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    \textbf{SYO013$\wedge$1.p} Formula valid with Boolean extentionality 2






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

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

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

$( a\mbox{ and }b\mbox{ and }p@ a ) \ \Rightarrow \ ( p@ b )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO382$\wedge$5.p} TPS problem THM407






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

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

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

$( ( \exists \mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xv} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{cQ}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cS}@ \mbox{xu} ) \ \Rightarrow \ ( ( \exists \mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xv} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\mbox{cQ}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cS}@ \mbox{xu} )\qquad\mbox{thf}(\mbox{cTHM}_{407},\mbox{conjecture})$

\medskip\textbf{SYO280$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} )\iff \neg\,a@ ( s@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG\_SV\_EO}_1,\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$3.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_m}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ p )@ ( \mbox{mdia\_m}@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO215$\wedge$5.p} TPS problem THM26






$\exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;x\ne y \ \Rightarrow \ \forall u{:}\;\mbox{\$i}{:}\;\exists z{:}\;\mbox{\$i}{:}\;z\ne u\qquad\mbox{thf}(\mbox{cTHM}_{26},\mbox{conjecture})$

\medskip\textbf{SYO457$\wedge$3.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) )@ ( \mbox{mdia\_m}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO301$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$( \mbox{cP}@ ( f@ b ) ) \ \Rightarrow \ \exists \mbox{xs}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }a@ ( f@ \mbox{xs} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE}_5,\mbox{conjecture})$

\medskip\textbf{SYO456$\wedge$6.p} Ted Sider's propositional modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO307$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}\mbox{ or }( a@ \mbox{xy}\mbox{ and }a@ \mbox{xz} ) ) \ \Rightarrow \ ( ( p@ \mbox{xx}\mbox{ and }\mbox{cR}@ \mbox{xy} )\mbox{ or }( q@ \mbox{xx}\mbox{ and }\mbox{cT}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBLEDSOE5F},\mbox{conjecture})$

\medskip\textbf{SYO573$\wedge$7.p} Quantified modal logics wwfs. problem 15.






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

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

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

$\forall v{:}\;\mbox{\$i}{:}\;( \mbox{exists\_in\_world}@ a@ v )\qquad\mbox{thf}(\mbox{existence\_of\_a\_ax},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mdia\_s}_4@ ( f@ a ) )@ ( \mbox{mdia\_s}_4@ ( g@ a ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ ( f@ a )@ ( g@ a ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO027$\wedge$1.p} Something is true






$\exists p{:}\;\mbox{\$o}{:}\;p\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO379$\wedge$5.p} TPS problem from QUANTDEPTH-THMS






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

$\mbox{cQDP}_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQDP0\_type},\mbox{type})$

$\mbox{cQDP}_1{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQDP1\_type},\mbox{type})$

$\mbox{cQDP}_0 = ( \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\mbox{xz} = c )\qquad\mbox{thf}(\mbox{cQDP0\_def},\mbox{definition})$

$\mbox{cQDP}_1 = ( \lambda \mbox{xz}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xz} = \mbox{cQDP}_0\mbox{ and }\exists \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xz}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cQDP1\_def},\mbox{definition})$

$\exists \mbox{xs}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cQDP}_1@ \mbox{xs} )\qquad\mbox{thf}(\mbox{cQDTHM}_1,\mbox{conjecture})$

\medskip\textbf{SYO477$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO527$\wedge$1.p} Skolem Property on two types

For every total relation r on a * b, there is a corresponding 
 function from a to b.




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

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

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

$\forall x{:}\;a{:}\;\exists y{:}\;b{:}\;( r@ x@ y )\qquad\mbox{thf}(\mbox{rtotal},\mbox{axiom})$

$\exists f{:}\;a\to b{:}\;\forall x{:}\;a{:}\;( r@ x@ ( f@ x ) )\qquad\mbox{thf}(\mbox{skolem},\mbox{conjecture})$

\medskip\textbf{SYO224$\wedge$5.p} TPS problem LING1






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

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

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

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

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

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

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

$( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{cUNIQUE}@ x ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{cWRH}@ z\mbox{ and }\mbox{cW}@ z ) \ \Rightarrow \ x = z ) )\mbox{ and }\mbox{cUNIQUE}@ \mbox{cS}\mbox{ and }\mbox{cW}@ \mbox{cS}\mbox{ and }\mbox{cWRH}@ \mbox{cS} ) \ \Rightarrow \ \exists \mbox{xan}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xan}@ \mbox{cP} ) = ( \mbox{cLIKE}@ \mbox{cP}@ \mbox{cS} )\mbox{ and }( \mbox{xan}@ \mbox{cJ} ) = ( \exists x{:}\;\mbox{\$i}{:}\;( \mbox{cUNIQUE}@ x\mbox{ and }\mbox{cW}@ x\mbox{ and }\mbox{cWRH}@ x\mbox{ and }\mbox{cLIKE}@ \mbox{cJ}@ x ) ) )\qquad\mbox{thf}(\mbox{cLING}_1,\mbox{conjecture})$

\medskip\textbf{SYO398$\wedge$1.p} Ted Sider's modal proposition logic theorem 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mnot}@ p ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO539$\wedge$1.p} Range of if-then-else on \$i defined from choice on \$i

A choice operator on \$i is used to define an if-then-else operator
 at \$i. Check that it always returns the then-part or the else-part.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{if}@ b@ x@ y ) = x\mbox{ or }( \mbox{if}@ b@ x@ y ) = y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$3.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mdia\_m}@ p ) )@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO263$\wedge$5.p} TPS problem THM125D

Trivial theorem to test flexible-flexible pairs.




$\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xn}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xn}@ \mbox{xa}\mbox{ or }\mbox{xm}@ \mbox{xa} )\mbox{ and }( p@ \mbox{xb}\mbox{ or }\mbox{xn}@ \mbox{xb} )\mbox{ and }( \mbox{xm}@ \mbox{xc}\mbox{ or }\neg\,p@ \mbox{xc} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xn}@ \mbox{xx} )\iff \neg\,p@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM125D},\mbox{conjecture})$

\medskip\textbf{SYO126$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\mbox{cP}@ c_0\mbox{ and }\forall x{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ ( \mbox{cP}@ ( \mbox{cS}@ x ) ) )\qquad\mbox{thf}(\mbox{cSUPERSET\_NAT},\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$4.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_b}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_b}@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_b}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO339$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

$\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},x_0{:}\;\mbox{\$i}{:}\;( ( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \exists \mbox{xl}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xl} )\mbox{ and }\exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ \mbox{xu} ) ) ) \ \Rightarrow \ \exists \mbox{xl}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ \mbox{xl} ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xl}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ x_0@ ( \mbox{xf}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}@ \mbox{xx}\mbox{ and }\forall \mbox{xs}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xt}@ \mbox{xs}\mbox{ and }\mbox{xr}@ \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xr}@ x_0@ ( \mbox{xf}@ \mbox{xs} ) ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ ( \mbox{xf}@ \mbox{xx} )@ x_0 ) \ \Rightarrow \ \exists \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xt}\mbox{ and }\forall \mbox{xs}{:}\;\mbox{\$i}{:}\;( ( \mbox{xr}@ \mbox{xs}@ \mbox{xt}\mbox{ and }\mbox{xr}@ \mbox{xx}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{xr}@ ( \mbox{xf}@ \mbox{xs} )@ x_0 ) ) ) )\mbox{ and }\mbox{xr}@ a@ b\mbox{ and }\mbox{xr}@ ( \mbox{xf}@ a )@ x_0\mbox{ and }\mbox{xr}@ x_0@ ( \mbox{xf}@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ a@ \mbox{xx}\mbox{ and }\mbox{xr}@ \mbox{xx}@ b\mbox{ and }\mbox{xr}@ ( \mbox{xf}@ \mbox{xx} )@ x_0\mbox{ and }\mbox{xr}@ x_0@ ( \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG\_SV\_IMV\_SOL},\mbox{conjecture})$

\medskip\textbf{SYO350$\wedge$5.p} TPS problem E1FUNC

Example from [Ben99] p.115.




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

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

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

$\forall x{:}\;\mbox{\$i}{:}\;( f@ x ) = ( g@ x ) \ \Rightarrow \ ( ( p@ f ) \ \Rightarrow \ ( p@ g ) )\qquad\mbox{thf}(\mbox{cE1FUNC},\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$5.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO487$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_5@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO034$\wedge$1.p} Formula not making use of projection






$\exists n{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}{:}\;( ( n@ p )\iff p )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO190$\wedge$5.p} TPS problem CT15






$\exists \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;( \mbox{xx}\mbox{ or }\mbox{xy} )\qquad\mbox{thf}(\mbox{cCT}_{15},\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$3.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mand}@ p@ ( \mbox{mdia\_m}@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ p ) )@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO493$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{mor}@ ( f@ x )@ ( g@ x ) ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO373$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$f{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$\forall \mbox{xx}{:}\;b{:}\;( ( f@ \mbox{xx} )\iff ( g@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;( b\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ f ) \ \Rightarrow \ ( \mbox{xq}@ g ) )\qquad\mbox{thf}(\mbox{cEXT\_SET},\mbox{conjecture})$

\medskip\textbf{SYO335$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

$( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ \forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ c_0@ c_0\mbox{ and }q@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ and }\mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{xv}@ \mbox{xu}\mbox{ or }\mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xv} )@ ( \mbox{cS}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cTHM300C},\mbox{conjecture})$

\medskip\textbf{SYO133$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

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

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}_1@ \mbox{xx}\mbox{ and }( ( \mbox{cP}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}_2@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \mbox{cP}_2@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBAFFLER}_2,\mbox{conjecture})$

\medskip\textbf{SYO209$\wedge$5.p} TPS problem CT29






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

$\neg\,\exists \mbox{xg}{:}\;a\to a\to \mbox{\$o}{:}\;\forall \mbox{xf}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xj}{:}\;a{:}\;\forall \mbox{xp}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ ( \mbox{xg}@ \mbox{xj} ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cCT}_{29},\mbox{conjecture})$

\medskip\textbf{SYO236$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

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

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

$( \forall q{:}\;( b\to a )\to \mbox{\$o}{:}\;( ( q@ f ) \ \Rightarrow \ ( q@ g ) )\mbox{ or }\forall \mbox{xx}{:}\;b{:}\;( f@ \mbox{xx} ) = ( g@ \mbox{xx} ) ) \ \Rightarrow \ f = g\qquad\mbox{thf}(\mbox{cTHM}_{504},\mbox{conjecture})$

\medskip\textbf{SYO260$\wedge$5.p} TPS problem THM125A

Trivial theorem to test flexible-flexible pairs.




$\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xn}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xm}@ \mbox{xa}\mbox{ or }\mbox{xn}@ \mbox{xa} )\mbox{ and }( p@ \mbox{xb}\mbox{ or }\mbox{xn}@ \mbox{xb} )\mbox{ and }( \mbox{xm}@ \mbox{xc}\mbox{ or }\neg\,p@ \mbox{xc} ) )\qquad\mbox{thf}(\mbox{cTHM125A},\mbox{conjecture})$

\medskip\textbf{SYO393$\wedge$1.p} Ted Sider's modal proposition logic theorem 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ ( \mbox{mor}@ p@ q ) ) )@ ( \mbox{mand}@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ p ) )@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO311$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\exists \mbox{xop}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xop}@ \mbox{\$true}@ \mbox{\$true}\mbox{ and }\neg\,\mbox{xop}@ \mbox{\$false}@ \mbox{\$false}\mbox{ and }\neg\,\mbox{xop}@ \mbox{\$false}@ \mbox{\$true}\mbox{ and }\neg\,\mbox{xop}@ \mbox{\$true}@ \mbox{\$false} )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xm}@ \mbox{\$true} )\iff ( \mbox{xm}@ ( \mbox{\$true} \ \Rightarrow \ \mbox{\$true} ) ) )\qquad\mbox{thf}(\mbox{cOMEGA\_EXAMPLE}_2,\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$2.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ p )@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO053$\wedge$1.p} Simple textbook example 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mdia}@ r@ \mbox{mtrue} ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO438$\wedge$1.p} Ted Sider's modal proposition logic theorem 47






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$4.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO065$\wedge$4.001.p} ILTP Problem SYJ201+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO417$\wedge$1.p} Ted Sider's modal proposition logic theorem 26






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mdia\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO560$\wedge$1.p} Choice on \$o$>$\$o applied to choice on \$o is identity or constant






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

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epso}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxo},\mbox{axiom})$

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

$\forall p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsoo}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxoo},\mbox{axiom})$

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

$q@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$true}\qquad\mbox{thf}(\mbox{qkt},\mbox{axiom})$

$q@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$false}\qquad\mbox{thf}(\mbox{qkf},\mbox{axiom})$

$q@ \lambda x{:}\;\mbox{\$o}{:}\;x\qquad\mbox{thf}(\mbox{qid},\mbox{axiom})$

$q@ ( \mbox{epsoo}@ \mbox{epso} )\qquad\mbox{thf}(c,\mbox{conjecture})$

\medskip\textbf{SYO155$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPELL26}_4,\mbox{conjecture})$

\medskip\textbf{SYO073$\wedge$4.003.p} ILTP Problem SYJ209+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ f ) ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO492$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( f@ x )@ ( g@ x ) ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( \mbox{mbox\_s}_5@ ( f@ x ) )@ ( \mbox{mbox\_s}_5@ ( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$6.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ p@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO433$\wedge$1.p} Ted Sider's modal proposition logic theorem 42






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mdia\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ p@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO029$\wedge$1.p} There is an identity unary connective






$\exists n{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}{:}\;( ( n@ p )\iff \neg\,p )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO368$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

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

$( ( a\iff b )\mbox{ and }\mbox{cP}@ a ) \ \Rightarrow \ ( \mbox{cP}@ b )\qquad\mbox{thf}(\mbox{cTRIVEXT}_1,\mbox{conjecture})$

\medskip\textbf{SYO096$\wedge$5.p} TPS problem LX1






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

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

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

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

$( \mbox{cR}@ \mbox{cA}@ \mbox{cB}\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \exists y{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ x@ y ) \ \Rightarrow \ ( \mbox{cQ}@ x@ x ) )\mbox{ and }\forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ u@ v ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ z@ v ) ) ) \ \Rightarrow \ \exists w{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{cB}@ w\mbox{ and }\mbox{cQ}@ w@ \mbox{cA} )\qquad\mbox{thf}(\mbox{cLX}_1,\mbox{conjecture})$

\medskip\textbf{SYO226$\wedge$5.p} TPS problem THM47B






$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall z{:}\;\mbox{\$i}{:}\;( r@ z@ z ) \ \Rightarrow \ ( r@ x@ y ) ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{cTHM47B},\mbox{conjecture})$

\medskip\textbf{SYO255$\wedge$5.p} TPS problem BLEDSOE3






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

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

$( \mbox{cP}@ a ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE}_3,\mbox{conjecture})$

\medskip\textbf{SYO463$\wedge$2.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO558$\wedge$1.p} Forall on \$i can be expressed in terms of choice on \$






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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( p@ ( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\neg\,p@ x ) ) ) = ( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;!! (p) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$6.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO259$\wedge$5.p} TPS problem THM125B

Trivial theorem to test flexible-flexible pairs.




$\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xn}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xn}@ \mbox{xa}\mbox{ or }\mbox{xm}@ \mbox{xa} )\mbox{ and }( p@ \mbox{xb}\mbox{ or }\mbox{xn}@ \mbox{xb} )\mbox{ and }( \mbox{xm}@ \mbox{xc}\mbox{ or }\neg\,p@ \mbox{xc} ) )\qquad\mbox{thf}(\mbox{cTHM125B},\mbox{conjecture})$

\medskip\textbf{SYO062$\wedge$4.003.p} ILTP Problem SYJ105+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

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

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO419$\wedge$1.p} Ted Sider's modal proposition logic theorem 28






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mbox\_d}@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO387$\wedge$5.p} TPS problem THM409-4






$\mbox{cQ}_5{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_5,\mbox{type})$

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

$\mbox{cP}_4{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_4,\mbox{type})$

$\mbox{cQ}_4{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_4,\mbox{type})$

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

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

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

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

$\mbox{cP}_3{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_3,\mbox{type})$

$\mbox{cQ}_3{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_3,\mbox{type})$

$\mbox{cP}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

$\mbox{cQ}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_2,\mbox{type})$

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

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

$\neg\,\mbox{cQ}_1@ a@ b@ c\mbox{ and }\mbox{cP}_1@ a@ a\mbox{ and }\mbox{cP}_1@ b@ b\mbox{ and }\mbox{cP}_1@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_1@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_1@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_1@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_2@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_2@ a@ a\mbox{ and }\mbox{cP}_2@ b@ b\mbox{ and }\mbox{cP}_2@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_2@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_2@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_2@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_3@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_3@ a@ a\mbox{ and }\mbox{cP}_3@ b@ b\mbox{ and }\mbox{cP}_3@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_3@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_3@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_3@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_3@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_4@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_4@ a@ a\mbox{ and }\mbox{cP}_4@ b@ b\mbox{ and }\mbox{cP}_4@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_4@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_4@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_4@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_4@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_4@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_4@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_4@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_5@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cQ}_5@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\qquad\mbox{thf}(\mbox{cTHM409}_4,\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$3.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_m}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_m}@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mdia\_m}@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO211$\wedge$5.p} TPS problem from BASIC-HO-PROP-THMS






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

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{xf}@ x ) = x\qquad\mbox{thf}(\mbox{cCT}_{24},\mbox{conjecture})$

\medskip\textbf{SYO563+2.p} Unequal numbers - rationals






$1/1\ne 2/1\qquad\mbox{fof}(\mbox{one\_not\_equal\_to}_2,\mbox{conjecture})$

\medskip\textbf{SYO193$\wedge$5.p} TPS problem CT20






$\neg\,\forall \mbox{xq}{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xq}@ \mbox{xp} )\qquad\mbox{thf}(\mbox{cCT}_{20},\mbox{conjecture})$

\medskip\textbf{SYO033$\wedge$1.p} There is a universal quantifer






$\exists q{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ p )\iff \forall x{:}\;\mbox{\$i}{:}\;( p@ x ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO495$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( \mbox{mbox\_s}_5@ ( f@ x ) )@ ( \mbox{mbox\_s}_5@ ( g@ x ) ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( f@ x )@ ( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO395$\wedge$1.p} Ted Sider's modal proposition logic theorem 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mnot}@ p )@ p ) )@ ( \mbox{mbox\_k}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO341$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

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

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists q_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},q_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( q_1@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )@ y\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_1@ x@ y_0\mbox{ or }\mbox{cP}@ x\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cP}@ x\mbox{ or }\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q_2@ x@ y_0 ) )\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_2@ x@ y_0\mbox{ or }\mbox{cP}@ x\mbox{ or }q_2@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_2@ x@ y_0\mbox{ or }\neg\,\mbox{cP}@ x\mbox{ or }q_1@ x@ ( \mbox{cG}@ y_0 ) ) ) )\iff \exists q_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cP}@ ( \mbox{cF}@ x@ ( \mbox{cG}@ y ) )\mbox{ or }q@ ( \mbox{cF}@ x@ ( \mbox{cG}@ y ) )@ ( \mbox{cG}@ y ) )\mbox{ and }( \neg\,\mbox{cP}@ ( \mbox{cF}@ x@ ( \mbox{cG}@ y ) )\mbox{ or }q_1@ ( \mbox{cF}@ x@ ( \mbox{cG}@ y ) )@ y )\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q_1@ x@ ( \mbox{cG}@ y_0 ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM74A},\mbox{conjecture})$

\medskip\textbf{SYO044$\wedge$1.p} Simple textbook example 1






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mbox}@ r@ \mbox{mtrue} ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO084$\wedge$5.p} TPS problem THM75

Related to THM87, which was used for CADE-6.




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

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

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

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

$\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xj}{:}\;\mbox{\$i}{:}\;\exists \mbox{xq}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ a@ \mbox{xj}@ ( h@ \mbox{xj} )\mbox{ or }\mbox{cP}@ \mbox{xv}@ \mbox{xj}@ ( k@ \mbox{xj} ) ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xv}@ \mbox{xj}@ \mbox{xq} ) )\qquad\mbox{thf}(\mbox{cTHM}_{75},\mbox{conjecture})$

\medskip\textbf{SYO212$\wedge$5.p} TPS problem from BASIC-HO-PROP-THMS






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

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

$\forall \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ a\mbox{ and }\mbox{xp}@ b ) \ \Rightarrow \ ( \mbox{xp}@ ( a\mbox{ and }b ) ) )\qquad\mbox{thf}(\mbox{cEMB}_2,\mbox{conjecture})$

\medskip\textbf{SYO041$\wedge$1.p} Unsatisfiable basic formula 3






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

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

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

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

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

$x\ne y\mbox{ and }( g@ x ) = y\mbox{ and }( g@ y ) = x\mbox{ and }( f@ ( f@ ( f@ x ) ) ) = ( g@ ( f@ x ) )\qquad\mbox{thf}(3,\mbox{axiom})$

\medskip\textbf{SYO285$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xp}@ y ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx} )\mbox{ and }\mbox{cQ}@ y )\qquad\mbox{thf}(\mbox{cTEST}_2,\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$2.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ p )@ ( \mbox{mdia\_d}@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO561\_2.p} Distinct objects






$\mbox{"Apple"}\ne \mbox{"Microsoft"}\qquad\mbox{tff}(\mbox{apple\_not\_microsoft},\mbox{conjecture})$

\medskip\textbf{SYO458$\wedge$5.p} Ted Sider's propositional modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO094$\wedge$5.p} TPS problem THM55A






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

$( \exists x{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ x@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ y@ y ) ) \ \Rightarrow \ \exists u{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ u@ u ) \ \Rightarrow \ ( \mbox{cR}@ v@ v ) )\qquad\mbox{thf}(\mbox{cTHM55A},\mbox{conjecture})$

\medskip\textbf{SYO051$\wedge$1.p} Simple textbook example 8






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO043$\wedge$1.p} Unsatisfiable basic formula 5






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

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

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

$q@ f@ x\mbox{ and }f@ ( f@ x )\mbox{ and }( f@ ( q@ f@ x ) )\ne ( f@ x )\qquad\mbox{thf}(5,\mbox{axiom})$

\medskip\textbf{SYO457$\wedge$4.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mdia\_b}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO356$\wedge$5.p} TPS problem E1LEIBEQ2

Example from [Ben9] about alternative defns of equality.




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

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

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

$\forall q{:}\;a\to a\to \mbox{\$o}{:}\;( \forall z{:}\;a{:}\;( q@ z@ z ) \ \Rightarrow \ ( q@ u@ v ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ u ) \ \Rightarrow \ ( \mbox{xq}@ v ) )\qquad\mbox{thf}(\mbox{cE1LEIBEQ2\_pme},\mbox{conjecture})$

\medskip\textbf{SYO429$\wedge$1.p} Ted Sider's modal proposition logic theorem 38






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO489$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mdia\_s}_5@ ( f@ a ) )@ ( \mbox{mdia\_s}_5@ ( g@ a ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ ( f@ a )@ ( g@ a ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$5.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO242$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

$\exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) \ \Rightarrow \ \exists r{:}\;a\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( r@ \mbox{xx}@ \mbox{xy}\mbox{ and }r@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall s{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xu}{:}\;a{:}\;( s@ \mbox{xu} ) \ \Rightarrow \ \exists \mbox{xv}{:}\;a{:}\;( s@ \mbox{xv}\mbox{ and }\forall \mbox{xz}{:}\;a{:}\;( r@ \mbox{xv}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{539},\mbox{conjecture})$

\medskip\textbf{SYO521=1.p} There are more than two integers






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

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

$\exists x{:}\;\mbox{\$int}{:}\;( \mbox{\$sum}(2,2) = x\mbox{ and }\forall y{:}\;\mbox{\$int}{:}\;( y = a\mbox{ or }y = b ) )\qquad\mbox{tff}(a,\mbox{conjecture})$

\medskip\textbf{SYO440$\wedge$1.p} Ted Sider's modal proposition logic theorem 49






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO015$\wedge$1.p} A is not equal to not A






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

$a\ne \neg\,a\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$4.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ p )@ ( \mbox{mdia\_b}@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO537$\wedge$1.p} Choice on binary relations between functions






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

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

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

$\mbox{epsa} = ( \lambda r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \mbox{epsii}@ \lambda x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

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

$\mbox{epsb} = ( \lambda r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \mbox{epsii}@ \lambda y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\forall r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO038-1.003.004.p} Boolos' Curious Inference, size f(3,4)






$f(n,1) = s(1)\qquad\mbox{cnf}(\mbox{ax}_1,\mbox{axiom})$

$f(1,s(x)) = s(s(f(1,x)))\qquad\mbox{cnf}(\mbox{ax}_2,\mbox{axiom})$

$f(s(n),s(x)) = f(n,f(s(n),x))\qquad\mbox{cnf}(\mbox{ax}_3,\mbox{axiom})$

$d(1)\qquad\mbox{cnf}(\mbox{ax}_4,\mbox{axiom})$

$d(x) \ \Rightarrow \ d(s(x))\qquad\mbox{cnf}(\mbox{ax}_5,\mbox{axiom})$

$\neg\,d(f(s(s(1)),s(s(s(1)))))\qquad\mbox{cnf}(\mbox{conj},\mbox{negated\_conjecture})$

\medskip\textbf{SYO390$\wedge$5.p} TPS problem UNKNOWN






$\forall \mbox{xp}{:}\;\mbox{\$o},\mbox{xq}{:}\;\mbox{\$o}{:}\;( ( \mbox{xp}\iff \mbox{xq} ) \ \Rightarrow \ \mbox{xp} = \mbox{xq} )\qquad\mbox{thf}(\mbox{cEXT\_eq}_0,\mbox{conjecture})$

\medskip\textbf{SYO007$\wedge$1.p} The non-trivial direction of Boolean extensionality






$\mbox{leibeq}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq1\_type},\mbox{type})$

$\mbox{leibeq} = ( \lambda u{:}\;\mbox{\$o},v{:}\;\mbox{\$o}{:}\;\forall q{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq}_1,\mbox{definition})$

$\forall a{:}\;\mbox{\$o},b{:}\;\mbox{\$o}{:}\;( ( a\iff b ) \ \Rightarrow \ ( \mbox{leibeq}@ a@ b ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO039$\wedge$2.p} Unsatisfiable basic formula 1






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

$( h@ ( h@ \mbox{\$false} ) = ( h@ \neg\,\mbox{\$false} ) ) = ( h@ \mbox{\$false} )\qquad\mbox{thf}(1,\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$4.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ p )@ q ) )@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO120$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

$\forall z_3{:}\;\mbox{\$i}{:}\;( ( \mbox{cA}@ z_3 ) \ \Rightarrow \ ( \mbox{cA}@ z_3 ) )\qquad\mbox{thf}(\mbox{cSET80\_pme},\mbox{conjecture})$

\medskip\textbf{SYO547$\wedge$1.p} Choice Complement

The choice operator applied to complements of predicates chooses 
 an element not in the predicate, if there is one.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{epscomp} = ( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\neg\,p@ x ) )\qquad\mbox{thf}(\mbox{epscompd},\mbox{definition})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;\neg\,p@ x \ \Rightarrow \ \neg\,p@ ( \mbox{epscomp}@ p ) )\qquad\mbox{thf}(\mbox{choicecomp},\mbox{conjecture})$

\medskip\textbf{SYO038-1.003.003.p} Boolos' Curious Inference, size f(3,3)






$f(n,1) = s(1)\qquad\mbox{cnf}(\mbox{ax}_1,\mbox{axiom})$

$f(1,s(x)) = s(s(f(1,x)))\qquad\mbox{cnf}(\mbox{ax}_2,\mbox{axiom})$

$f(s(n),s(x)) = f(n,f(s(n),x))\qquad\mbox{cnf}(\mbox{ax}_3,\mbox{axiom})$

$d(1)\qquad\mbox{cnf}(\mbox{ax}_4,\mbox{axiom})$

$d(x) \ \Rightarrow \ d(s(x))\qquad\mbox{cnf}(\mbox{ax}_5,\mbox{axiom})$

$\neg\,d(f(s(s(1)),s(s(1))))\qquad\mbox{cnf}(\mbox{conj},\mbox{negated\_conjecture})$

\medskip\textbf{SYO229$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

$\exists \mbox{xu}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\mbox{xu} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cA}@ \mbox{xx}\mbox{ and }\mbox{cB}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_3,\mbox{conjecture})$

\medskip\textbf{SYO394$\wedge$1.p} Ted Sider's modal proposition logic theorem 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mand}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO360$\wedge$5.p} TPS problem EDEC1

A version of EDEC from [Ben99] using = instead of LeibEq.




