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    \textbf{LCL001-0.ax} Wajsberg algebra






$\mbox{implies}(\mbox{truth},x) = x\qquad\mbox{cnf}(\mbox{wajsberg}_1,\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))) = \mbox{truth}\qquad\mbox{cnf}(\mbox{wajsberg}_2,\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(x,y),y) = \mbox{implies}(\mbox{implies}(y,x),x)\qquad\mbox{cnf}(\mbox{wajsberg}_3,\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)) = \mbox{truth}\qquad\mbox{cnf}(\mbox{wajsberg}_4,\mbox{axiom})$

\medskip\textbf{LCL007+4.ax} Axiomatization of S1-0






$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_modus\_ponens\_strict\_implies},\mbox{axiom})$

$\mbox{substitution\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_substitution\_strict\_equiv},\mbox{axiom})$

$\mbox{adjunction}\qquad\mbox{fof}(\mbox{s1\_0\_adjunction},\mbox{axiom})$

$\mbox{axiom\_m}_1\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_1,\mbox{axiom})$

$\mbox{axiom\_m}_2\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_2,\mbox{axiom})$

$\mbox{axiom\_m}_3\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_3,\mbox{axiom})$

$\mbox{axiom\_m}_4\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_4,\mbox{axiom})$

$\mbox{axiom\_m}_5\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_5,\mbox{axiom})$

\medskip\textbf{LCL006+2.ax} Hilbert's axiomatization of propositional logic






$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_ponens},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

\medskip\textbf{LCL007+3.ax} KM4B axiomatization of S5 based on Hilbert's PC






$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{km4b\_op\_possibly},\mbox{axiom})$

$\mbox{necessitation}\qquad\mbox{fof}(\mbox{km4b\_necessitation},\mbox{axiom})$

$\mbox{axiom\_K}\qquad\mbox{fof}(\mbox{km4b\_axiom\_K},\mbox{axiom})$

$\mbox{axiom\_M}\qquad\mbox{fof}(\mbox{km4b\_axiom\_M},\mbox{axiom})$

$\mbox{axiom}_4\qquad\mbox{fof}(\mbox{km4b\_axiom}_4,\mbox{axiom})$

$\mbox{axiom\_B}\qquad\mbox{fof}(\mbox{km4b\_axiom\_B},\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$5.ax} Modal logic S4

Embedding of monomodal logic S4 in simple type theory.




$\mbox{rel\_s}_4{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{rel\_s4\_type},\mbox{type})$

$\mbox{mbox\_s}_4{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{mbox\_s4\_type},\mbox{type})$

$\mbox{mbox\_s}_4 = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_s}_4@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_s}_4,\mbox{definition})$

$\mbox{mdia\_s}_4{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{mdia\_s4\_type},\mbox{type})$

$\mbox{mdia\_s}_4 = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_s}_4,\mbox{definition})$

$\mbox{mreflexive}@ \mbox{rel\_s}_4\qquad\mbox{thf}(a_1,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{rel\_s}_4\qquad\mbox{thf}(a_2,\mbox{axiom})$

\medskip\textbf{LCL006+4.ax} Principia's axiomatization of propositional logic






$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{principia\_modus\_ponens},\mbox{axiom})$

$r_1\qquad\mbox{fof}(\mbox{principia\_r}_1,\mbox{axiom})$

$r_2\qquad\mbox{fof}(\mbox{principia\_r}_2,\mbox{axiom})$

$r_3\qquad\mbox{fof}(\mbox{principia\_r}_3,\mbox{axiom})$

$r_4\qquad\mbox{fof}(\mbox{principia\_r}_4,\mbox{axiom})$

$r_5\qquad\mbox{fof}(\mbox{principia\_r}_5,\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$1.ax} Modal logic K

Embedding of monomodal logic K in simple type theory.




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

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

$\mbox{mbox\_k} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_k}@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_k},\mbox{definition})$

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

$\mbox{mdia\_k} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_k}@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_k},\mbox{definition})$

\medskip\textbf{LCL006+1.ax} Propositional logic definitions






$\mbox{op\_or} \ \Rightarrow \ \forall x,y{:}\;\mbox{or}(x,y) = \mbox{not}(\mbox{and}(\mbox{not}(x),\mbox{not}(y)))\qquad\mbox{fof}(\mbox{op\_or},\mbox{axiom})$

$\mbox{op\_and} \ \Rightarrow \ \forall x,y{:}\;\mbox{and}(x,y) = \mbox{not}(\mbox{or}(\mbox{not}(x),\mbox{not}(y)))\qquad\mbox{fof}(\mbox{op\_and},\mbox{axiom})$

$\mbox{op\_implies\_and} \ \Rightarrow \ \forall x,y{:}\;\mbox{implies}(x,y) = \mbox{not}(\mbox{and}(x,\mbox{not}(y)))\qquad\mbox{fof}(\mbox{op\_implies\_and},\mbox{axiom})$

$\mbox{op\_implies\_or} \ \Rightarrow \ \forall x,y{:}\;\mbox{implies}(x,y) = \mbox{or}(\mbox{not}(x),y)\qquad\mbox{fof}(\mbox{op\_implies\_or},\mbox{axiom})$

$\mbox{op\_equiv} \ \Rightarrow \ \forall x,y{:}\;\mbox{equiv}(x,y) = \mbox{and}(\mbox{implies}(x,y),\mbox{implies}(y,x))\qquad\mbox{fof}(\mbox{op\_equiv},\mbox{axiom})$

\medskip\textbf{LCL004-2.ax} Propositional logic deduction axioms for EQUIVALENT






$\mbox{equivalent}(p,q) = \mbox{and}(\mbox{implies}(p,q),\mbox{implies}(q,p))\qquad\mbox{cnf}(\mbox{equivalent\_defn},\mbox{axiom})$

\medskip\textbf{LCL007+0.ax} Propositional modal logic rules and axioms






$\mbox{necessitation}\iff \forall x{:}\;( \mbox{is\_a\_theorem}(x) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{necessarily}(x)) )\qquad\mbox{fof}(\mbox{necessitation},\mbox{axiom})$

$\mbox{modus\_ponens\_strict\_implies}\iff \forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(\mbox{strict\_implies}(x,y)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) )\qquad\mbox{fof}(\mbox{modus\_ponens\_strict\_implies},\mbox{axiom})$

$\mbox{adjunction}\iff \forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(y) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{and}(x,y)) )\qquad\mbox{fof}(\mbox{adjunction},\mbox{axiom})$

$\mbox{substitution\_strict\_equiv}\iff \forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{strict\_equiv}(x,y)) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{substitution\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_K}\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessarily}(\mbox{implies}(x,y)),\mbox{implies}(\mbox{necessarily}(x),\mbox{necessarily}(y))))\qquad\mbox{fof}(\mbox{axiom\_K},\mbox{axiom})$

