessf},\mbox{type})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;\neg\,( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy}\qquad\mbox{thf}(\mbox{satz41b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessf}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62c},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \neg\,\mbox{eq}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \neg\,\mbox{moref}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ ( \mbox{lessf}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{satz41a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz62a},\mbox{axiom})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(\mbox{satz63b},\mbox{conjecture})$

\medskip\textbf{NUM916=1.p} Every sum left exists






$\forall u{:}\;\mbox{\$int},v{:}\;\mbox{\$int}{:}\;\exists w{:}\;\mbox{\$int}{:}\;\mbox{\$sum}(w,u) = v\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM348+1.p} X (127 + 1 = X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;n_{127} + n_1{=}x\qquad\mbox{fof}(\mbox{anti\_upper\_boundary},\mbox{conjecture})$

\medskip\textbf{NUM021-1.p} If a $\leq$ b $<$ c, then c cannot divide a






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM001-2.ax'})$

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

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

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

$\mbox{less}(b,c)\qquad\mbox{cnf}(\mbox{b\_less\_than\_c},\mbox{hypothesis})$

$\neg\,\mbox{less}(b,a)\qquad\mbox{cnf}(\mbox{b\_greater\_equal\_a},\mbox{hypothesis})$

$\mbox{divides}(c,a)\qquad\mbox{cnf}(\mbox{impossible\_c\_divides\_a},\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{prove\_a\_contradiction},\mbox{negated\_conjecture})$

\medskip\textbf{NUM286-2.p} Number theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/NUM002-0.ax'})$

\medskip\textbf{NUM921=1.p} Increasing function property






$f{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$\forall u{:}\;\mbox{\$int}{:}\;\mbox{\$greater}(f(u),u) \ \Rightarrow \ \forall v{:}\;\mbox{\$int}{:}\;\mbox{\$less}(\mbox{\$difference}(v,f(v)),0)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM531+2.p} Ramsey's Infinite Theorem 01, 01 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0)\mbox{ and }w_0 = \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM903=1.p} - - something is something






$\forall x{:}\;\mbox{\$rat}{:}\;\mbox{\$uminus}(\mbox{\$uminus}(x)) = x\qquad\mbox{tff}(\mbox{rat\_uminus\_problem}_7,\mbox{conjecture})$

\medskip\textbf{NUM155-1.p} There is no ordinal between x and x + 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_no\_ordinal\_between}_1,\mbox{negated\_conjecture})$

$y\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_no\_ordinal\_between}_2,\mbox{negated\_conjecture})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_no\_ordinal\_between}_3,\mbox{negated\_conjecture})$

$y\in \mbox{successor}(x)\qquad\mbox{cnf}(\mbox{prove\_no\_ordinal\_between}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM898=1.p} Sum and difference and less






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$difference}(x,y) = z_2\mbox{ and }\mbox{\$less}(z_2,z_1) )\qquad\mbox{tff}(\mbox{anti\_sum\_diff\_less}_2,\mbox{conjecture})$

\medskip\textbf{NUM213-1.p} Alternate 3 for transfinite induction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{null\_class}\in x\qquad\mbox{cnf}(\mbox{prove\_alternate\_3\_transfinite\_induction}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},x),x)\qquad\mbox{cnf}(\mbox{prove\_alternate\_3\_transfinite\_induction}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{intersection}(\mbox{power\_class}(x),\mbox{limit\_ordinals}),x)\qquad\mbox{cnf}(\mbox{prove\_alternate\_3\_transfinite\_induction}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{ordinal\_numbers},x)\qquad\mbox{cnf}(\mbox{prove\_alternate\_3\_transfinite\_induction}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM858=1.p} Basic upper bound replace maximum

This is an abstraction of a problem to show equivalence of two 
 given constraint models. More precisely, the task is to prove that
 the minimal solutions of a certain constraint model are preserved
 if the applications of the "maximum" function in it are replaced
 by "upper bounds" only.




$\mbox{summation}{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{summation\_type},\mbox{type})$

$\mbox{ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{ub\_type},\mbox{type})$

$\mbox{model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_max\_type},\mbox{type})$

$\mbox{model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{model\_ub\_type},\mbox{type})$

$\mbox{minsol\_model\_max}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_max\_type},\mbox{type})$

$\mbox{minsol\_model\_ub}{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{minsol\_model\_ub\_type},\mbox{type})$

$\mbox{max}{:}\;( \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{max\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$lesseq}(x,y)\iff \mbox{\$lesseq}(\mbox{summation}(x),\mbox{summation}(y)) )\qquad\mbox{tff}(\mbox{summation\_monotone},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = x\mbox{ or }\neg\,\mbox{\$lesseq}(y,x) )\qquad\mbox{tff}(\mbox{max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{max}(x,y) = y\mbox{ or }\neg\,\mbox{\$lesseq}(x,y) )\qquad\mbox{tff}(\mbox{max}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{ub}(x,y,z)\iff ( \mbox{\$lesseq}(x,z)\mbox{ and }\mbox{\$lesseq}(y,z) ) )\qquad\mbox{tff}(\mbox{ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,n)\iff n = \mbox{max}(x,y) )\qquad\mbox{tff}(\mbox{model\_max}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,n)\iff \mbox{ub}(x,y,n) )\qquad\mbox{tff}(\mbox{model\_ub}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_max}(x,y,n)\iff ( \mbox{model\_max}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_max}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_max},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},n{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,n)\iff ( \mbox{model\_ub}(x,y,n)\mbox{ and }\forall z{:}\;\mbox{\$int}{:}\;( \mbox{model\_ub}(x,y,z) \ \Rightarrow \ \mbox{\$lesseq}(n,z) ) ) )\qquad\mbox{tff}(\mbox{minsol\_model\_ub},\mbox{axiom})$

$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{minsol\_model\_ub}(x,y,z)\iff \mbox{minsol\_model\_max}(x,y,z) )\qquad\mbox{tff}(\mbox{max\_is\_ub}_1,\mbox{conjecture})$

\medskip\textbf{NUM019-1.p} Symmetry of equality can be derived






$\mbox{include}(\mbox{'Axioms/NUM001-0.ax'})$

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

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

$\neg\,\mbox{successor}(a){=}n_0\qquad\mbox{cnf}(\mbox{zero\_is\_the\_first\_number},\mbox{hypothesis})$

$a{=}\mbox{aa}\qquad\mbox{cnf}(\mbox{a\_equals\_aa},\mbox{hypothesis})$

$\neg\,\mbox{aa}{=}a\qquad\mbox{cnf}(\mbox{prove\_b\_equals\_a},\mbox{negated\_conjecture})$

\medskip\textbf{NUM090-1.p} Corollary to sections property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{section}(\mbox{element\_relation},y,z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property1}_1,\mbox{negated\_conjecture})$

$u\in \mbox{intersection}(v,z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property1}_2,\mbox{negated\_conjecture})$

$v\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property1}_3,\mbox{negated\_conjecture})$

$\neg\,u\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM106-1.p} Ordinal property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property7}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property7}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_ordinal\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM150-1.p} Corollary 1 to successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{omega},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_successor\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM343+1.p} -1 $<$ ? \& ? $<$ 1 =$>$ 21 + ? = 21






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;( ( \mbox{less}(\mbox{nn}_1,x)\mbox{ and }\mbox{less}(x,n_1) ) \ \Rightarrow \ n_{21} + x{=}n_{21} )\qquad\mbox{fof}(\mbox{sum\_something\_n0\_samething},\mbox{conjecture})$

\medskip\textbf{NUM694$\wedge$1.p} Landau theorem 24a

lessis n\_1 x




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

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

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{moreis}@ \mbox{xx}@ n_1 )\qquad\mbox{thf}(\mbox{satz}_{24},\mbox{axiom})$

$\mbox{lessis}@ n_1@ x\qquad\mbox{thf}(\mbox{satz24a},\mbox{conjecture})$

\medskip\textbf{NUM707$\wedge$1.p} Landau theorem 28f

pl (ts x y) x = ts x (suc y)




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

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

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

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ ( \mbox{suc}@ \mbox{xy} ) ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz28b},\mbox{axiom})$

$( \mbox{pl}@ ( \mbox{ts}@ x@ y )@ x ) = ( \mbox{ts}@ x@ ( \mbox{suc}@ y ) )\qquad\mbox{thf}(\mbox{satz28f},\mbox{conjecture})$