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

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

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

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

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

$( \forall x{:}\;a\to a,y{:}\;a{:}\;( f@ x@ y ) = ( g@ x@ y )\mbox{ and }\forall z{:}\;a{:}\;( h@ z ) = ( j@ z ) ) \ \Rightarrow \ ( f@ h ) = ( g@ j )\qquad\mbox{thf}(\mbox{cEDEC}_1,\mbox{conjecture})$

\medskip\textbf{SYO248$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

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

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

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

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

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

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

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

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

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

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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( ( ( ( p@ a ) = ( p@ \mbox{aa} )\mbox{ and }( p@ b ) = ( p@ \mbox{bb} )\mbox{ and }( p@ e ) = ( p@ \mbox{hh} ) ) \ \Rightarrow \ ( p@ c ) = ( p@ \mbox{dd} ) )\mbox{ and }( ( ( p@ a ) = ( p@ \mbox{aa} )\mbox{ and }( p@ h ) = ( p@ \mbox{hh} )\mbox{ and }( p@ b ) = ( p@ \mbox{cc} ) ) \ \Rightarrow \ ( p@ d )\ne ( p@ \mbox{ee} ) )\mbox{ and }( ( ( p@ c ) = ( p@ \mbox{cc} )\mbox{ and }( p@ \mbox{cc} ) = ( p@ d )\mbox{ and }( p@ d ) = ( p@ \mbox{dd} )\mbox{ and }( p@ a )\ne ( p@ \mbox{bb} ) ) \ \Rightarrow \ ( p@ e )\ne ( p@ \mbox{hh} ) )\mbox{ and }( ( ( p@ a ) = ( p@ \mbox{aa} )\mbox{ and }( p@ d ) = ( p@ \mbox{dd} )\mbox{ and }( p@ b )\ne ( p@ \mbox{cc} ) ) \ \Rightarrow \ ( p@ e ) = ( p@ \mbox{hh} ) )\mbox{ and }( ( ( p@ e ) = ( p@ \mbox{ee} )\mbox{ and }( p@ h ) = ( p@ \mbox{hh} )\mbox{ and }( p@ c ) = ( p@ \mbox{dd} ) ) \ \Rightarrow \ ( p@ a )\ne ( p@ \mbox{bb} ) )\mbox{ and }( ( ( p@ b ) = ( p@ \mbox{bb} )\mbox{ and }( p@ \mbox{bb} ) = ( p@ c )\mbox{ and }( p@ c ) = ( p@ \mbox{cc} )\mbox{ and }( p@ e )\ne ( p@ \mbox{hh} ) ) \ \Rightarrow \ ( p@ d ) = ( p@ \mbox{ee} ) ) ) \ \Rightarrow \ ( ( p@ a )\ne ( p@ \mbox{aa} )\mbox{ or }( p@ b )\ne ( p@ \mbox{bb} )\mbox{ or }( p@ c )\ne ( p@ \mbox{cc} )\mbox{ or }( p@ d )\ne ( p@ \mbox{dd} )\mbox{ or }( p@ e )\ne ( p@ \mbox{ee} )\mbox{ or }( p@ h )\ne ( p@ \mbox{hh} ) ) )\qquad\mbox{thf}(\mbox{cSIXFRIENDS\_AGAIN},\mbox{conjecture})$

\medskip\textbf{SYO160$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

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

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

$( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\mbox{ or }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \mbox{cR}@ a\mbox{ and }\mbox{cR}@ b\mbox{ and }\mbox{cR}@ c )\qquad\mbox{thf}(\mbox{cDISJ\_THIRD},\mbox{conjecture})$

\medskip\textbf{SYO554$\wedge$1.p} Teucke's example






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

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

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

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

$p@ s\mbox{ or }p@ t\qquad\mbox{thf}(\mbox{pst},\mbox{axiom})$

$\neg\,p@ u\mbox{ or }\neg\,p@ t\qquad\mbox{thf}(\mbox{puv},\mbox{axiom})$

$s = t\qquad\mbox{thf}(\mbox{st},\mbox{axiom})$

$t = u\qquad\mbox{thf}(\mbox{tu},\mbox{axiom})$

\medskip\textbf{SYO370$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$( \mbox{cP}@ \mbox{cA} ) \ \Rightarrow \ ( \mbox{cP}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\neg\,\mbox{cA}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cTRIVEXT}_3,\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$3.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ p ) )@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO199$\wedge$5.p} TPS problem CT7






$\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cCT}_7,\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.007.p} Seven function variant of the Kaminski equation






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

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$f_4{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_4,\mbox{type})$

$f_5{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_5,\mbox{type})$

$f_6{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_6,\mbox{type})$

$( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ ( f_3@ ( f_3@ ( f_3@ ( f_4@ ( f_5@ ( f_5@ ( f_5@ ( f_6@ x ) ) ) ) ) ) ) ) ) ) ) ) ) = ( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ ( f_3@ ( f_4@ ( f_4@ ( f_4@ ( f_5@ ( f_6@ ( f_6@ ( f_6@ x ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_7,\mbox{conjecture})$

\medskip\textbf{SYO496$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( g@ x ) ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( \mbox{mbox\_s}_5@ ( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO050$\wedge$2.p} Simple textbook example 7






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ a ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO030$\wedge$1.p} Not every unary connective is the identity






$\mbox{leibeq}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\neg\,\forall f{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\exists x{:}\;\mbox{\$o}{:}\;( \mbox{leibeq}@ ( f@ x )@ x )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO105$\wedge$5.p} TPS problem X2201TEST






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

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

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

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

$\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}@ y ) \ \Rightarrow \ \neg\,\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xu}@ y@ \mbox{xz} ) \ \Rightarrow \ \neg\,\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xv} ) )\iff \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\forall \mbox{xv}{:}\;\mbox{\$i}{:}\;\neg\,( \mbox{cQ}@ \mbox{xx}@ y ) \ \Rightarrow \ \neg\,( \mbox{cP}@ \mbox{xu}@ y@ \mbox{xz} ) \ \Rightarrow \ \neg\,\mbox{cR}@ \mbox{xv}\qquad\mbox{thf}(\mbox{cX2201TEST},\mbox{conjecture})$

\medskip\textbf{SYO104$\wedge$5.p} TPS problem TTTP2129






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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ y\mbox{ and }\mbox{cQ}@ \mbox{xx}@ y )\iff ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ y )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}@ y ) )\qquad\mbox{thf}(\mbox{cTTTP}_{2129},\mbox{conjecture})$

\medskip\textbf{SYO220$\wedge$5.p} TPS problem THM47A






$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x = y \ \Rightarrow \ \forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall z{:}\;\mbox{\$i}{:}\;( r@ z@ z ) \ \Rightarrow \ ( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{cTHM47A},\mbox{conjecture})$

\medskip\textbf{SYO032$\wedge$1.p} There is a disjunction connective






$\exists d{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o},q{:}\;\mbox{\$o}{:}\;( ( d@ p@ q )\iff ( p\mbox{ or }q ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO282$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

$\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;( \exists \mbox{xx}_0{:}\;b{:}\;( \mbox{xx}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cL}_{51},\mbox{conjecture})$

\medskip\textbf{SYO289$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xp}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTRIV}_2,\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$6.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ p ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO503$\wedge$1.p} Tableau with two branches






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

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

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

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

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

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

$a = b\mbox{ or }\neg\,f@ a\mbox{ or }\neg\,f@ b\mbox{ or }\neg\,g@ a\mbox{ or }\neg\,g@ b\mbox{ or }\neg\,p@ f\mbox{ or }p@ g\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO549$\wedge$1.p} The eta double negation problem






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

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

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

$p@ \lambda x{:}\;\mbox{\$o}{:}\;( f@ \neg\,\neg\,x )@ g\qquad\mbox{thf}(\mbox{pfg},\mbox{axiom})$

$p@ f@ \lambda x{:}\;\mbox{\$o}{:}\;( g@ \neg\,\neg\,x )\qquad\mbox{thf}(\mbox{pfgc},\mbox{conjecture})$

\medskip\textbf{SYO185$\wedge$5.p} TPS problem CT17






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

$\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;( y \ \Rightarrow \ y )\qquad\mbox{thf}(\mbox{cCT}_{17},\mbox{conjecture})$

\medskip\textbf{SYO446$\wedge$1.p} Ted Sider's modal proposition logic theorem 55






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ p@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mand}@ ( \mbox{mdia\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO067$\wedge$4.002.p} ILTP Problem SYJ203+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO506$\wedge$1.p} (if (X = Y) then X else Y) = Y






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

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( c@ x = y@ x@ y ) = y\mbox{ or }\neg\,\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( c@ \mbox{\$true}@ x@ y ) = x\mbox{ or }\neg\,\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( c@ \mbox{\$false}@ x@ y ) = y\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO468$\wedge$2.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ p )@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO058$\wedge$4.p} ILTP problem SYJ101+1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$\mbox{ivalid}@ ( \mbox{iatom}@ a )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ a )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO463$\wedge$5.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO038$\wedge$1.005.005.p} Boolos' Curious Inference, size f(5,5)






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

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

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

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

$\forall n{:}\;\mbox{\$i}{:}\;( f@ n@ 1 ) = ( s@ 1 )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( f@ 1@ ( s@ x ) ) = ( s@ ( s@ ( f@ 1@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\forall n{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( f@ ( s@ n )@ ( s@ x ) ) = ( f@ n@ ( f@ ( s@ n )@ x ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$d@ 1\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( d@ x ) \ \Rightarrow \ ( d@ ( s@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$d@ ( f@ ( s@ ( s@ ( s@ ( s@ 1 ) ) ) )@ ( s@ ( s@ ( s@ ( s@ 1 ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO498$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( n@ x )@ ( \mbox{mand}@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( n@ y )@ ( \mbox{meq\_ind}@ y@ x ) ) )@ ( \mbox{mbox\_s}_5@ ( o@ x ) ) ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( n@ x )@ ( \mbox{mand}@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( n@ y )@ ( \mbox{meq\_ind}@ y@ x ) ) )@ ( o@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO068$\wedge$4.005.p} ILTP Problem SYJ204+1.005






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p5\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_5 )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_3 ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_4 ) ) )\qquad\mbox{thf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO176$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

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

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

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

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

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

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

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

$( ( \mbox{cL}\mbox{ and }\mbox{cP} ) \ \Rightarrow \ \mbox{cM} )\mbox{ and }( ( \mbox{cG}\mbox{ and }\neg\,\mbox{cR} ) \ \Rightarrow \ \mbox{cM} )\mbox{ and }( ( \neg\,\mbox{cK}\mbox{ and }\mbox{cN}\mbox{ and }\mbox{cM} ) \ \Rightarrow \ \mbox{cF} )\mbox{ and }( ( \neg\,\mbox{cG}\mbox{ and }\neg\,\mbox{cP} ) \ \Rightarrow \ \mbox{cR} )\mbox{ and }( ( \mbox{cK}\mbox{ and }\mbox{cB} ) \ \Rightarrow \ \mbox{cC} )\mbox{ and }( ( \mbox{cR}\mbox{ and }\neg\,\mbox{cN}\mbox{ and }\neg\,\mbox{cF} ) \ \Rightarrow \ \mbox{cP} )\mbox{ and }( ( \mbox{cL}\mbox{ and }\mbox{cM} ) \ \Rightarrow \ \mbox{cC} )\mbox{ and }( ( \mbox{cE}\mbox{ and }\neg\,\mbox{cK}\mbox{ and }\mbox{cG}\mbox{ and }\neg\,\mbox{cN} ) \ \Rightarrow \ \neg\,\mbox{cM} )\mbox{ and }( ( \neg\,\mbox{cG}\mbox{ and }\neg\,\mbox{cR} ) \ \Rightarrow \ \mbox{cK} )\mbox{ and }( ( \mbox{cK}\mbox{ and }\mbox{cL}\mbox{ and }\mbox{cE} ) \ \Rightarrow \ \neg\,\mbox{cM} )\mbox{ and }( ( \mbox{cR}\mbox{ and }\mbox{cE} ) \ \Rightarrow \ \neg\,\mbox{cC} )\mbox{ and }( ( \mbox{cG}\mbox{ and }\neg\,\mbox{cK}\mbox{ and }\neg\,\mbox{cM} ) \ \Rightarrow \ \neg\,\mbox{cB} )\mbox{ and }( ( \mbox{cN}\mbox{ and }\neg\,\mbox{cP}\mbox{ and }\neg\,\mbox{cF} ) \ \Rightarrow \ \mbox{cC} )\mbox{ and }( ( \mbox{cG}\mbox{ and }\mbox{cB}\mbox{ and }\neg\,\mbox{cR} ) \ \Rightarrow \ \neg\,\mbox{cC} )\mbox{ and }( ( \mbox{cR}\mbox{ and }\neg\,\mbox{cK}\mbox{ and }\neg\,\mbox{cM} ) \ \Rightarrow \ \mbox{cG} )\qquad\mbox{thf}(\mbox{cPORKCHOP},\mbox{conjecture})$

\medskip\textbf{SYO536$\wedge$1.p} Choice on relations between functions and individuals






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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

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

$\mbox{epsa} = ( \lambda r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{epsii}@ \lambda x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

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

$\mbox{epsb} = ( \lambda r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda y{:}\;\mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\forall r{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO066$\wedge$4.001.p} ILTP Problem SYJ202+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ o_{11} )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ o_{21} )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO122$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

$\forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cEQ}@ \mbox{xu}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cA\_4A}_3,\mbox{conjecture})$

\medskip\textbf{SYO038$\wedge$1.002.304.p} Boolos' Curious Inference, size f(2,f(3,4))






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

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

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

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

$\forall n{:}\;\mbox{\$i}{:}\;( f@ n@ 1 ) = ( s@ 1 )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( f@ 1@ ( s@ x ) ) = ( s@ ( s@ ( f@ 1@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\forall n{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( f@ ( s@ n )@ ( s@ x ) ) = ( f@ n@ ( f@ ( s@ n )@ x ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$d@ 1\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( d@ x ) \ \Rightarrow \ ( d@ ( s@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$d@ ( f@ ( s@ 1 )@ ( f@ ( s@ ( s@ 1 ) )@ ( s@ ( s@ ( s@ 1 ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO293$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}\mbox{ and }a@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE5A},\mbox{conjecture})$

\medskip\textbf{SYO264$\wedge$5.p} TPS problem THM125C

Trivial theorem to test flexible-flexible pairs.




$\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xn}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xm}@ \mbox{xa}\mbox{ or }\mbox{xn}@ \mbox{xa} )\mbox{ and }( p@ \mbox{xb}\mbox{ or }\mbox{xn}@ \mbox{xb} )\mbox{ and }( \mbox{xm}@ \mbox{xc}\mbox{ or }\neg\,p@ \mbox{xc} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xn}@ \mbox{xx} )\iff \neg\,p@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM125C},\mbox{conjecture})$

\medskip\textbf{SYO055$\wedge$1.p} Simple textbook example 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mor}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ b ) )@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ b )@ a ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$1.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_k}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO235$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

$( \forall q{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( q@ \mbox{cA} ) \ \Rightarrow \ ( q@ \mbox{cB} ) )\mbox{ or }( \mbox{cA}\iff \mbox{cB} ) ) \ \Rightarrow \ \mbox{cA} = \mbox{cB}\qquad\mbox{thf}(\mbox{cTHM}_{505},\mbox{conjecture})$

\medskip\textbf{SYO090$\wedge$5.p} TPS problem DMG8

A propositional problem suggested by Jay Hunschel Kim.




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

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

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

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

$( \neg\,\neg\,( \neg\,( \neg\,p\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t\mbox{ or }\neg\,q\mbox{ or }\neg\,s\mbox{ or }\neg\,t ) \ \Rightarrow \ ( \neg\,( \neg\,( \neg\,( \neg\,p\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\qquad\mbox{thf}(\mbox{cDMG}_8,\mbox{conjecture})$

\medskip\textbf{SYO258$\wedge$5.p} TPS problem BLEDSOE-FENG-6






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

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

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

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

$( \mbox{cP}@ a@ b\mbox{ and }\forall e{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cQ}@ b@ e ) ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xg} ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xg}@ \mbox{xx}\mbox{ and }\mbox{cQ}@ \mbox{xx}@ a )\mbox{ and }a@ a ) )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG}_6,\mbox{conjecture})$

\medskip\textbf{SYO556$\wedge$1.p} Relationship between if-then-else and choice on \$






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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ p ) = ( \mbox{if}@ \exists x{:}\;\mbox{\$i}{:}\;( p@ x )@ ( \mbox{eps}@ p )@ ( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO443$\wedge$1.p} Ted Sider's modal proposition logic theorem 52






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ ( \mbox{mbox\_s}_5@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_5@ ( \mbox{mnot}@ p ) )@ ( \mbox{mbox\_s}_5@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO457$\wedge$5.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO064$\wedge$4.004.p} ILTP Problem SYJ107+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$a_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a4\_type},\mbox{type})$

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

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$b_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ a_4 )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ b_3 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ b_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ b_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a ) )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ a_4 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO062$\wedge$4.002.p} ILTP Problem SYJ105+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO381$\wedge$5.p} TPS problem X-2002-12-17

Suggested by a student's answer on a test.




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

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xz}@ \mbox{xy} )\iff \forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx}@ \mbox{xw} ) )\qquad\mbox{thf}(\mbox{cX\_2002\_12}_{17},\mbox{conjecture})$

\medskip\textbf{SYO357$\wedge$5.p} TPS problem E2LEIBEQ2

Example from [Ben99] about alternative defns of equality.




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

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

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

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

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

$\forall p{:}\;\mbox{atype}\to \mbox{\$o}{:}\;( ( ( a\mbox{ or }\neg\,a )\mbox{ and }p@ u ) \ \Rightarrow \ ( ( b\mbox{ or }\neg\,b )\mbox{ and }p@ v ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;\mbox{atype}\to \mbox{\$o}{:}\;( ( \mbox{xq}@ u ) \ \Rightarrow \ ( \mbox{xq}@ v ) )\qquad\mbox{thf}(\mbox{cE2LEIBEQ2\_pme},\mbox{conjecture})$

\medskip\textbf{SYO112$\wedge$5.p} TPS problem THM53






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

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x )\iff \exists y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ y ) )\iff ( \forall x{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ x )\iff \exists y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ y ) )\qquad\mbox{thf}(\mbox{cTHM}_{53},\mbox{conjecture})$

\medskip\textbf{SYO121$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

$\exists e{:}\;\mbox{\$i}{:}\;\forall k{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ k@ e@ k )\qquad\mbox{thf}(\mbox{cTHM}_{27},\mbox{conjecture})$

\medskip\textbf{SYO066$\wedge$4.003.p} ILTP Problem SYJ202+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{12}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o12\_type},\mbox{type})$

$o_{13}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o13\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$o_{22}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o22\_type},\mbox{type})$

$o_{23}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o23\_type},\mbox{type})$

$o_{31}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o31\_type},\mbox{type})$

$o_{32}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o32\_type},\mbox{type})$

$o_{33}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o33\_type},\mbox{type})$

$o_{41}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o41\_type},\mbox{type})$

$o_{42}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o42\_type},\mbox{type})$

$o_{43}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o43\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{13} ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{23} ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{32} )@ ( \mbox{iatom}@ o_{33} ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{41} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{42} )@ ( \mbox{iatom}@ o_{43} ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{22} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{32} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{23} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{33} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{43} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{23} )@ ( \mbox{iatom}@ o_{33} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{23} )@ ( \mbox{iatom}@ o_{43} ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{33} )@ ( \mbox{iatom}@ o_{43} ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO159$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\neg\,\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cA}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cB}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cC}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cA}@ \mbox{xu}\mbox{ and }\neg\,\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;\neg\,( \mbox{cB}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ \neg\,\mbox{cC}@ \mbox{xv} )\qquad\mbox{thf}(\mbox{cNNF\_EXAMPLE},\mbox{conjecture})$

\medskip\textbf{SYO353$\wedge$5.p} TPS problem E1LEIBEQ1

Example about alternative defns of equality.




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

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

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

$\forall \mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ u ) \ \Rightarrow \ ( \mbox{xq}@ v ) ) \ \Rightarrow \ \forall q{:}\;a\to a\to \mbox{\$o}{:}\;( \forall z{:}\;a{:}\;( q@ z@ z ) \ \Rightarrow \ ( q@ u@ v ) )\qquad\mbox{thf}(\mbox{cE1LEIBEQ1\_pme},\mbox{conjecture})$

\medskip\textbf{SYO109$\wedge$5.p} TPS problem THM271






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

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

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

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cN}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;\exists \mbox{xu}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ \mbox{xu}@ \mbox{xv} )\mbox{ or }\mbox{cN}@ \mbox{xu}@ \mbox{xw} )\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cM}@ \mbox{xw}@ \mbox{xz}\mbox{ or }\mbox{cN}@ \mbox{xw}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cM}@ \mbox{xz}@ \mbox{xw}\mbox{ or }\mbox{cN}@ \mbox{xz}@ \mbox{xw}\mbox{ or }\mbox{cM}@ \mbox{xz}@ \mbox{xz}\mbox{ or }\mbox{cN}@ \mbox{xz}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cTHM}_{271},\mbox{conjecture})$

\medskip\textbf{SYO217$\wedge$5.p} TPS problem THM174

Principle of extensionality for binary relations.




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

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

$\forall \mbox{xr}{:}\;b\to a\to \mbox{\$o},\mbox{xs}{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;a{:}\;( ( \mbox{xr}@ \mbox{xx}@ \mbox{xy} )\iff ( \mbox{xs}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xr} = \mbox{xs} )\qquad\mbox{thf}(\mbox{cTHM}_{174},\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$1.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO239$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(c_2,\mbox{type})$

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

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\mbox{xx} = ( \mbox{c\_star}@ c_2@ \mbox{xu} )\iff \neg\,\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( s@ \mbox{xx} ) = ( \mbox{c\_star}@ c_2@ \mbox{xv} ) ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} )\iff \neg\,a@ ( s@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG\_SV\_EO1\_W\_LEM},\mbox{conjecture})$

\medskip\textbf{SYO086$\wedge$5.p} TPS problem THM50-11

Simple formula for debugging.




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

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

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

$( \neg\,\mbox{cP}\mbox{ and }( ( \mbox{cQ}\mbox{ and }\neg\,\mbox{cR} )\mbox{ or }( \neg\,\mbox{cQ}\mbox{ and }\mbox{cR} ) ) ) \ \Rightarrow \ ( ( ( \mbox{cP}\mbox{ and }\mbox{cQ} )\mbox{ or }( \neg\,\mbox{cP}\mbox{ and }\neg\,\mbox{cQ} ) ) \ \Rightarrow \ \mbox{cR} )\qquad\mbox{thf}(\mbox{cTHM50}_{11},\mbox{conjecture})$

\medskip\textbf{SYO456$\wedge$1.p} Ted Sider's propositional modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO269$\wedge$5.p} TPS problem THM112D






$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}_9{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}_{10}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}@ \mbox{xw}_1 )\mbox{ and }\forall g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ g@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ g@ \mbox{xw}_1 )\mbox{ and }\forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ g@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ g@ \mbox{xw}_1 )\mbox{ and }\forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ h@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ h@ \mbox{xw}_1 ) ) \ \Rightarrow \ ( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( h@ \mbox{xx} ) )@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( h@ \mbox{xx} ) )@ \mbox{xw}_1 )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ ( g@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM112D},\mbox{conjecture})$

\medskip\textbf{SYO552$\wedge$1.p} The first projection exists

There is a binary function that returns its first argument




$\exists f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( f@ x@ y ) = x\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO533$\wedge$1.p} Binary choice on individuals 4

There is an Epsa such that Epsa and (epsb Epsa) work together to 
 give an a and b such that R a b holds, if such an a and b exist 
 for a binary relation R on \$i. A choice operator on i can be used
 to define a choice operator on i*i (Curried). In this version the
 prover must synthesize both parts of the solution.




$\exists \mbox{epsa}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i},\mbox{epsb}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO366$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$( \mbox{cP}@ ( \mbox{cA} \ \Rightarrow \ \mbox{cA} ) ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{\$true} )\qquad\mbox{thf}(\mbox{cTRIVEXT}_4,\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$1.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_k}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO175$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$\mbox{cGRAIN}{:}\;p\qquad\mbox{thf}(\mbox{cGRAIN},\mbox{type})$

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

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

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

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

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

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

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

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

$( \forall \mbox{xx}{:}\;a{:}\;( \forall \mbox{xy}{:}\;p{:}\;( \mbox{cVEG\_EATS}@ \mbox{xx}@ \mbox{xy} )\mbox{ or }\forall \mbox{xz}{:}\;a{:}\;( ( \mbox{cSMALLER}@ \mbox{xz}@ \mbox{xx}\mbox{ and }\exists \mbox{xw}{:}\;p{:}\;( \mbox{cVEG\_EATS}@ \mbox{xz}@ \mbox{xw} ) ) \ \Rightarrow \ ( \mbox{cMEAT\_EATS}@ \mbox{xx}@ \mbox{xz} ) ) )\mbox{ and }\mbox{cSMALLER}@ \mbox{cCATERPILLAR}@ \mbox{cBIRD}\mbox{ and }\mbox{cSMALLER}@ \mbox{cSNAIL}@ \mbox{cBIRD}\mbox{ and }\mbox{cSMALLER}@ \mbox{cBIRD}@ \mbox{cFOX}\mbox{ and }\mbox{cSMALLER}@ \mbox{cFOX}@ \mbox{cWOLF}\mbox{ and }\neg\,\mbox{cMEAT\_EATS}@ \mbox{cWOLF}@ \mbox{cFOX}\mbox{ and }\neg\,\mbox{cVEG\_EATS}@ \mbox{cWOLF}@ \mbox{cGRAIN}\mbox{ and }\mbox{cMEAT\_EATS}@ \mbox{cBIRD}@ \mbox{cCATERPILLAR}\mbox{ and }\neg\,\mbox{cMEAT\_EATS}@ \mbox{cBIRD}@ \mbox{cSNAIL}\mbox{ and }\exists \mbox{xy}{:}\;p{:}\;( \mbox{cVEG\_EATS}@ \mbox{cCATERPILLAR}@ \mbox{xy} )\mbox{ and }\exists \mbox{xy}{:}\;p{:}\;( \mbox{cVEG\_EATS}@ \mbox{cSNAIL}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cMEAT\_EATS}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cVEG\_EATS}@ \mbox{xy}@ \mbox{cGRAIN} )\qquad\mbox{thf}(\mbox{cPUZ031\_1\_HO},\mbox{conjecture})$

\medskip\textbf{SYO148$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$\forall u{:}\;a{:}\;( \neg\,\mbox{cX}@ u\iff ( ( \mbox{cY}@ u )\iff ( \mbox{cZ}@ u ) ) ) \ \Rightarrow \ \mbox{cX} = ( \lambda u{:}\;a{:}\;\neg\,( \mbox{cY}@ u )\iff ( \mbox{cZ}@ u ) )\qquad\mbox{thf}(\mbox{cBOOL}_{25},\mbox{conjecture})$

\medskip\textbf{SYO555$\wedge$1.p} If-then-else defined from choice is independent of choice






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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_1@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_1,\mbox{axiom})$

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

$\mbox{if}_1 = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}_1@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{if1d},\mbox{definition})$

$\mbox{eps}_2{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps}_2,\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_2@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_2,\mbox{axiom})$

$\mbox{if}_2{:}\;\mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if}_2,\mbox{type})$

$\mbox{if}_2 = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}_2@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{if2d},\mbox{definition})$

$\mbox{if}_1 = \mbox{if}_2\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO279$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cReduct}@ ( \mbox{cAPP}@ ( \mbox{cLAM}@ \mbox{xx} )@ \mbox{xy} )@ ( \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBETA\_CONVERSION},\mbox{conjecture})$

\medskip\textbf{SYO407$\wedge$1.p} Ted Sider's modal proposition logic theorem 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ q@ r ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ r ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO459$\wedge$2.p} Ted Sider's propositional modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO069$\wedge$4.001.p} ILTP Problem SYJ205+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

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

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) ) ) )@ ( \mbox{iatom}@ f ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO166$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ z ) )@ ( i@ ( i@ \mbox{xx}@ \mbox{xy} )@ ( i@ \mbox{xx}@ z ) ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ ( n@ \mbox{xx} )@ ( n@ \mbox{xy} ) )@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cT}@ ( i@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cT}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ ( n@ ( n@ \mbox{xx} ) )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPELL}_{67},\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$5.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_s}_4@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO542$\wedge$1.p} If-then-else on \$i$>$\$i defined from choice on \$i$>$\$i

A choice operator on (\$i$>$\$i) is used to define an if-then-else 
 operator at (\$i$>$\$i). Check that it always returns the then-part 
 or the else-part.




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

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

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

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{epsii}@ \lambda z{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{if}@ b@ x@ y ) = x\mbox{ or }( \mbox{if}@ b@ x@ y ) = y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO409$\wedge$1.p} Ted Sider's modal proposition logic theorem 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_k}@ ( \mbox{mor}@ p@ q ) )@ ( \mbox{mor}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$6.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mdia\_s}_5@ p ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$3.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mor}@ p@ ( \mbox{mdia\_m}@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mdia\_m}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO384$\wedge$5.p} TPS problem THM408






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

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

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

$( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xn}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xw}@ \mbox{xx}@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xd}{:}\;\mbox{\$i}{:}\;\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;( ( \mbox{cD}@ \mbox{xa}@ \mbox{xb}@ \mbox{xd} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xa}@ \mbox{xy}\mbox{ and }\mbox{cF}@ \mbox{xb}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xy}@ \mbox{xz}@ \mbox{xe} ) ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xm}@ \mbox{xw} ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xw}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xz}@ \mbox{xy}@ \mbox{xe} ) ) ) ) \ \Rightarrow \ ( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xn}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xn}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xw}@ \mbox{xx}@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xd}{:}\;\mbox{\$i}{:}\;\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;( ( \mbox{cD}@ \mbox{xa}@ \mbox{xb}@ \mbox{xd} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xa}@ \mbox{xy}\mbox{ and }\mbox{cF}@ \mbox{xb}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xy}@ \mbox{xz}@ \mbox{xe} ) ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xe}{:}\;\mbox{\$i}{:}\;\exists \mbox{xm}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xm}@ \mbox{xw} ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xw}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cD}@ \mbox{xz}@ \mbox{xy}@ \mbox{xe} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{408},\mbox{conjecture})$

\medskip\textbf{SYO168$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ z ) )@ ( i@ ( i@ \mbox{xx}@ \mbox{xy} )@ ( i@ \mbox{xx}@ z ) ) )\mbox{ and }\forall \mbox{xx}_0{:}\;\mbox{\$i},\mbox{xy}_0{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ ( i@ \mbox{xy}_0@ \mbox{xx}_0 )@ ( i@ ( n@ \mbox{xx}_0 )@ ( n@ \mbox{xy}_0 ) ) ) )\mbox{ and }( ( \mbox{cT}@ ( i@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cT}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( n@ ( n@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{68},\mbox{conjecture})$

\medskip\textbf{SYO428$\wedge$1.p} Ted Sider's modal proposition logic theorem 37






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_m}@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mdia\_m}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO456$\wedge$2.p} Ted Sider's propositional modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) ) )@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO553$\wedge$1.p} The second projection exists.

There is a binary function that returns its second argument.




$\exists f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( f@ x@ y ) = y\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO107$\wedge$5.p} TPS problem THM66






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

$\exists x{:}\;\mbox{\$i}{:}\;\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ ( \mbox{cP}@ y ) )\iff \exists x{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ y ) )\qquad\mbox{thf}(\mbox{cTHM}_{66},\mbox{conjecture})$

\medskip\textbf{SYO189$\wedge$5.p} TPS problem CT5






$\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;\forall \mbox{xy}{:}\;\mbox{\$o}{:}\;( \mbox{xx} \ \Rightarrow \ \mbox{xy} )\qquad\mbox{thf}(\mbox{cCT}_5,\mbox{conjecture})$

\medskip\textbf{SYO432$\wedge$1.p} Ted Sider's modal proposition logic theorem 41






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ q ) ) ) ) )@ ( \mbox{mbox\_b}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO068$\wedge$4.010.p} ILTP Problem SYJ204+1.010






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_{10}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p10\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p5\_type},\mbox{type})$

$p_6{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p6\_type},\mbox{type})$

$p_7{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p7\_type},\mbox{type})$

$p_8{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p8\_type},\mbox{type})$

$p_9{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p9\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_{10} )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_3 ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_4 ) ) )\qquad\mbox{thf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_5 ) ) )\qquad\mbox{thf}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_7 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_7 )@ ( \mbox{iatom}@ p_6 ) ) )\qquad\mbox{thf}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_8 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_8 )@ ( \mbox{iatom}@ p_7 ) ) )\qquad\mbox{thf}(\mbox{axiom}_9,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_9 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_9 )@ ( \mbox{iatom}@ p_8 ) ) )\qquad\mbox{thf}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_{10} )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_{10} )@ ( \mbox{iatom}@ p_9 ) ) )\qquad\mbox{thf}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO302$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\mbox{c\_less\_}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_less\_},\mbox{type})$

$( \mbox{c\_less\_}@ a@ b\mbox{ and }\mbox{c\_less\_}@ b@ c ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,a@ a\mbox{ and }a@ b\mbox{ and }\neg\,a@ c )\qquad\mbox{thf}(\mbox{cBLEDSOE}_4,\mbox{conjecture})$

\medskip\textbf{SYO334$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

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

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

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

\medskip\textbf{SYO266$\wedge$5.p} TPS problem THM44






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

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

$\exists s{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( s@ x\mbox{ or }\mbox{cP}@ x )\mbox{ and }( \neg\,s@ x\mbox{ or }\mbox{cQ}@ x ) )\iff \forall y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ y\mbox{ or }\mbox{cQ}@ y )\qquad\mbox{thf}(\mbox{cTHM}_{44},\mbox{conjecture})$

\medskip\textbf{SYO562\_1.p} If-then-else






$a{:}\;\mbox{\$i}\qquad\mbox{tff}(\mbox{a\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

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

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

$\forall z{:}\;\mbox{\$i}{:}\;\mbox{\$ite\_f}(\exists x{:}\;p(f(x)),\forall x{:}\;q(x,x),q(z,a))\qquad\mbox{tff}(\mbox{ite\_f},\mbox{axiom})$

$p(\mbox{\$ite\_t}(q(a,f(a)),a,f(a)))\qquad\mbox{tff}(\mbox{ite\_t},\mbox{axiom})$

$q(a,a)\qquad\mbox{tff}(\mbox{fact},\mbox{axiom})$

$q(f(a),f(a))\qquad\mbox{tff}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO261$\wedge$5.p} TPS problem BLEDSOE-FENG-SV-I1






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

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

$\mbox{c1\_plus}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_plus},\mbox{type})$

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

$( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( a@ \mbox{cO}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( a@ ( \mbox{c1\_plus}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( a@ n ) )\mbox{ and }\mbox{cP}@ \mbox{cO}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ ( \mbox{c1\_plus}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{cP}@ n )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG\_SV\_I}_1,\mbox{conjecture})$

\medskip\textbf{SYO449$\wedge$1.p} Ted Sider's modal proposition logic theorem 58






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ p )@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_5@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO139$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}\mbox{ or }\neg\,\mbox{cP}@ \mbox{xx} )\mbox{ and }\mbox{cQ}@ \mbox{xy} )\mbox{ or }\neg\,\mbox{cQ}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cCOUNTER}_1,\mbox{conjecture})$

\medskip\textbf{SYO515$\wedge$1.p} A choice operator at type oo






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

$t = ( \lambda p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o},x{:}\;\mbox{\$o}{:}\;( \neg\,p@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$false}\mbox{ and }( p@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$true}\mbox{ or }( p@ \lambda x{:}\;\mbox{\$o}{:}\;\neg\,x ) = \neg\,x ) ) )\qquad\mbox{thf}(t,\mbox{definition})$

$\forall p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \exists y{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( p@ y ) \ \Rightarrow \ ( p@ ( t@ p ) ) )\qquad\mbox{thf}(\mbox{choiceoo}_1,\mbox{conjecture})$

\medskip\textbf{SYO183$\wedge$5.p} TPS problem CT2






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

$\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xp}@ a )\qquad\mbox{thf}(\mbox{cCT}_2,\mbox{conjecture})$

\medskip\textbf{SYO470$\wedge$5.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) )@ ( \mbox{mbox\_s}_4@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO426$\wedge$1.p} Ted Sider's modal proposition logic theorem 35






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) )@ ( \mbox{mdia\_m}@ ( \mbox{mor}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO218$\wedge$5.p} TPS problem THM7B

Half-proved version of thm7 for test purposes.