$\mbox{axiom\_M}\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessarily}(x),x))\qquad\mbox{fof}(\mbox{axiom\_M},\mbox{axiom})$

$\mbox{axiom}_4\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessarily}(x),\mbox{necessarily}(\mbox{necessarily}(x))))\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{axiom\_B}\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{necessarily}(\mbox{possibly}(x))))\qquad\mbox{fof}(\mbox{axiom\_B},\mbox{axiom})$

$\mbox{axiom}_5\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{possibly}(x),\mbox{necessarily}(\mbox{possibly}(x))))\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{axiom\_s}_1\iff \forall x,y,z{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{and}(\mbox{necessarily}(\mbox{implies}(x,y)),\mbox{necessarily}(\mbox{implies}(y,z))),\mbox{necessarily}(\mbox{implies}(x,z))))\qquad\mbox{fof}(\mbox{axiom\_s}_1,\mbox{axiom})$

$\mbox{axiom\_s}_2\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{possibly}(\mbox{and}(p,q)),\mbox{and}(\mbox{possibly}(p),\mbox{possibly}(q))))\qquad\mbox{fof}(\mbox{axiom\_s}_2,\mbox{axiom})$

$\mbox{axiom\_s}_3\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{strict\_implies}(x,y),\mbox{strict\_implies}(\mbox{not}(\mbox{possibly}(y)),\mbox{not}(\mbox{possibly}(x)))))\qquad\mbox{fof}(\mbox{axiom\_s}_3,\mbox{axiom})$

$\mbox{axiom\_s}_4\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{necessarily}(x),\mbox{necessarily}(\mbox{necessarily}(x))))\qquad\mbox{fof}(\mbox{axiom\_s}_4,\mbox{axiom})$

$\mbox{axiom\_m}_1\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{and}(x,y),\mbox{and}(y,x)))\qquad\mbox{fof}(\mbox{axiom\_m}_1,\mbox{axiom})$

$\mbox{axiom\_m}_2\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{and}(x,y),x))\qquad\mbox{fof}(\mbox{axiom\_m}_2,\mbox{axiom})$

$\mbox{axiom\_m}_3\iff \forall x,y,z{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{and}(\mbox{and}(x,y),z),\mbox{and}(x,\mbox{and}(y,z))))\qquad\mbox{fof}(\mbox{axiom\_m}_3,\mbox{axiom})$

$\mbox{axiom\_m}_4\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(x,\mbox{and}(x,x)))\qquad\mbox{fof}(\mbox{axiom\_m}_4,\mbox{axiom})$

$\mbox{axiom\_m}_5\iff \forall x,y,z{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{and}(\mbox{strict\_implies}(x,y),\mbox{strict\_implies}(y,z)),\mbox{strict\_implies}(x,z)))\qquad\mbox{fof}(\mbox{axiom\_m}_5,\mbox{axiom})$

$\mbox{axiom\_m}_6\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(x,\mbox{possibly}(x)))\qquad\mbox{fof}(\mbox{axiom\_m}_6,\mbox{axiom})$

$\mbox{axiom\_m}_7\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{possibly}(\mbox{and}(p,q)),p))\qquad\mbox{fof}(\mbox{axiom\_m}_7,\mbox{axiom})$

$\mbox{axiom\_m}_8\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{strict\_implies}(p,q),\mbox{strict\_implies}(\mbox{possibly}(p),\mbox{possibly}(q))))\qquad\mbox{fof}(\mbox{axiom\_m}_8,\mbox{axiom})$

$\mbox{axiom\_m}_9\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{possibly}(\mbox{possibly}(x)),\mbox{possibly}(x)))\qquad\mbox{fof}(\mbox{axiom\_m}_9,\mbox{axiom})$

$\mbox{axiom\_m}_{10}\iff \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{strict\_implies}(\mbox{possibly}(x),\mbox{necessarily}(\mbox{possibly}(x))))\qquad\mbox{fof}(\mbox{axiom\_m}_{10},\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$2.ax} Modal logic D

Embedding of monomodal logic D in simple type theory




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

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

$\mbox{mbox\_d} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_d}@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_d},\mbox{definition})$

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

$\mbox{mdia\_d} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_d}@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_d},\mbox{definition})$

$\mbox{mserial}@ \mbox{rel\_d}\qquad\mbox{thf}(a_1,\mbox{axiom})$

\medskip\textbf{LCL007+5.ax} M6S3M9B axiomatization of S5 based on S1-0






$\mbox{axiom\_m}_6\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_6,\mbox{axiom})$

$\mbox{axiom\_s}_3\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_s}_3,\mbox{axiom})$

$\mbox{axiom\_m}_9\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_9,\mbox{axiom})$

$\mbox{axiom\_b}\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_b},\mbox{axiom})$

\medskip\textbf{LCL006+0.ax} Propositional logic rules and axioms






$\mbox{modus\_ponens}\iff \forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(\mbox{implies}(x,y)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) )\qquad\mbox{fof}(\mbox{modus\_ponens},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\iff \forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{modus\_tollens}\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(y),\mbox{not}(x)),\mbox{implies}(x,y)))\qquad\mbox{fof}(\mbox{modus\_tollens},\mbox{axiom})$

$\mbox{implies}_1\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{fof}(\mbox{implies}_1,\mbox{axiom})$

$\mbox{implies}_2\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(x,y)),\mbox{implies}(x,y)))\qquad\mbox{fof}(\mbox{implies}_2,\mbox{axiom})$

$\mbox{implies}_3\iff \forall x,y,z{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{fof}(\mbox{implies}_3,\mbox{axiom})$

$\mbox{and}_1\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{and}(x,y),x))\qquad\mbox{fof}(\mbox{and}_1,\mbox{axiom})$

$\mbox{and}_2\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{and}(x,y),y))\qquad\mbox{fof}(\mbox{and}_2,\mbox{axiom})$

$\mbox{and}_3\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,\mbox{and}(x,y))))\qquad\mbox{fof}(\mbox{and}_3,\mbox{axiom})$

$\mbox{or}_1\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{or}(x,y)))\qquad\mbox{fof}(\mbox{or}_1,\mbox{axiom})$

$\mbox{or}_2\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(y,\mbox{or}(x,y)))\qquad\mbox{fof}(\mbox{or}_2,\mbox{axiom})$

$\mbox{or}_3\iff \forall x,y,z{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,z),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{or}(x,y),z))))\qquad\mbox{fof}(\mbox{or}_3,\mbox{axiom})$