\medskip\textbf{NUM166-1.p} Corollary to omega definition, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{successor}(\mbox{null\_class})\in \mbox{omega}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_omega\_defn2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM082-1.p} Uniqueness of the least element of a non-empty subset






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique\_in\_non\_empty\_set}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(u,y)\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique\_in\_non\_empty\_set}_2,\mbox{negated\_conjecture})$

$v\in u\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique\_in\_non\_empty\_set}_3,\mbox{negated\_conjecture})$

$\mbox{segment}(\mbox{xr},u,v) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique\_in\_non\_empty\_set}_4,\mbox{negated\_conjecture})$

$\mbox{least}(\mbox{xr},u)\ne v\qquad\mbox{cnf}(\mbox{prove\_least\_is\_unique\_in\_non\_empty\_set}_5,\mbox{negated\_conjecture})$

\medskip\textbf{NUM841+2.p} holds(214,352,0)






$\mbox{greater}(\mbox{vplus}(\mbox{vd}_{344},\mbox{vd}_{347}),\mbox{vplus}(\mbox{vd}_{345},\mbox{vd}_{348}))\qquad\mbox{fof}(\mbox{'holds(214, 352, 0)'},\mbox{conjecture})$

$\mbox{vplus}(\mbox{vd}_{348},\mbox{vd}_{345}) = \mbox{vplus}(\mbox{vd}_{345},\mbox{vd}_{348})\qquad\mbox{fof}(\mbox{'holds(213, 351, 2)'},\mbox{axiom})$

$\mbox{greater}(\mbox{vplus}(\mbox{vd}_{347},\mbox{vd}_{345}),\mbox{vplus}(\mbox{vd}_{348},\mbox{vd}_{345}))\qquad\mbox{fof}(\mbox{'holds(213, 351, 1)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vd}_{345},\mbox{vd}_{347}) = \mbox{vplus}(\mbox{vd}_{347},\mbox{vd}_{345})\qquad\mbox{fof}(\mbox{'holds(213, 351, 0)'},\mbox{axiom})$

$\mbox{greater}(\mbox{vplus}(\mbox{vd}_{344},\mbox{vd}_{347}),\mbox{vplus}(\mbox{vd}_{345},\mbox{vd}_{347}))\qquad\mbox{fof}(\mbox{'holds(212, 350, 0)'},\mbox{axiom})$

$\mbox{greater}(\mbox{vd}_{347},\mbox{vd}_{348})\qquad\mbox{fof}(\mbox{'holds(conjunct2(211), 349, 0)'},\mbox{axiom})$

$\mbox{greater}(\mbox{vd}_{344},\mbox{vd}_{345})\qquad\mbox{fof}(\mbox{'holds(conjunct1(211), 346, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{328},\mbox{vd}_{329},\mbox{vd}_{330}{:}\;( \mbox{less}(\mbox{vplus}(\mbox{vd}_{328},\mbox{vd}_{330}),\mbox{vplus}(\mbox{vd}_{329},\mbox{vd}_{330})) \ \Rightarrow \ \mbox{less}(\mbox{vd}_{328},\mbox{vd}_{329}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(202), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{328},\mbox{vd}_{329},\mbox{vd}_{330}{:}\;( \mbox{vplus}(\mbox{vd}_{328},\mbox{vd}_{330}) = \mbox{vplus}(\mbox{vd}_{329},\mbox{vd}_{330}) \ \Rightarrow \ \mbox{vd}_{328} = \mbox{vd}_{329} )\qquad\mbox{fof}(\mbox{'ass(cond(goal(202), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{328},\mbox{vd}_{329},\mbox{vd}_{330}{:}\;( \mbox{greater}(\mbox{vplus}(\mbox{vd}_{328},\mbox{vd}_{330}),\mbox{vplus}(\mbox{vd}_{329},\mbox{vd}_{330})) \ \Rightarrow \ \mbox{greater}(\mbox{vd}_{328},\mbox{vd}_{329}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(202), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{301},\mbox{vd}_{302},\mbox{vd}_{303}{:}\;( \mbox{less}(\mbox{vd}_{301},\mbox{vd}_{302}) \ \Rightarrow \ \mbox{less}(\mbox{vplus}(\mbox{vd}_{301},\mbox{vd}_{303}),\mbox{vplus}(\mbox{vd}_{302},\mbox{vd}_{303})) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(193), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{301},\mbox{vd}_{302},\mbox{vd}_{303}{:}\;( \mbox{vd}_{301} = \mbox{vd}_{302} \ \Rightarrow \ \mbox{vplus}(\mbox{vd}_{301},\mbox{vd}_{303}) = \mbox{vplus}(\mbox{vd}_{302},\mbox{vd}_{303}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(193), 0), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{301},\mbox{vd}_{302},\mbox{vd}_{303}{:}\;( \mbox{greater}(\mbox{vd}_{301},\mbox{vd}_{302}) \ \Rightarrow \ \mbox{greater}(\mbox{vplus}(\mbox{vd}_{301},\mbox{vd}_{303}),\mbox{vplus}(\mbox{vd}_{302},\mbox{vd}_{303})) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(193), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{295},\mbox{vd}_{296}{:}\;\mbox{greater}(\mbox{vplus}(\mbox{vd}_{295},\mbox{vd}_{296}),\mbox{vd}_{295})\qquad\mbox{fof}(\mbox{'ass(cond(189, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{226},\mbox{vd}_{227}{:}\;( \mbox{less}(\mbox{vd}_{226},\mbox{vd}_{227}) \ \Rightarrow \ \mbox{greater}(\mbox{vd}_{227},\mbox{vd}_{226}) )\qquad\mbox{fof}(\mbox{'ass(cond(147, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{208},\mbox{vd}_{209}{:}\;( \mbox{greater}(\mbox{vd}_{208},\mbox{vd}_{209}) \ \Rightarrow \ \mbox{less}(\mbox{vd}_{209},\mbox{vd}_{208}) )\qquad\mbox{fof}(\mbox{'ass(cond(140, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \mbox{vd}_{203} = \mbox{vd}_{204}\mbox{ or }\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{203},\mbox{vd}_{204}{:}\;( \neg\,\mbox{greater}(\mbox{vd}_{203},\mbox{vd}_{204})\mbox{ or }\neg\,\mbox{less}(\mbox{vd}_{203},\mbox{vd}_{204}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(130), 0), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{193},\mbox{vd}_{194}{:}\;( \mbox{greater}(\mbox{vd}_{194},\mbox{vd}_{193})\iff \exists \mbox{vd}_{196}{:}\;\mbox{vd}_{194} = \mbox{vplus}(\mbox{vd}_{193},\mbox{vd}_{196}) )\qquad\mbox{fof}(\mbox{'def(cond(conseq(axiom(3)), 11), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{120},\mbox{vd}_{121}{:}\;( \mbox{vd}_{120}\ne \mbox{vd}_{121}\mbox{ or }\neg\,\exists \mbox{vd}_{123}{:}\;\mbox{vd}_{120} = \mbox{vplus}(\mbox{vd}_{121},\mbox{vd}_{123}) )\qquad\mbox{fof}(\mbox{'ass(cond(goal(88), 0), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

\medskip\textbf{NUM067-1.p} Well ordering property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property2}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property2}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,u)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property2}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM418$\wedge$1.p} Find N such that N + 3 = 3






$\mbox{include}(\mbox{'Axioms/NUM006\string^0.ax'})$

$\exists n{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;(  + @ n@ \mbox{three} ) = \mbox{three}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{NUM032-1.p} Symmetrization property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{symmetrization\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\ne x'\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM239-1.p} Transfinite recursion lemma 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma8}_1,\mbox{negated\_conjecture})$

$y\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma8}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(x)\in \mbox{domain\_of}(y)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma8}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{rest\_of}(x),\mbox{rest\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_lemma8}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM536+2.p} Ramsey's Infinite Theorem 05, 01 expansion






$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mSetSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aElement}_0(w_0) \ \Rightarrow \ \mbox{\$true} )\qquad\mbox{fof}(\mbox{mElmSort},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{aElement}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mEOfElem},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isFinite}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mFinRel},\mbox{axiom})$