$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xq}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( \mbox{xx} = ( \mbox{xq}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\mbox{xx} = \mbox{xz} ) \ \Rightarrow \ \mbox{xy} = ( \mbox{xq}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\mbox{xy} = \mbox{xz} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM7B\_pme},\mbox{conjecture})$

\medskip\textbf{SYO146$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ or }\mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{cP}@ a@ b\mbox{ and }\mbox{cP}@ a@ c )\mbox{ or }\mbox{cQ}@ a )\qquad\mbox{thf}(\mbox{cDUP\_EXPL}_1,\mbox{conjecture})$

\medskip\textbf{SYO070$\wedge$4.003.p} ILTP Problem SYJ211+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$b_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b3\_type},\mbox{type})$

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

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_3 ) ) )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_3 ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_0 ) ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_2 ) ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO333$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

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

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

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

$( \mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{cSx}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\mbox{cHALF}@ c_0@ c_0\mbox{ and }\mbox{cHALF}@ c_0@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cHALF}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xu}_0{:}\;\mbox{\$i},\mbox{xv}_0{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}_0@ \mbox{xv}_0\mbox{ and }q@ c_0@ c_0\mbox{ and }q@ c_0@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}_0@ \mbox{xv}_0 ) ) ) \ \Rightarrow \ ( ( \mbox{cHALF}@ u@ v ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ v@ u ) )\qquad\mbox{thf}(\mbox{cHALF\_TO\_DOUBLE}_1,\mbox{conjecture})$

\medskip\textbf{SYO435$\wedge$1.p} Ted Sider's modal proposition logic theorem 44






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO412$\wedge$1.p} Ted Sider's modal proposition logic theorem 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_k}@ ( \mbox{mimplies}@ p@ q ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO083$\wedge$5.p} TPS problem THM62






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

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

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

$( \forall u{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{cA}@ u )\mbox{ or }\forall v{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ v@ \mbox{cB} ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ x@ x )\qquad\mbox{thf}(\mbox{cTHM}_{62},\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$6.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO445$\wedge$1.p} Ted Sider's modal proposition logic theorem 54






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ p@ ( \mbox{mdia\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO566$\wedge$7.p} Girle problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( g@ x ) ) ) )@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( g@ x ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO066$\wedge$4.002.p} ILTP Problem SYJ202+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{12}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o12\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$o_{22}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o22\_type},\mbox{type})$

$o_{31}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o31\_type},\mbox{type})$

$o_{32}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o32\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{12} ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{22} ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{iatom}@ o_{32} ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{22} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{32} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO035$\wedge$1.p} Higher-order unification does not always provide projection terms






$\mbox{leibeq}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq1\_type},\mbox{type})$

$\mbox{leibeq}_1 = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq}_1,\mbox{definition})$

$\mbox{leibeq}_2{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq2\_type},\mbox{type})$

$\mbox{leibeq}_2 = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall p{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq}_2,\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{leibeq}_2@ \lambda z{:}\;\mbox{\$i}{:}\;z = x@ \lambda z{:}\;\mbox{\$i}{:}\;z = y ) \ \Rightarrow \ ( \mbox{leibeq}_1@ x@ y ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$1.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO004$\wedge$1.p} Relating Leibniz equality to primitive equality






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

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{leibeq}@ x@ y ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$2.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_d}@ q ) ) )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO065$\wedge$4.003.p} ILTP Problem SYJ201+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p5\_type},\mbox{type})$

$p_6{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p6\_type},\mbox{type})$

$p_7{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p7\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_7 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_7 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO238$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

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

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

$( \mbox{\$true} \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( f@ \mbox{xx}_0 ) = \mbox{xx}_0 ) \ \Rightarrow \ ( ( ( \lambda \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( f@ ( f@ \mbox{xx}_0 ) ) ) = ( \lambda \mbox{xx}_0{:}\;\mbox{\$i}{:}\;\mbox{xx}_0 ) \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ ( ( \mbox{cP}@ x ) \ \Rightarrow \ \mbox{cB} ) )\qquad\mbox{thf}(\mbox{cADDHYP}_5,\mbox{conjecture})$

\medskip\textbf{SYO456$\wedge$4.p} Ted Sider's propositional modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) ) )@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO219$\wedge$5.p} TPS problem THM6






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

$\forall m{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ m )\ne m \ \Rightarrow \ \neg\,\exists g{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;\forall f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists j{:}\;\mbox{\$i}{:}\;( g@ j ) = f\qquad\mbox{thf}(\mbox{cTHM}_6,\mbox{conjecture})$

\medskip\textbf{SYO437$\wedge$1.p} Ted Sider's modal proposition logic theorem 46






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ r ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_s}_4@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$4.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mand}@ p@ ( \mbox{mdia\_b}@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) )@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO470$\wedge$3.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_m}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) )@ ( \mbox{mbox\_m}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO171$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\neg\,\forall a{:}\;\mbox{\$i}{:}\;( g@ a@ a\mbox{ or }g@ ( f@ a )@ a )\mbox{ and }\forall a{:}\;\mbox{\$i}{:}\;( g@ a@ a\mbox{ or }g@ a@ ( f@ a ) )\mbox{ and }\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,g@ a@ b\mbox{ or }g@ ( f@ b )@ b )\mbox{ and }\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,g@ a@ b\mbox{ or }g@ b@ ( f@ b ) )\mbox{ and }\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,g@ a@ b\mbox{ or }\neg\,g@ b@ a )\qquad\mbox{thf}(\mbox{cSYN031}_1,\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$5.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_s}_4@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$5.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO416$\wedge$1.p} Ted Sider's modal proposition logic theorem 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO207$\wedge$5.p} TPS problem CT26






$\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xy} = \mbox{xy} = \mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{26},\mbox{conjecture})$

\medskip\textbf{SYO026$\wedge$1.p} Four functions from truth values to truth values

In Henkin semantics there are exactly four functions from truth
 values to truth values.




$p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_decl},\mbox{type})$

$( p@ \lambda x{:}\;\mbox{\$o}{:}\;x\mbox{ and }p@ \lambda x{:}\;\mbox{\$o}{:}\;\neg\,x\mbox{ and }p@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$false}\mbox{ and }p@ \lambda x{:}\;\mbox{\$o}{:}\;\mbox{\$true} ) \ \Rightarrow \ \forall y{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( p@ y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO422$\wedge$1.p} Ted Sider's modal proposition logic theorem 31






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mdia\_d}@ ( \mbox{mimplies}@ ( \mbox{mimplies}@ ( \mbox{mimplies}@ p@ q )@ p )@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO038-1.005.005.p} Boolos' Curious Inference, size f(5,5)






$f(n,1) = s(1)\qquad\mbox{cnf}(\mbox{ax}_1,\mbox{axiom})$

$f(1,s(x)) = s(s(f(1,x)))\qquad\mbox{cnf}(\mbox{ax}_2,\mbox{axiom})$

$f(s(n),s(x)) = f(n,f(s(n),x))\qquad\mbox{cnf}(\mbox{ax}_3,\mbox{axiom})$

$d(1)\qquad\mbox{cnf}(\mbox{ax}_4,\mbox{axiom})$

$d(x) \ \Rightarrow \ d(s(x))\qquad\mbox{cnf}(\mbox{ax}_5,\mbox{axiom})$

$\neg\,d(f(s(s(s(s(1)))),s(s(s(s(1))))))\qquad\mbox{cnf}(\mbox{conj},\mbox{negated\_conjecture})$

\medskip\textbf{SYO365$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$\mbox{cEXT}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cEXT2\_type},\mbox{type})$

$\mbox{cEXT\_eq}_0{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cEXT\_eq\_0\_type},\mbox{type})$

$\mbox{cEXT}_2 = ( ( \mbox{cC}@ \mbox{cA} ) \ \Rightarrow \ ( \mbox{cC}@ \neg\,\neg\,\mbox{cA} ) )\qquad\mbox{thf}(\mbox{cEXT2\_def},\mbox{definition})$

$\mbox{cEXT\_eq}_0 = ( \forall \mbox{xp}{:}\;\mbox{\$o},\mbox{xq}{:}\;\mbox{\$o}{:}\;( ( \mbox{xp}\iff \mbox{xq} ) \ \Rightarrow \ \mbox{xp} = \mbox{xq} ) )\qquad\mbox{thf}(\mbox{cEXT\_eq\_0\_def},\mbox{definition})$

$\mbox{cEXT\_eq}_0 \ \Rightarrow \ \mbox{cEXT}_2\qquad\mbox{thf}(\mbox{cEXT2A},\mbox{conjecture})$

\medskip\textbf{SYO563+3.p} Unequal numbers - integers






$1\ne 2\qquad\mbox{fof}(\mbox{one\_not\_equal\_to}_2,\mbox{conjecture})$

\medskip\textbf{SYO299$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\forall p{:}\;\mbox{\$o},q{:}\;\mbox{\$o},r{:}\;\mbox{\$o}{:}\;\exists s{:}\;\mbox{\$o}{:}\;( ( ( p\iff q )\iff r ) \ \Rightarrow \ ( p\iff ( q\iff s ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{51},\mbox{conjecture})$

\medskip\textbf{SYO518$\wedge$1.p} There is an if-then-else operator at type i






$\exists i{:}\;\mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( i@ \mbox{\$true}@ x@ y ) = x\mbox{ and }( i@ \mbox{\$false}@ x@ y ) = y )\qquad\mbox{thf}(\mbox{ifi},\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$1.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_k}@ q ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO494$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( \mbox{mdia\_s}_5@ ( g@ x ) ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$2.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO378$\wedge$5.p} TPS problem from QUANTDEPTH-THMS






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

$\mbox{cQDP}_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQDP0\_type},\mbox{type})$

$\mbox{cQDP}_1{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQDP1\_type},\mbox{type})$

$\mbox{cQDP}_2{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQDP2\_type},\mbox{type})$

$\mbox{cQDP}_0 = ( \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\mbox{xz} = c )\qquad\mbox{thf}(\mbox{cQDP0\_def},\mbox{definition})$

$\mbox{cQDP}_1 = ( \lambda \mbox{xz}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xz} = \mbox{cQDP}_0\mbox{ and }\exists \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xz}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cQDP1\_def},\mbox{definition})$

$\mbox{cQDP}_2 = ( \lambda \mbox{xz}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \mbox{xz} = \mbox{cQDP}_1\mbox{ and }\exists \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xz}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cQDP2\_def},\mbox{definition})$

$\exists \mbox{xs}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \mbox{cQDP}_2@ \mbox{xs} )\qquad\mbox{thf}(\mbox{cQDTHM}_2,\mbox{conjecture})$

\medskip\textbf{SYO174$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} )\iff ( \mbox{cP}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xy} ) ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} )\iff ( \mbox{cQ}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xy} ) ) ) )\iff ( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ \mbox{xx} )\iff ( \mbox{cR}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} )\iff ( \mbox{cS}@ \mbox{xy} ) )\iff ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} )\iff \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{138},\mbox{conjecture})$

\medskip\textbf{SYO252$\wedge$5.p} TPS problem THM123B

Trivial theorem to test TPS.




$\forall \mbox{xp}{:}\;\mbox{\$o}{:}\;\exists \mbox{xf}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\neg\,\mbox{xf}@ \mbox{xp}\qquad\mbox{thf}(\mbox{cTHM123B},\mbox{conjecture})$

\medskip\textbf{SYO192$\wedge$5.p} TPS problem CT12






$\exists \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xy}\qquad\mbox{thf}(\mbox{cCT}_{12},\mbox{conjecture})$

\medskip\textbf{SYO548$\wedge$1.p} hoice complement

There is an operator that chooses an element not in the predicate,
 if there is one.




$\exists e{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;\neg\,p@ x \ \Rightarrow \ \neg\,p@ ( e@ p ) )\qquad\mbox{thf}(\mbox{choicecomp},\mbox{conjecture})$

\medskip\textbf{SYO550$\wedge$1.p} The identity function on individuals exists






$\exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( f@ x ) = x\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO140$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cADDHYP}_6,\mbox{conjecture})$

\medskip\textbf{SYO369$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

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

$( \mbox{cP}@ ( \mbox{cA} \ \Rightarrow \ \mbox{cB} ) ) \ \Rightarrow \ ( \mbox{cP}@ ( \neg\,\mbox{cA}\mbox{ or }\mbox{cB} ) )\qquad\mbox{thf}(\mbox{cTRIVEXT}_5,\mbox{conjecture})$

\medskip\textbf{SYO197$\wedge$5.p} TPS problem CT18






$\exists \mbox{xy}{:}\;\mbox{\$o}{:}\;\forall \mbox{xz}{:}\;\mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;( \mbox{xz} \ \Rightarrow \ \mbox{xx} )\qquad\mbox{thf}(\mbox{cCT}_{18},\mbox{conjecture})$

\medskip\textbf{SYO040$\wedge$2.p} Unsatisfiable basic formula 2

Variant of the Kaminski equation.




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

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

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

$( h@ ( f@ ( f@ ( f@ x ) ) ) ) = ( h@ ( f@ x ) )\qquad\mbox{thf}(2,\mbox{conjecture})$

\medskip\textbf{SYO359$\wedge$5.p} TPS problem EXT1

Theorem about extensionality.




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

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

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

$h{:}\;( b\to \mbox{\$o} )\to \mbox{gtype}\qquad\mbox{thf}(h,\mbox{type})$

$f{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$( \forall \mbox{xx}{:}\;b{:}\;( ( f@ \mbox{xx} )\iff ( g@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;( b\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ f ) \ \Rightarrow \ ( \mbox{xq}@ g ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;b{:}\;( ( f@ \mbox{xx} )\iff ( g@ \mbox{xx} ) ) \ \Rightarrow \ ( h@ f ) = ( h@ g ) )\qquad\mbox{thf}(\mbox{cEXT}_1,\mbox{conjecture})$

\medskip\textbf{SYO402$\wedge$1.p} Ted Sider's modal proposition logic theorem 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$1.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_k}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO463$\wedge$3.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$1.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$6.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_s}_5@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO497$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( \mbox{mbox\_s}_5@ ( f@ x ) ) ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO526$\wedge$1.p} The BQFQFE problem






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

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

$a\iff b\qquad\mbox{thf}(\mbox{ab},\mbox{axiom})$

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

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

$f = g\qquad\mbox{thf}(\mbox{fg},\mbox{axiom})$

$f = ( \lambda x{:}\;\mbox{\$i}{:}\;a )\qquad\mbox{thf}(\mbox{fa},\mbox{axiom})$

$g = ( \lambda x{:}\;\mbox{\$i}{:}\;b )\qquad\mbox{thf}(\mbox{gb},\mbox{conjecture})$

\medskip\textbf{SYO340$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

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

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists q_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},q_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( q_1@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )@ y\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_1@ x@ y_0\mbox{ or }\mbox{cP}@ x\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cP}@ x\mbox{ or }\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q_2@ x@ y_0 ) )\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_2@ x@ y_0\mbox{ or }\mbox{cP}@ x\mbox{ or }q_2@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_2@ x@ y_0\mbox{ or }\neg\,\mbox{cP}@ x\mbox{ or }q_1@ x@ ( \mbox{cG}@ y_0 ) ) ) )\iff \exists q_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cP}@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )\mbox{ or }q@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )@ ( \mbox{cG}@ y ) )\mbox{ and }( \neg\,\mbox{cP}@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )\mbox{ or }q_1@ ( \mbox{cF}@ x@ ( \mbox{cG}@ x ) )@ y )\mbox{ and }\forall y_0{:}\;\mbox{\$i}{:}\;( ( \neg\,q_1@ x@ y_0\mbox{ or }\neg\,\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q@ x@ ( \mbox{cG}@ y_0 ) )\mbox{ and }( \neg\,q_1@ x@ y_0\mbox{ or }\mbox{cR}@ ( \mbox{cG}@ y_0 )@ \mbox{cK}\mbox{ or }q_1@ x@ ( \mbox{cG}@ y_0 ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM74B}_1,\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$4.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_b}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_b}@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mdia\_b}@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO358$\wedge$5.p} TPS problem E2LEIBEQ1

Example from [Ben99] about alternative defns of equality.




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

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

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

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

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

$\forall \mbox{xq}{:}\;\mbox{atype}\to \mbox{\$o}{:}\;( ( \mbox{xq}@ u ) \ \Rightarrow \ ( \mbox{xq}@ v ) ) \ \Rightarrow \ \forall p{:}\;\mbox{atype}\to \mbox{\$o}{:}\;( ( ( a\mbox{ or }\neg\,a )\mbox{ and }p@ u ) \ \Rightarrow \ ( ( b\mbox{ or }\neg\,b )\mbox{ and }p@ v ) )\qquad\mbox{thf}(\mbox{cE2LEIBEQ1\_pme},\mbox{conjecture})$

\medskip\textbf{SYO468$\wedge$5.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO048$\wedge$1.p} Simple textbook example 5






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mequiv}@ ( \mbox{mbox}@ r@ ( \mbox{mand}@ a@ b ) )@ ( \mbox{mand}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$3.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_m}@ ( \mbox{mand}@ ( \mbox{mdia\_m}@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_m}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO410$\wedge$1.p} Ted Sider's modal proposition logic theorem 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mbox\_k}@ q ) )@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO003$\wedge$1.p} Leibniz equality obeys the congruence property under predicates






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

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{leibeq}@ x@ y\mbox{ and }p@ x ) \ \Rightarrow \ ( p@ y ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO214$\wedge$5.p} TPS problem THM12






$\forall r{:}\;\mbox{\$i}\to \mbox{\$o},s{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( r = s \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( s@ x ) \ \Rightarrow \ ( r@ x ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{12},\mbox{conjecture})$

\medskip\textbf{SYO439$\wedge$1.p} Ted Sider's modal proposition logic theorem 48






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO457$\wedge$1.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) )@ ( \mbox{mdia\_k}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$2.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO152$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$\exists \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( ( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\mbox{cP}\mbox{ or }\mbox{cQ} )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;\mbox{cR} ) \ \Rightarrow \ \neg\,( \neg\,\mbox{cP}\mbox{ and }\neg\,\mbox{cS} )\mbox{ or }\neg\,\mbox{cR} )\qquad\mbox{thf}(\mbox{cQUANTIFIER\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO544$\wedge$1.p} Case operator from (\$o$>$\$o) to \$i defined from choice on \$i

A case operator from (\$o$>$\$o) (with 4 elements) to \$i is defined 
 using a choice operator on \$i. Check all 4 equations.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{case} = ( \lambda b{:}\;\mbox{\$o}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$false} )\mbox{ and }z = x )\mbox{ or }( b = \neg\,\mbox{ and }z = y )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;a )\mbox{ and }z = u )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$true} )\mbox{ and }z = v ) ) ) )\qquad\mbox{thf}(\mbox{caseood},\mbox{definition})$

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$( f_0@ \mbox{\$false} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{f0f},\mbox{axiom})$

$( f_0@ \mbox{\$true} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{f0t},\mbox{axiom})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$( f_1@ \mbox{\$false} ) = \mbox{\$true}\qquad\mbox{thf}(\mbox{f1f},\mbox{axiom})$

$( f_1@ \mbox{\$true} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{f1t},\mbox{axiom})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$( f_2@ \mbox{\$false} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{f2f},\mbox{axiom})$

$( f_2@ \mbox{\$true} ) = \mbox{\$true}\qquad\mbox{thf}(\mbox{f2t},\mbox{axiom})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$( f_3@ \mbox{\$false} ) = \mbox{\$true}\qquad\mbox{thf}(\mbox{f3f},\mbox{axiom})$

$( f_3@ \mbox{\$true} ) = \mbox{\$true}\qquad\mbox{thf}(\mbox{f3t},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( ( \mbox{case}@ f_0@ x@ y@ u@ v ) = x\mbox{ and }( \mbox{case}@ f_1@ x@ y@ u@ v ) = y\mbox{ and }( \mbox{case}@ f_2@ x@ y@ u@ v ) = u\mbox{ and }( \mbox{case}@ f_3@ x@ y@ u@ v ) = v )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO532$\wedge$1.p} Binary choice on individuals 3

Epsa and (epsb Epsa) work together to give an a and b such that R
 a b holds, if such an a and b exist for a binary relation R on \$i.
 A choice operator on i can be used to define a choice operator on
 i*i (Curried). In this version, the second half of the solution is
 given.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{epsb} = ( \lambda \mbox{epsa}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda y{:}\;\mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\exists \mbox{epsa}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ \mbox{epsa}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO134$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$( ( \mbox{cP}@ a ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ a ) )\qquad\mbox{thf}(\mbox{cADDHYP}_1,\mbox{conjecture})$

\medskip\textbf{SYO559$\wedge$1.p} Choice on \$o$>$\$o applied to choice on \$o cannot be negatio






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

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epso}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxo},\mbox{axiom})$

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

$\forall p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsoo}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxoo},\mbox{axiom})$

$( \mbox{epsoo}@ \mbox{epso}@ \mbox{\$false} ) \ \Rightarrow \ ( \mbox{epsoo}@ \mbox{epso}@ \mbox{\$true} )\qquad\mbox{thf}(c,\mbox{conjecture})$

\medskip\textbf{SYO572$\wedge$7.p} Quantified modal logics wwfs. problem 13.






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_4@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO401$\wedge$1.p} Ted Sider's modal proposition logic theorem 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mand}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO098$\wedge$5.p} TPS problem THM65






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

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

$\forall w{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cR}@ w@ w \ \Rightarrow \ \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cR}@ x@ y\mbox{ and }( ( \mbox{cQ}@ y@ x ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ z@ z ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{65},\mbox{conjecture})$

\medskip\textbf{SYO093$\wedge$5.p} TPS problem THM63






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

$\forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ u@ v\mbox{ or }\mbox{cP}@ v@ w ) \ \Rightarrow \ \exists x{:}\;\mbox{\$i}{:}\;\forall y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ x@ y )\qquad\mbox{thf}(\mbox{cTHM}_{63},\mbox{conjecture})$

\medskip\textbf{SYO507$\wedge$1.p} Example 4.1






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

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

$\neg\,p@ f\mbox{ and }\neg\,p@ \lambda x{:}\;\mbox{\$i}{:}\;\neg\,\neg\,f@ x\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO049$\wedge$1.p} Simple textbook example 6






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mequiv}@ ( \mbox{mdia}@ r@ ( \mbox{mor}@ a@ b ) )@ ( \mbox{mor}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO375$\wedge$5.p} TPS problem from EXTENSIONALITY






$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall w{:}\;\mbox{\$i}{:}\;( ( x@ w )\iff ( y@ w ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( x@ \mbox{xx} )\iff ( y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cEXT\_SET}_1,\mbox{conjecture})$

\medskip\textbf{SYO187$\wedge$5.p} TPS problem CT10






$\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;\forall \mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{10},\mbox{conjecture})$

\medskip\textbf{SYO481$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mnot}@ ( \mbox{meq\_ind}@ x@ y ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mnot}@ ( \mbox{meq\_ind}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO070$\wedge$4.001.p} ILTP Problem SYJ211+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

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

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_1 ) ) )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_1 ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_0 ) ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO423$\wedge$1.p} Ted Sider's modal proposition logic theorem 32






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mbox\_d}@ ( \mbox{mand}@ ( \mbox{mbox\_d}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ ( \mbox{mnot}@ q ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO434$\wedge$1.p} Ted Sider's modal proposition logic theorem 43






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO173$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\neg\,\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cT}@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xq} )\mbox{ or }\neg\,\mbox{cT}@ \mbox{xp}\mbox{ or }\mbox{cT}@ \mbox{xq} )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ \mbox{xp}@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xp} ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i},\mbox{xr}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ ( \mbox{imp}@ \mbox{xp}@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xr} ) )@ ( \mbox{imp}@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xq} )@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xr} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ ( \mbox{imp}@ ( \mbox{nt}@ \mbox{xp} )@ ( \mbox{nt}@ \mbox{xq} ) )@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xp} ) ) )\mbox{ and }\exists \mbox{xa}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cT}@ ( \mbox{imp}@ ( \mbox{nt}@ ( \mbox{nt}@ \mbox{xa} ) )@ \mbox{xa} )\qquad\mbox{thf}(\mbox{cLCL077}_1,\mbox{conjecture})$

\medskip\textbf{SYO074$\wedge$4.001.p} ILTP Problem SYJ210+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO138$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSIMPLE\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO038-1.004.004.p} Boolos' Curious Inference, size f(4,4)






$f(n,1) = s(1)\qquad\mbox{cnf}(\mbox{ax}_1,\mbox{axiom})$

$f(1,s(x)) = s(s(f(1,x)))\qquad\mbox{cnf}(\mbox{ax}_2,\mbox{axiom})$

$f(s(n),s(x)) = f(n,f(s(n),x))\qquad\mbox{cnf}(\mbox{ax}_3,\mbox{axiom})$

$d(1)\qquad\mbox{cnf}(\mbox{ax}_4,\mbox{axiom})$

$d(x) \ \Rightarrow \ d(s(x))\qquad\mbox{cnf}(\mbox{ax}_5,\mbox{axiom})$

$\neg\,d(f(s(s(s(1))),s(s(s(1)))))\qquad\mbox{cnf}(\mbox{conj},\mbox{negated\_conjecture})$

\medskip\textbf{SYO463$\wedge$1.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO208$\wedge$5.p} TPS problem CT31






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

$\neg\,\exists \mbox{xg}{:}\;a\to a\to \mbox{\$o}{:}\;\forall \mbox{xf}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xj}{:}\;a{:}\;( \mbox{xg}@ \mbox{xj} ) = \mbox{xf} = \mbox{xf} = \mbox{xf}\qquad\mbox{thf}(\mbox{cCT}_{31},\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$5.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_s}_4@ q ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO331$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_0{:}\;a{:}\;( ( s@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_0 ) )\mbox{ and }s@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cK}@ x@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cK}@ \lambda \mbox{xx}_0{:}\;a{:}\;\exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_1{:}\;a{:}\;( ( s@ \mbox{xx}_1 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_1 ) )\mbox{ and }s@ \mbox{xx}_0 )@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( \exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_0{:}\;a{:}\;( ( s@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_0 ) )\mbox{ and }s@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cK}@ \lambda \mbox{xx}_0{:}\;a{:}\;\exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_1{:}\;a{:}\;( ( s@ \mbox{xx}_1 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_1 ) )\mbox{ and }s@ \mbox{xx}_0 )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM116\_2SS\_pme},\mbox{conjecture})$

\medskip\textbf{SYO336$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

$( \mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xp}@ c_0@ c_0\mbox{ and }\mbox{xq}@ c_0@ c_0\mbox{ and }\mbox{xp}@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\mbox{xq}@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} )\mbox{ and }\mbox{xq}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xq}@ \mbox{xu}@ \mbox{xv} ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{xv}@ \mbox{xu}\mbox{ or }\mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xv} )@ ( \mbox{cS}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cTRY}_4,\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$4.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_b}@ ( \mbox{mand}@ ( \mbox{mdia\_b}@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_b}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO546$\wedge$1.p} Property of case from (\$o$>$\$o) to \$i defined from choice on \$i

A choice operator on \$i is used to define an if-then-else operator
 at \$i. Check that case always returns one of the four given 
 results.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{case} = ( \lambda b{:}\;\mbox{\$o}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$false} )\mbox{ and }z = x )\mbox{ or }( b = \neg\,\mbox{ and }z = y )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;a )\mbox{ and }z = u )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$true} )\mbox{ and }z = v ) ) ) )\qquad\mbox{thf}(\mbox{caseood},\mbox{definition})$

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

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{case}@ f@ x@ x@ x@ x ) = x\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO513$\wedge$1.p} There is a choice operator at type o






$\exists c{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( c@ p ) ) )\qquad\mbox{thf}(\mbox{choiceo},\mbox{conjecture})$

\medskip\textbf{SYO463$\wedge$4.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO349$\wedge$5.p} TPS problem THM617

Simple extensionality example.




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

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

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

$\neg\,\mbox{cA}\mbox{ or }\neg\,\mbox{cB}\mbox{ or }\neg\,\mbox{cP}@ \mbox{cA}\mbox{ or }\mbox{cP}@ \mbox{cB}\qquad\mbox{thf}(\mbox{cTHM}_{617},\mbox{conjecture})$

\medskip\textbf{SYO505$\wedge$1.p} Explosive confrontation

The Mensa Example at type oo: There are not 5 distinct values of
 type oo.