$\mbox{equivalence}_1\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{equiv}(x,y),\mbox{implies}(x,y)))\qquad\mbox{fof}(\mbox{equivalence}_1,\mbox{axiom})$

$\mbox{equivalence}_2\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{equiv}(x,y),\mbox{implies}(y,x)))\qquad\mbox{fof}(\mbox{equivalence}_2,\mbox{axiom})$

$\mbox{equivalence}_3\iff \forall x,y{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,x),\mbox{equiv}(x,y))))\qquad\mbox{fof}(\mbox{equivalence}_3,\mbox{axiom})$

$\mbox{kn}_1\iff \forall p{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(p,\mbox{and}(p,p)))\qquad\mbox{fof}(\mbox{kn}_1,\mbox{axiom})$

$\mbox{kn}_2\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{and}(p,q),p))\qquad\mbox{fof}(\mbox{kn}_2,\mbox{axiom})$

$\mbox{kn}_3\iff \forall p,q,r{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{not}(\mbox{and}(q,r)),\mbox{not}(\mbox{and}(r,p)))))\qquad\mbox{fof}(\mbox{kn}_3,\mbox{axiom})$

$\mbox{cn}_1\iff \forall p,q,r{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(p,r))))\qquad\mbox{fof}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{cn}_2\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(p,\mbox{implies}(\mbox{not}(p),q)))\qquad\mbox{fof}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{cn}_3\iff \forall p{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(p),p),p))\qquad\mbox{fof}(\mbox{cn}_3,\mbox{axiom})$

$r_1\iff \forall p{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{or}(p,p),p))\qquad\mbox{fof}(r_1,\mbox{axiom})$

$r_2\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(q,\mbox{or}(p,q)))\qquad\mbox{fof}(r_2,\mbox{axiom})$

$r_3\iff \forall p,q{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{or}(q,p)))\qquad\mbox{fof}(r_3,\mbox{axiom})$

$r_4\iff \forall p,q,r{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{or}(p,\mbox{or}(q,r)),\mbox{or}(q,\mbox{or}(p,r))))\qquad\mbox{fof}(r_4,\mbox{axiom})$

$r_5\iff \forall p,q,r{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{or}(p,q),\mbox{or}(p,r))))\qquad\mbox{fof}(r_5,\mbox{axiom})$

\medskip\textbf{LCL005-0.ax} Propositional logic






$\mbox{c\_PropLog\_Opl\_Ofalse}\ne \mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a\_H},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a\_H},\mbox{t\_a})\ne \mbox{c\_PropLog\_Opl\_Ofalse}\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Ofalse}\ne \mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_3\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a})\ne \mbox{c\_PropLog\_Opl\_Ofalse}\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_4\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a})\ne \mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_5\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a})\ne \mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Odistinct\_\_6\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}) = \mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a\_H},\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_a} = \mbox{v\_a\_H}\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Oinject\_\_1\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{v\_pl}_2,\mbox{t\_a}) = \mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_pl}_1 = \mbox{v\_pl1\_H}\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Oinject\_\_2\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{v\_pl}_2,\mbox{t\_a}) = \mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl1\_H},\mbox{v\_pl2\_H},\mbox{t\_a}) \ \Rightarrow \ \mbox{v\_pl}_2 = \mbox{v\_pl2\_H}\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Opl\_Oinject\_\_2\_\_iff1}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

\medskip\textbf{LCL014$\wedge$0.ax} Region Connection Calculus






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

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

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

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

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

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

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

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

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

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

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

$\forall x{:}\;\mbox{reg}{:}\;( c@ x@ x )\qquad\mbox{thf}(\mbox{c\_reflexive},\mbox{axiom})$

$\forall x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( ( c@ x@ y ) \ \Rightarrow \ ( c@ y@ x ) )\qquad\mbox{thf}(\mbox{c\_symmetric},\mbox{axiom})$

$\mbox{dc} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;\neg\,c@ x@ y )\qquad\mbox{thf}(\mbox{dc},\mbox{definition})$

$p = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;\forall z{:}\;\mbox{reg}{:}\;( ( c@ z@ x ) \ \Rightarrow \ ( c@ z@ y ) ) )\qquad\mbox{thf}(p,\mbox{definition})$

$\mbox{eq} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( p@ x@ y\mbox{ and }p@ y@ x ) )\qquad\mbox{thf}(\mbox{eq},\mbox{definition})$

$o = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;\exists z{:}\;\mbox{reg}{:}\;( p@ z@ x\mbox{ and }p@ z@ y ) )\qquad\mbox{thf}(o,\mbox{definition})$

$\mbox{po} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( o@ x@ y\mbox{ and }\neg\,p@ x@ y\mbox{ and }\neg\,p@ y@ x ) )\qquad\mbox{thf}(\mbox{po},\mbox{definition})$

$\mbox{ec} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( c@ x@ y\mbox{ and }\neg\,o@ x@ y ) )\qquad\mbox{thf}(\mbox{ec},\mbox{definition})$

$\mbox{pp} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( p@ x@ y\mbox{ and }\neg\,p@ y@ x ) )\qquad\mbox{thf}(\mbox{pp},\mbox{definition})$

$\mbox{tpp} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( \mbox{pp}@ x@ y\mbox{ and }\exists z{:}\;\mbox{reg}{:}\;( \mbox{ec}@ z@ x\mbox{ and }\mbox{ec}@ z@ y ) ) )\qquad\mbox{thf}(\mbox{tpp},\mbox{definition})$

$\mbox{ntpp} = ( \lambda x{:}\;\mbox{reg},y{:}\;\mbox{reg}{:}\;( \mbox{pp}@ x@ y\mbox{ and }\neg\,\exists z{:}\;\mbox{reg}{:}\;( \mbox{ec}@ z@ x\mbox{ and }\mbox{ec}@ z@ y ) ) )\qquad\mbox{thf}(\mbox{ntpp},\mbox{definition})$

\medskip\textbf{LCL007+1.ax} Propositional modal logic definitions






$\mbox{op\_possibly} \ \Rightarrow \ \forall x{:}\;\mbox{possibly}(x) = \mbox{not}(\mbox{necessarily}(\mbox{not}(x)))\qquad\mbox{fof}(\mbox{op\_possibly},\mbox{axiom})$

$\mbox{op\_necessarily} \ \Rightarrow \ \forall x{:}\;\mbox{necessarily}(x) = \mbox{not}(\mbox{possibly}(\mbox{not}(x)))\qquad\mbox{fof}(\mbox{op\_necessarily},\mbox{axiom})$