$\forall w_0{:}\;( w_0 = \mbox{slcrc}_0\iff ( \mbox{aSet}_0(w_0)\mbox{ and }\neg\,\exists w_1{:}\;\mbox{aElementOf}_0(w_1,w_0) ) )\qquad\mbox{fof}(\mbox{mDefEmp},\mbox{definition})$

$\mbox{isFinite}_0(\mbox{slcrc}_0)\qquad\mbox{fof}(\mbox{mEmpFin},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ ( \mbox{isCountable}_0(w_0) \ \Rightarrow \ \mbox{\$true} ) )\qquad\mbox{fof}(\mbox{mCntRel},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ \neg\,\mbox{isFinite}_0(w_0) )\qquad\mbox{fof}(\mbox{mCountNFin},\mbox{axiom})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isCountable}_0(w_0) ) \ \Rightarrow \ w_0\ne \mbox{slcrc}_0 )\qquad\mbox{fof}(\mbox{mCountNFin}_{01},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0)\iff ( \mbox{aSet}_0(w_1)\mbox{ and }\forall w_2{:}\;( \mbox{aElementOf}_0(w_2,w_1) \ \Rightarrow \ \mbox{aElementOf}_0(w_2,w_0) ) ) ) )\qquad\mbox{fof}(\mbox{mDefSub},\mbox{definition})$

$\forall w_0{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{isFinite}_0(w_0) ) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aSubsetOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{isFinite}_0(w_1) ) )\qquad\mbox{fof}(\mbox{mSubFSet},\mbox{axiom})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_0) )\qquad\mbox{fof}(\mbox{mSubRefl},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_0) ) \ \Rightarrow \ w_0 = w_1 ) )\qquad\mbox{fof}(\mbox{mSubASymm},\mbox{axiom})$

$\forall w_0,w_1,w_2{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aSet}_0(w_1)\mbox{ and }\mbox{aSet}_0(w_2) ) \ \Rightarrow \ ( ( \mbox{aSubsetOf}_0(w_0,w_1)\mbox{ and }\mbox{aSubsetOf}_0(w_1,w_2) ) \ \Rightarrow \ \mbox{aSubsetOf}_0(w_0,w_2) ) )\qquad\mbox{fof}(\mbox{mSubTrans},\mbox{axiom})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtpldt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }( \mbox{aElementOf}_0(w_3,w_0)\mbox{ or }w_3 = w_1 ) ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefCons},\mbox{definition})$

$\forall w_0,w_1{:}\;( ( \mbox{aSet}_0(w_0)\mbox{ and }\mbox{aElement}_0(w_1) ) \ \Rightarrow \ \forall w_2{:}\;( w_2 = \mbox{sdtmndt}_0(w_0,w_1)\iff ( \mbox{aSet}_0(w_2)\mbox{ and }\forall w_3{:}\;( \mbox{aElementOf}_0(w_3,w_2)\iff ( \mbox{aElement}_0(w_3)\mbox{ and }\mbox{aElementOf}_0(w_3,w_0)\mbox{ and }w_3\ne w_1 ) ) ) ) )\qquad\mbox{fof}(\mbox{mDefDiff},\mbox{definition})$

$\forall w_0{:}\;( \mbox{aSet}_0(w_0) \ \Rightarrow \ \forall w_1{:}\;( \mbox{aElementOf}_0(w_1,w_0) \ \Rightarrow \ \mbox{sdtpldt}_0(\mbox{sdtmndt}_0(w_0,w_1),w_1) = w_0 ) )\qquad\mbox{fof}(\mbox{mConsDiff},\mbox{axiom})$

$\mbox{aElement}_0(\mbox{xx})\mbox{ and }\mbox{aSet}_0(\mbox{xS})\qquad\mbox{fof}(\mbox{m\_}_{679},\mbox{hypothesis})$

$\neg\,\mbox{aElementOf}_0(\mbox{xx},\mbox{xS})\qquad\mbox{fof}(\mbox{m\_\_679}_{02},\mbox{hypothesis})$

$( \mbox{aSet}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}))\mbox{ and }\forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}))\iff ( \mbox{aElement}_0(w_0)\mbox{ and }( \mbox{aElementOf}_0(w_0,\mbox{xS})\mbox{ or }w_0 = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( ( \mbox{aSet}_0(\mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\mbox{ and }\forall w_0{:}\;( \mbox{aElementOf}_0(w_0,\mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}))\iff ( \mbox{aElement}_0(w_0)\mbox{ and }\mbox{aElementOf}_0(w_0,\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}))\mbox{ and }w_0\ne \mbox{xx} ) ) ) \ \Rightarrow \ \mbox{sdtmndt}_0(\mbox{sdtpldt}_0(\mbox{xS},\mbox{xx}),\mbox{xx}) = \mbox{xS} )\qquad\mbox{fof}(\mbox{m\_\_},\mbox{conjecture})$

\medskip\textbf{NUM265-1.p} Ordinal addition property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_add}(x,\mbox{null\_class})\ne x\qquad\mbox{cnf}(\mbox{prove\_ordinal\_addition\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM833$\wedge$5.p} TPS problem from PETER-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cIND}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIND\_type},\mbox{type})$

$\mbox{cIND} = ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cIND\_def},\mbox{definition})$

$( \mbox{cIND}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} ) = ( \mbox{cS}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xn} )\ne c_0 ) \ \Rightarrow \ \exists \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ c_0@ \mbox{xn}@ ( \mbox{cS}@ \mbox{xn} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xm}@ ( \mbox{cS}@ c_0 )@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ c_0@ \mbox{xk} ) )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i},\mbox{xl}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ \mbox{xn}@ \mbox{xl} ) \ \Rightarrow \ ( ( \mbox{xr}@ \mbox{xm}@ \mbox{xl}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xr}@ ( \mbox{cS}@ \mbox{xm} )@ ( \mbox{cS}@ \mbox{xn} )@ \mbox{xk} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xx}@ \mbox{xy}@ y ) \ \Rightarrow \ x = y ) ) )\qquad\mbox{thf}(\mbox{cTHM605\_pme},\mbox{conjecture})$

\medskip\textbf{NUM877=1.p} Difference identity






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$difference}(x,y) = 0 \ \Rightarrow \ x = y )\qquad\mbox{tff}(\mbox{diff\_identity},\mbox{conjecture})$

\medskip\textbf{NUM785$\wedge$1.p} Landau theorem 81d

~(more x0 y0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\neg\,\mbox{less}@ x_0@ y_0 \ \Rightarrow \ ( \mbox{is}@ x_0@ y_0 )\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;\neg\,( ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\neg\,( ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) \ \Rightarrow \ \neg\,\mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0\qquad\mbox{thf}(\mbox{satz81b},\mbox{axiom})$

$\neg\,\mbox{more}@ x_0@ y_0\qquad\mbox{thf}(\mbox{satz81d},\mbox{conjecture})$

\medskip\textbf{NUM846+2.p} holds(286,441,2)






$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 2)'},\mbox{conjecture})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(286, 441, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vsucc}(\mbox{vd}_{439}))) = \mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})))\qquad\mbox{fof}(\mbox{'holds(286, 441, 0)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},\mbox{vd}_{439})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{439}))\qquad\mbox{fof}(\mbox{'holds(285, 440, 0)'},\mbox{axiom})$

$\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vmul}(\mbox{vd}_{436},v_1))\qquad\mbox{fof}(\mbox{'holds(284, 438, 2)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{437})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{436},\mbox{vd}_{437}),\mbox{vd}_{436})\qquad\mbox{fof}(\mbox{'holds(284, 438, 1)'},\mbox{axiom})$