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

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

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

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

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

$f_0{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_0,\mbox{type})$

$g_0{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_0,\mbox{type})$

$f_1{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_1,\mbox{type})$

$g_1{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_1,\mbox{type})$

$f_2{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_2,\mbox{type})$

$g_2{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_2,\mbox{type})$

$f_3{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_3,\mbox{type})$

$g_3{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_3,\mbox{type})$

$f_4{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_4,\mbox{type})$

$g_4{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_4,\mbox{type})$

$f_5{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_5,\mbox{type})$

$g_5{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_5,\mbox{type})$

$f_6{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_6,\mbox{type})$

$g_6{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_6,\mbox{type})$

$f_7{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_7,\mbox{type})$

$g_7{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_7,\mbox{type})$

$f_8{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_8,\mbox{type})$

$g_8{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_8,\mbox{type})$

$f_9{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(f_9,\mbox{type})$

$g_9{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(g_9,\mbox{type})$

$( f_0@ a ) = ( g_0@ b )\qquad\mbox{thf}(\mbox{anotb}_1,\mbox{axiom})$

$( f_0@ b )\ne ( g_0@ a )\qquad\mbox{thf}(\mbox{anotb}_2,\mbox{axiom})$

$( f_1@ a ) = ( g_1@ c )\qquad\mbox{thf}(\mbox{anotc}_1,\mbox{axiom})$

$( f_1@ c )\ne ( g_1@ a )\qquad\mbox{thf}(\mbox{anotc}_2,\mbox{axiom})$

$( f_2@ a ) = ( g_2@ d )\qquad\mbox{thf}(\mbox{anotd}_1,\mbox{axiom})$

$( f_2@ d )\ne ( g_2@ a )\qquad\mbox{thf}(\mbox{anotd}_2,\mbox{axiom})$

$( f_3@ a ) = ( g_3@ e )\qquad\mbox{thf}(\mbox{anote}_1,\mbox{axiom})$

$( f_3@ e )\ne ( g_3@ a )\qquad\mbox{thf}(\mbox{anote}_2,\mbox{axiom})$

$( f_4@ b ) = ( g_4@ c )\qquad\mbox{thf}(\mbox{bnotc}_1,\mbox{axiom})$

$( f_4@ c )\ne ( g_4@ b )\qquad\mbox{thf}(\mbox{bnotc}_2,\mbox{axiom})$

$( f_5@ b ) = ( g_5@ d )\qquad\mbox{thf}(\mbox{bnotd}_1,\mbox{axiom})$

$( f_5@ d )\ne ( g_5@ b )\qquad\mbox{thf}(\mbox{bnotd}_2,\mbox{axiom})$

$( f_6@ b ) = ( g_6@ e )\qquad\mbox{thf}(\mbox{bnote}_1,\mbox{axiom})$

$( f_6@ e )\ne ( g_6@ b )\qquad\mbox{thf}(\mbox{bnote}_2,\mbox{axiom})$

$( f_7@ c ) = ( g_7@ d )\qquad\mbox{thf}(\mbox{cnotd}_1,\mbox{axiom})$

$( f_7@ d )\ne ( g_7@ c )\qquad\mbox{thf}(\mbox{cnotd}_2,\mbox{axiom})$

$( f_8@ c ) = ( g_8@ e )\qquad\mbox{thf}(\mbox{cnote}_1,\mbox{axiom})$

$( f_8@ e )\ne ( g_8@ c )\qquad\mbox{thf}(\mbox{cnote}_2,\mbox{axiom})$

$( f_9@ d ) = ( g_9@ e )\qquad\mbox{thf}(\mbox{dnote}_1,\mbox{axiom})$

$( f_9@ e )\ne ( g_9@ d )\qquad\mbox{thf}(\mbox{dnote}_2,\mbox{axiom})$

$\mbox{\$false}\qquad\mbox{thf}(\mbox{mensaoo},\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$6.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ p@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ p ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO313$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \exists \mbox{xr}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xr} )\mbox{ and }\exists \mbox{xu}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cELESS}@ \mbox{xx}@ \mbox{xu} ) ) ) \ \Rightarrow \ \exists \mbox{xl}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cELESS}@ \mbox{xx}@ \mbox{xl} ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cELESS}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{cELESS}@ \mbox{xl}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cELUB},\mbox{conjecture})$

\medskip\textbf{SYO079$\wedge$5.p} TPS problem THM50-A

Associativity of equivalence.




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

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

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

$( ( \mbox{cP}\iff \mbox{cQ} )\iff \mbox{cR} ) \ \Rightarrow \ ( \mbox{cP}\iff ( \mbox{cQ}\iff \mbox{cR} ) )\qquad\mbox{thf}(\mbox{cTHM50\_A},\mbox{conjecture})$

\medskip\textbf{SYO052$\wedge$2.p} Simple textbook example 9






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

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

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

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

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ a@ b ) )@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO565$\wedge$7.p} Fitting and Mendelsohn problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( a@ x ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( a@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$6.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ ( \mbox{mdia\_s}_5@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO202$\wedge$5.p} TPS problem CT22






$\neg\,\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xp}@ \mbox{xp} = \mbox{xp}\mbox{ and }\neg\,\mbox{xp}@ \mbox{xp} = \mbox{xp} )\qquad\mbox{thf}(\mbox{cCT}_{22},\mbox{conjecture})$

\medskip\textbf{SYO346$\wedge$5.p} TPS problem THM621

Simple extensionality example.




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

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

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

$\mbox{cA}\ne \mbox{cB}\mbox{ or }\neg\,\mbox{cP}@ \mbox{cA}\mbox{ or }\mbox{cP}@ \mbox{cB}\qquad\mbox{thf}(\mbox{cTHM}_{621},\mbox{conjecture})$

\medskip\textbf{SYO480$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_5@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO060$\wedge$4.p} ILTP problem SYJ103+1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

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

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ b ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ b ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO457$\wedge$6.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO364$\wedge$5.p} TPS problem EDEC

Example from [Ben99] about decomposition (using Leibniz equality).




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

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

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

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

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

$( \forall x{:}\;a\to a,y{:}\;a,\mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( f@ x@ y ) ) \ \Rightarrow \ ( \mbox{xq}@ ( g@ x@ y ) ) )\mbox{ and }\forall z{:}\;a,\mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( h@ z ) ) \ \Rightarrow \ ( \mbox{xq}@ ( j@ z ) ) ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( f@ h ) ) \ \Rightarrow \ ( \mbox{xq}@ ( g@ j ) ) )\qquad\mbox{thf}(\mbox{cEDEC\_pme},\mbox{conjecture})$

\medskip\textbf{SYO530$\wedge$1.p} Binary choice on individuals

epsa and epsb work together to give an a and b such that R a b 
 holds, if such an a and b exist for a binary relation R on \$i.
 A choice operator on i can be used to define a choice operator on
 i*i (Curried). In this version, the solution is given and the 
 goal is to check that it works.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{epsa} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

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

$\mbox{epsb} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda y{:}\;\mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO021$\wedge$1.p} De Morgan by equality






$\forall x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;( x\mbox{ and }y ) = \neg\,\neg\,x\mbox{ or }\neg\,y\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO061$\wedge$4.p} ILTP Problem SYJ104+1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

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

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO009$\wedge$1.p} Eta-equality using Leibniz equality






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

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

$( p@ \lambda x{:}\;\mbox{\$i}{:}\;( f@ x ) ) \ \Rightarrow \ ( p@ f )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.004.p} Four function variant of the Kaminski equation






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

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ ( f_3@ x ) ) ) ) ) ) ) ) = ( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ ( f_3@ ( f_3@ ( f_3@ x ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_4,\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$4.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mdia\_b}@ p ) )@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO011$\wedge$1.p} Invalid formula in model classes not requiring f






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

$\mbox{leibeq} = ( \lambda u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;\forall q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

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

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

$( \forall x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}@ ( f@ x )@ x )\mbox{ and }p@ \lambda x{:}\;\mbox{\$i}{:}\;x ) \ \Rightarrow \ ( p@ f )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO413$\wedge$1.p} Ted Sider's modal proposition logic theorem 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ p )@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ q )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO461$\wedge$2.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$2.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_d}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_d}@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mdia\_d}@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$5.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ ( \mbox{mdia\_s}_4@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$1.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_k}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_k}@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO564$\wedge$7.p} Barcan scheme instance. (Ted Sider's qml wwf 1)






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( f@ x ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO288$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\exists \mbox{xv}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xv}@ \mbox{xx}@ c_0 )\iff ( \mbox{cA}@ \mbox{xx}\mbox{ and }\mbox{cB}@ \mbox{xx}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_{10},\mbox{conjecture})$

\medskip\textbf{SYO543$\wedge$1.p} If-then-else on \$i$>$\$i defined from choice on \$i$>$\$i

A choice operator on (\$i$>$\$i) is used to define an if-then-else 
 operator at (\$i$>$\$i). Check that if the then-part and else-part 
 are both X, then it returns X.




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

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

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

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{epsii}@ \lambda z{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{if}@ b@ x@ x ) = x\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO459$\wedge$6.p} Ted Sider's propositional modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO005$\wedge$1.p} The trivial direction of functional extensionality






$\mbox{leibeq}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq1\_type},\mbox{type})$

$\mbox{leibeq}_1 = ( \lambda u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;\forall q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq}_1,\mbox{definition})$

$\mbox{leibeq}_2{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq2\_type},\mbox{type})$

$\mbox{leibeq}_2 = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq}_2,\mbox{definition})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{leibeq}_2@ f@ g ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}_1@ ( f@ x )@ ( g@ x ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO069$\wedge$4.002.p} ILTP Problem SYJ205+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

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

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) ) ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) ) ) ) )@ ( \mbox{iatom}@ f ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO195$\wedge$5.p} TPS problem CT25






$\neg\,\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xy}\qquad\mbox{thf}(\mbox{cCT}_{25},\mbox{conjecture})$

\medskip\textbf{SYO362$\wedge$5.p} TPS problem THM631A

If a set function preserves unions, then it is monotone.




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

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cK}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz} ) ) = ( \lambda \mbox{xw}{:}\;\mbox{\$i}{:}\;( \mbox{cK}@ x@ \mbox{xw}\mbox{ or }\mbox{cK}@ y@ \mbox{xw} ) ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cK}@ x@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cK}@ y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM631A\_pme},\mbox{conjecture})$

\medskip\textbf{SYO053$\wedge$2.p} Simple textbook example 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mdia}@ r@ \mbox{mtrue} )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO415$\wedge$1.p} Ted Sider's modal proposition logic theorem 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ p ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) ) ) )@ ( \mbox{mdia\_k}@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO216$\wedge$5.p} TPS problem THM107






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

$\forall x{:}\;b\to b,y{:}\;b\to b,z{:}\;b\to b{:}\;( \lambda w{:}\;b{:}\;( x@ ( y@ ( z@ w ) ) ) ) = ( \lambda w{:}\;b{:}\;( x@ ( y@ ( z@ w ) ) ) )\qquad\mbox{thf}(\mbox{cTHM107\_pme},\mbox{conjecture})$

\medskip\textbf{SYO137$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cM}@ \mbox{xx} )\iff ( \mbox{cN}@ \mbox{xx} ) )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ \mbox{xx} ) )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cN}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cX}_{2304},\mbox{conjecture})$

\medskip\textbf{SYO123$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ a\mbox{ and }\mbox{cP}@ b )\qquad\mbox{thf}(\mbox{cALLCONJ}_2,\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$1.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_k}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$1.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mdia\_k}@ p ) )@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO458$\wedge$6.p} Ted Sider's propositional modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO291$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

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

$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xp}@ y ) \ \Rightarrow \ ( \mbox{cR}@ x\mbox{ and }\mbox{cR}@ ( f@ x )\mbox{ and }\mbox{cQ}@ y )\qquad\mbox{thf}(\mbox{cTEST}_3,\mbox{conjecture})$

\medskip\textbf{SYO196$\wedge$5.p} TPS problem CT21






$\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xp}@ \forall \mbox{xy}{:}\;\mbox{\$o}{:}\;( \mbox{xy}\mbox{ and }\neg\,\mbox{xy} ) )\qquad\mbox{thf}(\mbox{cCT}_{21},\mbox{conjecture})$

\medskip\textbf{SYO130$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

$( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cP}@ \mbox{xx}\qquad\mbox{thf}(\mbox{cSIMPLER\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO482$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{meq\_ind}@ x@ y ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO017$\wedge$1.p} Formula valid in MBb, but not in model classes not requiring b






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

$( h@ ( h@ \mbox{\$true} ) = ( h@ \mbox{\$false} ) ) = ( h@ \mbox{\$false} )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO056$\wedge$2.p} Simple textbook example 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mor}@ a@ b ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO164$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) )\mbox{ and }( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cPELL26}_2,\mbox{conjecture})$

\medskip\textbf{SYO231$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

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

$\exists \mbox{xu}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\mbox{xu} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cA}@ \mbox{xx}\mbox{ and }\mbox{cB}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_6,\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$5.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_s}_4@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mdia\_s}_4@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO351$\wedge$5.p} TPS problem E6EXT

Example from [Ben99] p.116.




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

$( \lambda r{:}\;a\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;a{:}\;( ( r@ \mbox{xx} ) \ \Rightarrow \ \mbox{\$false} ) ) = ( \lambda \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cE6EXT\_pme},\mbox{conjecture})$

\medskip\textbf{SYO461$\wedge$6.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$2.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mand}@ p@ ( \mbox{mdia\_d}@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO337$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

$( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( ( \mbox{xp}@ c_0@ c_0\mbox{ or }\mbox{xq}@ c_0@ c_0 )\mbox{ and }( \mbox{xp}@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ or }\mbox{xq}@ ( \mbox{cS}@ c_0 )@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{xq}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} )\mbox{ or }\mbox{xq}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xu}@ \mbox{xv}\mbox{ or }\mbox{xq}@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ or }( \neg\,\mbox{cHALF}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ c_0@ c_0\mbox{ and }q@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) ) ) )\mbox{ and }\mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{xv}@ \mbox{xu}\mbox{ or }\mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xv} )@ ( \mbox{cS}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cTHM300A\_EXPAND},\mbox{conjecture})$

\medskip\textbf{SYO010$\wedge$1.p} Something requiring Xi but not Eta






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

$\mbox{leibeq} = ( \lambda u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;\forall q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

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

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

$( \forall x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}@ ( f@ x )@ x )\mbox{ and }p@ \lambda x{:}\;\mbox{\$i}{:}\;x ) \ \Rightarrow \ ( p@ \lambda x{:}\;\mbox{\$i}{:}\;( f@ x ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO327$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

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

$( \forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xu}@ \mbox{xv}\mbox{ and }q@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) )\mbox{ and }\mbox{cHALF}@ c_0@ c_0\mbox{ and }\mbox{cHALF}@ c_0@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cHALF}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cHALF}@ \mbox{xv}@ \mbox{xu} ) )\qquad\mbox{thf}(\mbox{cDOUBLE\_TO\_HALF}_6,\mbox{conjecture})$

\medskip\textbf{SYO287$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\exists \mbox{xv}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xv}@ \mbox{xx}@ c_0 )\iff ( \mbox{cA}@ \mbox{xx}\mbox{ or }\mbox{cB}@ \mbox{xx}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_9,\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$4.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) )@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO448$\wedge$1.p} Ted Sider's modal proposition logic theorem 57






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ q )@ ( \mbox{mbox\_s}_5@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO461$\wedge$5.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO069$\wedge$4.004.p} ILTP Problem SYJ205+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$a_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a4\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$b_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b3\_type},\mbox{type})$

$b_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b4\_type},\mbox{type})$

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

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_4 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iatom}@ a_3 ) ) ) ) ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_4 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) ) ) ) ) ) )@ ( \mbox{iatom}@ f ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$6.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO246$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






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

$\forall \mbox{xs}{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a\to \mbox{\$o}{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) ) \ \Rightarrow \ \forall a{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \exists x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( a@ x )\mbox{ and }\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ x ) \ \Rightarrow \ \exists \mbox{xu}{:}\;a\to \mbox{\$o}{:}\;( x@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;a{:}\;\forall \mbox{xa}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ \mbox{xa} ) \ \Rightarrow \ \exists \mbox{xb}{:}\;a\to \mbox{\$o}{:}\;( \mbox{xa}@ \mbox{xb}\mbox{ and }\mbox{xb}@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\exists \mbox{xf}{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\forall \mbox{xa}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ \mbox{xa} ) \ \Rightarrow \ ( \mbox{xa}@ ( \mbox{xf}@ \mbox{xa} )\mbox{ and }\mbox{xf}@ \mbox{xa}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{535},\mbox{conjecture})$

\medskip\textbf{SYO227$\wedge$5.p} TPS problem BLEDSOE4-W-AX






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

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

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

$\mbox{c\_less\_}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_less\_},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{c\_less\_}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}\ne \mbox{xy} ) \ \Rightarrow \ ( ( \mbox{c\_less\_}@ a@ b\mbox{ and }\mbox{c\_less\_}@ b@ c ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,a@ a\mbox{ and }a@ b\mbox{ and }\neg\,a@ c ) )\qquad\mbox{thf}(\mbox{cBLEDSOE4\_W\_AX},\mbox{conjecture})$

\medskip\textbf{SYO442$\wedge$1.p} Ted Sider's modal proposition logic theorem 51






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO111$\wedge$5.p} TPS problem THM80






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

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

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

$( \forall r{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ r )\mbox{ or }\exists x{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cG}@ x\mbox{ or }\neg\,\forall y{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ y )\mbox{ or }\exists s{:}\;\mbox{\$i}{:}\;( \mbox{cN}@ s )\mbox{ or }\neg\,\forall z{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cN}@ z\mbox{ or }\neg\,\forall t{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ t ) ) \ \Rightarrow \ ( \forall r{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ r )\mbox{ or }\exists x{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cG}@ x\mbox{ or }\neg\,\forall y{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ y\mbox{ or }\exists s{:}\;\mbox{\$i}{:}\;( \mbox{cN}@ s ) )\mbox{ or }\neg\,\forall z{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cN}@ z\mbox{ or }\neg\,\forall t{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ t ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{80},\mbox{conjecture})$

\medskip\textbf{SYO523=1.p} Injective pigeon hole function






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

$( \forall x{:}\;\mbox{\$int},y{:}\;\mbox{\$int}{:}\;( f(x) = f(y) \ \Rightarrow \ x = y )\mbox{ and }\mbox{\$less}(6,f(3))\mbox{ and }\mbox{\$less}(f(3),9)\mbox{ and }\mbox{\$less}(6,f(4))\mbox{ and }\mbox{\$less}(f(4),9) ) \ \Rightarrow \ ( \mbox{\$lesseq}(f(5),6)\mbox{ or }\mbox{\$lesseq}(9,f(5)) )\qquad\mbox{tff}(\mbox{injective\_f\_pigeonhole},\mbox{conjecture})$

\medskip\textbf{SYO067$\wedge$4.003.p} ILTP Problem SYJ203+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ f ) ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO347$\wedge$5.p} TPS problem THM620

Simple extensionality example.




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

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

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

$\mbox{cC}\ne \mbox{cD}\mbox{ or }\neg\,\mbox{cR}@ \mbox{cC}\mbox{ or }\mbox{cR}@ \mbox{cD}\qquad\mbox{thf}(\mbox{cTHM}_{620},\mbox{conjecture})$

\medskip\textbf{SYO499$\wedge$1.p} Explosive confrontation

The Mensa Example: There are not 3 distinct values of type \$o.




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

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

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

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

$f_1{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(f_2,\mbox{type})$

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

$g_1{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(g_1,\mbox{type})$

$g_2{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(g_2,\mbox{type})$

$( f@ a ) = ( g@ b )\mbox{ or }( f@ b )\ne ( g@ a )\mbox{ or }( f_1@ a ) = ( g_1@ c )\mbox{ or }( f_1@ c )\ne ( g_1@ a )\mbox{ or }( f_2@ b ) = ( g_2@ c )\mbox{ or }( f_2@ c )\ne ( g_2@ b )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO502$\wedge$1.p} Rules sym and con handle positive equations at i






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

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

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

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

$a\ne b\mbox{ or }( f@ a )\ne ( g@ b )\mbox{ or }( f@ b ) = ( g@ a )\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO377$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;( ( \mbox{xx}\iff \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xq}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ a\mbox{ and }\mbox{xp}@ b ) \ \Rightarrow \ ( \mbox{xp}@ ( a\mbox{ and }b ) ) )\qquad\mbox{thf}(\mbox{cEXT\_IMP\_EMB}_2,\mbox{conjecture})$

\medskip\textbf{SYO514$\wedge$1.p} A choice operator at type oo






$\exists c{:}\;( ( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( c@ p ) ) )\qquad\mbox{thf}(\mbox{choiceoo},\mbox{conjecture})$

\medskip\textbf{SYO102$\wedge$5.p} TPS problem THM101






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

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cP}@ \mbox{xz}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{cP}@ \mbox{xv}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xu}@ \mbox{xw} ) )\qquad\mbox{thf}(\mbox{cTHM}_{101},\mbox{conjecture})$

\medskip\textbf{SYO459$\wedge$4.p} Ted Sider's propositional modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO421$\wedge$1.p} Ted Sider's modal proposition logic theorem 30






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mand}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ ( \mbox{mnot}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO047$\wedge$1.p} Simple textbook example 4






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ a@ b ) )@ ( \mbox{mimplies}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO068$\wedge$4.001.p} ILTP Problem SYJ204+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_1 )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO392$\wedge$1.p} Ted Sider's modal proposition logic theorem 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mnot}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ ( \mbox{mnot}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO490$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_5@ ( r@ a@ x ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( r@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$5.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO080$\wedge$5.p} TPS problem THM200

Nepejvoda's problem; supposedly a difficult ND problem.




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

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

$( ( ( ( \mbox{cA} \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ \mbox{cA} ) \ \Rightarrow \ \mbox{cA} ) \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ \mbox{cB}\qquad\mbox{thf}(\mbox{cTHM}_{200},\mbox{conjecture})$

\medskip\textbf{SYO325$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cK}@ x@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cK}@ y@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cK}@ \lambda \mbox{xx}_0{:}\;a{:}\;\exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_1{:}\;a{:}\;( ( s@ \mbox{xx}_1 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_1 ) )\mbox{ and }s@ \mbox{xx}_0 )@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cK}@ ( \mbox{cK}@ \lambda \mbox{xx}_0{:}\;a{:}\;\exists s{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_1{:}\;a{:}\;( ( s@ \mbox{xx}_1 ) \ \Rightarrow \ ( \mbox{cK}@ s@ \mbox{xx}_1 ) )\mbox{ and }s@ \mbox{xx}_0 ) )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM116\_1SS},\mbox{conjecture})$

\medskip\textbf{SYO072$\wedge$4.001.p} ILTP Problem SYJ208+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{11} ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{21} ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO414$\wedge$1.p} Ted Sider's modal proposition logic theorem 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mand}@ q@ r ) ) )@ ( \mbox{mand}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ q ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO551$\wedge$1.p} The identity function on functions from \$i to \$i exists






$\exists f{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( f@ x ) = x\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO136$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$( ( \exists \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0 ) \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ ( ( \mbox{cQ}@ x ) \ \Rightarrow \ \exists \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{cADDHYP}_3,\mbox{conjecture})$

\medskip\textbf{SYO051$\wedge$2.p} Simple textbook example 8






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO571$\wedge$7.p} Quantified modal logics wwfs. problem 9.






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\forall v{:}\;\mbox{\$i}{:}\;( \mbox{exists\_in\_world}@ a@ v )\qquad\mbox{thf}(\mbox{existence\_of\_a\_ax},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{qmltpeq}@ x@ a ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO458$\wedge$3.p} Ted Sider's propositional modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ q ) )@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO020$\wedge$1.p} De Morgan by Leibnitz






$\mbox{leibeq}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;( \mbox{leibeq}@ ( x\mbox{ and }y )@ \neg\,\neg\,x\mbox{ or }\neg\,y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO067$\wedge$4.001.p} ILTP Problem SYJ203+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

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

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ f ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO147$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_3,\mbox{type})$

$\mbox{cP}_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

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

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}_1@ \mbox{xx}\mbox{ and }( ( \mbox{cP}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}_2@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cP}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}_3@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \mbox{cP}_3@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBAFFLER}_3,\mbox{conjecture})$

\medskip\textbf{SYO228$\wedge$5.p} TPS problem THM126-EXPANDED






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

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

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

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} )@ ( \mbox{xh}_1@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} )@ ( \mbox{xh}_2@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy}\mbox{ and }\mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) )@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM126\_EXPANDED\_pme},\mbox{conjecture})$

\medskip\textbf{SYO113$\wedge$5.p} TPS problem THM350






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

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

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

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

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

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }( ( \mbox{cNUMBER}@ c_0\mbox{ and }\mbox{cNUMBER}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cNUMBER}@ \mbox{xx}\mbox{ and }\mbox{cNUMBER}@ ( \mbox{cS}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{cNUMBER}@ ( \mbox{cS}@ \mbox{xx} )\mbox{ and }\mbox{cNUMBER}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cNUMBER}@ \mbox{xx}\mbox{ and }\mbox{cNUMBER}@ ( \mbox{cS}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cNUMBER}@ \mbox{xn} )\iff ( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cNUMBER}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cTHM}_{350},\mbox{conjecture})$

\medskip\textbf{SYO153$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cReduct}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cReduct}@ ( \mbox{cAPP}@ \mbox{xx}@ \mbox{xz} )@ ( \mbox{cApp}@ \mbox{xy}@ \mbox{xz} )\mbox{ and }\mbox{cReduct}@ ( \mbox{cAPP}@ \mbox{xz}@ \mbox{xx} )@ ( \mbox{cAPP}@ \mbox{xz}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cBETA\_STABLE}_1,\mbox{conjecture})$

\medskip\textbf{SYO352$\wedge$5.p} TPS problem E5EXT

Nontrivial direction of functional extensionality using Leibniz 
 equality.




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

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

$\forall x{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ ( m@ x ) ) \ \Rightarrow \ ( p@ ( n@ x ) ) ) \ \Rightarrow \ \forall q{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( q@ m ) \ \Rightarrow \ ( q@ n ) )\qquad\mbox{thf}(\mbox{cE5EXT},\mbox{conjecture})$

\medskip\textbf{SYO431$\wedge$1.p} Ted Sider's modal proposition logic theorem 40






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mdia\_b}@ ( \mbox{mand}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) ) )@ ( \mbox{mbox\_b}@ ( \mbox{mnot}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO129$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ a\mbox{ and }\mbox{cP}@ b\mbox{ and }\mbox{cP}@ c )\qquad\mbox{thf}(\mbox{cALLCONJ}_3,\mbox{conjecture})$

\medskip\textbf{SYO074$\wedge$4.003.p} ILTP Problem SYJ210+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_3 ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO411$\wedge$1.p} Ted Sider's modal proposition logic theorem 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) )@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ q@ r ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO371$\wedge$5.p} TPS problem from EXTENSIONALITY






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

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

$\forall r{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( r@ \mbox{cP} ) \ \Rightarrow \ ( r@ \mbox{cQ} ) ) \ \Rightarrow \ ( \mbox{cP}\iff \mbox{cQ} )\qquad\mbox{thf}(\mbox{cEXT\_O\_LEIB},\mbox{conjecture})$

\medskip\textbf{SYO354$\wedge$5.p} TPS problem E4EXT






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

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

$\forall x{:}\;\mbox{\$i},p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ ( m@ x ) ) \ \Rightarrow \ ( p@ ( n@ x ) ) ) \ \Rightarrow \ \forall q{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( q@ \lambda x{:}\;\mbox{\$i}{:}\;( m@ x ) ) \ \Rightarrow \ ( q@ \lambda x{:}\;\mbox{\$i}{:}\;( n@ x ) ) )\qquad\mbox{thf}(\mbox{cE4EXT},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$3.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) )@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$6.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO424$\wedge$1.p} Ted Sider's modal proposition logic theorem 33






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mor}@ ( \mbox{mdia\_d}@ ( \mbox{mnot}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mdia\_d}@ ( \mbox{mand}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO074$\wedge$4.004.p} ILTP Problem SYJ210+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_4 ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_3 ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO461$\wedge$3.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO286$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

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

$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xp}@ y ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ y ) )\qquad\mbox{thf}(\mbox{cTEST}_1,\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$1.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ p@ ( \mbox{mdia\_k}@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ p ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO570$\wedge$7.p} Forbes problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( r@ x@ y ) ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( r@ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$4.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_b}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO315$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

$\exists \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\neg\,\mbox{xp}@ \mbox{xy}\mbox{ and }\mbox{xq}@ \mbox{xy}\mbox{ and }\neg\,\mbox{xq}@ z ) \ \Rightarrow \ \exists \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{xq}@ z\mbox{ and }\mbox{xq}@ \mbox{xy}\mbox{ and }\mbox{xp}@ \mbox{xx}\mbox{ and }\neg\,\mbox{xp}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cSILLYWFF},\mbox{conjecture})$

\medskip\textbf{SYO281$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTRIV}_3,\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$3.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_m}@ q ) ) )@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO161$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy} ) )\mbox{ and }( ( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) )\mbox{ and }( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cX2125\_HALF},\mbox{conjecture})$

\medskip\textbf{SYO483$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{meq\_ind}@ x@ a ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO245$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\forall \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( \exists \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}@ ( \mbox{xh}@ \mbox{xt} )\mbox{ and }( \mbox{xh}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xt}_0{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}_0@ ( \mbox{xh}@ \mbox{xt}_0 )\mbox{ and }\mbox{xz} = ( \mbox{xh}@ \mbox{xt}_0 ) ) ) = ( \mbox{xh}@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}@ ( \mbox{xh}@ \mbox{xt} )\mbox{ and }( \mbox{xh}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xt}_0{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}_0@ ( \mbox{xh}@ \mbox{xt}_0 )\mbox{ and }\mbox{xz} = ( \mbox{xh}@ \mbox{xt}_0 ) ) ) = ( \mbox{xh}@ \mbox{xt} ) ) ) \ \Rightarrow \ \neg\,\exists \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xy}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xh}@ \mbox{xx} ) = ( \mbox{xh}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM193A},\mbox{conjecture})$

\medskip\textbf{SYO464$\wedge$5.p} Ted Sider's propositional modal logic wff 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ p )@ ( \mbox{mdia\_s}_4@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO031$\wedge$1.p} Not every unary connective is the identity






$\neg\,\forall f{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;\exists x{:}\;\mbox{\$o}{:}\;( f@ x ) = x\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO271$\wedge$5.p} TPS problem X5500






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

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

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

$\forall p{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;b{:}\;( p@ \mbox{xx} ) \ \Rightarrow \ ( p@ ( \mbox{cJ}@ p ) ) ) \ \Rightarrow \ \forall \mbox{xf}{:}\;b\to a,\mbox{xg}{:}\;b\to a{:}\;( ( \mbox{xf}@ ( \mbox{cJ}@ \lambda \mbox{xx}{:}\;b{:}\;( \mbox{xf}@ \mbox{xx} )\ne ( \mbox{xg}@ \mbox{xx} ) ) ) = ( \mbox{xg}@ ( \mbox{cJ}@ \lambda \mbox{xx}{:}\;b{:}\;( \mbox{xf}@ \mbox{xx} )\ne ( \mbox{xg}@ \mbox{xx} ) ) ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} )\qquad\mbox{thf}(\mbox{cX}_{5500},\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.006.p} Six function variant of the Kaminski equation






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

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$f_4{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_4,\mbox{type})$

$f_5{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_5,\mbox{type})$

$( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ ( f_3@ ( f_4@ ( f_4@ ( f_4@ ( f_5@ x ) ) ) ) ) ) ) ) ) ) ) ) = ( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ ( f_3@ ( f_3@ ( f_3@ ( f_4@ ( f_5@ ( f_5@ ( f_5@ x ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_6,\mbox{conjecture})$

\medskip\textbf{SYO470$\wedge$2.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_d}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) ) )@ ( \mbox{mbox\_d}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO296$\wedge$5.p} TPS problem from BASIC-HO-THMS






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

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

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}\mbox{ or }a@ \mbox{xy} ) \ \Rightarrow \ ( p@ \mbox{xx}\mbox{ and }q@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE5D},\mbox{conjecture})$

\medskip\textbf{SYO344$\wedge$5.p} TPS problem THM618

Simple E-unification example.




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

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

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

$\mbox{cC}\ne \mbox{cD}\mbox{ or }\neg\,\mbox{cQ}@ \mbox{cC}\mbox{ or }\mbox{cQ}@ \mbox{cD}\qquad\mbox{thf}(\mbox{cTHM}_{618},\mbox{conjecture})$

\medskip\textbf{SYO531$\wedge$1.p} Binary choice on individuals 2

There is an Epsb such that epsa and Epsb work together to give an 
 a and b such that R a b holds, if such an a and b exist for a 
 binary relation R on \$i. A choice operator on i can be used to
 define a choice operator on i*i (Curried). In this version, the
 first half of the solution is given.