$\mbox{op\_strict\_implies} \ \Rightarrow \ \forall x,y{:}\;\mbox{strict\_implies}(x,y) = \mbox{necessarily}(\mbox{implies}(x,y))\qquad\mbox{fof}(\mbox{op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_strict\_equiv} \ \Rightarrow \ \forall x,y{:}\;\mbox{strict\_equiv}(x,y) = \mbox{and}(\mbox{strict\_implies}(x,y),\mbox{strict\_implies}(y,x))\qquad\mbox{fof}(\mbox{op\_strict\_equiv},\mbox{axiom})$

\medskip\textbf{LCL009$\wedge$0.ax} Translating constructive S4 (CS4) to bimodal classical S4 (BS4)






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

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

$\mbox{cs4\_atom}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_atom\_decl},\mbox{type})$

$\mbox{cs4\_and}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_and\_decl},\mbox{type})$

$\mbox{cs4\_or}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_or\_decl},\mbox{type})$

$\mbox{cs4\_impl}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_impl\_decl},\mbox{type})$

$\mbox{cs4\_true}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_true\_decl},\mbox{type})$

$\mbox{cs4\_false}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_false\_decl},\mbox{type})$

$\mbox{cs4\_all}{:}\;( \mbox{individuals}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_all\_decl},\mbox{type})$

$\mbox{cs4\_box}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_box\_decl},\mbox{type})$

$\mbox{cs4\_atom} = ( \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ \mbox{reli}@ p ) )\qquad\mbox{thf}(\mbox{cs4\_atom},\mbox{definition})$

$\mbox{cs4\_and} = ( \lambda a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mand}@ a@ b ) )\qquad\mbox{thf}(\mbox{cs4\_and},\mbox{definition})$

$\mbox{cs4\_or} = ( \lambda a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mor}@ a@ b ) )\qquad\mbox{thf}(\mbox{cs4\_or},\mbox{definition})$

$\mbox{cs4\_impl} = ( \lambda a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ \mbox{reli}@ ( \mbox{mimpl}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{cs4\_impl},\mbox{definition})$

$\mbox{cs4\_true} = \mbox{mtrue}\qquad\mbox{thf}(\mbox{cs4\_true},\mbox{definition})$

$\mbox{cs4\_false} = \mbox{mfalse}\qquad\mbox{thf}(\mbox{cs4\_false},\mbox{definition})$

$\mbox{cs4\_all} = ( \lambda a{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ \mbox{reli}@ ( \mbox{mall}@ a ) ) )\qquad\mbox{thf}(\mbox{cs4\_all},\mbox{definition})$

$\mbox{cs4\_box} = ( \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ \mbox{reli}@ ( \mbox{mbox}@ \mbox{relr}@ a ) ) )\qquad\mbox{thf}(\mbox{cs4\_box},\mbox{definition})$

$\mbox{cs4\_valid}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cs4\_valid\_decl},\mbox{type})$

$\mbox{cs4\_valid} = ( \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ a ) )\qquad\mbox{thf}(\mbox{cs4\_valid\_def},\mbox{definition})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ \mbox{reli}@ a )@ a ) )\qquad\mbox{thf}(\mbox{refl\_axiom\_i},\mbox{axiom})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ \mbox{relr}@ a )@ a ) )\qquad\mbox{thf}(\mbox{refl\_axiom\_r},\mbox{axiom})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ \mbox{reli}@ a )@ ( \mbox{mbox}@ \mbox{reli}@ ( \mbox{mbox}@ \mbox{reli}@ a ) ) ) )\qquad\mbox{thf}(\mbox{trans\_axiom\_i},\mbox{axiom})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ \mbox{relr}@ a )@ ( \mbox{mbox}@ \mbox{relr}@ ( \mbox{mbox}@ \mbox{relr}@ a ) ) ) )\qquad\mbox{thf}(\mbox{trans\_axiom\_r},\mbox{axiom})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ \mbox{reli}@ ( \mbox{mbox}@ \mbox{relr}@ a ) )@ ( \mbox{mbox}@ \mbox{relr}@ ( \mbox{mbox}@ \mbox{reli}@ a ) ) ) )\qquad\mbox{thf}(\mbox{ax\_i\_r\_commute},\mbox{axiom})$

\medskip\textbf{LCL006+3.ax} Lukasiewicz's axiomatization of propositional logic






$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{luka\_modus\_ponens},\mbox{axiom})$

$\mbox{cn}_1\qquad\mbox{fof}(\mbox{luka\_cn}_1,\mbox{axiom})$

$\mbox{cn}_2\qquad\mbox{fof}(\mbox{luka\_cn}_2,\mbox{axiom})$

$\mbox{cn}_3\qquad\mbox{fof}(\mbox{luka\_cn}_3,\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

\medskip\textbf{LCL001-2.ax} Wajsberg algebra AND and OR definitions






$\mbox{or}(x,y) = \mbox{implies}(\mbox{not}(x),y)\qquad\mbox{cnf}(\mbox{or\_definition},\mbox{axiom})$

$\mbox{or}(\mbox{or}(x,y),z) = \mbox{or}(x,\mbox{or}(y,z))\qquad\mbox{cnf}(\mbox{or\_associativity},\mbox{axiom})$

$\mbox{or}(x,y) = \mbox{or}(y,x)\qquad\mbox{cnf}(\mbox{or\_commutativity},\mbox{axiom})$

$\mbox{and}(x,y) = \mbox{not}(\mbox{or}(\mbox{not}(x),\mbox{not}(y)))\qquad\mbox{cnf}(\mbox{and\_definition},\mbox{axiom})$

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

$\mbox{and}(x,y) = \mbox{and}(y,x)\qquad\mbox{cnf}(\mbox{and\_commutativity},\mbox{axiom})$

\medskip\textbf{LCL007+2.ax} KM5 axiomatization of S5 based on Hilbert's PC






$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{km5\_op\_possibly},\mbox{axiom})$

$\mbox{necessitation}\qquad\mbox{fof}(\mbox{km5\_necessitation},\mbox{axiom})$

$\mbox{axiom\_K}\qquad\mbox{fof}(\mbox{km5\_axiom\_K},\mbox{axiom})$

$\mbox{axiom\_M}\qquad\mbox{fof}(\mbox{km5\_axiom\_M},\mbox{axiom})$

$\mbox{axiom}_5\qquad\mbox{fof}(\mbox{km5\_axiom}_5,\mbox{axiom})$

\medskip\textbf{LCL001-1.ax} Wajsberg algebra lattice structure definitions






$\mbox{big\_V}(x,y) = \mbox{implies}(\mbox{implies}(x,y),y)\qquad\mbox{cnf}(\mbox{big\_V\_definition},\mbox{axiom})$

$\mbox{big\_hat}(x,y) = \mbox{not}(\mbox{big\_V}(\mbox{not}(x),\mbox{not}(y)))\qquad\mbox{cnf}(\mbox{big\_hat\_definition},\mbox{axiom})$