$\mbox{vmul}(\mbox{vd}_{436},\mbox{vplus}(\mbox{vd}_{437},v_1)) = \mbox{vmul}(\mbox{vd}_{436},\mbox{vsucc}(\mbox{vd}_{437}))\qquad\mbox{fof}(\mbox{'holds(284, 438, 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{418},\mbox{vd}_{419}{:}\;\mbox{vmul}(\mbox{vd}_{418},\mbox{vd}_{419}) = \mbox{vmul}(\mbox{vd}_{419},\mbox{vd}_{418})\qquad\mbox{fof}(\mbox{'ass(cond(270, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{400}{:}\;\mbox{vmul}(v_1,\mbox{vd}_{400}) = \mbox{vd}_{400}\qquad\mbox{fof}(\mbox{'ass(cond(253, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{386},\mbox{vd}_{387}{:}\;( \mbox{less}(\mbox{vd}_{386},\mbox{vplus}(\mbox{vd}_{387},v_1)) \ \Rightarrow \ \mbox{leq}(\mbox{vd}_{386},\mbox{vd}_{387}) )\qquad\mbox{fof}(\mbox{'ass(cond(241, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{375},\mbox{vd}_{376}{:}\;( \mbox{greater}(\mbox{vd}_{375},\mbox{vd}_{376}) \ \Rightarrow \ \mbox{geq}(\mbox{vd}_{375},\mbox{vplus}(\mbox{vd}_{376},v_1)) )\qquad\mbox{fof}(\mbox{'ass(cond(234, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{369}{:}\;\mbox{geq}(\mbox{vd}_{369},v_1)\qquad\mbox{fof}(\mbox{'ass(cond(228, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{68},\mbox{vd}_{69}{:}\;\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{68}),\mbox{vd}_{69}) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{68},\mbox{vd}_{69}))\qquad\mbox{fof}(\mbox{'ass(cond(52, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{24}{:}\;( \mbox{vd}_{24}\ne v_1 \ \Rightarrow \ \mbox{vd}_{24} = \mbox{vsucc}(\mbox{vskolem}_2(\mbox{vd}_{24})) )\qquad\mbox{fof}(\mbox{'ass(cond(20, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{16}{:}\;\mbox{vsucc}(\mbox{vd}_{16})\ne \mbox{vd}_{16}\qquad\mbox{fof}(\mbox{'ass(cond(12, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_7,\mbox{vd}_8{:}\;( \mbox{vd}_7\ne \mbox{vd}_8 \ \Rightarrow \ \mbox{vsucc}(\mbox{vd}_7)\ne \mbox{vsucc}(\mbox{vd}_8) )\qquad\mbox{fof}(\mbox{'ass(cond(6, 0), 0)'},\mbox{axiom})$

\medskip\textbf{NUM250-1.p} Transfinite recursion property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{domain\_relation},\mbox{recursion\_equation\_functions}(z)),\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property6}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM783$\wedge$1.p} Landau theorem 81a

or3 (is x0 y0) (more x0 y0) (less x0 y0)




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

$x_0{:}\;\mbox{rat}\qquad\mbox{thf}(x_0,\mbox{type})$

$y_0{:}\;\mbox{rat}\qquad\mbox{thf}(y_0,\mbox{type})$

$\mbox{or}_3{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{or}_3,\mbox{type})$

$\mbox{is}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is},\mbox{type})$

$\mbox{more}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\mbox{less}{:}\;\mbox{rat}\to \mbox{rat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{ec}_3{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ec}_3,\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{rat},\mbox{xy}_0{:}\;\mbox{rat}{:}\;\neg\,( \mbox{or}_3@ ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 )@ ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 )@ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 ) ) \ \Rightarrow \ \neg\,\mbox{ec}_3@ ( \mbox{is}@ \mbox{xx}_0@ \mbox{xy}_0 )@ ( \mbox{more}@ \mbox{xx}_0@ \mbox{xy}_0 )@ ( \mbox{less}@ \mbox{xx}_0@ \mbox{xy}_0 )\qquad\mbox{thf}(\mbox{satz}_{81},\mbox{axiom})$

$\mbox{or}_3@ ( \mbox{is}@ x_0@ y_0 )@ ( \mbox{more}@ x_0@ y_0 )@ ( \mbox{less}@ x_0@ y_0 )\qquad\mbox{thf}(\mbox{satz81a},\mbox{conjecture})$

\medskip\textbf{NUM863$\wedge$1.p} A property of cardinal numbers.

|A|=|A'| \& |B|=|B'| \& A' disjoint B', then |A U B| $\leq$ |A' U B'|.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{is\_function}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is\_function\_type},\mbox{type})$

$\mbox{is\_function} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall e{:}\;\mbox{\$i}{:}\;( ( x@ e ) \ \Rightarrow \ ( y@ ( f@ e ) ) ) )\qquad\mbox{thf}(\mbox{is\_function},\mbox{definition})$

$\mbox{injection}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{injection\_type},\mbox{type})$

$\mbox{injection} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{is\_function}@ x@ f@ y\mbox{ and }\forall e_1{:}\;\mbox{\$i},e_2{:}\;\mbox{\$i}{:}\;( ( x@ e_1\mbox{ and }x@ e_2\mbox{ and }( f@ e_1 ) = ( f@ e_2 ) ) \ \Rightarrow \ e_1 = e_2 ) ) )\qquad\mbox{thf}(\mbox{injection},\mbox{definition})$

$\mbox{surjection}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{surjection\_type},\mbox{type})$

$\mbox{surjection} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{is\_function}@ x@ f@ y\mbox{ and }\forall e_1{:}\;\mbox{\$i}{:}\;( ( y@ e_1 ) \ \Rightarrow \ \exists e_2{:}\;\mbox{\$i}{:}\;( x@ e_2\mbox{ and }( f@ e_2 ) = e_1 ) ) ) )\qquad\mbox{thf}(\mbox{surjection},\mbox{definition})$

$\mbox{bijection}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{bijection\_type},\mbox{type})$

$\mbox{bijection} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{injection}@ x@ f@ y\mbox{ and }\mbox{surjection}@ x@ f@ y ) )\qquad\mbox{thf}(\mbox{bijection},\mbox{definition})$

$\mbox{equinumerous}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{equinumerous\_type},\mbox{type})$

$\mbox{equinumerous} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{bijection}@ x@ f@ y ) )\qquad\mbox{thf}(\mbox{equinumerous},\mbox{definition})$

$\mbox{embedding}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{embedding\_type},\mbox{type})$

$\mbox{embedding} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{injection}@ x@ f@ y ) )\qquad\mbox{thf}(\mbox{embedding},\mbox{definition})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{bp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{equinumerous}@ a@ \mbox{ap}\mbox{ and }\mbox{equinumerous}@ b@ \mbox{bp}\mbox{ and }( \mbox{intersection}@ \mbox{ap}@ \mbox{bp} ) = \mbox{emptyset} ) \ \Rightarrow \ ( \mbox{embedding}@ ( \mbox{union}@ a@ b )@ ( \mbox{union}@ \mbox{ap}@ \mbox{bp} ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{NUM263-1.p} Alternate 4 for transfinite induction, part 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$u\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction4}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{recursion}(x,y,z),\mbox{successor}(u))\ne \mbox{apply}(y,\mbox{apply}(\mbox{recursion}(x,y,z),u))\qquad\mbox{cnf}(\mbox{prove\_alternate\_4\_transfinite\_induction4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM007-1.p} Least Common Multiple

If LCM(a,b) is the least common multiple of two positive
 integers a, b, then LCM(a,b) = a*b/GCD(a,b).




$\mbox{divides}(x,x)\qquad\mbox{cnf}(\mbox{reflexivity\_of\_divides},\mbox{axiom})$

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

$\mbox{divides}(x,x\cdot y)\qquad\mbox{cnf}(\mbox{operand\_divides\_product},\mbox{axiom})$

$\mbox{divides}(y,z) \ \Rightarrow \ \mbox{divides}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{divides\_and\_multiply},\mbox{axiom})$

$\mbox{divides}(\mbox{quotient}(x,x),y)\qquad\mbox{cnf}(\mbox{one\_divides\_everything},\mbox{axiom})$

$( \mbox{divides}(y,x)\mbox{ and }\mbox{divides}(\mbox{quotient}(x,y),z) ) \ \Rightarrow \ \mbox{divides}(x,y\cdot z)\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_1,\mbox{axiom})$

$( \mbox{divides}(z,y)\mbox{ and }\mbox{divides}(x\cdot z,y) ) \ \Rightarrow \ \mbox{divides}(x,\mbox{quotient}(y,z))\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_2,\mbox{axiom})$

$( \mbox{divides}(y,x)\mbox{ and }\mbox{divides}(x,y\cdot z) ) \ \Rightarrow \ \mbox{divides}(\mbox{quotient}(x,y),z)\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_3,\mbox{axiom})$

$\mbox{gcd}(x,y,u) \ \Rightarrow \ \mbox{divides}(u,y)\qquad\mbox{cnf}(\mbox{gcd\_divides}_1,\mbox{axiom})$