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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

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

$\mbox{epsa} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

$\exists \mbox{epsb}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO091$\wedge$5.p} TPS problem THM50Q






$\forall p{:}\;\mbox{\$o},q{:}\;\mbox{\$o},r{:}\;\mbox{\$o}{:}\;( ( ( p\iff q )\iff r )\iff ( p\iff ( q\iff r ) ) )\qquad\mbox{thf}(\mbox{cTHM50Q},\mbox{conjecture})$

\medskip\textbf{SYO274$\wedge$5.p} TPS problem THM48-EXPD






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

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

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

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

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

$( \forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( \forall \mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( \mbox{cF}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cF}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;c,\mbox{xy}{:}\;c{:}\;( \forall \mbox{xq}{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( \mbox{cG}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cG}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;c\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;c,\mbox{xy}{:}\;c{:}\;( \forall \mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( \mbox{cF}@ ( \mbox{cG}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \mbox{xq}@ ( \mbox{cF}@ ( \mbox{cG}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;c\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xq}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cTHM48\_EXPD},\mbox{conjecture})$

\medskip\textbf{SYO078$\wedge$5.p} TPS problem THM49






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

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

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

$( \mbox{cP}\iff \mbox{cQ} )\mbox{ or }( \mbox{cQ}\iff \mbox{cR} )\mbox{ or }( \mbox{cP}\iff \mbox{cR} )\qquad\mbox{thf}(\mbox{cTHM}_{49},\mbox{conjecture})$

\medskip\textbf{SYO389$\wedge$5.p} TPS problem from MISC

Reflexivity of constants.




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

$\mbox{cA}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\mbox{cA} = \mbox{cA}\qquad\mbox{thf}(\mbox{cEQ\_THM},\mbox{conjecture})$

\medskip\textbf{SYO154$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

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

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$( \mbox{cP}@ a\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ ( f@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cR}@ ( g@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \exists \mbox{xw}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xw} )\qquad\mbox{thf}(\mbox{cDEMO\_TRIV}_1,\mbox{conjecture})$

\medskip\textbf{SYO064$\wedge$4.001.p} ILTP Problem SYJ107+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ c )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a ) )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{iand}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ c ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$2.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mor}@ p@ ( \mbox{mdia\_d}@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mdia\_d}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO326$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cC}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cB}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\exists \mbox{xu}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cA}@ \mbox{xw}\mbox{ and }\mbox{cB}@ \mbox{xz}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xu}@ ( f@ \mbox{xw} )@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cC}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xu}@ \mbox{xz}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xv}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \forall \mbox{xw}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cA}@ \mbox{xw}\mbox{ and }\mbox{cB}@ \mbox{xz}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xv}@ ( f@ \mbox{xw} )@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cC}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xv}@ \mbox{xz}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xu}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xv}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cSV}_8,\mbox{conjecture})$

\medskip\textbf{SYO534$\wedge$1.p} 3-ary choice on individuals

epsa, epsb and epsc work together to give an a, b and c such that 
 R a b c holds, if such an a, b and c exist for a 3-ary relation R 
 on \$i. A choice operator on i can be used to define a choice
 operator on i*i*i (Curried). In this version, the solution is
 given and the goal is to check that it works.




$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$\mbox{epsa}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsa},\mbox{type})$

$\mbox{epsa} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r@ x@ y@ z ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

$\mbox{epsb}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsb},\mbox{type})$

$\mbox{epsb} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda y{:}\;\mbox{\$i}{:}\;\exists z{:}\;\mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y@ z ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\mbox{epsc}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsc},\mbox{type})$

$\mbox{epsc} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r )@ z ) ) )\qquad\mbox{thf}(\mbox{epscd},\mbox{definition})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( r@ x@ y@ z ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r )@ ( \mbox{epsc}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO070$\wedge$4.004.p} ILTP Problem SYJ211+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$a_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a4\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$b_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b3\_type},\mbox{type})$

$b_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b4\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_4 ) ) )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_4 ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_0 ) ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_2 ) ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_3 ) ) )@ ( \mbox{iatom}@ a_4 ) )@ ( \mbox{iatom}@ a_3 ) )\qquad\mbox{thf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$3.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ ( \mbox{mdia\_m}@ p )@ ( \mbox{mdia\_m}@ q ) ) )@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_m}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO292$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}\mbox{ or }a@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE5E},\mbox{conjecture})$

\medskip\textbf{SYO118$\wedge$5.p} TPS problem from BASIC-FO-THMS






$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$q{:}\;\mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$( ( p \ \Rightarrow \ q ) \ \Rightarrow \ p ) \ \Rightarrow \ p\qquad\mbox{thf}(\mbox{cPEIRCE},\mbox{conjecture})$

\medskip\textbf{SYO081$\wedge$5.p} TPS problem THM137

Trivial theorem for logic lessons.




$\mbox{cB}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ x@ \mbox{cA} ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{cB}@ y )\qquad\mbox{thf}(\mbox{cTHM}_{137},\mbox{conjecture})$

\medskip\textbf{SYO330$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cNUMBER}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cNUMBER},\mbox{type})$

$\mbox{cODD}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cODD},\mbox{type})$

$\mbox{cEVEN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEVEN},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \mbox{cEVEN}@ c_0\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEVEN}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cEVEN}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\mbox{cODD}@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cODD}@ \mbox{xn} ) \ \Rightarrow \ ( \mbox{cODD}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xn} ) ) ) )\mbox{ and }\forall p{:}\;\mbox{\$i}\to \mbox{\$o},q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ c_0\mbox{ and }q@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( p@ \mbox{xx}\mbox{ and }q@ \mbox{xx} ) \ \Rightarrow \ ( p@ ( \mbox{cS}@ \mbox{xx} )\mbox{ and }q@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx}\mbox{ and }q@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( ( \mbox{cNUMBER}@ \mbox{xn} )\iff ( \mbox{cEVEN}@ \mbox{xn}\mbox{ or }\mbox{cODD}@ \mbox{xn} ) ) ) \ \Rightarrow \ \forall \mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cNUMBER}@ \mbox{xn} )\qquad\mbox{thf}(\mbox{cEVEN\_ODD}_4,\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$6.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ p )@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO380$\wedge$5.p} TPS problem X-2002-12-17-B

Suggested by a student's answer on a test.




$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ x@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xy}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{cX\_2002\_12\_17\_B},\mbox{conjecture})$

\medskip\textbf{SYO087$\wedge$5.p} TPS problem THM50-13

Simple formula for debugging.




$\mbox{cR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \neg\,\mbox{cP}\mbox{ and }( ( \mbox{cQ}\mbox{ and }\neg\,\mbox{cR} )\mbox{ or }( \neg\,\mbox{cQ}\mbox{ and }\mbox{cR} ) ) ) \ \Rightarrow \ ( ( ( \mbox{cP}\mbox{ and }\mbox{cQ} )\mbox{ or }( \neg\,\mbox{cP}\mbox{ and }\neg\,\mbox{cQ} ) ) \ \Rightarrow \ \mbox{cR} ) )\qquad\mbox{thf}(\mbox{cTHM50}_{13},\mbox{conjecture})$

\medskip\textbf{SYO290$\wedge$5.p} TPS problem from BASIC-HO-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$g_2{:}\;a\to a\qquad\mbox{thf}(g_2,\mbox{type})$

$g_1{:}\;a\to a\qquad\mbox{thf}(g_1,\mbox{type})$

$p{:}\;( a\to a )\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\neg\,p@ \lambda \mbox{xw}{:}\;a{:}\;( g_1@ ( g_2@ \mbox{xw} ) )\mbox{ or }p@ \lambda \mbox{xx}{:}\;a{:}\;( g_1@ ( g_2@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTRANS\_BUG\_EX}_2,\mbox{conjecture})$

\medskip\textbf{SYO457$\wedge$2.p} Ted Sider's propositional modal logic wff 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mdia\_d}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO397$\wedge$1.p} Ted Sider's modal proposition logic theorem 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO561+1.p} Distinct objects






$\mbox{"Apple"}\ne \mbox{"Microsoft"}\qquad\mbox{fof}(\mbox{apple\_not\_microsoft},\mbox{conjecture})$

\medskip\textbf{SYO463$\wedge$6.p} Ted Sider's propositional modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO071$\wedge$4.003.p} ILTP Problem SYJ207+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p5\_type},\mbox{type})$

$p_6{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p6\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_6 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_0 )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_6 ) ) ) ) ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_0 ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO458$\wedge$4.p} Ted Sider's propositional modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ q ) )@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO462$\wedge$2.p} Ted Sider's propositional modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO082$\wedge$5.p} TPS problem BAFFLER-VARIANT






$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ ( f@ \mbox{xy}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBAFFLER\_VARIANT},\mbox{conjecture})$

\medskip\textbf{SYO312$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cE}_2{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cE}_2,\mbox{type})$

$\mbox{cE}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cE}_1,\mbox{type})$

$\neg\,\exists \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cE}_1@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cE}_2@ \mbox{xp}@ \mbox{xq} ) \ \Rightarrow \ ( ( \mbox{xp}@ ( \mbox{xh}@ \mbox{xp} ) )\iff ( \mbox{xq}@ ( \mbox{xh}@ \mbox{xq} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cE}_1@ ( \mbox{xh}@ \mbox{xp} )@ ( \mbox{xh}@ \mbox{xq} ) ) \ \Rightarrow \ ( \mbox{cE}_2@ \mbox{xp}@ \mbox{xq} ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{118},\mbox{conjecture})$

\medskip\textbf{SYO338$\wedge$5.p} TPS problem from BASIC-HO-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall r{:}\;a\to a\to \mbox{\$o},u{:}\;( a\to \mbox{\$o} )\to a{:}\;( \exists x{:}\;a,y{:}\;a,z{:}\;a{:}\;( ( r@ x@ y\mbox{ or }r@ x@ y )\mbox{ and }( r@ y@ z\mbox{ or }r@ y@ z )\mbox{ and }( \neg\,r@ x@ z\mbox{ or }\neg\,r@ x@ z ) )\mbox{ or }\exists \mbox{xs}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xz}\mbox{ or }\mbox{xs}@ \mbox{xz} )\mbox{ and }( \neg\,r@ \mbox{xz}@ ( u@ \mbox{xs} )\mbox{ or }\neg\,r@ \mbox{xz}@ ( u@ \mbox{xs} ) ) )\mbox{ or }\exists \mbox{xj}{:}\;a{:}\;( \forall \mbox{xk}{:}\;a{:}\;( \neg\,\mbox{xs}@ \mbox{xk}\mbox{ or }\neg\,\mbox{xs}@ \mbox{xk}\mbox{ or }r@ \mbox{xk}@ \mbox{xj}\mbox{ or }r@ \mbox{xk}@ \mbox{xj} )\mbox{ and }( \neg\,r@ ( u@ \mbox{xs} )@ \mbox{xj}\mbox{ or }\neg\,r@ ( u@ \mbox{xs} )@ \mbox{xj} ) ) )\mbox{ or }\forall \mbox{xf}{:}\;a\to a{:}\;( \exists \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( r@ \mbox{xx}@ \mbox{xy}\mbox{ or }r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }( \neg\,r@ ( \mbox{xf}@ \mbox{xx} )@ ( \mbox{xf}@ \mbox{xy} )\mbox{ or }\neg\,r@ ( \mbox{xf}@ \mbox{xx} )@ ( \mbox{xf}@ \mbox{xy} ) ) )\mbox{ or }\exists \mbox{xw}{:}\;a{:}\;( ( r@ \mbox{xw}@ ( \mbox{xf}@ \mbox{xw} )\mbox{ or }r@ \mbox{xw}@ ( \mbox{xf}@ \mbox{xw} ) )\mbox{ and }( r@ ( \mbox{xf}@ \mbox{xw} )@ \mbox{xw}\mbox{ or }r@ ( \mbox{xf}@ \mbox{xw} )@ \mbox{xw} ) ) ) )\qquad\mbox{thf}(\mbox{cT145\_DOUBLE},\mbox{conjecture})$

\medskip\textbf{SYO056$\wedge$1.p} Simple textbook example 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mor}@ a@ b ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO128$\wedge$5.p} TPS problem from BASIC-FO-THMS






$r{:}\;\mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$q{:}\;\mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$( p\mbox{ and }( q\mbox{ or }r ) ) \ \Rightarrow \ ( ( p\mbox{ and }q )\mbox{ or }( p\mbox{ and }r ) )\qquad\mbox{thf}(\mbox{cDISJ\_BUG}_2,\mbox{conjecture})$

\medskip\textbf{SYO459$\wedge$3.p} Ted Sider's propositional modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) )@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO222$\wedge$5.p} TPS problem THM115A

Sunil's example.




$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ ( f@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cP}@ a\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cTHM115A},\mbox{conjecture})$

\medskip\textbf{SYO376$\wedge$5.p} TPS problem from EXTENSIONALITY






$\mbox{cG}{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cP}{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \mbox{cP}@ \mbox{cF}\mbox{ and }\mbox{cP}@ \mbox{cG} ) \ \Rightarrow \ ( \mbox{cP}@ \lambda \mbox{xx}{:}\;\mbox{\$o}{:}\;( \mbox{cF}@ \mbox{xx}\mbox{ or }\mbox{cG}@ \mbox{xx} )\mbox{ or }\mbox{cP}@ \lambda \mbox{xx}{:}\;\mbox{\$o}{:}\;( \neg\,\mbox{cF}@ \mbox{xx}\mbox{ or }\mbox{cG}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM}_{627},\mbox{conjecture})$

\medskip\textbf{SYO237$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( f@ \mbox{xx}_0 ) = ( g@ \mbox{xx}_0 ) \ \Rightarrow \ ( ( p@ \lambda \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( f@ \mbox{xx}_0 ) ) \ \Rightarrow \ ( ( q@ x ) \ \Rightarrow \ ( p@ \lambda \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( g@ \mbox{xx}_0 ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{508},\mbox{conjecture})$

\medskip\textbf{SYO522=1.p} Functions are either odd or even






$f{:}\;( \mbox{\$int}\times \mbox{\$int}\times \mbox{\$int} )\to \mbox{\$int}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;f(x,x,y) = \mbox{\$product}(2,z)\mbox{ or }\exists x{:}\;\mbox{\$int},y{:}\;\mbox{\$int},z{:}\;\mbox{\$int}{:}\;f(x,y,y) = \mbox{\$sum}(\mbox{\$product}(2,z),1)\qquad\mbox{tff}(\mbox{fxxx\_is\_either\_even\_or\_odd},\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$2.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_d}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_d}@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_d}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO569$\wedge$7.p} Fitting and Mendelsohn problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$g{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mbox\_s}_4@ ( f@ x ) ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( g@ x ) ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( f@ x )@ ( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO149$\wedge$5.p} TPS problem from BASIC-FO-THMS






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cQ}@ ( f@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy} ) )\mbox{ and }\mbox{cP}@ a ) \ \Rightarrow \ ( \mbox{cP}@ ( f@ a ) )\qquad\mbox{thf}(\mbox{cSIMPLEPQ},\mbox{conjecture})$

\medskip\textbf{SYO418$\wedge$1.p} Ted Sider's modal proposition logic theorem 27






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mdia\_d}@ ( \mbox{mimplies}@ q@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO073$\wedge$4.001.p} ILTP Problem SYJ209+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )@ ( \mbox{iatom}@ f ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO462$\wedge$5.p} Ted Sider's propositional modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO072$\wedge$4.003.p} ILTP Problem SYJ208+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{12}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o12\_type},\mbox{type})$

$o_{13}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o13\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$o_{22}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o22\_type},\mbox{type})$

$o_{23}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o23\_type},\mbox{type})$

$o_{31}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o31\_type},\mbox{type})$

$o_{32}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o32\_type},\mbox{type})$

$o_{33}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o33\_type},\mbox{type})$

$o_{41}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o41\_type},\mbox{type})$

$o_{42}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o42\_type},\mbox{type})$

$o_{43}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o43\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{13} ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{23} ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{32} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{33} ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{41} )@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{42} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{43} ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{iatom}@ o_{41} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{22} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{32} )@ ( \mbox{iatom}@ o_{42} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{23} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{33} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{13} )@ ( \mbox{iatom}@ o_{43} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{23} )@ ( \mbox{iatom}@ o_{33} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{23} )@ ( \mbox{iatom}@ o_{43} ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{33} )@ ( \mbox{iatom}@ o_{43} ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$5.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_4@ p ) ) )@ ( \mbox{mbox\_s}_4@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) )@ ( \mbox{mbox\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO028$\wedge$1.p} Not all things are false






$\neg\,\forall p{:}\;\mbox{\$o}{:}\;p\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO150$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cC}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cR}_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_4,\mbox{type})$

$\mbox{cR}_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_3,\mbox{type})$

$\mbox{cR}_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_2,\mbox{type})$

$\mbox{cR}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_1,\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}_0 ) ) \ \Rightarrow \ ( ( \mbox{cR}_1@ x\mbox{ or }\mbox{cR}_2@ x\mbox{ or }\mbox{cR}_3@ x\mbox{ or }\mbox{cR}_4@ x ) \ \Rightarrow \ \mbox{cC} )\qquad\mbox{thf}(\mbox{cADDHYP}_8,\mbox{conjecture})$

\medskip\textbf{SYO076$\wedge$5.p} TPS problem THM114






$\mbox{cQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \mbox{cP}\mbox{ and }( \mbox{cP} \ \Rightarrow \ \mbox{cQ} ) ) \ \Rightarrow \ \mbox{cQ}\qquad\mbox{thf}(\mbox{cTHM}_{114},\mbox{conjecture})$

\medskip\textbf{SYO225$\wedge$5.p} TPS problem THM126-CORRECTED






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b\to b,\mbox{xh}_{10}{:}\;g\to b,\mbox{xh}_{20}{:}\;b\to a,\mbox{xs}_{10}{:}\;g\to \mbox{\$o},\mbox{xf}_{10}{:}\;g\to g\to g,\mbox{xs}_{20}{:}\;b\to \mbox{\$o},\mbox{xf}_{20}{:}\;b\to b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx}\mbox{ and }\mbox{xs}_{10}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_{10}@ ( \mbox{xf}_{10}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx}\mbox{ and }\mbox{xs}_{10}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_{10}@ ( \mbox{xf}_{10}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} )@ ( \mbox{xh}_{10}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_{20}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_{20}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_{20}@ \mbox{xx}\mbox{ and }\mbox{xs}_{20}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_{20}@ ( \mbox{xf}_{20}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_{20}@ \mbox{xx} )@ ( \mbox{xh}_{20}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx}\mbox{ and }\mbox{xs}_{10}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_{10}@ ( \mbox{xf}_{10}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_{10}@ \mbox{xx}\mbox{ and }\mbox{xs}_{10}@ \mbox{xy}\mbox{ and }\mbox{xs}_{10}@ \mbox{xx}\mbox{ and }\mbox{xs}_{10}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_{20}@ ( \mbox{xh}_{10}@ ( \mbox{xf}_{10}@ \mbox{xx}@ \mbox{xy} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_{20}@ ( \mbox{xh}_{10}@ \mbox{xx} ) )@ ( \mbox{xh}_{20}@ ( \mbox{xh}_{10}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM126\_CORRECTED\_pme},\mbox{conjecture})$

\medskip\textbf{SYO101$\wedge$5.p} TPS problem THM83






$\mbox{cW}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cW},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ x@ y\mbox{ and }\mbox{cP}@ z ) ) \ \Rightarrow \ \exists u{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ v ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{cW}@ u ) )\qquad\mbox{thf}(\mbox{cTHM}_{83},\mbox{conjecture})$

\medskip\textbf{SYO323$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cHALF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHALF},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE},\mbox{type})$

$\forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xu}@ \mbox{xv}\mbox{ and }q@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) )\mbox{ and }\mbox{cHALF}@ c_0@ c_0\mbox{ and }\mbox{cHALF}@ c_0@ ( \mbox{cS}@ c_0 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cHALF}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cDOUBLE\_TO\_HALF}_1,\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$2.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO270$\wedge$5.p} TPS problem THM85






$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ ( \mbox{xf}@ \mbox{xx} ) )\mbox{ and }\mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy}@ ( g@ ( h@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{85},\mbox{conjecture})$

\medskip\textbf{SYO262$\wedge$5.p} TPS problem THM19SK1






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cE}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\neg\,\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ ( \mbox{cE}@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\neg\,\mbox{cP}@ ( \mbox{cE}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cP}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) )@ ( \mbox{xf}@ ( \mbox{cE}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cP}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM19SK}_1,\mbox{conjecture})$

\medskip\textbf{SYO511$\wedge$1.p} Two different choice operators at type i






$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{epschoice},\mbox{axiom})$

$\mbox{eps}_2{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps}_2,\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_2@ p ) ) )\qquad\mbox{thf}(\mbox{eps2choice},\mbox{axiom})$

$\mbox{eps} = \mbox{eps}_2\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO163$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(i,\mbox{type})$

$n{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(n,\mbox{type})$

$z{:}\;\mbox{\$i}\qquad\mbox{thf}(z,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ z ) )@ ( i@ ( i@ \mbox{xx}@ \mbox{xy} )@ ( i@ \mbox{xx}@ z ) ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ ( n@ \mbox{xx} )@ ( n@ \mbox{xy} ) )@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cT}@ ( i@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cT}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cMORGAN\_AXIOMS},\mbox{conjecture})$

\medskip\textbf{SYO561\_1.p} Distinct objects






$\mbox{company}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{company\_type},\mbox{type})$

$\mbox{apple}{:}\;\mbox{company}\qquad\mbox{tff}(\mbox{apple\_company},\mbox{type})$

$\mbox{microsoft}{:}\;\mbox{company}\qquad\mbox{tff}(\mbox{microsoft\_company},\mbox{type})$

$\mbox{\$distinct}(\mbox{apple},\mbox{microsoft})\qquad\mbox{tff}(\mbox{distinct\_companies},\mbox{axiom})$

$\mbox{apple}\ne \mbox{microsoft}\qquad\mbox{tff}(\mbox{apple\_not\_microsoft},\mbox{conjecture})$

\medskip\textbf{SYO297$\wedge$5.p} TPS problem from BASIC-HO-THMS






$n{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{n\_type},\mbox{type})$

$\mbox{cS}{:}\;n\to n\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$c_0{:}\;n\qquad\mbox{thf}(c_0,\mbox{type})$

$\forall \mbox{xp}{:}\;n\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_0\mbox{ and }\forall \mbox{xx}{:}\;n{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cS}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;n{:}\;( \mbox{xp}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cPA\_IND},\mbox{conjecture})$

\medskip\textbf{SYO244$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\exists \mbox{xf}{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a\to \mbox{\$o}{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) \ \Rightarrow \ \forall a{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \exists x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( a@ x )\mbox{ and }\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ x ) \ \Rightarrow \ \exists \mbox{xu}{:}\;a\to \mbox{\$o}{:}\;( x@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;a{:}\;\forall \mbox{xa}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ \mbox{xa} ) \ \Rightarrow \ \exists \mbox{xb}{:}\;a\to \mbox{\$o}{:}\;( \mbox{xa}@ \mbox{xb}\mbox{ and }\mbox{xb}@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\exists \mbox{xf}{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\forall \mbox{xa}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( a@ \mbox{xa} ) \ \Rightarrow \ ( \mbox{xa}@ ( \mbox{xf}@ \mbox{xa} )\mbox{ and }\mbox{xf}@ \mbox{xa}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM535B},\mbox{conjecture})$

\medskip\textbf{SYO541$\wedge$1.p} If-then-else on \$i$>$\$i defined from choice on \$i$>$\$i

A choice operator on (\$i$>$\$i) is used to define an if-then-else 
 operator at (\$i$>$\$i). Check that it works.




$\mbox{epsii}{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsii},\mbox{type})$

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

$\mbox{if}{:}\;\mbox{\$o}\to ( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if},\mbox{type})$

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{epsii}@ \lambda z{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{if}@ \mbox{\$true}@ x@ y ) = x\mbox{ and }( \mbox{if}@ \mbox{\$false}@ x@ y ) = y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO023$\wedge$1.p} De Morgan lambda terms by Leibnitz






$( \lambda u{:}\;\mbox{\$o},v{:}\;\mbox{\$o}{:}\;( u\mbox{ and }v ) ) = ( \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\neg\,\neg\,x\mbox{ or }\neg\,y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO103$\wedge$5.p} TPS problem THM147

Theorem 211 on page 120 of [Chu56].




$\mbox{imp}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{imp},\mbox{type})$

$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$\mbox{nt}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{nt},\mbox{type})$

$\neg\,\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cT}@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xq} )\mbox{ or }\neg\,\mbox{cT}@ \mbox{xp}\mbox{ or }\mbox{cT}@ \mbox{xq} )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ \mbox{xp}@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xp} ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i},\mbox{xr}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ ( \mbox{imp}@ \mbox{xp}@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xr} ) )@ ( \mbox{imp}@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xq} )@ ( \mbox{imp}@ \mbox{xp}@ \mbox{xr} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;\mbox{\$i},\mbox{xq}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( \mbox{imp}@ ( \mbox{imp}@ ( \mbox{nt}@ \mbox{xp} )@ ( \mbox{nt}@ \mbox{xq} ) )@ ( \mbox{imp}@ \mbox{xq}@ \mbox{xp} ) ) )\mbox{ and }\exists \mbox{xa}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cT}@ ( \mbox{imp}@ \mbox{xa}@ \mbox{xa} )\qquad\mbox{thf}(\mbox{cTHM}_{147},\mbox{conjecture})$

\medskip\textbf{SYO444$\wedge$1.p} Ted Sider's modal proposition logic theorem 53






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mnot}@ ( \mbox{mdia\_s}_5@ p ) )@ ( \mbox{mdia\_s}_5@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO462$\wedge$4.p} Ted Sider's propositional modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mbox\_b}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO265$\wedge$5.p} TPS problem X5210






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$x{:}\;a\qquad\mbox{thf}(x,\mbox{type})$

$( \lambda \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;\mbox{xx} = \mbox{xy}@ x ) = ( \lambda \mbox{xz}{:}\;a{:}\;\exists \mbox{xy}{:}\;a{:}\;( \mbox{xy} = x\mbox{ and }\mbox{xz} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cX}_{5210},\mbox{conjecture})$

\medskip\textbf{SYO038-1.002.304.p} Boolos' Curious Inference, size f(2,f(3,4))






$f(n,1) = s(1)\qquad\mbox{cnf}(\mbox{ax}_1,\mbox{axiom})$

$f(1,s(x)) = s(s(f(1,x)))\qquad\mbox{cnf}(\mbox{ax}_2,\mbox{axiom})$

$f(s(n),s(x)) = f(n,f(s(n),x))\qquad\mbox{cnf}(\mbox{ax}_3,\mbox{axiom})$

$d(1)\qquad\mbox{cnf}(\mbox{ax}_4,\mbox{axiom})$

$d(x) \ \Rightarrow \ d(s(x))\qquad\mbox{cnf}(\mbox{ax}_5,\mbox{axiom})$

$\neg\,d(f(s(1),f(s(s(1)),s(s(s(1))))))\qquad\mbox{cnf}(\mbox{conj},\mbox{negated\_conjecture})$

\medskip\textbf{SYO461$\wedge$4.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO184$\wedge$5.p} TPS problem CT9






$\exists \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_9,\mbox{conjecture})$

\medskip\textbf{SYO484$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{meq\_ind}@ x@ y ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO504$\wedge$1.p} Hoeschele p.21






$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$h{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(h,\mbox{type})$

$i{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(i,\mbox{type})$

$g{:}\;( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$f{:}\;( ( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$( ( h@ a@ b ) = ( a\mbox{ and }b )\mbox{ and }( i@ a@ b ) = ( a\mbox{ or }b )\mbox{ and }( f@ g ) = ( i@ a@ b )\mbox{ and }h@ a@ b ) \ \Rightarrow \ ( f@ g )\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO169$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{ab}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{ab},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cPx}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cPx},\mbox{type})$

$e{:}\;\mbox{\$i}\qquad\mbox{thf}(e,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ e@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy}@ e@ \mbox{xy} )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ \mbox{xz}@ e )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xxy}{:}\;\mbox{\$i},\mbox{xyz}{:}\;\mbox{\$i},\mbox{xxyz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy}@ \mbox{xxy}\mbox{ and }\mbox{cP}@ \mbox{xy}@ \mbox{xz}@ \mbox{xyz} ) \ \Rightarrow \ ( ( \mbox{cP}@ \mbox{xxy}@ \mbox{xz}@ \mbox{xxyz} )\iff ( \mbox{cPx}@ \mbox{xyz}@ \mbox{xxyz} ) ) ) ) \ \Rightarrow \ ( ( \mbox{cP}@ a@ b@ \mbox{ab} ) \ \Rightarrow \ ( \mbox{cP}@ b@ a@ \mbox{ab} ) )\qquad\mbox{thf}(\mbox{cGRP\_COMM},\mbox{conjecture})$

\medskip\textbf{SYO470$\wedge$1.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_k}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) )@ ( \mbox{mbox\_k}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO001$\wedge$1.p} Leibniz equality is transitive






$\mbox{leibeq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( \mbox{leibeq}@ x@ y\mbox{ and }\mbox{leibeq}@ y@ z ) \ \Rightarrow \ ( \mbox{leibeq}@ x@ z ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO059$\wedge$4.p} ILTP problem SYJ102+1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ a )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO042$\wedge$2.p} Unsatisfiable basic formula 4

Negation is the unique function g such that g x = y and g y = x
 for x,y:o distinct.




$g{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$y{:}\;\mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$\neg\,x\ne y\mbox{ and }( g@ x ) = y\mbox{ and }( g@ y ) = x\mbox{ and }p@ g\mbox{ and }\neg\,p@ \neg\,\qquad\mbox{thf}(4,\mbox{conjecture})$

\medskip\textbf{SYO524=1.p} Monotone function






$f{:}\;\mbox{\$int}\to \mbox{\$int}\qquad\mbox{tff}(\mbox{f\_type},\mbox{type})$

$( \forall u{:}\;\mbox{\$int}{:}\;\mbox{\$lesseq}(f(\mbox{\$sum}(u,1)),f(\mbox{\$sum}(u,2)))\mbox{ and }\mbox{\$lesseq}(f(7),3) ) \ \Rightarrow \ \mbox{\$lesseq}(f(4),3)\qquad\mbox{tff}(\mbox{co}_1,\mbox{conjecture})$

\medskip\textbf{SYO089$\wedge$5.p} TPS problem DMG7

A propositional problem suggested by Jay Hunschel Kim.