$\mbox{ordered}(x,y) \ \Rightarrow \ \mbox{implies}(x,y) = \mbox{truth}\qquad\mbox{cnf}(\mbox{partial\_order\_definition}_1,\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{truth} \ \Rightarrow \ \mbox{ordered}(x,y)\qquad\mbox{cnf}(\mbox{partial\_order\_definition}_2,\mbox{axiom})$

\medskip\textbf{LCL003-0.ax} Propositional logic deduction






$\mbox{axiom}(\mbox{or}(\mbox{not}(\mbox{or}(a,a)),a))\qquad\mbox{cnf}(\mbox{axiom\_1}_2,\mbox{axiom})$

$\mbox{axiom}(\mbox{or}(\mbox{not}(a),\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{axiom\_1}_3,\mbox{axiom})$

$\mbox{axiom}(\mbox{or}(\mbox{not}(\mbox{or}(a,b)),\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{axiom\_1}_4,\mbox{axiom})$

$\mbox{axiom}(\mbox{or}(\mbox{not}(\mbox{or}(a,\mbox{or}(b,c))),\mbox{or}(b,\mbox{or}(a,c))))\qquad\mbox{cnf}(\mbox{axiom\_1}_5,\mbox{axiom})$

$\mbox{axiom}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(a),b)),\mbox{or}(\mbox{not}(\mbox{or}(c,a)),\mbox{or}(c,b))))\qquad\mbox{cnf}(\mbox{axiom\_1}_6,\mbox{axiom})$

$\mbox{axiom}(x) \ \Rightarrow \ \mbox{theorem}(x)\qquad\mbox{cnf}(\mbox{rule}_1,\mbox{axiom})$

$( \mbox{axiom}(\mbox{or}(\mbox{not}(y),x))\mbox{ and }\mbox{theorem}(y) ) \ \Rightarrow \ \mbox{theorem}(x)\qquad\mbox{cnf}(\mbox{rule}_2,\mbox{axiom})$

$( \mbox{axiom}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{theorem}(\mbox{or}(\mbox{not}(y),z)) ) \ \Rightarrow \ \mbox{theorem}(\mbox{or}(\mbox{not}(x),z))\qquad\mbox{cnf}(\mbox{rule}_3,\mbox{axiom})$

\medskip\textbf{LCL002-1.ax} Alternative Wajsberg algebra definitions






$\mbox{xor}(x,y) = \mbox{or}(\mbox{and}(x,\mbox{not}(y)),\mbox{and}(\mbox{not}(x),y))\qquad\mbox{cnf}(\mbox{xor\_definition},\mbox{axiom})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

$\mbox{and\_star}(x,y) = \mbox{not}(\mbox{or}(\mbox{not}(x),\mbox{not}(y)))\qquad\mbox{cnf}(\mbox{and\_star\_definition},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

\medskip\textbf{LCL015$\wedge$1.ax} Cumulative domain specific axioms






$\forall x{:}\;\mbox{mu},v{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( ( \mbox{exists\_in\_world}@ x@ v\mbox{ and }\mbox{rel\_s}_4@ v@ w ) \ \Rightarrow \ ( \mbox{exists\_in\_world}@ x@ w ) )\qquad\mbox{thf}(\mbox{cumulative\_ax},\mbox{axiom})$

\medskip\textbf{LCL002-0.ax} Alternative Wajsberg algebra






$\mbox{not}(x) = \mbox{xor}(x,\mbox{truth})\qquad\mbox{cnf}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{xor}(x,\mbox{falsehood}) = x\qquad\mbox{cnf}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{xor}(x,x) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{and\_star}(x,\mbox{truth}) = x\qquad\mbox{cnf}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{and\_star}(x,\mbox{falsehood}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{and\_star}(\mbox{xor}(\mbox{truth},x),x) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{xor}(x,\mbox{xor}(\mbox{truth},y)) = \mbox{xor}(\mbox{xor}(x,\mbox{truth}),y)\qquad\mbox{cnf}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{and\_star}(\mbox{xor}(\mbox{and\_star}(\mbox{xor}(\mbox{truth},x),y),\mbox{truth}),y) = \mbox{and\_star}(\mbox{xor}(\mbox{and\_star}(\mbox{xor}(\mbox{truth},y),x),\mbox{truth}),x)\qquad\mbox{cnf}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{LCL006+5.ax} Rosser's axiomatization of propositional logic






$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{rosser\_modus\_ponens},\mbox{axiom})$

$\mbox{kn}_1\qquad\mbox{fof}(\mbox{rosser\_kn}_1,\mbox{axiom})$

$\mbox{kn}_2\qquad\mbox{fof}(\mbox{rosser\_kn}_2,\mbox{axiom})$

$\mbox{kn}_3\qquad\mbox{fof}(\mbox{rosser\_kn}_3,\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$4.ax} Modal logic B

Embedding of monomodal logic B in simple type theory.




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

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

$\mbox{mbox\_b} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_b}@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_b},\mbox{definition})$

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

$\mbox{mdia\_b} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_b}@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_b},\mbox{definition})$

$\mbox{mreflexive}@ \mbox{rel\_b}\qquad\mbox{thf}(a_1,\mbox{axiom})$

$\mbox{msymmetric}@ \mbox{rel\_b}\qquad\mbox{thf}(a_2,\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$3.ax} Modal logic M

Embedding of monomodal logic M in simple type theory.




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

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

$\mbox{mbox\_m} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_m}@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_m},\mbox{definition})$

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

$\mbox{mdia\_m} = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_m}@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_m},\mbox{definition})$

$\mbox{mreflexive}@ \mbox{rel\_m}\qquad\mbox{thf}(a_1,\mbox{axiom})$

\medskip\textbf{LCL016$\wedge$1.ax} Embedding of second order modal logic in simple type theory

Extends K to KB by adding symmetric of rel.




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

$\mbox{msymmetric} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall s{:}\;\mbox{\$i},t{:}\;\mbox{\$i}{:}\;( ( r@ s@ t ) \ \Rightarrow \ ( r@ t@ s ) ) )\qquad\mbox{thf}(\mbox{msymmetric},\mbox{definition})$

$\mbox{msymmetric}@ \mbox{rel}\qquad\mbox{thf}(\mbox{sym},\mbox{axiom})$

\medskip\textbf{LCL013$\wedge$6.ax} Modal logic S5

Embedding of monomodal logic S5 in simple type theory.