$\mbox{gcd}(x,y,u) \ \Rightarrow \ \mbox{divides}(u,x)\qquad\mbox{cnf}(\mbox{gcd\_divides}_2,\mbox{axiom})$

$( \mbox{divides}(v,x)\mbox{ and }\mbox{divides}(v,y)\mbox{ and }\mbox{gcd}(x,y,u) ) \ \Rightarrow \ \mbox{divides}(v,u)\qquad\mbox{cnf}(\mbox{gcd}_1,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y) ) \ \Rightarrow \ ( \mbox{gcd}(x,y,u)\mbox{ or }\mbox{divides}(h(y,x,u),x) )\qquad\mbox{cnf}(\mbox{gcd}_2,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y) ) \ \Rightarrow \ ( \mbox{gcd}(x,y,u)\mbox{ or }\mbox{divides}(h(y,x,u),y) )\qquad\mbox{cnf}(\mbox{gcd}_3,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y)\mbox{ and }\mbox{divides}(h(y,x,u),u) ) \ \Rightarrow \ \mbox{gcd}(x,y,u)\qquad\mbox{cnf}(\mbox{gcd}_4,\mbox{axiom})$

$\mbox{gcd}(x,y,u) \ \Rightarrow \ \mbox{gcd}(z\cdot x,z\cdot y,z\cdot u)\qquad\mbox{cnf}(\mbox{property\_of\_gcd},\mbox{axiom})$

$( \mbox{divides}(x,u)\mbox{ and }\mbox{divides}(y,u) ) \ \Rightarrow \ ( \mbox{lcm}(x,y,u)\mbox{ or }\mbox{divides}(x,k(y,x,u)) )\qquad\mbox{cnf}(\mbox{lcm}_1,\mbox{axiom})$

$( \mbox{divides}(x,u)\mbox{ and }\mbox{divides}(y,u) ) \ \Rightarrow \ ( \mbox{lcm}(x,y,u)\mbox{ or }\mbox{divides}(y,k(y,x,u)) )\qquad\mbox{cnf}(\mbox{lcm}_2,\mbox{axiom})$

$( \mbox{divides}(x,u)\mbox{ and }\mbox{divides}(y,u)\mbox{ and }\mbox{divides}(u,k(y,x,u)) ) \ \Rightarrow \ \mbox{lcm}(x,y,u)\qquad\mbox{cnf}(\mbox{lcm}_3,\mbox{axiom})$

$k(x,y,z) = k(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_k},\mbox{axiom})$

$h(x,y,z) = h(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_h},\mbox{axiom})$

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

$\mbox{lcm}(x,y,z) \ \Rightarrow \ \mbox{lcm}(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_lcm},\mbox{axiom})$

$\mbox{gcd}(x,y,z) \ \Rightarrow \ \mbox{gcd}(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_gcd},\mbox{axiom})$

$\mbox{gcd}(a,b,c)\qquad\mbox{cnf}(\mbox{c\_is\_gcd\_of\_a\_and\_b},\mbox{negated\_conjecture})$

$\neg\,\mbox{lcm}(a,b,\mbox{quotient}(a\cdot b,c))\qquad\mbox{cnf}(\mbox{prove\_lcm},\mbox{negated\_conjecture})$

\medskip\textbf{NUM115-1.p} The class of ordinals is not a set.






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{ordinal\_numbers}\in x\qquad\mbox{cnf}(\mbox{prove\_class\_of\_ordinals\_is\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM664$\wedge$1.p} Landau theorem 16b

less x z




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\mbox{less}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\neg\,\mbox{less}@ y@ z \ \Rightarrow \ y = z\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{less}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{15},\mbox{axiom})$

$\mbox{less}@ x@ z\qquad\mbox{thf}(\mbox{satz16b},\mbox{conjecture})$

\medskip\textbf{NUM097-1.p} Corollary to sections property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{section}(\mbox{element\_relation},w,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property5}_2,\mbox{negated\_conjecture})$

$\mbox{intersection}(y,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(w',y)))\ne w\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property5}_3,\mbox{negated\_conjecture})$

$y\ne w\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sections\_property5}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM364+1.p} !XY (X + Y $>$ X - Y)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x,y,z_1,z_2{:}\;( x + y{=}z_1\mbox{ and }x\setminus y{=}z_2\mbox{ and }\mbox{less}(z_1,z_2) )\qquad\mbox{fof}(\mbox{anti\_sum\_diff\_less}_1,\mbox{conjecture})$

\medskip\textbf{NUM033-1.p} Symmetrization property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{symmetrization\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property6}_2,\mbox{negated\_conjecture})$

$x'\ne x\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM358+1.p} !X (X + X = X)






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;x + x{=}x\qquad\mbox{fof}(\mbox{anti\_sum\_idempotence},\mbox{conjecture})$

\medskip\textbf{NUM922=1.p} Universal predicate






$p{:}\;\mbox{\$int}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{p\_type},\mbox{type})$

$( p(0)\mbox{ and }\forall u{:}\;\mbox{\$int}{:}\;( p(u) \ \Rightarrow \ p(\mbox{\$sum}(u,1)) )\mbox{ and }\forall v{:}\;\mbox{\$int}{:}\;( p(v) \ \Rightarrow \ p(\mbox{\$difference}(v,1)) ) ) \ \Rightarrow \ \forall w{:}\;\mbox{\$int}{:}\;p(w)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{NUM253-1.p} Transfinite recursion property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{recursion\_equation\_functions}(z)\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property9}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{union}(\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{apply}(z,x))),x))\qquad\mbox{cnf}(\mbox{prove\_transfinite\_recursion\_property9}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM187-1.p} Lemma for successor property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_lemma\_for\_successor\_property8}_1,\mbox{negated\_conjecture})$

$y\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_lemma\_for\_successor\_property8}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{intersection}(\mbox{power\_class}(\mbox{kind\_1\_ordinals}),\mbox{kind\_1\_ordinals})\qquad\mbox{cnf}(\mbox{prove\_lemma\_for\_successor\_property8}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM704$\wedge$1.p} Landau theorem 27

~(forall x:nat.~(~((forall x\_0\_0:nat.p x\_0\_0 $\to$ ~(less x x\_0\_0) $\to$
 x = x\_0\_0) $\to$ ~(p x))))




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

$p{:}\;\mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\neg\,\forall \mbox{xx}{:}\;\mbox{nat}{:}\;\neg\,p@ \mbox{xx}\qquad\mbox{thf}(s,\mbox{axiom})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\mbox{more}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{more},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{more}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\neg\,\mbox{less}@ \mbox{xx}@ \mbox{xy} \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{satz10g},\mbox{axiom})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{more}@ ( \mbox{pl}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{18},\mbox{axiom})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \neg\,\mbox{less}@ n_1@ \mbox{xx} \ \Rightarrow \ n_1 = \mbox{xx} )\qquad\mbox{thf}(\mbox{satz24a},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ ( \neg\,\mbox{less}@ ( \mbox{pl}@ \mbox{xy}@ n_1 )@ \mbox{xx} \ \Rightarrow \ ( \mbox{pl}@ \mbox{xy}@ n_1 ) = \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz25b},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{pl}@ \mbox{xx}@ n_1 ) = ( \mbox{suc}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz4a},\mbox{axiom})$

$\neg\,\forall \mbox{xx}{:}\;\mbox{nat}{:}\;\neg\,\neg\,\forall \mbox{xx\_0}_0{:}\;\mbox{nat}{:}\;( ( p@ \mbox{xx\_0}_0 ) \ \Rightarrow \ ( \neg\,\mbox{less}@ \mbox{xx}@ \mbox{xx\_0}_0 \ \Rightarrow \ \mbox{xx} = \mbox{xx\_0}_0 ) ) \ \Rightarrow \ \neg\,p@ \mbox{xx}\qquad\mbox{thf}(\mbox{satz}_{27},\mbox{conjecture})$

\medskip\textbf{NUM899=1.p} Difference less than sum






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$difference}(x,y) = z_2 ) \ \Rightarrow \ \mbox{\$less}(z_2,z_1) )\qquad\mbox{tff}(\mbox{anti\_x\_sum\_y\_greater\_x\_difference\_y},\mbox{conjecture})$

\medskip\textbf{NUM073-1.p} Corollary to well ordering property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{element\_relation},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_2,\mbox{negated\_conjecture})$

$u\in v\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,w)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_4,\mbox{negated\_conjecture})$