$t{:}\;\mbox{\$o}\qquad\mbox{thf}(t,\mbox{type})$

$s{:}\;\mbox{\$o}\qquad\mbox{thf}(s,\mbox{type})$

$q{:}\;\mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$( \neg\,\neg\,( \neg\,p\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t\mbox{ or }\neg\,q\mbox{ or }\neg\,s\mbox{ or }\neg\,t ) \ \Rightarrow \ ( \neg\,( \neg\,( \neg\,p\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\mbox{ and }q\mbox{ or }\neg\,s\mbox{ and }t )\qquad\mbox{thf}(\mbox{cDMG}_7,\mbox{conjecture})$

\medskip\textbf{SYO186$\wedge$5.p} TPS problem CT11






$\forall \mbox{xy}{:}\;\mbox{\$o}{:}\;\exists \mbox{xy}_0{:}\;\mbox{\$o}{:}\;\mbox{xy}_0\qquad\mbox{thf}(\mbox{cCT}_{11},\mbox{conjecture})$

\medskip\textbf{SYO516$\wedge$1.p} Every functional relation corresponds to a function






$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x ) = ( \lambda z{:}\;\mbox{\$i}{:}\;y = z ) \ \Rightarrow \ \exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( r@ x@ ( f@ x ) )\qquad\mbox{thf}(\mbox{descr}_2,\mbox{conjecture})$

\medskip\textbf{SYO095$\wedge$5.p} TPS problem THM81






$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$( \forall s{:}\;\mbox{\$i},t{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ s@ s )\iff ( \mbox{cR}@ s@ t ) )\mbox{ and }\forall w{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ w@ w )\iff ( \mbox{cR}@ z@ w ) ) ) \ \Rightarrow \ ( \exists x{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ x@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ y@ y ) )\qquad\mbox{thf}(\mbox{cTHM}_{81},\mbox{conjecture})$

\medskip\textbf{SYO465$\wedge$1.p} Ted Sider's propositional modal logic wff 16






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ p ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO272$\wedge$5.p} TPS problem THM301A






$\mbox{cHALF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHALF},\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xu}@ \mbox{xv} )\iff \forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ and }\mbox{cHALF}@ c_0@ c_0\mbox{ and }\mbox{cHALF}@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cHALF}@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cHALF}@ \mbox{xv}@ \mbox{xu} ) )\qquad\mbox{thf}(\mbox{cTHM301A},\mbox{conjecture})$

\medskip\textbf{SYO018$\wedge$1.p} Formula requiring b and Eta






$f{:}\;\mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$a{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$( p@ \lambda x{:}\;\mbox{\$i}{:}\;( f@ ( a@ \lambda x{:}\;\mbox{\$i}{:}\;( f@ b@ x )\mbox{ and }b )@ x ) ) \ \Rightarrow \ ( p@ ( f@ ( b\mbox{ and }a@ ( f@ b ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.p} The Kaminski equation






$f{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$( f@ ( f@ ( f@ x ) ) ) = ( f@ x )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO070$\wedge$4.002.p} ILTP Problem SYJ211+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_2 ) ) )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_2 ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_0 ) ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ b_1 ) ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO470$\wedge$4.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_b}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) )@ ( \mbox{mbox\_b}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO045$\wedge$1.p} Simple textbook example 2






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ a@ b ) )@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO557$\wedge$1.p} Exists on \$i can be expressed in terms of choice on \$






$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( p@ ( \mbox{eps}@ p ) ) ) = ( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;?? (p) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO363$\wedge$5.p} TPS problem EDEC2

EDEC from [Ben99], using Primeq in the antecedent instead of 
 Leibeq.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$j{:}\;a\to a\qquad\mbox{thf}(j,\mbox{type})$

$g{:}\;( a\to a )\to a\to a\qquad\mbox{thf}(g,\mbox{type})$

$h{:}\;a\to a\qquad\mbox{thf}(h,\mbox{type})$

$f{:}\;( a\to a )\to a\to a\qquad\mbox{thf}(f,\mbox{type})$

$( \forall x{:}\;a\to a,y{:}\;a{:}\;( f@ x@ y ) = ( g@ x@ y )\mbox{ and }\forall z{:}\;a{:}\;( h@ z ) = ( j@ z ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ ( f@ h ) ) \ \Rightarrow \ ( \mbox{xq}@ ( g@ j ) ) )\qquad\mbox{thf}(\mbox{cEDEC2\_pme},\mbox{conjecture})$

\medskip\textbf{SYO309$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx}@ ( \mbox{cQ}@ \mbox{xu} ) )\iff \neg\,\exists \mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cQ}@ \mbox{xv} ) ) ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} )\iff \neg\,a@ ( \mbox{cS}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cEO}_1,\mbox{conjecture})$

\medskip\textbf{SYO405$\wedge$1.p} Ted Sider's modal proposition logic theorem 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mnot}@ q ) ) ) )@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO046$\wedge$1.p} Simple textbook example 3






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mdia}@ r@ ( \mbox{mimplies}@ a@ b ) )@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO276$\wedge$5.p} TPS problem BLEDSOE-FENG-SV-I2






$m{:}\;\mbox{\$i}\qquad\mbox{thf}(m,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$n{:}\;\mbox{\$i}\qquad\mbox{thf}(n,\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(s,\mbox{type})$

$\mbox{cO}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cO},\mbox{type})$

$( \forall a{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( a@ \mbox{cO}@ \mbox{cO}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( a@ ( s@ \mbox{xx} )@ ( s@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( a@ n@ m ) )\mbox{ and }\mbox{cP}@ n ) \ \Rightarrow \ ( \mbox{cP}@ m )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG\_SV\_I}_2,\mbox{conjecture})$

\medskip\textbf{SYO213$\wedge$5.p} TPS problem from BASIC-HO-PROP-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\neg\,\exists \mbox{xg}{:}\;a\to a\to \mbox{\$o}{:}\;\forall \mbox{xf}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xj}{:}\;a{:}\;\forall \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ ( \mbox{xg}@ \mbox{xj} ) = \mbox{xf} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xf} = \mbox{xf} ) )\qquad\mbox{thf}(\mbox{cCT}_{30},\mbox{conjecture})$

\medskip\textbf{SYO403$\wedge$1.p} Ted Sider's modal proposition logic theorem 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mor}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO568$\wedge$7.p} Girle problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$a{:}\;\mbox{mu}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall v{:}\;\mbox{\$i}{:}\;( \mbox{exists\_in\_world}@ a@ v )\qquad\mbox{thf}(\mbox{existence\_of\_a\_ax},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( f@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( f@ x ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_4@ ( f@ a ) )@ ( f@ a ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO191$\wedge$5.p} TPS problem CT14






$\exists \mbox{xz}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{14},\mbox{conjecture})$

\medskip\textbf{SYO540$\wedge$1.p} Property of if-then-else on \$i defined from choice on \$i

A choice operator on \$i is used to define an if-then-else operator
 at \$i. Check that if the then-part and the else-part are both X, 
 it returns X.




$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$\mbox{if}{:}\;\mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if},\mbox{type})$

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall b{:}\;\mbox{\$o},x{:}\;\mbox{\$i}{:}\;( \mbox{if}@ b@ x@ x ) = x\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO318$\wedge$5.p} TPS problem from BASIC-HO-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(c_2,\mbox{type})$

$\mbox{c\_star}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{c\_star},\mbox{type})$

$\mbox{c\_less\_}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_less\_},\mbox{type})$

$\mbox{div}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{div},\mbox{type})$

$( \neg\,\mbox{c\_less\_}@ c_0@ c_0\mbox{ and }\neg\,\mbox{c\_less\_}@ ( \mbox{div}@ c_0@ c_2 )@ c_0\mbox{ and }\neg\,\mbox{c\_less\_}@ ( \mbox{c\_star}@ c_2@ c_0 )@ c_0 ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx}\mbox{ and }a@ \mbox{xy} ) \ \Rightarrow \ ( \neg\,\mbox{c\_less\_}@ ( \mbox{div}@ \mbox{xx}@ c_2 )@ c_0\mbox{ and }\neg\,\mbox{c\_less\_}@ \mbox{xy}@ ( \mbox{div}@ \mbox{xx}@ c_2 )\mbox{ and }\neg\,\mbox{c\_less\_}@ ( \mbox{c\_star}@ c_2@ \mbox{xx} )@ \mbox{xy} ) )\mbox{ and }a@ c_0 )\qquad\mbox{thf}(\mbox{cBLEDSOE\_FENG}_9,\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.005.p} Five function variant of the Kaminski equation






$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$f_4{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_4,\mbox{type})$

$( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ ( f_3@ ( f_3@ ( f_3@ ( f_4@ x ) ) ) ) ) ) ) ) ) = ( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ ( f_3@ ( f_4@ ( f_4@ ( f_4@ x ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_5,\mbox{conjecture})$

\medskip\textbf{SYO367$\wedge$5.p} TPS problem from EXTENSIONALITY






$\mbox{cA}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\mbox{cC}{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$( \mbox{cC}@ \mbox{cA} ) \ \Rightarrow \ ( \mbox{cC}@ \neg\,\neg\,\mbox{cA} )\qquad\mbox{thf}(\mbox{cEXT}_2,\mbox{conjecture})$

\medskip\textbf{SYO114$\wedge$5.p} TPS problem THM119






$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$d{:}\;\mbox{\$i}\qquad\mbox{thf}(d,\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$c{:}\;\mbox{\$i}\qquad\mbox{thf}(c,\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\neg\,\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xz}\mbox{ or }\mbox{cR}@ \mbox{xz} )\mbox{ and }\mbox{cQ}@ \mbox{xz} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}\mbox{ or }\neg\,\mbox{cQ}@ \mbox{xx}\mbox{ or }\neg\,\mbox{cQ}@ \mbox{xy}\mbox{ or }\neg\,\mbox{cQ}@ c\mbox{ or }\neg\,\mbox{cQ}@ d )\mbox{ and }( \neg\,\mbox{cP}@ a\mbox{ or }\neg\,\mbox{cP}@ b )\qquad\mbox{thf}(\mbox{cTHM}_{119},\mbox{conjecture})$

\medskip\textbf{SYO022$\wedge$1.p} De Morgan lambda terms by Leibnitz






$\mbox{leibeq}{:}\;( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to ( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o},y{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\mbox{leibeq}@ \lambda u{:}\;\mbox{\$o},v{:}\;\mbox{\$o}{:}\;( u\mbox{ and }v )@ \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\neg\,\neg\,x\mbox{ or }\neg\,y\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO065$\wedge$4.002.p} ILTP Problem SYJ201+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$p_5{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p5\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_5 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_5 ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO233$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\mbox{cN}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cM}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ ( \mbox{cM}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cN}@ \mbox{xx} ) ) ) \ \Rightarrow \ \mbox{cM} = \mbox{cN}\qquad\mbox{thf}(\mbox{cLEIBNIZ},\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$2.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_d}@ p ) ) )@ ( \mbox{mbox\_d}@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mbox\_d}@ p ) )@ ( \mbox{mbox\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO300$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\forall p{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i} )\to ( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i} )\to \mbox{\$o}{:}\;\exists x{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}{:}\;( \forall \mbox{xz}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}{:}\;( p@ \mbox{xz}@ \mbox{xz} ) \ \Rightarrow \ ( p@ \lambda u{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( u@ ( x@ \lambda v{:}\;\mbox{\$i}{:}\;v ) )@ x ) )\qquad\mbox{thf}(\mbox{cUNIFTHM}_1,\mbox{conjecture})$

\medskip\textbf{SYO230$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\mbox{cB}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\exists \mbox{xu}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\mbox{xu} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cA}@ \mbox{xx}\mbox{ or }\mbox{cB}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_2,\mbox{conjecture})$

\medskip\textbf{SYO441$\wedge$1.p} Ted Sider's modal proposition logic theorem 50






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) ) )@ ( \mbox{mbox\_s}_5@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO462$\wedge$6.p} Ted Sider's propositional modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO057$\wedge$1.p} Simple textbook example 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ a ) )@ ( \mbox{mbox}@ r@ a ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO073$\wedge$4.002.p} ILTP Problem SYJ209+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO383$\wedge$5.p} TPS problem THM409-1






$\mbox{cQ}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_2,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cP}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_1,\mbox{type})$

$\mbox{cQ}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_1,\mbox{type})$

$d{:}\;\mbox{\$i}\qquad\mbox{thf}(d,\mbox{type})$

$c{:}\;\mbox{\$i}\qquad\mbox{thf}(c,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\neg\,\mbox{cQ}_1@ a@ b@ c\mbox{ and }\mbox{cP}_1@ a@ a\mbox{ and }\mbox{cP}_1@ b@ b\mbox{ and }\mbox{cP}_1@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_1@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_1@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_1@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_2@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cQ}_2@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\qquad\mbox{thf}(\mbox{cTHM409}_1,\mbox{conjecture})$

\medskip\textbf{SYO462$\wedge$3.p} Ted Sider's propositional modal logic wff 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mequiv}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO183$\wedge$6.p} TPS problem THM123

Trivial theorem to test TPS.




$\forall \mbox{xp}{:}\;\mbox{\$o}{:}\;\exists \mbox{xf}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xf}@ \mbox{xp} )\qquad\mbox{thf}(\mbox{cTHM}_{123},\mbox{conjecture})$

\medskip\textbf{SYO188$\wedge$5.p} TPS problem CT19






$\exists \mbox{xq}{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xq}@ \mbox{xp} )\qquad\mbox{thf}(\mbox{cCT}_{19},\mbox{conjecture})$

\medskip\textbf{SYO469$\wedge$3.p} Ted Sider's propositional modal logic wff 20






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ q ) )@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO170$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{ab}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{ab},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$e{:}\;\mbox{\$i}\qquad\mbox{thf}(e,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ e@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy}@ e@ \mbox{xy} )\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ \mbox{xz}@ e )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xxy}{:}\;\mbox{\$i},\mbox{xyz}{:}\;\mbox{\$i},\mbox{xxyz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx}@ \mbox{xy}@ \mbox{xxy}\mbox{ and }\mbox{cP}@ \mbox{xy}@ \mbox{xz}@ \mbox{xyz} ) \ \Rightarrow \ ( ( \mbox{cP}@ \mbox{xxy}@ \mbox{xz}@ \mbox{xxyz} )\iff ( \mbox{cP}@ \mbox{xx}@ \mbox{xyz}@ \mbox{xxyz} ) ) ) ) \ \Rightarrow \ ( ( \mbox{cP}@ a@ b@ \mbox{ab} ) \ \Rightarrow \ ( \mbox{cP}@ b@ a@ \mbox{ab} ) )\qquad\mbox{thf}(\mbox{cTHM}_{105},\mbox{conjecture})$

\medskip\textbf{SYO456$\wedge$3.p} Ted Sider's propositional modal logic wff 07






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) ) )@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO567$\wedge$7.p} Girle problem






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$g{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( \mbox{mbox\_s}_4@ ( g@ x ) ) ) )@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( g@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO125$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( ( \mbox{cP}\iff \mbox{cQ} )\mbox{ and }( \mbox{cQ}\iff \mbox{cR} ) ) \ \Rightarrow \ ( \mbox{cP}\iff \mbox{cR} )\qquad\mbox{thf}(\mbox{cTRIV}_4,\mbox{conjecture})$

\medskip\textbf{SYO577$\wedge$7.p} Mixed Modal Propositional Logic WFFs. problem 19






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO071$\wedge$4.002.p} ILTP Problem SYJ207+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_3 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_0 )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_0 ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO040$\wedge$1.p} Unsatisfiable basic formula 2






$f{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$h{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$( h@ ( f@ ( f@ ( f@ x ) ) ) )\ne ( h@ ( f@ x ) )\qquad\mbox{thf}(2,\mbox{axiom})$

\medskip\textbf{SYO145$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xy}@ \mbox{xz}\mbox{ and }\mbox{cP}@ \mbox{xx}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cEXPVAR\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO528$\wedge$1.p} There can be 4 distinct choice operators on type \$o






$\mbox{eps}_1{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_1,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_1@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_1,\mbox{axiom})$

$\mbox{eps}_2{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_2,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_2@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_2,\mbox{axiom})$

$\mbox{eps}_3{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_3,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_3@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_3,\mbox{axiom})$

$\mbox{eps}_4{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_4,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_4@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_4,\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_2\qquad\mbox{thf}(\mbox{choiceax}_{12},\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_3\qquad\mbox{thf}(\mbox{choiceax}_{13},\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{14},\mbox{axiom})$

$\mbox{eps}_2\ne \mbox{eps}_3\qquad\mbox{thf}(\mbox{choiceax}_{23},\mbox{axiom})$

$\mbox{eps}_2\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{24},\mbox{axiom})$

$\mbox{eps}_3\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{34},\mbox{axiom})$

\medskip\textbf{SYO200$\wedge$5.p} TPS problem CT4






$\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cCT}_4,\mbox{conjecture})$

\medskip\textbf{SYO203$\wedge$5.p} TPS problem PROP-2003-3-13






$r{:}\;\mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$q{:}\;\mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$( q \ \Rightarrow \ r ) \ \Rightarrow \ ( ( \neg\,q \ \Rightarrow \ \neg\,p ) \ \Rightarrow \ ( p \ \Rightarrow \ r ) )\qquad\mbox{thf}(\mbox{cPROP\_2003\_3}_{13},\mbox{conjecture})$

\medskip\textbf{SYO074$\wedge$4.002.p} ILTP Problem SYJ210+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_0 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ p_0 )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO535$\wedge$1.p} Choice on relations between individuals and functions






$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$\mbox{epsii}{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsii},\mbox{type})$

$\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{epsii}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceaxii},\mbox{axiom})$

$\mbox{epsa}{:}\;( \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsa},\mbox{type})$

$\mbox{epsa} = ( \lambda r{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{epsad},\mbox{definition})$

$\mbox{epsb}{:}\;( \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{epsb},\mbox{type})$

$\mbox{epsb} = ( \lambda r{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \mbox{epsii}@ \lambda y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ ( \mbox{epsa}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{epsbd},\mbox{definition})$

$\forall r{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( \mbox{epsa}@ r )@ ( \mbox{epsb}@ r ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO450$\wedge$6.p} Ted Sider's propositional modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ ( \mbox{mdia\_s}_5@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mimplies}@ q@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mdia\_s}_5@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.008.p} Eight function variant of the Kaminski equation






$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$f_3{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_3,\mbox{type})$

$f_4{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_4,\mbox{type})$

$f_5{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_5,\mbox{type})$

$f_6{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_6,\mbox{type})$

$f_7{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_7,\mbox{type})$

$( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ ( f_3@ ( f_4@ ( f_4@ ( f_4@ ( f_5@ ( f_6@ ( f_6@ ( f_6@ ( f_7@ x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ ( f_3@ ( f_3@ ( f_3@ ( f_4@ ( f_5@ ( f_5@ ( f_5@ ( f_6@ ( f_7@ ( f_7@ ( f_7@ x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_8,\mbox{conjecture})$

\medskip\textbf{SYO468$\wedge$1.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ p )@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO400$\wedge$1.p} Ted Sider's modal proposition logic theorem 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mnot}@ q ) ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mnot}@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$4.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO303$\wedge$5.p} TPS problem from BASIC-HO-THMS






$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,p@ c_0\mbox{ or }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx} ) ) \ \Rightarrow \ ( \neg\,\mbox{cS}@ c_0\mbox{ and }\mbox{cT}@ c_0\mbox{ or }( \mbox{cT}@ a\mbox{ and }\mbox{cS}@ b ) )\qquad\mbox{thf}(\mbox{cMIN\_QUAN\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO468$\wedge$4.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ p )@ q ) )@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO043$\wedge$2.p} Unsatisfiable basic formula 5

Variant of the Kaminski equation.




$f{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$q{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$q@ f@ x\mbox{ and }f@ ( f@ x )\mbox{ and }( f@ ( q@ f@ x ) ) = ( f@ x )\qquad\mbox{thf}(5,\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$1.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_k}@ p ) ) )@ ( \mbox{mbox\_k}@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) )@ ( \mbox{mbox\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO206$\wedge$5.p} TPS problem CT265






$\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;\forall \mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xy} = \mbox{xy} = \mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{265},\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$2.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_d}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO529$\wedge$1.p} There cannot be 5 distinct choice operators on type \$o






$\mbox{eps}_1{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_1,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_1@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_1,\mbox{axiom})$

$\mbox{eps}_2{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_2,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_2@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_2,\mbox{axiom})$

$\mbox{eps}_3{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_3,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_3@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_3,\mbox{axiom})$

$\mbox{eps}_4{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_4,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_4@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_4,\mbox{axiom})$

$\mbox{eps}_5{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{eps}_5,\mbox{type})$

$\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$o}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}_5@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax}_5,\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_2\qquad\mbox{thf}(\mbox{choiceax}_{12},\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_3\qquad\mbox{thf}(\mbox{choiceax}_{13},\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{14},\mbox{axiom})$

$\mbox{eps}_1\ne \mbox{eps}_5\qquad\mbox{thf}(\mbox{choiceax}_{15},\mbox{axiom})$

$\mbox{eps}_2\ne \mbox{eps}_3\qquad\mbox{thf}(\mbox{choiceax}_{23},\mbox{axiom})$

$\mbox{eps}_2\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{24},\mbox{axiom})$

$\mbox{eps}_2\ne \mbox{eps}_5\qquad\mbox{thf}(\mbox{choiceax}_{25},\mbox{axiom})$

$\mbox{eps}_3\ne \mbox{eps}_4\qquad\mbox{thf}(\mbox{choiceax}_{34},\mbox{axiom})$

$\mbox{eps}_3\ne \mbox{eps}_5\qquad\mbox{thf}(\mbox{choiceax}_{35},\mbox{axiom})$

$\mbox{eps}_4\ne \mbox{eps}_5\qquad\mbox{thf}(\mbox{choiceax}_{45},\mbox{axiom})$

\medskip\textbf{SYO468$\wedge$3.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ q ) )@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$6.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_5@ p ) ) )@ ( \mbox{mbox\_s}_5@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) )@ ( \mbox{mbox\_s}_5@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO253$\wedge$5.p} TPS problem THM124

Example to show the need for primitive substitutions that
 introduce negation.




$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\exists \mbox{xf}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xf}@ ( p\mbox{ and }\neg\,p )\mbox{ and }\neg\,\mbox{xf}@ ( p\mbox{ or }\neg\,p ) )\qquad\mbox{thf}(\mbox{cTHM}_{124},\mbox{conjecture})$

\medskip\textbf{SYO517$\wedge$1.p} A description operator at type i






$\exists d{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ x = y ) ) \ \Rightarrow \ ( p@ ( d@ p ) ) )\qquad\mbox{thf}(\mbox{descri},\mbox{conjecture})$

\medskip\textbf{SYO454$\wedge$2.p} Ted Sider's propositional modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mdia\_d}@ ( \mbox{mand}@ ( \mbox{mdia\_d}@ p )@ ( \mbox{mnot}@ q ) ) )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_d}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO071$\wedge$4.001.p} ILTP Problem SYJ207+1.001






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p0\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iequiv}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ p_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ p_0 )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ p_2 ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ p_0 ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO256$\wedge$5.p} TPS problem THM121

Test theorem for nested primitive substitutions.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xr}{:}\;\mbox{\$o}\to \mbox{\$o},\mbox{xp}{:}\;a\to \mbox{\$o},\mbox{xq}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$o}{:}\;( ( \mbox{xr}@ \mbox{xh} )\iff ( \mbox{xr}@ \forall \mbox{xx}{:}\;a{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ or }\mbox{xq}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{121},\mbox{conjecture})$

\medskip\textbf{SYO460$\wedge$3.p} Ted Sider's propositional modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mdia\_m}@ ( \mbox{mand}@ p@ q ) )@ ( \mbox{mdia\_m}@ ( \mbox{mand}@ p@ r ) ) )@ ( \mbox{mdia\_m}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO088$\wedge$5.p} TPS problem ARR-COM-DMG5

A propositional problem suggested by Jay Hunschel Kim.




$t{:}\;\mbox{\$o}\qquad\mbox{thf}(t,\mbox{type})$

$s{:}\;\mbox{\$o}\qquad\mbox{thf}(s,\mbox{type})$

$p{:}\;\mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$d{:}\;\mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$c{:}\;\mbox{\$o}\qquad\mbox{thf}(c,\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$( ( p\mbox{ and }( \neg\,a\mbox{ or }\neg\,b\mbox{ or }\neg\,c\mbox{ or }\neg\,d ) )\mbox{ or }( s \ \Rightarrow \ t ) ) \ \Rightarrow \ ( ( ( \neg\,a\mbox{ and }b\mbox{ or }\neg\,c\mbox{ and }d )\mbox{ and }p )\mbox{ or }( s \ \Rightarrow \ t ) )\qquad\mbox{thf}(\mbox{cARR\_COM\_DMG}_5,\mbox{conjecture})$

\medskip\textbf{SYO436$\wedge$1.p} Ted Sider's modal proposition logic theorem 45






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$5.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_4@ p )@ ( \mbox{mdia\_s}_4@ q ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_s}_4@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO002$\wedge$1.p} Leibniz equality obeys the congruence property under functions






$\mbox{leibeq}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{leibeq}@ x@ y ) \ \Rightarrow \ ( \mbox{leibeq}@ ( f@ x )@ ( f@ y ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO545$\wedge$1.p} Property of case from (\$o$>$\$o) to \$i defined from choice on \$i

A choice operator on \$i is used to define an if-then-else operator
 at \$i. Check that case always returns one of the four given 
 results.




$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$\mbox{case}{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{caseoo},\mbox{type})$

$\mbox{case} = ( \lambda b{:}\;\mbox{\$o}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$false} )\mbox{ and }z = x )\mbox{ or }( b = \neg\,\mbox{ and }z = y )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;a )\mbox{ and }z = u )\mbox{ or }( b = ( \lambda a{:}\;\mbox{\$o}{:}\;\mbox{\$true} )\mbox{ and }z = v ) ) ) )\qquad\mbox{thf}(\mbox{caseood},\mbox{definition})$

$f{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( ( \mbox{case}@ f@ x@ y@ u@ v ) = x\mbox{ or }( \mbox{case}@ f@ x@ y@ u@ v ) = y\mbox{ or }( \mbox{case}@ f@ x@ y@ u@ v ) = u\mbox{ or }( \mbox{case}@ f@ x@ y@ u@ v ) = v )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO194$\wedge$5.p} TPS problem CT23






$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{23},\mbox{conjecture})$

\medskip\textbf{SYO234$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$d{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{d\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\mbox{cB}{:}\;d\to c\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;c\to d\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\exists \mbox{xu}{:}\;c\to d\to \mbox{\$o}{:}\;\mbox{xu} = ( \lambda \mbox{xx}{:}\;c,\mbox{xy}{:}\;d{:}\;( \mbox{cA}@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cB}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cSV}_4,\mbox{conjecture})$

\medskip\textbf{SYO054$\wedge$2.p} Simple textbook example 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mbox}@ r@ a ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO452$\wedge$5.p} Ted Sider's propositional modal logic wff 03






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mand}@ p@ ( \mbox{mdia\_s}_4@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO478$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( f@ x ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$4.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_b}@ q ) ) )@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO073$\wedge$4.004.p} ILTP Problem SYJ209+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p_1 ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ f ) ) ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$5.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{mor}@ p@ ( \mbox{mdia\_s}_4@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mdia\_s}_4@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO038$\wedge$1.003.004.p} Boolos' Curious Inference, size f(3,4)






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(s,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$d{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$\forall n{:}\;\mbox{\$i}{:}\;( f@ n@ 1 ) = ( s@ 1 )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( f@ 1@ ( s@ x ) ) = ( s@ ( s@ ( f@ 1@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\forall n{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( f@ ( s@ n )@ ( s@ x ) ) = ( f@ n@ ( f@ ( s@ n )@ x ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$d@ 1\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( d@ x ) \ \Rightarrow \ ( d@ ( s@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$d@ ( f@ ( s@ ( s@ 1 ) )@ ( s@ ( s@ ( s@ 1 ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO025$\wedge$1.p} De Morgan by connectives and equality






$\mbox{and} = ( \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\neg\,\neg\,x\mbox{ or }\neg\,y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO385$\wedge$5.p} TPS problem THM409-2






$\mbox{cQ}_3{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_3,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cP}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

$\mbox{cQ}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_2,\mbox{type})$

$d{:}\;\mbox{\$i}\qquad\mbox{thf}(d,\mbox{type})$

$c{:}\;\mbox{\$i}\qquad\mbox{thf}(c,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cP}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_1,\mbox{type})$

$\mbox{cQ}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_1,\mbox{type})$

$\neg\,\mbox{cQ}_1@ a@ b@ c\mbox{ and }\mbox{cP}_1@ a@ a\mbox{ and }\mbox{cP}_1@ b@ b\mbox{ and }\mbox{cP}_1@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_1@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_1@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_1@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_2@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_2@ a@ a\mbox{ and }\mbox{cP}_2@ b@ b\mbox{ and }\mbox{cP}_2@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_2@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_2@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_2@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_3@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cQ}_3@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\qquad\mbox{thf}(\mbox{cTHM409}_2,\mbox{conjecture})$

\medskip\textbf{SYO054$\wedge$1.p} Simple textbook example 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mbox}@ r@ a ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO223$\wedge$5.p} TPS problem LING2






$\mbox{cJ}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cJ},\mbox{type})$

$\mbox{cLIKE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cLIKE},\mbox{type})$

$\mbox{cUNIQUE}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cUNIQUE},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{cUNIQUE}@ x ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;x = z )\mbox{ and }\mbox{cUNIQUE}@ \mbox{cS} ) \ \Rightarrow \ \exists \mbox{xan}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xan}@ \mbox{cP} ) = ( \mbox{cLIKE}@ \mbox{cP}@ \mbox{cS} )\mbox{ and }( \mbox{xan}@ \mbox{cJ} ) = ( \exists x{:}\;\mbox{\$i}{:}\;( \mbox{cUNIQUE}@ x\mbox{ and }\mbox{cLIKE}@ \mbox{cJ}@ x ) ) )\qquad\mbox{thf}(\mbox{cLING}_2,\mbox{conjecture})$

\medskip\textbf{SYO399$\wedge$1.p} Ted Sider's modal proposition logic theorem 08






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ ( \mbox{mand}@ q@ r ) ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ q@ ( \mbox{mnot}@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO131$\wedge$5.p} TPS problem from BASIC-FO-THMS






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$( \mbox{cR}@ ( g@ ( f@ a ) )@ ( f@ ( g@ ( f@ a ) ) ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xx}@ ( f@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cEXAMPLE}_1,\mbox{conjecture})$

\medskip\textbf{SYO267$\wedge$5.p} TPS problem THM111






$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;\forall g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( m@ g\mbox{ and }m@ h ) \ \Rightarrow \ ( m@ \lambda z{:}\;\mbox{\$i}{:}\;( g@ ( h@ z ) )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ ( g@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{111},\mbox{conjecture})$

\medskip\textbf{SYO268$\wedge$5.p} TPS problem X5308






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$r{:}\;a\to b\to \mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$\exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;b{:}\;( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( r@ \mbox{xx}@ \mbox{xy} )\iff \exists \mbox{xf}{:}\;a\to b{:}\;\forall \mbox{xx}{:}\;a{:}\;( r@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cX}_{5308},\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$3.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_m}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO519$\wedge$1.p} For any X,Y:i, there is a function swapping X and Y






$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;\exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( f@ x ) = y\mbox{ and }( f@ y ) = x )\qquad\mbox{thf}(\mbox{ifi},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$1.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mor}@ p@ ( \mbox{mdia\_k}@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO275$\wedge$5.p} TPS problem THM300A






$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE},\mbox{type})$

$\mbox{cHALF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHALF},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} )\iff \forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ c_0@ c_0\mbox{ and }q@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ and }\mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{xv}@ \mbox{xu}\mbox{ or }\mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xv} )@ ( \mbox{cS}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cTHM300A},\mbox{conjecture})$

\medskip\textbf{SYO455$\wedge$6.p} Ted Sider's propositional modal logic wff 06






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ ( \mbox{mor}@ ( \mbox{mnot}@ p )@ q ) ) )@ ( \mbox{mdia\_s}_5@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO427$\wedge$1.p} Ted Sider's modal proposition logic theorem 36






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ q ) ) ) )@ ( \mbox{mdia\_m}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO468$\wedge$6.p} Ted Sider's propositional modal logic wff 19






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mor}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ p )@ q ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ q )@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO520$\wedge$1.p} A simple problem with a choice operator






$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{epschoice},\mbox{axiom})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$p@ ( \mbox{eps}@ \lambda x{:}\;\mbox{\$i}{:}\;\neg\,p@ x )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\neg\,p@ ( \mbox{eps}@ p )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

\medskip\textbf{SYO165$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$s{:}\;\mbox{\$i}\qquad\mbox{thf}(s,\mbox{type})$