$\mbox{rel\_s}_5{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{rel\_s5\_type},\mbox{type})$

$\mbox{mbox\_s}_5{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{mbox\_s5\_type},\mbox{type})$

$\mbox{mbox\_s}_5 = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{rel\_s}_5@ w@ v\mbox{ or }\mbox{phi}@ v ) )\qquad\mbox{thf}(\mbox{mbox\_s}_5,\mbox{definition})$

$\mbox{mdia\_s}_5{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{mdia\_s5\_type},\mbox{type})$

$\mbox{mdia\_s}_5 = ( \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mnot}@ ( \mbox{mbox\_s}_5@ ( \mbox{mnot}@ \mbox{phi} ) ) ) )\qquad\mbox{thf}(\mbox{mdia\_s}_5,\mbox{definition})$

$\mbox{mreflexive}@ \mbox{rel\_s}_5\qquad\mbox{thf}(a_1,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{rel\_s}_5\qquad\mbox{thf}(a_2,\mbox{axiom})$

$\mbox{msymmetric}@ \mbox{rel\_s}_5\qquad\mbox{thf}(a_3,\mbox{axiom})$

\medskip\textbf{LCL004-1.ax} Propositional logic deduction axioms for AND






$\mbox{and}(p,q) = \mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))\qquad\mbox{cnf}(\mbox{and\_defn},\mbox{axiom})$

\medskip\textbf{LCL007+6.ax} M10 axiomatization of S5 based on S1-0






$\mbox{axiom\_m}_{10}\qquad\mbox{fof}(\mbox{s1\_0\_m10\_axiom\_m}_{10},\mbox{axiom})$

\medskip\textbf{LCL004-0.ax} Propositional logic deduction






$\mbox{axiom}(\mbox{implies}(\mbox{or}(a,a),a))\qquad\mbox{cnf}(\mbox{axiom\_1}_2,\mbox{axiom})$

$\mbox{axiom}(\mbox{implies}(a,\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{axiom\_1}_3,\mbox{axiom})$

$\mbox{axiom}(\mbox{implies}(\mbox{or}(a,b),\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{axiom\_1}_4,\mbox{axiom})$

$\mbox{axiom}(\mbox{implies}(\mbox{or}(a,\mbox{or}(b,c)),\mbox{or}(b,\mbox{or}(a,c))))\qquad\mbox{cnf}(\mbox{axiom\_1}_5,\mbox{axiom})$

$\mbox{axiom}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{or}(c,a),\mbox{or}(c,b))))\qquad\mbox{cnf}(\mbox{axiom\_1}_6,\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{or}(\mbox{not}(x),y)\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{axiom}(x) \ \Rightarrow \ \mbox{theorem}(x)\qquad\mbox{cnf}(\mbox{rule}_1,\mbox{axiom})$

$( \mbox{theorem}(\mbox{implies}(y,x))\mbox{ and }\mbox{theorem}(y) ) \ \Rightarrow \ \mbox{theorem}(x)\qquad\mbox{cnf}(\mbox{rule}_2,\mbox{axiom})$

\medskip\textbf{LCL405-1.p} CN-71 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-71 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{not}(\mbox{implies}(\mbox{implies}(x,x),\mbox{not}(\mbox{implies}(y,y)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{71},\mbox{negated\_conjecture})$

\medskip\textbf{LCL469+1.p} Prove Hilbert's or\_1 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{conjecture})$

\medskip\textbf{LCL613$\wedge$1.p} Simple theorem of K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mand}@ x@ y ) )@ ( \mbox{mbox}@ r@ x ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL257-1.p} XHN depends on YQL

Show that XHN can be derived from the single Lukasiewicz
 axiom YQL.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(x,z))))\qquad\mbox{cnf}(\mbox{yql},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,x),y))))\qquad\mbox{cnf}(\mbox{prove\_xhn},\mbox{negated\_conjecture})$

\medskip\textbf{LCL326-3.p} Principia Mathematica 5.19






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{not}(\mbox{equivalent}(p,\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL911+1.p} KM5 axiomatization of S5 based on Hilbert's PC






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

\medskip\textbf{LCL163-1.p} The 3rd Wajsberg algebra axiom, from the alternative axioms






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{xor}(\mbox{truth},\mbox{and\_star}(x,\mbox{xor}(\mbox{truth},y)))\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(x,y),y)\ne \mbox{implies}(\mbox{implies}(y,x),x)\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL476+1.p} Prove Principia's r2 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_2\qquad\mbox{fof}(\mbox{principia\_r}_2,\mbox{conjecture})$

\medskip\textbf{LCL230-2.p} Principia Mathematica 2.85

Judged by [SRM73] to be the 'hardest' of the first 67 theorems
 of [WR27].




$q \ \Rightarrow \ ( p\mbox{ or }r )\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,r\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL438-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_pa},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_pa},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL181$\wedge$4.p} Principia Mathematica 2.15






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

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

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

$\mbox{ivalid}@ ( \mbox{iequiv}@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iatom}@ p ) )@ ( \mbox{iatom}@ q ) )@ ( \mbox{iimplies}@ ( \mbox{inot}@ ( \mbox{iatom}@ q ) )@ ( \mbox{iatom}@ p ) ) )\qquad\mbox{thf}(\mbox{pel}_4,\mbox{conjecture})$

\medskip\textbf{LCL629$\wedge$1.p} Simple theorem about knowledge






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )\qquad\mbox{thf}(\mbox{knowledge\_implies\_truth},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(\mbox{positive\_introspection},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) )@ ( \mbox{mbox}@ r@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{negitive\_introspection},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mand}@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ y ) ) ) ) ) )@ ( \mbox{mbox}@ a@ x ) )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ ( \mbox{mbox}@ b@ ( \mbox{mimpl}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL067-1.p} CN-1 depends on the second Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-1 depends on the second Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),z),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),z))))\qquad\mbox{cnf}(\mbox{cn}_{59},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL581$\wedge$1.p} Popkorn problem 2






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ a ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL241-1.p} Principia Mathematica 3.27






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),q))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL343-3.p} Principia Mathematica 5.53






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{or}(\mbox{or}(p,q),r),s),\mbox{and}(\mbox{and}(\mbox{implies}(p,s),\mbox{implies}(q,s)),\mbox{implies}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL609$\wedge$1.p} LAMBDA$\wedge$mm\_1 validates the generalization rule






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall p{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall x{:}\;\mbox{individuals}{:}\;( \mbox{mvalid}@ ( p@ x ) ) \ \Rightarrow \ ( \mbox{mvalid}@ ( \mbox{mall}@ \lambda x{:}\;\mbox{individuals}{:}\;( p@ x ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL312-3.p} Principia Mathematica 4.81






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{not}(p),p),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL692$\wedge$1.p} Prove transitivity in the CS4 translation






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cs4\_valid}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ a ) )@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{cs4\_trans},\mbox{conjecture})$