$v\in w\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_5,\mbox{negated\_conjecture})$

$\neg\,u\in w\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_well\_ordering\_property4}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM057-1.p} Segments property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{segment}(\mbox{xr},y,z),y)\qquad\mbox{cnf}(\mbox{prove\_segments\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM692$\wedge$1.p} Landau theorem 23a

lessis (pl x z) (pl y u)




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

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

$\mbox{lessis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{lessis},\mbox{type})$

$\mbox{lessis}@ x@ y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{lessis}@ z@ u\qquad\mbox{thf}(k,\mbox{axiom})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\mbox{moreis}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moreis},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{lessis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{13},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat},\mbox{xu}{:}\;\mbox{nat}{:}\;( ( \mbox{moreis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{moreis}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moreis}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{ts}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{23},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{lessis}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{moreis}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{14},\mbox{axiom})$

$\mbox{lessis}@ ( \mbox{ts}@ x@ z )@ ( \mbox{ts}@ y@ u )\qquad\mbox{thf}(\mbox{satz23a},\mbox{conjecture})$

\medskip\textbf{NUM129-1.p} Alternate 2 for transfinite induction, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{least}(\mbox{element\_relation},y)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction1}_2,\mbox{negated\_conjecture})$

$y\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_alternate\_2\_transfinite\_induction1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM781$\wedge$2.p} Landau theorem 79






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

$\forall a{:}\;a,b{:}\;a{:}\;( a = b \ \Rightarrow \ b = a )\qquad\mbox{thf}(\mbox{cES\_eq\_},\mbox{conjecture})$

\medskip\textbf{NUM865=1.p} Associativity of sum






$\forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int},z_1{:}\;\mbox{\$int},z_2{:}\;\mbox{\$int},z_3{:}\;\mbox{\$int},z_4{:}\;\mbox{\$int}{:}\;( ( \mbox{\$sum}(x,y) = z_1\mbox{ and }\mbox{\$sum}(z_1,z) = z_2\mbox{ and }\mbox{\$sum}(y,z) = z_3\mbox{ and }\mbox{\$sum}(x,z_3) = z_4 ) \ \Rightarrow \ z_2 = z_4 )\qquad\mbox{tff}(\mbox{associative\_sum\_forall},\mbox{conjecture})$

\medskip\textbf{NUM817$\wedge$5.p} TPS problem from IND-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

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

$\mbox{cEVEN}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN1\_type},\mbox{type})$

$\mbox{cODD}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD1\_type},\mbox{type})$

$\mbox{cEVEN}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{cEVEN1\_def},\mbox{definition})$

$\mbox{cODD}_1 = ( \lambda \mbox{xn}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cEVEN}_1@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cODD1\_def},\mbox{definition})$

$( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xu} )\ne c_0\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xv} ) = ( \mbox{cS}@ \mbox{xw} ) \ \Rightarrow \ \mbox{xv} = \mbox{xw} ) ) \ \Rightarrow \ \exists \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cODD}_1@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM}_{405},\mbox{conjecture})$

\medskip\textbf{NUM806$\wedge$5.p} TPS problem from NATS






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

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

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

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_star}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{c\_plus}@ ( \mbox{c\_star}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA}_4,\mbox{conjecture})$

\medskip\textbf{NUM873=1.p} Sum something and 0 is another thing






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,0) = y\mbox{ and }y\ne x )\qquad\mbox{tff}(\mbox{anti\_sum\_identity}_2,\mbox{conjecture})$

\medskip\textbf{NUM094-1.p} Sections property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{section}(\mbox{xr},w,y)\qquad\mbox{cnf}(\mbox{prove\_sections\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{least}(\mbox{xr},\mbox{intersection}(w',y))\in y\qquad\mbox{cnf}(\mbox{prove\_sections\_property3}_3,\mbox{negated\_conjecture})$

$y\ne w\qquad\mbox{cnf}(\mbox{prove\_sections\_property3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{NUM337+1.p} 5 - only 3 = 2






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\forall x{:}\;( n_5\setminus x{=}n_2 \ \Rightarrow \ x = n_3 )\qquad\mbox{fof}(\mbox{diff\_n5\_only\_n3\_n}_2,\mbox{conjecture})$

\medskip\textbf{NUM635$\wedge$1.p} Landau theorem 1

~(suc x = suc y)




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

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

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

$x\ne y\qquad\mbox{thf}(n,\mbox{axiom})$

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( ( \mbox{suc}@ \mbox{xx} ) = ( \mbox{suc}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$( \mbox{suc}@ x )\ne ( \mbox{suc}@ y )\qquad\mbox{thf}(\mbox{satz}_1,\mbox{conjecture})$

\medskip\textbf{NUM034-1.p} Symmetrization is idempotent






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{symmetrization\_of}(\mbox{symmetrization\_of}(x))\ne \mbox{symmetrization\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_idempotency\_of\_symmetrization}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM079-1.p} Well ordering property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{intersection}(y,\mbox{ordinal\_numbers})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property8}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{least}(\mbox{element\_relation},\mbox{intersection}(y,\mbox{ordinal\_numbers}))\in \mbox{intersection}(y,\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM049-1.p} Connected class property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(y,y),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{connected}(x,y)\qquad\mbox{cnf}(\mbox{prove\_connect\_class\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM893=1.p} Sum and difference






$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(x,y) = z\iff ( \mbox{\$difference}(z,y) = x\mbox{ and }\mbox{\$difference}(z,x) = y ) )\qquad\mbox{tff}(\mbox{sum\_same\_as\_difference},\mbox{conjecture})$

\medskip\textbf{NUM845+2.p} qu(ind(267),imp(267))






$\forall \mbox{vd}_{416}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{416}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{416}),\mbox{vd}_{416}) \ \Rightarrow \ \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vsucc}(\mbox{vd}_{416})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vsucc}(\mbox{vd}_{416})),\mbox{vsucc}(\mbox{vd}_{416})) )\qquad\mbox{fof}(\mbox{'qu(ind(267), imp(267))'},\mbox{conjecture})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{411}),\mbox{vsucc}(\mbox{vd}_{413})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vsucc}(\mbox{vd}_{413})),\mbox{vsucc}(\mbox{vd}_{413})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{411},\mbox{vsucc}(\mbox{vd}_{413}))) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{411}),\mbox{vsucc}(\mbox{vd}_{413})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 1)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{411},\mbox{vd}_{413}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{411},\mbox{vsucc}(\mbox{vd}_{413}))) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{411},\mbox{vd}_{413}))) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 3)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{413},\mbox{vsucc}(\mbox{vd}_{411}))) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 4)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vplus}(\mbox{vd}_{413},\mbox{vsucc}(\mbox{vd}_{411}))) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 5)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vplus}(\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) = \mbox{vplus}(\mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 6)'},\mbox{axiom})$

$\forall \mbox{vd}_{413}{:}\;( \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},\mbox{vd}_{413}),\mbox{vd}_{413}) \ \Rightarrow \ \mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vsucc}(\mbox{vd}_{413})) = \mbox{vplus}(\mbox{vmul}(\mbox{vsucc}(\mbox{vd}_{411}),\mbox{vd}_{413}),\mbox{vsucc}(\mbox{vd}_{411})) )\qquad\mbox{fof}(\mbox{'ass(cond(conseq(263), 1), 7)'},\mbox{axiom})$

$\mbox{vsucc}(\mbox{vmul}(\mbox{vd}_{411},v_1)) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{411},v_1),v_1)\qquad\mbox{fof}(\mbox{'holds(264, 412, 2)'},\mbox{axiom})$

$\forall \mbox{vd}_{396},\mbox{vd}_{397}{:}\;( \mbox{vmul}(\mbox{vd}_{396},\mbox{vsucc}(\mbox{vd}_{397})) = \mbox{vplus}(\mbox{vmul}(\mbox{vd}_{396},\mbox{vd}_{397}),\mbox{vd}_{396})\mbox{ and }\mbox{vmul}(\mbox{vd}_{396},v_1) = \mbox{vd}_{396} )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 32), and(holds(definiens(249), 399, 0), holds(definiens(249), 398, 0)))'},\mbox{axiom})$