$\mbox{cE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cE}@ \mbox{xz}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cE}@ \mbox{xy}@ \mbox{xx}\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cR}@ \mbox{xy}@ \mbox{xw} ) \ \Rightarrow \ \neg\,\mbox{cE}@ \mbox{xw}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xs}_0{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cE}@ \mbox{xy}@ \mbox{xs}_0\mbox{ and }\mbox{cR}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cE}@ \mbox{xx}@ s ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM117A},\mbox{conjecture})$

\medskip\textbf{SYO488$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mnot}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{mnot}@ ( f@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO016$\wedge$1.p} Formula valid in MBb, but not in model classes not requiring b






$\mbox{leibeq}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$h{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(h,\mbox{type})$

$\mbox{leibeq}@ ( h@ ( \mbox{leibeq}@ ( h@ \mbox{\$true} )@ ( h@ \mbox{\$false} ) ) )@ ( h@ \mbox{\$false} )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO472$\wedge$3.p} Ted Sider's propositional modal logic wff 23






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ q@ r ) ) ) )@ ( \mbox{mimplies}@ q@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ r ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO077$\wedge$5.p} TPS problem THM64






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists x{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ y ) )\qquad\mbox{thf}(\mbox{cTHM}_{64},\mbox{conjecture})$

\medskip\textbf{SYO277$\wedge$5.p} TPS problem THM47D






$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \forall \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xq}@ x ) \ \Rightarrow \ ( \mbox{xq}@ y ) )\iff \forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall z{:}\;\mbox{\$i}{:}\;( r@ z@ z ) \ \Rightarrow \ ( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{cTHM47D},\mbox{conjecture})$

\medskip\textbf{SYO064$\wedge$4.003.p} ILTP Problem SYJ107+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ a_3 )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ b_2 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ b_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a ) )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO124$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \mbox{cP}\mbox{ and }( \mbox{cR}\mbox{ or }\mbox{cQ} ) ) \ \Rightarrow \ ( ( \mbox{cP}\mbox{ and }\mbox{cR} )\mbox{ or }\mbox{cQ} )\qquad\mbox{thf}(\mbox{cDISJ\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO158$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i},u{:}\;\mbox{\$i},v{:}\;\mbox{\$i},\mbox{vV}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ x@ y@ u\mbox{ and }\mbox{cQ}@ y@ z@ v ) \ \Rightarrow \ ( ( \mbox{cQ}@ x@ v@ \mbox{vV} )\iff ( \mbox{cQ}@ u@ z@ \mbox{vV} ) ) )\qquad\mbox{thf}(\mbox{cHYP}_1,\mbox{conjecture})$

\medskip\textbf{SYO204$\wedge$5.p} TPS problem CT313






$\exists \mbox{xz}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$o}{:}\;\forall \mbox{xy}{:}\;\mbox{\$o}{:}\;( \mbox{xx} \ \Rightarrow \ \mbox{xy} )\qquad\mbox{thf}(\mbox{cCT}_{313},\mbox{conjecture})$

\medskip\textbf{SYO459$\wedge$1.p} Ted Sider's propositional modal logic wff 10






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) )@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO467$\wedge$1.p} Ted Sider's propositional modal logic wff 18






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_k}@ ( \mbox{mbox\_k}@ p ) )@ ( \mbox{mbox\_k}@ ( \mbox{mdia\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO151$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \exists g{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ g@ x@ y )\mbox{ and }\exists j{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ x@ j@ y )\mbox{ and }\exists f{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ x@ y@ f ) )\qquad\mbox{thf}(\mbox{cHYP}_2,\mbox{conjecture})$

\medskip\textbf{SYO479$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_5@ ( f@ x ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO408$\wedge$1.p} Ted Sider's modal proposition logic theorem 17






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mbox\_k}@ q ) )@ ( \mbox{mbox\_k}@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO316$\wedge$5.p} TPS problem from BASIC-HO-THMS






$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE},\mbox{type})$

$\mbox{cHALF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHALF},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( ( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} )\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ c_0@ c_0@ \mbox{xz} ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xu}@ \mbox{xv}@ \mbox{xz} ) ) ) \ \Rightarrow \ \neg\,\mbox{cDOUBLE}@ c_0@ c_0 )\qquad\mbox{thf}(\mbox{cTHM300\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO037$\wedge$1.p} Injective Cantor theorem






$\neg\,\exists h{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$o},q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( h@ p ) = ( h@ q ) \ \Rightarrow \ p = q )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO038$\wedge$1.003.003.p} Boolos' Curious Inference, size f(3,3)






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(s,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$d{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$\forall n{:}\;\mbox{\$i}{:}\;( f@ n@ 1 ) = ( s@ 1 )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( f@ 1@ ( s@ x ) ) = ( s@ ( s@ ( f@ 1@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\forall n{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( f@ ( s@ n )@ ( s@ x ) ) = ( f@ n@ ( f@ ( s@ n )@ x ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$d@ 1\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( d@ x ) \ \Rightarrow \ ( d@ ( s@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$d@ ( f@ ( s@ ( s@ 1 ) )@ ( s@ ( s@ 1 ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO308$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cB}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall z_1{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \forall z_3{:}\;\mbox{\$i}{:}\;( ( z_1@ z_3 ) \ \Rightarrow \ ( \mbox{cA}@ z_3 ) )\mbox{ or }\forall z_4{:}\;\mbox{\$i}{:}\;( ( z_1@ z_4 ) \ \Rightarrow \ ( \mbox{cB}@ z_4 ) ) ) \ \Rightarrow \ \forall z_5{:}\;\mbox{\$i}{:}\;( ( z_1@ z_5 ) \ \Rightarrow \ ( \mbox{cA}@ z_5\mbox{ or }\mbox{cB}@ z_5 ) ) )\qquad\mbox{thf}(\mbox{cSET81\_pme},\mbox{conjecture})$

\medskip\textbf{SYO329$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cDOUBLE}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cDOUBLE},\mbox{type})$

$\mbox{cHALF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHALF},\mbox{type})$

$c_0{:}\;\mbox{\$i}\qquad\mbox{thf}(c_0,\mbox{type})$

$( \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ \forall q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ c_0@ c_0\mbox{ and }q@ ( \mbox{cS}@ c_0 )@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( q@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( q@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xx} ) )@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( q@ \mbox{xu}@ \mbox{xv} ) ) )\mbox{ and }\mbox{cDOUBLE}@ c_0@ c_0\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cDOUBLE}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xx} )@ ( \mbox{cS}@ ( \mbox{cS}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cHALF}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cDOUBLE}@ \mbox{xv}@ \mbox{xu}\mbox{ or }\mbox{cDOUBLE}@ ( \mbox{cS}@ \mbox{xv} )@ ( \mbox{cS}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cTHM300B},\mbox{conjecture})$

\medskip\textbf{SYO052$\wedge$1.p} Simple textbook example 9






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ a@ b ) )@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO406$\wedge$1.p} Ted Sider's modal proposition logic theorem 15






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mor}@ p@ q ) )@ ( \mbox{mor}@ ( \mbox{mbox\_k}@ p )@ ( \mbox{mdia\_k}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO576$\wedge$7.p} Mixed Modal Propositional Logic WFFs. problem 7






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) )@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO132$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cX}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cX},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$\mbox{cP}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \mbox{cP}@ \mbox{cS}\mbox{ and }\mbox{cT}@ \mbox{cX}\mbox{ and }\mbox{cS} = \mbox{cT} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cT}\mbox{ and }\mbox{cS}@ \mbox{cX} )\qquad\mbox{thf}(\mbox{cTHM}_{503},\mbox{conjecture})$

\medskip\textbf{SYO119$\wedge$5.p} TPS problem from BASIC-FO-THMS






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ a )\qquad\mbox{thf}(\mbox{cEXX}_1,\mbox{conjecture})$

\medskip\textbf{SYO386$\wedge$5.p} TPS problem THM409-3






$\mbox{cQ}_4{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_4,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cP}_3{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_3,\mbox{type})$

$\mbox{cQ}_3{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_3,\mbox{type})$

$d{:}\;\mbox{\$i}\qquad\mbox{thf}(d,\mbox{type})$

$c{:}\;\mbox{\$i}\qquad\mbox{thf}(c,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cP}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

$\mbox{cQ}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_2,\mbox{type})$

$\mbox{cP}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_1,\mbox{type})$

$\mbox{cQ}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ}_1,\mbox{type})$

$\neg\,\mbox{cQ}_1@ a@ b@ c\mbox{ and }\mbox{cP}_1@ a@ a\mbox{ and }\mbox{cP}_1@ b@ b\mbox{ and }\mbox{cP}_1@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_1@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_1@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_1@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_1@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_2@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_2@ a@ a\mbox{ and }\mbox{cP}_2@ b@ b\mbox{ and }\mbox{cP}_2@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_2@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_2@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_2@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_2@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_3@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\mbox{cP}_3@ a@ a\mbox{ and }\mbox{cP}_3@ b@ b\mbox{ and }\mbox{cP}_3@ c@ c\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cP}_3@ d@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cP}_3@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{cP}_3@ ( f@ \mbox{xx} )@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cQ}_3@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xx} )@ \mbox{xu}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xy} )@ \mbox{xv}\mbox{ or }\neg\,\mbox{cP}_3@ ( f@ \mbox{xz} )@ \mbox{xw}\mbox{ or }\mbox{cQ}_4@ \mbox{xu}@ \mbox{xv}@ \mbox{xw} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cQ}_4@ \mbox{xx}@ \mbox{xy}@ \mbox{xz}\qquad\mbox{thf}(\mbox{cTHM409}_3,\mbox{conjecture})$

\medskip\textbf{SYO157$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) )\mbox{ and }( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cQ}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cPELL26}_1,\mbox{conjecture})$

\medskip\textbf{SYO278$\wedge$5.p} TPS problem from BASIC-HO-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$q{:}\;\mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$r{:}\;( ( a\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$\forall \mbox{xp}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( r@ \mbox{xp} ) \ \Rightarrow \ q\qquad\mbox{thf}(\mbox{cPRIMQWFF},\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$5.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mdia\_s}_4@ p ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mdia\_s}_4@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO108$\wedge$5.p} TPS problem THM79






$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\forall r{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ r )\mbox{ or }\exists x{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cG}@ x\mbox{ or }\neg\,\forall y{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ y\mbox{ or }\exists s{:}\;\mbox{\$i}{:}\;( \mbox{cN}@ s ) )\mbox{ or }\neg\,\forall z{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{cN}@ z\mbox{ or }\neg\,\forall t{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ t ) )\qquad\mbox{thf}(\mbox{cTHM}_{79},\mbox{conjecture})$

\medskip\textbf{SYO156$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$( \neg\,\mbox{cP}\mbox{ or }\mbox{cQ} )\mbox{ and }( \mbox{cP}\mbox{ or }\neg\,\mbox{cQ} )\mbox{ and }( \neg\,\mbox{cQ}\mbox{ or }\mbox{cR} )\mbox{ and }( \mbox{cQ}\mbox{ or }\neg\,\mbox{cR} )\mbox{ and }( \neg\,\mbox{cP}\mbox{ or }\neg\,\mbox{cR} )\mbox{ and }( \mbox{cP}\mbox{ or }\mbox{cR} )\qquad\mbox{thf}(\mbox{cCNF\_NTRIV}_4,\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$4.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ ( \mbox{mdia\_b}@ p )@ ( \mbox{mdia\_b}@ q ) ) )@ ( \mbox{mbox\_b}@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_b}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO328$\wedge$5.p} TPS problem from BASIC-HO-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cG2}_0{:}\;a\to a\qquad\mbox{thf}(\mbox{cG2}_0,\mbox{type})$

$\mbox{cG1}_0{:}\;a\to a\qquad\mbox{thf}(\mbox{cG1}_0,\mbox{type})$

$\mbox{cP}_0{:}\;( a\to a )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_0,\mbox{type})$

$j_7{:}\;a\to a\qquad\mbox{thf}(j_7,\mbox{type})$

$\mbox{cF}_0{:}\;a\to a\qquad\mbox{thf}(\mbox{cF}_0,\mbox{type})$

$j_6{:}\;a\to a\qquad\mbox{thf}(j_6,\mbox{type})$

$p_6{:}\;( a\to a )\to \mbox{\$o}\qquad\mbox{thf}(p_6,\mbox{type})$

$\mbox{cJ}_1{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ}_1,\mbox{type})$

$p_4{:}\;( a\to a )\to \mbox{\$o}\qquad\mbox{thf}(p_4,\mbox{type})$

$\mbox{cJ}_0{:}\;a\to a\qquad\mbox{thf}(\mbox{cJ}_0,\mbox{type})$

$( ( ( p_4@ \lambda \mbox{xu}_3{:}\;a{:}\;\mbox{xu}_3\mbox{ and }( ( p_4@ \mbox{cJ}_0 ) \ \Rightarrow \ ( p_4@ \lambda \mbox{xx}_4{:}\;a{:}\;( \mbox{cF}_0@ ( \mbox{cJ}_0@ \mbox{xx}_4 ) ) ) ) ) \ \Rightarrow \ ( p_4@ \mbox{cG1}_0 ) )\mbox{ and }( ( p_6@ \lambda \mbox{xu}_4{:}\;a{:}\;\mbox{xu}_4\mbox{ and }( ( p_6@ \mbox{cJ}_1 ) \ \Rightarrow \ ( p_6@ \lambda \mbox{xx}_5{:}\;a{:}\;( \mbox{cF}_0@ ( \mbox{cJ}_1@ \mbox{xx}_5 ) ) ) ) ) \ \Rightarrow \ ( p_6@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{cG2}_0@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( ( \mbox{cP}_0@ \lambda \mbox{xu}_5{:}\;a{:}\;\mbox{xu}_5\mbox{ and }( ( \mbox{cP}_0@ j_6 ) \ \Rightarrow \ ( \mbox{cP}_0@ \lambda \mbox{xx}_7{:}\;a{:}\;( \mbox{cF}_0@ ( j_6@ \mbox{xx}_7 ) ) ) )\mbox{ and }( ( \mbox{cP}_0@ j_7 ) \ \Rightarrow \ ( \mbox{cP}_0@ \lambda \mbox{xx}_7{:}\;a{:}\;( \mbox{cF}_0@ ( j_7@ \mbox{xx}_7 ) ) ) ) ) \ \Rightarrow \ ( \mbox{cP}_0@ \lambda \mbox{xx}_6{:}\;a{:}\;( \mbox{cG1}_0@ ( \mbox{cG2}_0@ \mbox{xx}_6 ) ) ) )\qquad\mbox{thf}(\mbox{cTHM135A\_EXP},\mbox{conjecture})$

\medskip\textbf{SYO485$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 11






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mdia\_s}_5@ ( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{meq\_ind}@ x@ y ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO453$\wedge$2.p} Ted Sider's propositional modal logic wff 04






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ ( \mbox{mdia\_d}@ p )@ ( \mbox{mdia\_d}@ q ) ) )@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ ( \mbox{mbox\_d}@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO085$\wedge$5.p} TPS problem COM-DMG02

A propositional problem suggested by Jay Hunschel Kim.




$d{:}\;\mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$c{:}\;\mbox{\$o}\qquad\mbox{thf}(c,\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$s{:}\;\mbox{\$o}\qquad\mbox{thf}(s,\mbox{type})$

$( \neg\,c\mbox{ and }d\mbox{ or }\neg\,a\mbox{ and }b\mbox{ or }s ) \ \Rightarrow \ ( s\mbox{ or }\neg\,a\mbox{ and }b\mbox{ or }\neg\,c\mbox{ and }d )\qquad\mbox{thf}(\mbox{cCOM\_DMG}_{02},\mbox{conjecture})$

\medskip\textbf{SYO574$\wedge$7.p} Modal Propositional Logic Theorems. problem 37






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_s}_4@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_s}_4@ q ) ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ p )@ ( \mbox{mdia\_s}_4@ q ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO458$\wedge$2.p} Ted Sider's propositional modal logic wff 09






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ q ) )@ ( \mbox{mbox\_d}@ ( \mbox{mequiv}@ p@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO317$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cEQ}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cEQ},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cEQ}@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEQ}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cP}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy} ) )\mbox{ and }\mbox{cP}@ a\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cEQ}@ ( f@ \mbox{xy} )@ ( f@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{cEQ}@ \mbox{xy}@ \mbox{xz} ) ) ) \ \Rightarrow \ \exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ ( f@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cTHM}_{304},\mbox{conjecture})$

\medskip\textbf{SYO420$\wedge$1.p} Ted Sider's modal proposition logic theorem 29






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_d}@ p )@ ( \mbox{mbox\_d}@ ( \mbox{mimplies}@ p@ q ) ) )@ ( \mbox{mdia\_d}@ q ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO324$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\forall \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( \exists \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xq}_{31}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xq}_{31}@ ( \mbox{xh}@ \mbox{xp} )\mbox{ or }\mbox{xq}_{31}@ ( \mbox{xh}@ \mbox{xq} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,( \mbox{xp}@ \mbox{xx} )\iff ( \mbox{xq}@ \mbox{xx} ) )\mbox{ or }\forall \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xt}@ ( \mbox{xh}@ \mbox{xt} )\mbox{ or }\exists \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{xq}@ ( \mbox{xh}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xt}_0{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}_0@ ( \mbox{xh}@ \mbox{xt}_0 )\mbox{ and }\forall \mbox{xq}_0{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xq}_0@ \mbox{xz}\mbox{ or }\mbox{xq}_0@ ( \mbox{xh}@ \mbox{xt}_0 ) ) ) )\mbox{ and }\neg\,\mbox{xq}@ ( \mbox{xh}@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM143\_EXPANDB},\mbox{conjecture})$

\medskip\textbf{SYO294$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cN}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cS}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$c_0{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(c_0,\mbox{type})$

$\forall p{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ c_0\mbox{ and }\forall x{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{cS}@ x ) ) ) ) \ \Rightarrow \ ( p@ \mbox{cN} ) )\qquad\mbox{thf}(\mbox{cNATN}_1,\mbox{conjecture})$

\medskip\textbf{SYO232$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cP}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$x{:}\;a\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cB}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$y{:}\;\mbox{\$i}\qquad\mbox{thf}(y,\mbox{type})$

$( y = y \ \Rightarrow \ \mbox{cB} ) \ \Rightarrow \ ( ( \mbox{cP}@ x ) \ \Rightarrow \ \exists \mbox{xx}_0{:}\;a{:}\;( \mbox{cP}@ \mbox{xx}_0 ) )\qquad\mbox{thf}(\mbox{cADDHYP}_4,\mbox{conjecture})$

\medskip\textbf{SYO486$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO491$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 17






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$g{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( f@ x )@ ( g@ x ) ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( f@ x )@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( g@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO332$\wedge$5.p} TPS problem from BASIC-HO-THMS






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$( \forall \mbox{xs}{:}\;( b\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xy}{:}\;b{:}\;( x@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall x{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) ) \ \Rightarrow \ \forall \mbox{xr}{:}\;a\to b\to \mbox{\$o}{:}\;\exists \mbox{xg}{:}\;a\to b{:}\;\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xy}{:}\;b{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ ( \mbox{xg}@ \mbox{xx} ) ) ) )\mbox{ and }( \forall \mbox{xr}{:}\;( a\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\exists \mbox{xg}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xy}{:}\;a{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ ( \mbox{xg}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xs}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{561},\mbox{conjecture})$

\medskip\textbf{SYO064$\wedge$4.002.p} ILTP Problem SYJ107+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ a_2 )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b )@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ b_1 ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{ior}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a ) )@ ( \mbox{iatom}@ b ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ a )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO179$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cE}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cD}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cD},\mbox{type})$

$\mbox{cC}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$\mbox{cB}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$( ( \mbox{cR}@ \mbox{cA}@ \mbox{cB}\mbox{ or }\mbox{cG}@ \mbox{cA}@ \mbox{cB} )\mbox{ and }( \mbox{cR}@ \mbox{cA}@ \mbox{cC}\mbox{ or }\mbox{cG}@ \mbox{cA}@ \mbox{cC} )\mbox{ and }( \mbox{cR}@ \mbox{cA}@ \mbox{cD}\mbox{ or }\mbox{cG}@ \mbox{cA}@ \mbox{cD} )\mbox{ and }( \mbox{cR}@ \mbox{cA}@ \mbox{cE}\mbox{ or }\mbox{cG}@ \mbox{cA}@ \mbox{cE} )\mbox{ and }( \mbox{cR}@ \mbox{cA}@ \mbox{cF}\mbox{ or }\mbox{cG}@ \mbox{cA}@ \mbox{cF} )\mbox{ and }( \mbox{cR}@ \mbox{cB}@ \mbox{cC}\mbox{ or }\mbox{cG}@ \mbox{cB}@ \mbox{cC} )\mbox{ and }( \mbox{cR}@ \mbox{cB}@ \mbox{cD}\mbox{ or }\mbox{cG}@ \mbox{cB}@ \mbox{cD} )\mbox{ and }( \mbox{cR}@ \mbox{cB}@ \mbox{cE}\mbox{ or }\mbox{cG}@ \mbox{cB}@ \mbox{cE} )\mbox{ and }( \mbox{cR}@ \mbox{cB}@ \mbox{cF}\mbox{ or }\mbox{cG}@ \mbox{cB}@ \mbox{cF} )\mbox{ and }( \mbox{cR}@ \mbox{cC}@ \mbox{cD}\mbox{ or }\mbox{cG}@ \mbox{cC}@ \mbox{cD} )\mbox{ and }( \mbox{cR}@ \mbox{cC}@ \mbox{cE}\mbox{ or }\mbox{cG}@ \mbox{cC}@ \mbox{cE} )\mbox{ and }( \mbox{cR}@ \mbox{cC}@ \mbox{cF}\mbox{ or }\mbox{cG}@ \mbox{cC}@ \mbox{cF} )\mbox{ and }( \mbox{cR}@ \mbox{cD}@ \mbox{cE}\mbox{ or }\mbox{cG}@ \mbox{cD}@ \mbox{cE} )\mbox{ and }( \mbox{cR}@ \mbox{cD}@ \mbox{cF}\mbox{ or }\mbox{cG}@ \mbox{cD}@ \mbox{cF} )\mbox{ and }( \mbox{cR}@ \mbox{cE}@ \mbox{cF}\mbox{ or }\mbox{cG}@ \mbox{cE}@ \mbox{cF} ) ) \ \Rightarrow \ ( \exists \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i}{:}\;( \mbox{cR}@ \mbox{xa}@ \mbox{xb}\mbox{ and }\mbox{cR}@ \mbox{xa}@ \mbox{xc}\mbox{ and }\mbox{cR}@ \mbox{xb}@ \mbox{xc} )\mbox{ or }\exists \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ \mbox{xa}@ \mbox{xb}\mbox{ and }\mbox{cG}@ \mbox{xa}@ \mbox{xc}\mbox{ and }\mbox{cG}@ \mbox{xb}@ \mbox{xc} ) )\qquad\mbox{thf}(\mbox{cSIX\_THEOREM\_B},\mbox{conjecture})$

\medskip\textbf{SYO241$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\forall \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xh}@ \mbox{xp} ) = ( \mbox{xh}@ \mbox{xq} ) \ \Rightarrow \ \mbox{xp} = \mbox{xq} ) \ \Rightarrow \ \neg\,\exists \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}@ ( \mbox{xh}@ \mbox{xt} )\mbox{ and }( \mbox{xh}@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xt}_0{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}_0@ ( \mbox{xh}@ \mbox{xt}_0 )\mbox{ and }\mbox{xz} = ( \mbox{xh}@ \mbox{xt}_0 ) ) ) = ( \mbox{xh}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cTHM143\_EXPAND},\mbox{conjecture})$

\medskip\textbf{SYO396$\wedge$1.p} Ted Sider's modal proposition logic theorem 05






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mnot}@ ( \mbox{mdia\_k}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO092$\wedge$5.p} TPS problem Y2141






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}\mbox{ and }( \mbox{cQ}@ \mbox{xy}\mbox{ or }\mbox{cQ}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}\mbox{ and }\mbox{cQ}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cY}_{2141},\mbox{conjecture})$

\medskip\textbf{SYO012$\wedge$1.p} Formula valid with Boolean extentionality 1






$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$p{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$( p@ ( a\mbox{ and }b ) ) \ \Rightarrow \ ( p@ ( b\mbox{ and }a ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$3.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_m}@ p ) ) )@ ( \mbox{mbox\_m}@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_m}@ ( \mbox{mbox\_m}@ p ) )@ ( \mbox{mbox\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO374$\wedge$5.p} TPS problem from EXTENSIONALITY






$\mbox{cN}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cM}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ ( \mbox{cM}@ \mbox{xx} ) )\iff ( \mbox{xp}@ ( \mbox{cN}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \mbox{cM} )\iff ( \mbox{xq}@ \mbox{cN} ) )\qquad\mbox{thf}(\mbox{cEXT\_A\_LEIB}_2,\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.002.p} Two function variant of the Kaminski equation






$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$( f_0@ ( f_0@ ( f_0@ ( f_1@ x ) ) ) ) = ( f_0@ ( f_1@ ( f_1@ ( f_1@ x ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_2,\mbox{conjecture})$

\medskip\textbf{SYO247$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\mbox{cP}_6{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_6,\mbox{type})$

$\mbox{cP}_5{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_5,\mbox{type})$

$\mbox{cP}_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_4,\mbox{type})$

$\mbox{cP}_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_3,\mbox{type})$

$\mbox{cP}_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_2,\mbox{type})$

$\mbox{cP}_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cP}_1,\mbox{type})$

$\forall k{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},s{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( s@ \mbox{cP}_1\mbox{ and }s@ \mbox{cP}_2\mbox{ and }s@ \mbox{cP}_3\mbox{ and }s@ \mbox{cP}_4\mbox{ and }s@ \mbox{cP}_5\mbox{ and }s@ \mbox{cP}_6\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( k@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( k@ \mbox{xy}@ \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( s@ \mbox{xx}\mbox{ and }s@ \mbox{xy}\mbox{ and }s@ \mbox{xz}\mbox{ and }\mbox{xx}\ne \mbox{xy}\mbox{ and }\mbox{xy}\ne \mbox{xz}\mbox{ and }\mbox{xz}\ne \mbox{xx}\mbox{ and }( ( k@ \mbox{xx}@ \mbox{xy}\mbox{ and }k@ \mbox{xy}@ \mbox{xz}\mbox{ and }k@ \mbox{xx}@ \mbox{xz} )\mbox{ or }( \neg\,k@ \mbox{xx}@ \mbox{xy}\mbox{ and }\neg\,k@ \mbox{xy}@ \mbox{xz}\mbox{ and }\neg\,k@ \mbox{xx}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cSIXPEOPLE}_2,\mbox{conjecture})$

\medskip\textbf{SYO305$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cB}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall z_1{:}\;\mbox{\$i}{:}\;( ( \mbox{cA}@ z_1 ) \ \Rightarrow \ ( \mbox{cB}@ z_1 ) ) \ \Rightarrow \ \forall z_1{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall z_3{:}\;\mbox{\$i}{:}\;( ( z_1@ z_3 ) \ \Rightarrow \ ( \mbox{cA}@ z_3 ) ) \ \Rightarrow \ \forall z_3{:}\;\mbox{\$i}{:}\;( ( z_1@ z_3 ) \ \Rightarrow \ ( \mbox{cB}@ z_3 ) ) )\qquad\mbox{thf}(\mbox{cSET79\_pme},\mbox{conjecture})$

\medskip\textbf{SYO284$\wedge$5.p} TPS problem from BASIC-HO-THMS






$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( r@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTRIV}_1,\mbox{conjecture})$

\medskip\textbf{SYO471$\wedge$2.p} Ted Sider's propositional modal logic wff 22






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^2.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_d}@ ( \mbox{mdia\_d}@ p ) )@ ( \mbox{mbox\_d}@ ( \mbox{mdia\_d}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO476$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO135$\wedge$5.p} TPS problem from BASIC-FO-THMS






$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$( \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0 )\mbox{ or }\mbox{cQ}@ x ) \ \Rightarrow \ ( ( \mbox{cP}@ a\mbox{ and }\mbox{cP}@ b )\mbox{ or }\mbox{cQ}@ x )\qquad\mbox{thf}(\mbox{cDUP\_BUG},\mbox{conjecture})$

\medskip\textbf{SYO475$\wedge$6.p} Ted Sider's S5 quantified modal logic wff 01






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$f{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_5@ ( f@ x ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO240$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\mbox{cK}{:}\;( a\to b\to \mbox{\$o} )\to a\to b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cK},\mbox{type})$

$\forall \mbox{xu}{:}\;a\to b\to \mbox{\$o},\mbox{xv}{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;b{:}\;( ( \mbox{xu}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xv}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;b{:}\;( ( \mbox{cK}@ \mbox{xu}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cK}@ \mbox{xv}@ \mbox{xx}@ \mbox{xy} ) ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;a\to b\to \mbox{\$o}{:}\;( \mbox{cK}@ \mbox{xu} ) = \mbox{xu}\qquad\mbox{thf}(\mbox{cTHM2B},\mbox{conjecture})$

\medskip\textbf{SYO447$\wedge$1.p} Ted Sider's modal proposition logic theorem 56






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mand}@ ( \mbox{mdia\_s}_5@ p )@ ( \mbox{mbox\_s}_5@ q ) )@ ( \mbox{mdia\_s}_5@ ( \mbox{mand}@ p@ ( \mbox{mbox\_s}_5@ q ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO067$\wedge$4.004.p} ILTP Problem SYJ203+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$p_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p1\_type},\mbox{type})$

$p_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p2\_type},\mbox{type})$

$p_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p3\_type},\mbox{type})$

$p_4{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p4\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iand}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ p_4 ) ) ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_1 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_2 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{ior}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_3 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p_4 )@ ( \mbox{iatom}@ f ) ) ) ) ) )@ ( \mbox{iatom}@ f ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iatom}@ f )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO055$\wedge$2.p} Simple textbook example 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mforall\_prop}@ \lambda b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mor}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ b ) )@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ b )@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO538$\wedge$1.p} If-then-else on \$i defined from choice on \$i

A choice operator on \$i is used to define an if-then-else operator
 at \$i. Check that it works.