\medskip\textbf{LCL200-1.p} Principia Mathematica 2.46






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(p,q))),\mbox{not}(q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL504+1.p} Prove Hilbert's implies\_2 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{conjecture})$

\medskip\textbf{LCL736$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}_{41}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xr}_{42}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b,\mbox{xw}_{11}{:}\;b{:}\;( \mbox{xr}_{41}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_{11}\mbox{ or }\mbox{xr}_{42}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_{11} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xw}_{11}{:}\;b{:}\;( \mbox{xr}_{41}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_{11}\mbox{ or }\mbox{xr}_{42}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_{11} ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310D},\mbox{conjecture})$

\medskip\textbf{LCL700$\wedge$1.p} Accessibility relation implies axiom for symmetry






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

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

\medskip\textbf{LCL336-3.p} Principia Mathematica 5.36






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{equivalent}(p,q)),\mbox{and}(q,\mbox{equivalent}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL015-1.p} WN depends on YRM

Show that the single Meredith axiom WN can be derived from
 the single Meredith axiom YRM.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(z,\mbox{equivalent}(\mbox{equivalent}(y,z),x))))\qquad\mbox{cnf}(\mbox{yrm},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(c,\mbox{equivalent}(a,b))))\qquad\mbox{cnf}(\mbox{prove\_wn},\mbox{negated\_conjecture})$

\medskip\textbf{LCL543+1.p} Prove axiom m3 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_3\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_3,\mbox{conjecture})$

\medskip\textbf{LCL300-3.p} Principia Mathematica 4.66






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{not}(p),\mbox{not}(q)),\mbox{or}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL414+1.p} Peter Andrews Problem THM147






$\neg\,\forall p,q{:}\;( \neg\,\mbox{a\_truth}(\mbox{implies}(p,q))\mbox{ or }\neg\,\mbox{a\_truth}(p)\mbox{ or }\mbox{a\_truth}(q) )\mbox{ and }\forall p,q{:}\;\mbox{a\_truth}(\mbox{implies}(p,\mbox{implies}(q,p)))\mbox{ and }\forall p,q,r{:}\;\mbox{a\_truth}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(p,r))))\mbox{ and }\forall p,q{:}\;\mbox{a\_truth}(\mbox{implies}(\mbox{implies}(\mbox{not}(p),\mbox{not}(q)),\mbox{implies}(q,p)))\mbox{ and }\exists a{:}\;\neg\,\mbox{a\_truth}(\mbox{implies}(a,a))\qquad\mbox{fof}(\mbox{thm}_{147},\mbox{conjecture})$

\medskip\textbf{LCL415-1.p} Non-axiom for intuitionistic implication

Show that the candidate formula is not a single axiom for
 intuitionistic implication by finding a model in which the
 required property fails.




$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,\mbox{implies}(\mbox{implies}(c,a),d)),\mbox{implies}(a,d))))\qquad\mbox{cnf}(\mbox{candidate},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_required\_property},\mbox{negated\_conjecture})$

\medskip\textbf{LCL879$\wedge$1.p} Correspondence for axiom 4s






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

$\forall i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},j{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ i@ \mbox{phi} )@ ( \mbox{mbox}@ j@ ( \mbox{mbox}@ i@ \mbox{phi} ) ) ) ) )\iff \forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( ( j@ u@ v\mbox{ and }i@ v@ w ) \ \Rightarrow \ ( i@ u@ w ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL675+1.001.p} In S4, the branching formula, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL350-3.p} Principia Mathematica 5.7






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{or}(p,r),\mbox{or}(q,r)),\mbox{or}(r,\mbox{equivalent}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL117-1.p} QYF depends on YQM

Single axioms for the R calculus are QYF, YQM, WO, all
 by Meredith and XGJ by Winker. Show that QYF depends on YQM.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(z,x))))\qquad\mbox{cnf}(\mbox{yqm},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(a,c)),\mbox{equivalent}(c,b)))\qquad\mbox{cnf}(\mbox{prove\_qyf},\mbox{negated\_conjecture})$

\medskip\textbf{LCL082-1.p} IC-2 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-2 depends on the first Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_ic}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL229-3.p} Principia Mathematica 2.83






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(\mbox{implies}(p,\mbox{implies}(r,s)),\mbox{implies}(p,\mbox{implies}(q,s)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL107-1.p} P-1 depends on the single McCune axiom

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that P-1 depends on the single McCune axiom.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(\mbox{equivalent}(u,v),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(w,v),\mbox{equivalent}(w,u)),v_6))),\mbox{equivalent}(z,\mbox{equivalent}(\mbox{equivalent}(y,x),v_6))))\qquad\mbox{cnf}(\mbox{lg\_27}_{1690},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(\mbox{equivalent}(e,b),\mbox{equivalent}(\mbox{equivalent}(a,e),c))))\qquad\mbox{cnf}(\mbox{prove\_p}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL667+1.001.p} In KT, pigeonhole formulae missing a conjunct, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }\neg\,p_{101}(y) )\mbox{ or }\neg\,p_{201}(x)\mbox{ and }p_{101}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL401-1.p} CN-67 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-67 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(x,y)),\mbox{not}(y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{67},\mbox{negated\_conjecture})$

\medskip\textbf{LCL582$\wedge$1.p} Popkorn problem 3






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ s@ a ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL098-1.p} LG-4 depends on LG-3

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-4 depends on LG-3.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),u),\mbox{equivalent}(\mbox{equivalent}(y,z),u)),v),v))\qquad\mbox{cnf}(\mbox{lg}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),e),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{falsehood}),c),\mbox{equivalent}(\mbox{equivalent}(b,\mbox{falsehood}),e))))\qquad\mbox{cnf}(\mbox{prove\_lg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL262-3.p} Principia Mathematica 4.10






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,q),\mbox{implies}(\mbox{not}(q),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL198-1.p} Principia Mathematica 2.42 and 2.43






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(p),q))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL254-1.p} Principia Mathematica 3.47






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),r)),\mbox{not}(\mbox{or}(\mbox{not}(q),s))))),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),\mbox{not}(\mbox{or}(\mbox{not}(r),\mbox{not}(s))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL051-1.p} CN-22 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-22 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{22},\mbox{negated\_conjecture})$

\medskip\textbf{LCL449+1.p} Congruence of equiv, to admit substitution of equivalents






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_ponens},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{axiom})$

$( \forall x{:}\;\mbox{is\_a\_theorem}(\mbox{equiv}(x,x))\mbox{ and }\forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{not}(x),\mbox{not}(y))) )\mbox{ and }\forall x_1,x_2,y_1,y_2{:}\;( ( \mbox{is\_a\_theorem}(\mbox{equiv}(x_1,x_2))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{equiv}(y_1,y_2)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{and}(x_1,y_1),\mbox{and}(x_2,y_2))) )\mbox{ and }\forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) ) ) \ \Rightarrow \ \forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{make\_subs\_of\_equiv},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{subs\_of\_equiv},\mbox{conjecture})$