$\forall \mbox{vd}_{78},\mbox{vd}_{79}{:}\;\mbox{vplus}(\mbox{vd}_{79},\mbox{vd}_{78}) = \mbox{vplus}(\mbox{vd}_{78},\mbox{vd}_{79})\qquad\mbox{fof}(\mbox{'ass(cond(61, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{59}{:}\;\mbox{vplus}(v_1,\mbox{vd}_{59}) = \mbox{vsucc}(\mbox{vd}_{59})\qquad\mbox{fof}(\mbox{'ass(cond(43, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{46},\mbox{vd}_{47},\mbox{vd}_{48}{:}\;\mbox{vplus}(\mbox{vplus}(\mbox{vd}_{46},\mbox{vd}_{47}),\mbox{vd}_{48}) = \mbox{vplus}(\mbox{vd}_{46},\mbox{vplus}(\mbox{vd}_{47},\mbox{vd}_{48}))\qquad\mbox{fof}(\mbox{'ass(cond(33, 0), 0)'},\mbox{axiom})$

$\forall \mbox{vd}_{42},\mbox{vd}_{43}{:}\;( \mbox{vplus}(\mbox{vd}_{42},\mbox{vsucc}(\mbox{vd}_{43})) = \mbox{vsucc}(\mbox{vplus}(\mbox{vd}_{42},\mbox{vd}_{43}))\mbox{ and }\mbox{vplus}(\mbox{vd}_{42},v_1) = \mbox{vsucc}(\mbox{vd}_{42}) )\qquad\mbox{fof}(\mbox{'qu(cond(conseq(axiom(3)), 3), and(holds(definiens(29), 45, 0), holds(definiens(29), 44, 0)))'},\mbox{axiom})$

\medskip\textbf{NUM038-1.p} Symmetrization property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\ne \mbox{restrict}(x,\mbox{domain\_of}(\mbox{symmetrization\_of}(x)),\mbox{domain\_of}(\mbox{symmetrization\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_symmetrization\_property9}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM891=1.p} X = - X means X is 0






$\forall x{:}\;\mbox{\$int}{:}\;( x = \mbox{\$uminus}(x)\iff x = 0 )\qquad\mbox{tff}(\mbox{uminus\_equal},\mbox{conjecture})$

\medskip\textbf{NUM758$\wedge$1.p} Landau theorem 63d

moref x y




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{moref}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{moref},\mbox{type})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\mbox{moref}@ ( \mbox{pf}@ z@ x )@ ( \mbox{pf}@ z@ y )\qquad\mbox{thf}(m,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz63a},\mbox{axiom})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{moref}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xx}@ \mbox{xz} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xy}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{moref}@ \mbox{xz}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{44},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{satz}_{58},\mbox{axiom})$

$\mbox{moref}@ x@ y\qquad\mbox{thf}(\mbox{satz63d},\mbox{conjecture})$

\medskip\textbf{NUM713$\wedge$1.p} Landau theorem 32a

~(forall x\_0:nat.~(ts x z = pl (ts y z) x\_0))




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{pl}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{pl},\mbox{type})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;x\ne ( \mbox{pl}@ y@ \mbox{xx}_0 )\qquad\mbox{thf}(m,\mbox{axiom})$

$\mbox{ts}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{ts},\mbox{type})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{ts}@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{29},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( \mbox{ts}@ \mbox{xx}@ ( \mbox{pl}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{pl}@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{ts}@ \mbox{xx}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{satz}_{30},\mbox{axiom})$

$\neg\,\forall \mbox{xx}_0{:}\;\mbox{nat}{:}\;( \mbox{ts}@ x@ z )\ne ( \mbox{pl}@ ( \mbox{ts}@ y@ z )@ \mbox{xx}_0 )\qquad\mbox{thf}(\mbox{satz32a},\mbox{conjecture})$

\medskip\textbf{NUM072-1.p} Well ordering property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{well\_ordering}(\mbox{xr},y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,w)\in \mbox{cross\_product}(y,y)\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_4,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(v,w)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,w)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_well\_ordering\_property4}_6,\mbox{negated\_conjecture})$

\medskip\textbf{NUM153-1.p} Corollary 4 to successor property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{image}(\mbox{successor\_relation},\mbox{ordinal\_numbers})\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_successor\_property1}_1,\mbox{negated\_conjecture})$

$x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_successor\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM005-1.p} Greatest Common Divisor

If GCD(a,b) is the greatest common divisor of two positive
 a, b, then for any positive integer d, GCD(a*d,b*d)
 = GCD(a,b)*d.




$\mbox{divides}(x,x)\qquad\mbox{cnf}(\mbox{reflexivity\_of\_divides},\mbox{axiom})$

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

$\mbox{divides}(x,x\cdot y)\qquad\mbox{cnf}(\mbox{operand\_divides\_product},\mbox{axiom})$

$\mbox{divides}(y,z) \ \Rightarrow \ \mbox{divides}(x\cdot y,x\cdot z)\qquad\mbox{cnf}(\mbox{divides\_and\_multiply},\mbox{axiom})$

$\mbox{divides}(\mbox{quotient}(x,x),y)\qquad\mbox{cnf}(\mbox{one\_divides\_everything},\mbox{axiom})$

$( \mbox{divides}(y,x)\mbox{ and }\mbox{divides}(\mbox{quotient}(x,y),z) ) \ \Rightarrow \ \mbox{divides}(x,y\cdot z)\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_1,\mbox{axiom})$

$( \mbox{divides}(z,y)\mbox{ and }\mbox{divides}(x\cdot z,y) ) \ \Rightarrow \ \mbox{divides}(x,\mbox{quotient}(y,z))\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_2,\mbox{axiom})$

$( \mbox{divides}(y,x)\mbox{ and }\mbox{divides}(x,y\cdot z) ) \ \Rightarrow \ \mbox{divides}(\mbox{quotient}(x,y),z)\qquad\mbox{cnf}(\mbox{divides\_quotient\_multiply}_3,\mbox{axiom})$

$\mbox{gcd}(x,y,u) \ \Rightarrow \ \mbox{divides}(u,y)\qquad\mbox{cnf}(\mbox{gcd\_divides}_1,\mbox{axiom})$

$\mbox{gcd}(x,y,u) \ \Rightarrow \ \mbox{divides}(u,x)\qquad\mbox{cnf}(\mbox{gcd\_divides}_2,\mbox{axiom})$

$\mbox{divides}(k(y,x),x)\qquad\mbox{cnf}(\mbox{divides\_k}_1,\mbox{axiom})$

$\mbox{divides}(k(y,x),y)\qquad\mbox{cnf}(\mbox{divides\_k}_2,\mbox{axiom})$

$( \mbox{divides}(v,x)\mbox{ and }\mbox{divides}(v,y) ) \ \Rightarrow \ \mbox{divides}(v,k(y,x))\qquad\mbox{cnf}(\mbox{divides\_k}_3,\mbox{axiom})$

$( \mbox{divides}(v,x)\mbox{ and }\mbox{divides}(v,y)\mbox{ and }\mbox{gcd}(x,y,u) ) \ \Rightarrow \ \mbox{divides}(v,u)\qquad\mbox{cnf}(\mbox{gcd}_1,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y) ) \ \Rightarrow \ ( \mbox{gcd}(x,y,u)\mbox{ or }\mbox{divides}(h(y,x,u),x) )\qquad\mbox{cnf}(\mbox{gcd}_2,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y) ) \ \Rightarrow \ ( \mbox{gcd}(x,y,u)\mbox{ or }\mbox{divides}(h(y,x,u),y) )\qquad\mbox{cnf}(\mbox{gcd}_3,\mbox{axiom})$

$( \mbox{divides}(u,x)\mbox{ and }\mbox{divides}(u,y)\mbox{ and }\mbox{divides}(h(y,x,u),u) ) \ \Rightarrow \ \mbox{gcd}(x,y,u)\qquad\mbox{cnf}(\mbox{gcd}_4,\mbox{axiom})$

$h(x,y,z) = h(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_h},\mbox{axiom})$

$k(x,y) = k(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_k},\mbox{axiom})$

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

$\mbox{gcd}(x,y,z) \ \Rightarrow \ \mbox{gcd}(y,x,z)\qquad\mbox{cnf}(\mbox{commutativity\_of\_gcd},\mbox{axiom})$

$\mbox{gcd}(a,b,e)\qquad\mbox{cnf}(\mbox{e\_is\_gcd\_of\_a\_and\_b},\mbox{hypothesis})$