$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{choiceax},\mbox{axiom})$

$\mbox{if}{:}\;\mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if},\mbox{type})$

$\mbox{if} = ( \lambda b{:}\;\mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{eps}@ \lambda z{:}\;\mbox{\$i}{:}\;( ( b\mbox{ and }z = x )\mbox{ or }( \neg\,b\mbox{ and }z = y ) ) ) )\qquad\mbox{thf}(\mbox{ifd},\mbox{definition})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{if}@ \mbox{\$true}@ x@ y ) = x\mbox{ and }( \mbox{if}@ \mbox{\$false}@ x@ y ) = y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO180$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\forall l{:}\;\mbox{\$o},p{:}\;\mbox{\$o},m{:}\;\mbox{\$o},g{:}\;\mbox{\$o},r{:}\;\mbox{\$o},e{:}\;\mbox{\$o},n{:}\;\mbox{\$o},f{:}\;\mbox{\$o},k{:}\;\mbox{\$o},b{:}\;\mbox{\$o},c{:}\;\mbox{\$o}{:}\;( ( ( ( l\mbox{ and }p ) \ \Rightarrow \ m )\mbox{ and }( ( g\mbox{ and }\neg\,r ) \ \Rightarrow \ m )\mbox{ and }( ( \neg\,k\mbox{ and }n\mbox{ and }m ) \ \Rightarrow \ f )\mbox{ and }( ( \neg\,g\mbox{ and }\neg\,p ) \ \Rightarrow \ r )\mbox{ and }( ( k\mbox{ and }b ) \ \Rightarrow \ c )\mbox{ and }( ( r\mbox{ and }\neg\,n\mbox{ and }\neg\,f ) \ \Rightarrow \ p )\mbox{ and }( ( l\mbox{ and }m ) \ \Rightarrow \ c )\mbox{ and }( ( e\mbox{ and }\neg\,k\mbox{ and }g\mbox{ and }\neg\,n ) \ \Rightarrow \ \neg\,m )\mbox{ and }( ( \neg\,g\mbox{ and }\neg\,r ) \ \Rightarrow \ k )\mbox{ and }( ( k\mbox{ and }l\mbox{ and }e ) \ \Rightarrow \ \neg\,m )\mbox{ and }( ( r\mbox{ and }e ) \ \Rightarrow \ \neg\,c )\mbox{ and }( ( g\mbox{ and }\neg\,k\mbox{ and }\neg\,m ) \ \Rightarrow \ \neg\,b )\mbox{ and }( ( n\mbox{ and }\neg\,p\mbox{ and }\neg\,f ) \ \Rightarrow \ c )\mbox{ and }( ( g\mbox{ and }b\mbox{ and }\neg\,r ) \ \Rightarrow \ \neg\,c )\mbox{ and }( ( r\mbox{ and }\neg\,k\mbox{ and }\neg\,m ) \ \Rightarrow \ g ) ) \ \Rightarrow \ ( ( e\mbox{ and }l ) \ \Rightarrow \ ( f\mbox{ and }\neg\,b ) ) )\qquad\mbox{thf}(\mbox{cPORKCHOP}_2,\mbox{conjecture})$

\medskip\textbf{SYO295$\wedge$5.p} TPS problem from BASIC-HO-THMS






$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$y{:}\;\mbox{\$i}\qquad\mbox{thf}(y,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$x\ne y \ \Rightarrow \ \exists \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xf}@ x ) = a\mbox{ and }( \mbox{xf}@ y ) = b )\qquad\mbox{thf}(\mbox{cX}_{5311},\mbox{conjecture})$

\medskip\textbf{SYO512$\wedge$1.p} Choice operator used to obtain functions from total relations

A choice operator can be used to obtain functions from total
 relations. 




$\mbox{eps}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{eps},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( p@ x ) \ \Rightarrow \ ( p@ ( \mbox{eps}@ p ) ) )\qquad\mbox{thf}(\mbox{epschoice},\mbox{axiom})$

$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( r@ x@ y )\qquad\mbox{thf}(\mbox{rtotal},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( r@ x@ ( \mbox{eps}@ ( r@ x ) ) )\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO501$\wedge$1.p} An unsatisfiable normal set with embedded formulas






$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$y{:}\;\mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\neg\,p@ ( f@ x@ \neg\,\neg\,y )\mbox{ or }p@ ( f@ x@ y )\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO243$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ ( \mbox{xh}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = \mbox{xy} \ \Rightarrow \ ( \mbox{xh}@ \mbox{xy} ) = ( f@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM588LEM}_1,\mbox{conjecture})$

\medskip\textbf{SYO509$\wedge$1.p} Existence of choice functions for binary relations






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(b,\mbox{type})$

$\exists c_1{:}\;( a\to b\to \mbox{\$o} )\to a,c_2{:}\;( a\to b\to \mbox{\$o} )\to b{:}\;\forall r{:}\;a\to b\to \mbox{\$o}{:}\;( \exists x{:}\;a,y{:}\;b{:}\;( r@ x@ y ) \ \Rightarrow \ ( r@ ( c_1@ r )@ ( c_2@ r ) ) )\qquad\mbox{thf}(\mbox{claim},\mbox{conjecture})$

\medskip\textbf{SYO062$\wedge$4.004.p} ILTP Problem SYJ105+1.004






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ a )@ ( \mbox{iand}@ ( \mbox{iatom}@ b )@ ( \mbox{iatom}@ c ) ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ a ) )@ ( \mbox{ior}@ ( \mbox{inot}@ ( \mbox{iatom}@ b ) )@ ( \mbox{inot}@ ( \mbox{iatom}@ c ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO474$\wedge$4.p} Ted Sider's propositional modal logic wff 25






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mimplies}@ p@ ( \mbox{mbox\_b}@ p ) ) )@ ( \mbox{mbox\_b}@ p ) ) )@ ( \mbox{mimplies}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mbox\_b}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO210$\wedge$5.p} TPS problem from BASIC-HO-PROP-THMS






$\exists \mbox{xx}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \neg\,\mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_6,\mbox{conjecture})$

\medskip\textbf{SYO575$\wedge$7.p} Modal Propositional Logic Theorems. problem 50






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mdia\_s}_4@ ( \mbox{mbox\_s}_4@ p ) ) ) )@ ( \mbox{mbox\_s}_4@ p ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO144$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cC}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cR}_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_3,\mbox{type})$

$\mbox{cR}_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_2,\mbox{type})$

$\mbox{cR}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR}_1,\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}_0 ) ) \ \Rightarrow \ ( ( \mbox{cR}_1@ x\mbox{ or }\mbox{cR}_2@ x\mbox{ or }\mbox{cR}_3@ x ) \ \Rightarrow \ \mbox{cC} )\qquad\mbox{thf}(\mbox{cADDHYP}_7,\mbox{conjecture})$

\medskip\textbf{SYO072$\wedge$4.002.p} ILTP Problem SYJ208+1.002






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$o_{11}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o11\_type},\mbox{type})$

$o_{12}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o12\_type},\mbox{type})$

$o_{21}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o21\_type},\mbox{type})$

$o_{22}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o22\_type},\mbox{type})$

$o_{31}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o31\_type},\mbox{type})$

$o_{32}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{o32\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{12} ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{22} ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iatom}@ o_{31} )@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ o_{32} ) ) ) )\qquad\mbox{thf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{21} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{11} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{21} )@ ( \mbox{iatom}@ o_{31} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{22} ) )@ ( \mbox{ior}@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{12} )@ ( \mbox{iatom}@ o_{32} ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ o_{22} )@ ( \mbox{iatom}@ o_{32} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO430$\wedge$1.p} Ted Sider's modal proposition logic theorem 39






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) )@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ ( \mbox{mbox\_b}@ p ) ) ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO167$\wedge$5.p} TPS problem from BASIC-FO-THMS






$n{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(n,\mbox{type})$

$i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(i,\mbox{type})$

$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$z{:}\;\mbox{\$i}\qquad\mbox{thf}(z,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ \mbox{xx}@ ( i@ \mbox{xy}@ z ) )@ ( i@ ( i@ \mbox{xx}@ \mbox{xy} )@ ( i@ \mbox{xx}@ z ) ) )\mbox{ and }\mbox{cT}@ ( i@ ( i@ ( n@ \mbox{xx} )@ ( n@ \mbox{xy} ) )@ ( i@ \mbox{xy}@ \mbox{xx} ) )\mbox{ and }( ( \mbox{cT}@ ( i@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cT}@ \mbox{xy} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cT}@ ( i@ \mbox{xx}@ ( n@ ( n@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{66},\mbox{conjecture})$

\medskip\textbf{SYO304$\wedge$5.p} TPS problem from UNKNOWN






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xq}{:}\;( a\to a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xq}@ \lambda x{:}\;a,y{:}\;a{:}\;\forall \mbox{xq}_0{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xq}_0@ x ) \ \Rightarrow \ ( \mbox{xq}_0@ y ) ) ) \ \Rightarrow \ ( \mbox{xq}@ \lambda x{:}\;a,y{:}\;a{:}\;x = y ) )\qquad\mbox{thf}(\mbox{cE2\_eq\_\_pme},\mbox{conjecture})$

\medskip\textbf{SYO451$\wedge$4.p} Ted Sider's propositional modal logic wff 02






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^4.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ ( \mbox{mor}@ p@ ( \mbox{mdia\_b}@ q ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_b}@ p )@ ( \mbox{mdia\_b}@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO019$\wedge$1.p} De Morgan by equivalance






$\forall x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;( ( x\mbox{ and }y )\iff \neg\,\neg\,x\mbox{ or }\neg\,y )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO142$\wedge$5.p} TPS problem from BASIC-FO-THMS






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( ( p@ \mbox{xx} ) \ \Rightarrow \ ( p@ ( f@ ( f@ \mbox{xy} ) ) ) )\mbox{ and }( ( p@ \mbox{xy} ) \ \Rightarrow \ ( p@ ( f@ ( f@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{cTEST}_4,\mbox{conjecture})$

\medskip\textbf{SYO473$\wedge$6.p} Ted Sider's propositional modal logic wff 24






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ q ) )@ ( \mbox{mdia\_s}_5@ q ) ) )@ ( \mbox{mbox\_s}_5@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ ( \mbox{mequiv}@ p@ q ) )@ q ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO141$\wedge$5.p} TPS problem from BASIC-FO-THMS






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$x{:}\;\mbox{\$i}\qquad\mbox{thf}(x,\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( ( \mbox{cP}@ a ) \ \Rightarrow \ \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xx}_0 ) ) \ \Rightarrow \ ( \forall \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xx}_0 ) \ \Rightarrow \ ( ( \mbox{cR}@ x ) \ \Rightarrow \ ( \mbox{cQ}@ a ) ) )\qquad\mbox{thf}(\mbox{cADDHYP}_2,\mbox{conjecture})$

\medskip\textbf{SYO500$\wedge$1.003.p} Three function variant of the Kaminski equation






$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$f_0{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_0,\mbox{type})$

$f_1{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_1,\mbox{type})$

$f_2{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f_2,\mbox{type})$

$( f_0@ ( f_1@ ( f_1@ ( f_1@ ( f_2@ x ) ) ) ) ) = ( f_0@ ( f_0@ ( f_0@ ( f_1@ ( f_2@ ( f_2@ ( f_2@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kaminski}_3,\mbox{conjecture})$

\medskip\textbf{SYO039$\wedge$1.p} Unsatisfiable basic formula 1






$h{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$( h@ ( h@ \mbox{\$false} ) = ( h@ \neg\,\mbox{\$false} ) )\ne ( h@ \mbox{\$false} )\qquad\mbox{thf}(1,\mbox{axiom})$

\medskip\textbf{SYO461$\wedge$1.p} Ted Sider's propositional modal logic wff 12






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mdia\_k}@ ( \mbox{mor}@ p@ ( \mbox{mnot}@ p ) ) )@ ( \mbox{mand}@ p@ ( \mbox{mnot}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO198$\wedge$5.p} TPS problem CT8






$\forall \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\exists \mbox{xp}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cCT}_8,\mbox{conjecture})$

\medskip\textbf{SYO249$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\mbox{cBIGPHI}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cBIGPHI},\mbox{type})$

$\mbox{cHOM\_FROM\_HH}_1{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHOM\_FROM\_HH}_1,\mbox{type})$

$\mbox{cPHI}_2{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cPHI}_2,\mbox{type})$

$\mbox{cHOM\_FROM\_SS\_PRIME}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHOM\_FROM\_SS\_PRIME},\mbox{type})$

$\mbox{cPHI}_1{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cPHI}_1,\mbox{type})$

$\mbox{cSS\_PRIME}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cSS\_PRIME},\mbox{type})$

$\mbox{el}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{el}_1,\mbox{type})$

$\mbox{cHH}_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHH}_1,\mbox{type})$

$\mbox{cBIGHI}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cBIGHI},\mbox{type})$

$\mbox{cHH}_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cHH}_2,\mbox{type})$

$\mbox{cTIMES}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cTIMES},\mbox{type})$

$( \forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}_1@ \mbox{cHH}_2\mbox{ and }\mbox{el}_1@ \mbox{xx}_2@ \mbox{cHH}_2\mbox{ and }\mbox{el}_1@ \mbox{xy}_1@ \mbox{cHH}_2\mbox{ and }\mbox{el}_1@ \mbox{xy}_2@ \mbox{cHH}_2\mbox{ and }\mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) \ \Rightarrow \ ( \mbox{cTIMES}@ \mbox{xx}_1@ \mbox{xy}_1 ) = ( \mbox{cTIMES}@ \mbox{xx}_2@ \mbox{xy}_2 ) )\mbox{ and }\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}_1@ \mbox{cSS\_PRIME}\mbox{ and }\mbox{el}_1@ \mbox{xx}_2@ \mbox{cSS\_PRIME} ) \ \Rightarrow \ ( \mbox{el}_1@ ( \mbox{cTIMES}@ \mbox{xx}_1@ \mbox{xx}_2 )@ \mbox{cSS\_PRIME} ) )\mbox{ and }\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}_1@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xx}_2@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xy}_1@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xy}_2@ \mbox{cHH}_1\mbox{ and }\mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) \ \Rightarrow \ ( \mbox{cTIMES}@ \mbox{xx}_1@ \mbox{xy}_1 ) = ( \mbox{cTIMES}@ \mbox{xx}_2@ \mbox{xy}_2 ) )\mbox{ and }\forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{cHOM\_FROM\_SS\_PRIME}@ \mbox{xphi} )\iff \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cSS\_PRIME}\mbox{ and }\mbox{el}_1@ \mbox{xy}@ \mbox{cSS\_PRIME} ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{cTIMES}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{cTIMES}@ ( \mbox{xphi}@ \mbox{xx} )@ ( \mbox{xphi}@ \mbox{xy} ) ) ) )\mbox{ and }( ( \mbox{cHOM\_FROM\_HH}_1@ \mbox{cBIGPHI} )\iff \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xy}@ \mbox{cHH}_1 ) \ \Rightarrow \ ( \mbox{cBIGPHI}@ ( \mbox{cTIMES}@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{cTIMES}@ ( \mbox{cBIGPHI}@ \mbox{xx} )@ ( \mbox{cBIGPHI}@ \mbox{xy} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xy}@ \mbox{cHH}_1 ) \ \Rightarrow \ ( \mbox{el}_1@ ( \mbox{cTIMES}@ \mbox{xx}@ \mbox{xy} )@ \mbox{cHH}_1 ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cSS\_PRIME} ) \ \Rightarrow \ ( \mbox{el}_1@ ( \mbox{cPHI}_2@ \mbox{xx} )@ \mbox{cHH}_2 ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cHH}_1 ) \ \Rightarrow \ ( \mbox{el}_1@ ( \mbox{cBIGPHI}@ \mbox{xx} )@ \mbox{cHH}_2 ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cSS\_PRIME} ) \ \Rightarrow \ ( \mbox{el}_1@ ( \mbox{cPHI}_1@ \mbox{xx} )@ \mbox{cHH}_1 ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cHH}_1\mbox{ and }\mbox{el}_1@ \mbox{xy}@ \mbox{cHH}_1\mbox{ and }\mbox{xx} = \mbox{xy} ) \ \Rightarrow \ ( \mbox{cBIGHI}@ \mbox{xx} ) = ( \mbox{cBIGPHI}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xx}@ \mbox{cSS\_PRIME} ) \ \Rightarrow \ ( \mbox{cBIGPHI}@ ( \mbox{cPHI}_1@ \mbox{xx} ) ) = ( \mbox{cPHI}_2@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{el}_1@ \mbox{xy}@ \mbox{cHH}_1 ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{el}_1@ \mbox{xx}@ \mbox{cSS\_PRIME}\mbox{ and }( \mbox{cPHI}_1@ \mbox{xx} ) = \mbox{xy} ) )\mbox{ and }\mbox{cHOM\_FROM\_SS\_PRIME}@ \mbox{cPHI}_1\mbox{ and }\mbox{cHOM\_FROM\_SS\_PRIME}@ \mbox{cPHI}_2 ) \ \Rightarrow \ ( \mbox{cHOM\_FROM\_HH}_1@ \mbox{cBIGPHI} )\qquad\mbox{thf}(\mbox{cPROB}_{757},\mbox{conjecture})$

\medskip\textbf{SYO038$\wedge$1.004.004.p} Boolos' Curious Inference, size f(4,4)






$1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{one},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(s,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$d{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$\forall n{:}\;\mbox{\$i}{:}\;( f@ n@ 1 ) = ( s@ 1 )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( f@ 1@ ( s@ x ) ) = ( s@ ( s@ ( f@ 1@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\forall n{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( f@ ( s@ n )@ ( s@ x ) ) = ( f@ n@ ( f@ ( s@ n )@ x ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$d@ 1\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( d@ x ) \ \Rightarrow \ ( d@ ( s@ x ) ) )\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$d@ ( f@ ( s@ ( s@ ( s@ 1 ) ) )@ ( s@ ( s@ ( s@ 1 ) ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO348$\wedge$5.p} TPS problem E1EXT

Example from [Ben99] p.115.




$b{:}\;\mbox{\$o}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$( a\iff b ) \ \Rightarrow \ \forall p{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( p@ a ) \ \Rightarrow \ ( p@ b ) )\qquad\mbox{thf}(\mbox{cE1EXT},\mbox{conjecture})$

\medskip\textbf{SYO069$\wedge$4.003.p} ILTP Problem SYJ205+1.003






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$a_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a0\_type},\mbox{type})$

$a_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a1\_type},\mbox{type})$

$a_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a2\_type},\mbox{type})$

$a_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a3\_type},\mbox{type})$

$b_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b0\_type},\mbox{type})$

$b_1{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b1\_type},\mbox{type})$

$b_2{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b2\_type},\mbox{type})$

$b_3{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b3\_type},\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{f\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) ) ) ) ) )@ ( \mbox{iatom}@ f ) )@ ( \mbox{iimplies}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_2 )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_1 )@ ( \mbox{iatom}@ a_2 ) )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_0 )@ ( \mbox{iatom}@ a_1 ) )@ ( \mbox{iatom}@ a_0 ) )@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ b_3 )@ ( \mbox{iatom}@ b_0 ) )@ ( \mbox{iatom}@ a_3 ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ a_0 )@ ( \mbox{iatom}@ f ) ) ) ) ) )@ ( \mbox{iatom}@ f ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{SYO372$\wedge$5.p} TPS problem from EXTENSIONALITY






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\mbox{gtype}{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$g{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$h{:}\;( b\to \mbox{\$o} )\to \mbox{gtype}\qquad\mbox{thf}(h,\mbox{type})$

$f{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$\forall \mbox{xx}{:}\;b{:}\;( ( f@ \mbox{xx} )\iff ( g@ \mbox{xx} ) ) \ \Rightarrow \ ( h@ f ) = ( h@ g )\qquad\mbox{thf}(\mbox{cEXT\_SET}_2,\mbox{conjecture})$

\medskip\textbf{SYO314$\wedge$5.p} TPS problem from BASIC-HO-THMS






$z{:}\;\mbox{\$i}\qquad\mbox{thf}(z,\mbox{type})$

$\exists \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\neg\,\mbox{xp}@ \mbox{xy}\mbox{ and }\mbox{xq}@ \mbox{xy}\mbox{ and }\neg\,\mbox{xq}@ z )\iff \exists \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{xq}@ z\mbox{ and }\mbox{xq}@ \mbox{xy}\mbox{ and }\mbox{xp}@ \mbox{xx}\mbox{ and }\neg\,\mbox{xp}@ \mbox{xy} )\qquad\mbox{thf}(\mbox{cSILLYWFF}_2,\mbox{conjecture})$

\medskip\textbf{SYO099$\wedge$5.p} TPS problem THM78






$\mbox{cG}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cN}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\forall r{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ r )\mbox{ or }\exists x{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cG}@ x\mbox{ or }\neg\,\forall y{:}\;\mbox{\$i}{:}\;( \mbox{cM}@ y )\mbox{ or }\exists s{:}\;\mbox{\$i}{:}\;( \mbox{cN}@ s )\mbox{ or }\neg\,\forall z{:}\;\mbox{\$i}{:}\;\neg\,\mbox{cN}@ z\mbox{ or }\neg\,\forall t{:}\;\mbox{\$i}{:}\;( \mbox{cG}@ t )\qquad\mbox{thf}(\mbox{cTHM}_{78},\mbox{conjecture})$

\medskip\textbf{SYO257$\wedge$5.p} TPS problem THM84






$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( \forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cP}@ \mbox{xz}@ ( \mbox{xf}@ \mbox{xx} )\mbox{ and }\mbox{cP}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xy}@ ( g@ ( h@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{84},\mbox{conjecture})$

\medskip\textbf{SYO221$\wedge$5.p} TPS problem BLEDSOE6






$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cP}@ a\mbox{ and }a\ne b ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( a@ \mbox{xy} )\mbox{ and }\neg\,a@ b ) )\qquad\mbox{thf}(\mbox{cBLEDSOE}_6,\mbox{conjecture})$

\medskip\textbf{SYO563+1.p} Unequal numbers - reals






$\mbox{1.}_0\ne \mbox{2.}_0\qquad\mbox{fof}(\mbox{one\_not\_equal\_to}_2,\mbox{conjecture})$

\medskip\textbf{SYO050$\wedge$1.p} Simple textbook example 7






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ a ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO298$\wedge$5.p} TPS problem from BASIC-HO-THMS






$q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(q,\mbox{type})$

$p{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall l{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( a@ \mbox{xz} ) \ \Rightarrow \ ( p@ \mbox{xz}@ \mbox{xy}\mbox{ and }q@ l@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cBLEDSOE5B},\mbox{conjecture})$

\medskip\textbf{SYO041$\wedge$2.p} Unsatisfiable basic formula 3

Variant of the Kaminski equation.




$a{:}\;\mbox{\$o}\qquad\mbox{thf}(a,\mbox{type})$

$f{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$y{:}\;\mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$\neg\,x\ne y\mbox{ and }( g@ x ) = y\mbox{ and }( g@ y ) = x\mbox{ and }( f@ ( f@ ( f@ x ) ) ) = ( g@ ( f@ x ) )\qquad\mbox{thf}(3,\mbox{conjecture})$

\medskip\textbf{SYO205$\wedge$5.p} TPS problem CT27






$\exists \mbox{xx}{:}\;\mbox{\$o},\mbox{xy}{:}\;\mbox{\$o}{:}\;\mbox{xx} = \mbox{xy} = \mbox{xy} = \mbox{xx}\qquad\mbox{thf}(\mbox{cCT}_{27},\mbox{conjecture})$

\medskip\textbf{SYO178$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cM}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cM},\mbox{type})$

$\mbox{cN}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cN},\mbox{type})$

$\mbox{cG}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cK}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cK},\mbox{type})$

$\mbox{cE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cF}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cC}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$\mbox{cB}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\neg\,\mbox{cL}\mbox{ and }\mbox{cE}\mbox{ and }( \neg\,\mbox{cF}\mbox{ or }\mbox{cB} )\mbox{ and }( \neg\,\mbox{cL}\mbox{ or }\neg\,\mbox{cP}\mbox{ or }\mbox{cM} )\mbox{ and }( \neg\,\mbox{cG}\mbox{ or }\mbox{cR}\mbox{ or }\mbox{cM} )\mbox{ and }( \mbox{cG}\mbox{ or }\mbox{cP}\mbox{ or }\mbox{cR} )\mbox{ and }( \neg\,\mbox{cK}\mbox{ or }\neg\,\mbox{cB}\mbox{ or }\mbox{cC} )\mbox{ and }( \neg\,\mbox{cL}\mbox{ or }\neg\,\mbox{cM}\mbox{ or }\mbox{cC} )\mbox{ and }( \mbox{cG}\mbox{ or }\mbox{cR}\mbox{ or }\mbox{cK} )\mbox{ and }( \neg\,\mbox{cR}\mbox{ or }\neg\,\mbox{cE}\mbox{ or }\neg\,\mbox{cC} )\mbox{ and }( \neg\,\mbox{cR}\mbox{ or }\mbox{cN}\mbox{ or }\mbox{cF}\mbox{ or }\mbox{cP} )\mbox{ and }( \neg\,\mbox{cK}\mbox{ or }\neg\,\mbox{cL}\mbox{ or }\neg\,\mbox{cE}\mbox{ or }\neg\,\mbox{cM} )\mbox{ and }( \neg\,\mbox{cG}\mbox{ or }\mbox{cK}\mbox{ or }\mbox{cM}\mbox{ or }\neg\,\mbox{cB} )\mbox{ and }( \neg\,\mbox{cN}\mbox{ or }\mbox{cP}\mbox{ or }\mbox{cF}\mbox{ or }\mbox{cC} )\mbox{ and }( \neg\,\mbox{cG}\mbox{ or }\neg\,\mbox{cB}\mbox{ or }\mbox{cR}\mbox{ or }\neg\,\mbox{cC} )\mbox{ and }( \mbox{cK}\mbox{ or }\neg\,\mbox{cN}\mbox{ or }\neg\,\mbox{cM}\mbox{ or }\mbox{cF} )\mbox{ and }( \neg\,\mbox{cR}\mbox{ or }\mbox{cK}\mbox{ or }\mbox{cM}\mbox{ or }\mbox{cG} )\mbox{ and }( \neg\,\mbox{cE}\mbox{ or }\mbox{cK}\mbox{ or }\neg\,\mbox{cG}\mbox{ or }\mbox{cN}\mbox{ or }\neg\,\mbox{cM} )\qquad\mbox{thf}(\mbox{cPORKCHOP1C},\mbox{conjecture})$

\medskip\textbf{SYO404$\wedge$1.p} Ted Sider's modal proposition logic theorem 13






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^1.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mand}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ q@ p ) )@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ ( \mbox{mnot}@ q )@ p ) ) )@ ( \mbox{mbox\_k}@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO042$\wedge$1.p} Unsatisfiable basic formula 4






$g{:}\;\mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$p{:}\;( \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(p,\mbox{type})$

$x{:}\;\mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$y{:}\;\mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$x\ne y\mbox{ and }( g@ x ) = y\mbox{ and }( g@ y ) = x\mbox{ and }p@ g\mbox{ and }\neg\,p@ \neg\,\qquad\mbox{thf}(4,\mbox{axiom})$

\medskip\textbf{SYO470$\wedge$6.p} Ted Sider's propositional modal logic wff 21






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^6.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mequiv}@ ( \mbox{mdia\_s}_5@ ( \mbox{mdia\_s}_5@ ( \mbox{mbox\_s}_5@ p ) ) )@ ( \mbox{mbox\_s}_5@ p ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO425$\wedge$1.p} Ted Sider's modal proposition logic theorem 34






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^3.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ p )@ ( \mbox{mbox\_m}@ ( \mbox{mdia\_m}@ p ) ) )\qquad\mbox{thf}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{SYO143$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cQ}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cQ},\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \neg\,\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cP}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{xx} ) )\mbox{ or }\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xz} ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cQ}@ \mbox{xz} )\qquad\mbox{thf}(\mbox{cDUP\_BUG}_1,\mbox{conjecture})$

\medskip\textbf{SYO008$\wedge$1.p} The non-trivial direction of functional extensionality






$\mbox{leibeq}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq1\_type},\mbox{type})$

$\mbox{leibeq}_1 = ( \lambda u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;\forall q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq}_1,\mbox{definition})$

$\mbox{leibeq}_2{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq2\_type},\mbox{type})$

$\mbox{leibeq}_2 = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$i},y{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq}_2,\mbox{definition})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall x{:}\;\mbox{\$i}{:}\;( \mbox{leibeq}_1@ ( f@ x )@ ( g@ x ) ) \ \Rightarrow \ ( \mbox{leibeq}_2@ f@ g ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO310$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\exists \mbox{xop}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}{:}\;( \mbox{xop}@ \mbox{\$true}@ \mbox{\$true}\mbox{ and }\mbox{xop}@ \mbox{\$false}@ \mbox{\$false}\mbox{ and }\neg\,\mbox{xop}@ \mbox{\$false}@ \mbox{\$true}\mbox{ and }\neg\,\mbox{xop}@ \mbox{\$true}@ \mbox{\$false} )\mbox{ and }\forall \mbox{xm}{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( \mbox{xm}@ \mbox{\$true} )\iff ( \mbox{xm}@ ( \mbox{\$true} \ \Rightarrow \ \mbox{\$true} ) ) )\qquad\mbox{thf}(\mbox{cOMEGA\_EXAMPLE},\mbox{conjecture})$

\medskip\textbf{SYO306$\wedge$5.p} TPS problem from BASIC-HO-THMS






$\mbox{cT}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx} ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xx}\mbox{ and }\mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cS}@ \mbox{xx}\mbox{ and }\mbox{cT}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cDUP\_BUG}_2,\mbox{conjecture})$

\medskip\textbf{SYO006$\wedge$1.p} The trivial direction of Boolean extensionality






$\mbox{leibeq}{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq1\_type},\mbox{type})$

$\mbox{leibeq} = ( \lambda u{:}\;\mbox{\$o},v{:}\;\mbox{\$o}{:}\;\forall q{:}\;\mbox{\$o}\to \mbox{\$o}{:}\;( ( q@ u ) \ \Rightarrow \ ( q@ v ) ) )\qquad\mbox{thf}(\mbox{leibeq}_1,\mbox{definition})$

$\forall a{:}\;\mbox{\$o},b{:}\;\mbox{\$o}{:}\;( ( \mbox{leibeq}@ a@ b ) \ \Rightarrow \ ( a\iff b ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO345$\wedge$5.p} TPS problem THM615






$\mbox{cH}{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cH},\mbox{type})$

$( \mbox{cH}@ ( \mbox{cH}@ \mbox{\$true} ) = ( \mbox{cH}@ \mbox{\$false} ) ) = ( \mbox{cH}@ \mbox{\$false} )\qquad\mbox{thf}(\mbox{cTHM}_{615},\mbox{conjecture})$

\medskip\textbf{SYO057$\wedge$2.p} Simple textbook example 14






$\mbox{include}(\mbox{'Axioms/LCL013\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ a )@ a ) )@ ( \mbox{mbox}@ r@ a ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO355$\wedge$5.p} TPS problem THM613

If there's one individual, then there's only one function.




$\forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;\mbox{xa} = \mbox{xb} \ \Rightarrow \ \forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\mbox{xf} = \mbox{xg}\qquad\mbox{thf}(\mbox{cTHM}_{613},\mbox{conjecture})$

\medskip\textbf{SYO024$\wedge$1.p} De Morgan by connectives and Leibnitz






$\mbox{leibeq}{:}\;( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to ( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_decl},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o},y{:}\;\mbox{\$o}\to \mbox{\$o}\to \mbox{\$o}{:}\;\forall p{:}\;( \mbox{\$o}\to \mbox{\$o}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$\mbox{leibeq}@ \mbox{and}@ \lambda x{:}\;\mbox{\$o},y{:}\;\mbox{\$o}{:}\;\neg\,\neg\,x\mbox{ or }\neg\,y\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SYO361$\wedge$5.p} TPS problem THM47

Shows equivalence of two definitions of =.




$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( x = y\iff \forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall z{:}\;\mbox{\$i}{:}\;( r@ z@ z ) \ \Rightarrow \ ( r@ x@ y ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{47},\mbox{conjecture})$

\medskip\textbf{SYO063$\wedge$4.p} ILTP Problem SYJ106+1






$\mbox{include}(\mbox{'Axioms/LCL010\string^0.ax'})$

$p{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{p\_type},\mbox{type})$

$q{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q\_type},\mbox{type})$

$r{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{r\_type},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{s\_type},\mbox{type})$

$t{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{t\_type},\mbox{type})$

$\mbox{ivalid}@ ( \mbox{iatom}@ s )\qquad\mbox{thf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ t )@ ( \mbox{iatom}@ r ) ) )@ ( \mbox{iatom}@ p ) )\qquad\mbox{thf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iand}@ ( \mbox{iimplies}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ t )@ ( \mbox{iatom}@ r ) ) ) )@ ( \mbox{iand}@ ( \mbox{inot}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) ) )@ ( \mbox{iand}@ ( \mbox{iatom}@ s )@ ( \mbox{iatom}@ s ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip
\end{document}