$\neg\,\mbox{gcd}(a\cdot c,b\cdot c,e\cdot c)\qquad\mbox{cnf}(\mbox{prove\_gcd},\mbox{negated\_conjecture})$

\medskip\textbf{NUM752$\wedge$1.p} Landau theorem 62e

eq (pf z x) (pf z y)




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

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

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

$z{:}\;\mbox{frac}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{eq}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eq},\mbox{type})$

$\mbox{eq}@ x@ y\qquad\mbox{thf}(e,\mbox{axiom})$

$\mbox{pf}{:}\;\mbox{frac}\to \mbox{frac}\to \mbox{frac}\qquad\mbox{thf}(\mbox{pf},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{frac},\mbox{xy}{:}\;\mbox{frac},\mbox{xz}{:}\;\mbox{frac},\mbox{xu}{:}\;\mbox{frac}{:}\;( ( \mbox{eq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{eq}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{eq}@ ( \mbox{pf}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{pf}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{satz}_{56},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{frac}{:}\;( \mbox{eq}@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{satz}_{37},\mbox{axiom})$

$\mbox{eq}@ ( \mbox{pf}@ z@ x )@ ( \mbox{pf}@ z@ y )\qquad\mbox{thf}(\mbox{satz62e},\mbox{conjecture})$

\medskip\textbf{NUM388+1.p} Ordinal numbers, theorem 19






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{epsilon\_transitive}(a)\iff \forall b{:}\;( \mbox{in}(b,a) \ \Rightarrow \ b\subseteq a ) )\qquad\mbox{fof}(\mbox{d2\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{ordinal}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{epsilon\_transitive}(c) \ \Rightarrow \ ( ( \mbox{in}(c,a)\mbox{ and }\mbox{in}(a,b) ) \ \Rightarrow \ \mbox{in}(c,b) ) ) ) )\qquad\mbox{fof}(\mbox{t19\_ordinal}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{NUM035-1.p} Domain equals range of symmetrization






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{domain\_of}(\mbox{symmetrization\_of}(x))\ne \mbox{range\_of}(\mbox{symmetrization\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_domain\_equals\_range\_of\_symmetrization}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM339+1.p} Show lower boundary






$\mbox{include}(\mbox{'Axioms/NUM005+0.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM005+2.ax'})$

$\exists x{:}\;\mbox{nn}_{127}\setminus n_1{=}x\qquad\mbox{fof}(\mbox{show\_lower\_boundary},\mbox{conjecture})$

\medskip\textbf{NUM218-1.p} Rest of is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\neg\,\mbox{function}(\mbox{rest\_of}(u))\qquad\mbox{cnf}(\mbox{prove\_rest\_of\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM636$\wedge$1.p} Landau theorem 2

~(suc x = x)




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

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

$\mbox{suc}{:}\;\mbox{nat}\to \mbox{nat}\qquad\mbox{thf}(\mbox{suc},\mbox{type})$

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

$\mbox{esti}{:}\;\mbox{nat}\to \mbox{set}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{esti},\mbox{type})$

$\mbox{setof}{:}\;( \mbox{nat}\to \mbox{\$o} )\to \mbox{set}\qquad\mbox{thf}(\mbox{setof},\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{estie},\mbox{axiom})$

$n_1{:}\;\mbox{nat}\qquad\mbox{thf}(n_1,\mbox{type})$

$\forall \mbox{xs}{:}\;\mbox{set}{:}\;( ( \mbox{esti}@ n_1@ \mbox{xs} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( ( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ ( \mbox{suc}@ \mbox{xx} )@ \mbox{xs} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{esti}@ \mbox{xx}@ \mbox{xs} ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\forall \mbox{xp}{:}\;\mbox{nat}\to \mbox{\$o},\mbox{xs}{:}\;\mbox{nat}{:}\;( ( \mbox{xp}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{esti}@ \mbox{xs}@ ( \mbox{setof}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{estii},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat}{:}\;( \mbox{suc}@ \mbox{xx} )\ne n_1\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ ( \mbox{suc}@ \mbox{xx} )\ne ( \mbox{suc}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{satz}_1,\mbox{axiom})$

$( \mbox{suc}@ x )\ne x\qquad\mbox{thf}(\mbox{satz}_2,\mbox{conjecture})$

\medskip\textbf{NUM134-1.p} Successor relation of an ordinal is an ordinal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_ordinal}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_successor\_of\_ordinal}_2,\mbox{negated\_conjecture})$

\medskip\textbf{NUM829$\wedge$5.p} TPS problem from PA-THMS






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

$c_0{:}\;n\qquad\mbox{thf}(\mbox{c0\_type},\mbox{type})$

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS\_type},\mbox{type})$

$\mbox{c\_plus}{:}\;n\to n\to n\qquad\mbox{thf}(\mbox{c\_plus\_type},\mbox{type})$

$\mbox{cPA}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_1\_type},\mbox{type})$

$\mbox{cPA}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_2\_type},\mbox{type})$

$\mbox{cPA\_IND\_EQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_type},\mbox{type})$

$\mbox{cPA}_1 = ( \forall \mbox{xx}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ c_0 ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cPA\_1\_def},\mbox{definition})$

$\mbox{cPA}_2 = ( \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ \mbox{xx}@ ( \mbox{cS}@ \mbox{xy} ) ) = ( \mbox{cS}@ ( \mbox{c\_plus}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cPA\_2\_def},\mbox{definition})$

$\mbox{cPA\_IND\_EQ} = ( \forall \mbox{xp}{:}\;n\to n,\mbox{xq}{:}\;n\to n{:}\;( ( ( \mbox{xp}@ c_0 ) = ( \mbox{xq}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;n{:}\;( ( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) = ( \mbox{xq}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{xp}@ \mbox{xx} ) = ( \mbox{xq}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cPA\_IND\_EQ\_def},\mbox{definition})$

$( \mbox{cPA}_1\mbox{ and }\mbox{cPA}_2\mbox{ and }\mbox{cPA\_IND\_EQ} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n,\mbox{xy}{:}\;n{:}\;( \mbox{c\_plus}@ ( \mbox{cS}@ \mbox{xx} )@ \mbox{xy} ) = ( \mbox{c\_plus}@ \mbox{xy}@ ( \mbox{cS}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPA\_THM}_3,\mbox{conjecture})$

\medskip\textbf{NUM116-1.p} Corollary to the class of ordinals is not set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{successor}(\mbox{ordinal\_numbers})\ne \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_class\_of\_ordinals\_is\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{NUM207-1.p} Least upper bound property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$x\in \mbox{ordinal\_numbers}\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(x)\in x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property6}_2,\mbox{negated\_conjecture})$

$\mbox{successor}(\mbox{sum\_class}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{NUM663$\wedge$1.p} Landau theorem 16a

less x z




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

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

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

$z{:}\;\mbox{nat}\qquad\mbox{thf}(z,\mbox{type})$

$\mbox{less}{:}\;\mbox{nat}\to \mbox{nat}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{less},\mbox{type})$

$\neg\,\mbox{less}@ x@ y \ \Rightarrow \ x = y\qquad\mbox{thf}(l,\mbox{axiom})$

$\mbox{less}@ y@ z\qquad\mbox{thf}(k,\mbox{axiom})$

$\forall \mbox{xa}{:}\;\mbox{\$o}{:}\;( \neg\,\neg\,\mbox{xa} \ \Rightarrow \ \mbox{xa} )\qquad\mbox{thf}(\mbox{et},\mbox{axiom})$

$\forall \mbox{xx}{:}\;\mbox{nat},\mbox{xy}{:}\;\mbox{nat},\mbox{xz}{:}\;\mbox{nat}{:}\;( ( \mbox{less}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{less}@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{less}@ \mbox{xx}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{satz}_{15},\mbox{axiom})$

$\mbox{less}@ x@ z\qquad\mbox{thf}(\mbox{satz16a},\mbox{conjecture})$

\medskip\textbf{NUM084-1.p} Alternate transitive class definition, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/NUM004-0.ax'})$

$\mbox{subclass}(\mbox{sum\_class}(x),w)\qquad\mbox{cnf}(\mbox{prove\_alternate\_trasitive\_class\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{power\_class}(w))\qquad\mbox{cnf}(\mbox{prove\_alternate\_trasitive\_class\_defn1}_2,\mbox{negated\_conjecture})$

\medskip
\end{document}