\medskip\textbf{LCL872$\wedge$1.p} Correspondence between box and diamond and a confluence property






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

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

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

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

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

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

$\mbox{confluence} = ( \lambda i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},j{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},k{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},l{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( ( i@ a@ b\mbox{ and }k@ a@ c ) \ \Rightarrow \ \exists d{:}\;\mbox{\$i}{:}\;( j@ b@ d\mbox{ and }l@ c@ d ) ) )\qquad\mbox{thf}(\mbox{confluence},\mbox{definition})$

$( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mdia}@ i@ ( \mbox{mbox}@ j@ p ) )@ ( \mbox{mbox}@ k@ ( \mbox{mdia}@ l@ p ) ) ) ) )\iff ( \mbox{confluence}@ i@ j@ k@ l )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL488+1.p} Prove Hilbert's and\_2 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{conjecture})$

\medskip\textbf{LCL373-1.p} CN-29 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-29 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(x,z),z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{LCL228-3.p} Principia Mathematica 2.82






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(\mbox{or}(p,q),r),\mbox{implies}(\mbox{or}(\mbox{or}(p,\mbox{not}(r)),s),\mbox{or}(\mbox{or}(p,q),s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL091-1.p} IC-2 depends on the 5th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-2 depends on the fifth Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(t,\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_ic}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL282-3.p} Principia Mathematica 4.41






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(p,\mbox{and}(q,r)),\mbox{and}(\mbox{or}(p,q),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL910+1.p} Rosser's axiomatization of propositional logic






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

\medskip\textbf{LCL614$\wedge$1.p} Regularity is a derived rule in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ ( \mbox{mimpl}@ x@ y ) ) \ \Rightarrow \ ( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(\mbox{regularity},\mbox{conjecture})$

\medskip\textbf{LCL171-1.p} Principia Mathematica 2.03






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))),\mbox{or}(\mbox{not}(q),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL575+1.p} Prove axiom B from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom\_B}\qquad\mbox{fof}(\mbox{km4b\_axiom\_B},\mbox{conjecture})$

\medskip\textbf{LCL607$\wedge$1.p} LAMBDA$\wedge$mm\_1 validates the axioms defining possibility






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{miff}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ ( \mbox{mnot}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL285-3.p} Principia Mathematica 4.44






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{or}(p,\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL432-2.p} Problem about propositional logic






$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_p},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ocompleteness\_\_0}_0,\mbox{axiom})$

\medskip\textbf{LCL744-1.p} Strong normalization of typed lambda calculus 019\_3






$\mbox{c\_Lambda\_Obeta}(\mbox{v\_s},\mbox{v\_t}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_s}),\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_t}))\qquad\mbox{cnf}(\mbox{cls\_abs}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_r}),\mbox{v\_s}) \ \Rightarrow \ \mbox{v\_s} = \mbox{c\_Lambda\_OdB\_OAbs}(\mbox{c\_Lambda\_Osko\_\_Lambda\_\_Xbeta\_\_cases\_\_2\_}_1(\mbox{v\_r},\mbox{v\_s}))\qquad\mbox{cnf}(\mbox{cls\_beta\_\_cases\_I2\_J}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{v\_s},\mbox{v\_t}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_s},\mbox{v\_u}),\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_t},\mbox{v\_u}))\qquad\mbox{cnf}(\mbox{cls\_appL}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{v\_s},\mbox{v\_t}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_u},\mbox{v\_s}),\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_u},\mbox{v\_t}))\qquad\mbox{cnf}(\mbox{cls\_appR}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{v\_r},\mbox{v\_s}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_Osubst}(\mbox{v\_r},\mbox{v\_t},\mbox{v\_i}),\mbox{c\_Lambda\_Osubst}(\mbox{v\_s},\mbox{v\_t},\mbox{v\_i}))\qquad\mbox{cnf}(\mbox{cls\_subst\_\_preserves\_\_beta}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_dB}) = \mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_dB\_H}) \ \Rightarrow \ \mbox{v\_dB} = \mbox{v\_dB\_H}\qquad\mbox{cnf}(\mbox{cls\_dB\_Osimps\_I3\_J}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OApp}(\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_s}),\mbox{v\_t}),\mbox{c\_Lambda\_Osubst}(\mbox{v\_s},\mbox{v\_t},\mbox{c\_HOL\_Ozero\_\_class\_Ozero}(\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_beta}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB}_1,\mbox{v\_dB}_2) = \mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB1\_H},\mbox{v\_dB2\_H}) \ \Rightarrow \ \mbox{v\_dB}_2 = \mbox{v\_dB2\_H}\qquad\mbox{cnf}(\mbox{cls\_dB\_Osimps\_I2\_J}_1,\mbox{axiom})$

$\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB}_1,\mbox{v\_dB}_2) = \mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB1\_H},\mbox{v\_dB2\_H}) \ \Rightarrow \ \mbox{v\_dB}_1 = \mbox{v\_dB1\_H}\qquad\mbox{cnf}(\mbox{cls\_dB\_Osimps\_I2\_J}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_r}),\mbox{v\_s}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{v\_r},\mbox{c\_Lambda\_Osko\_\_Lambda\_\_Xbeta\_\_cases\_\_2\_}_1(\mbox{v\_r},\mbox{v\_s}))\qquad\mbox{cnf}(\mbox{cls\_beta\_\_cases\_I2\_J}_1,\mbox{axiom})$

$\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_dB\_H})\ne \mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB}_1,\mbox{v\_dB}_2)\qquad\mbox{cnf}(\mbox{cls\_dB\_Osimps\_I9\_J}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_dB}_1,\mbox{v\_dB}_2)\ne \mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_dB\_H})\qquad\mbox{cnf}(\mbox{cls\_dB\_Osimps\_I8\_J}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Osubst}(\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_t},\mbox{v\_u}),\mbox{v\_s},\mbox{v\_k}) = \mbox{c\_Lambda\_OdB\_OApp}(\mbox{c\_Lambda\_Osubst}(\mbox{v\_t},\mbox{v\_s},\mbox{v\_k}),\mbox{c\_Lambda\_Osubst}(\mbox{v\_u},\mbox{v\_s},\mbox{v\_k}))\qquad\mbox{cnf}(\mbox{cls\_subst\_\_App}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Olift}(\mbox{c\_Lambda\_OdB\_OApp}(\mbox{v\_s},\mbox{v\_t}),\mbox{v\_k}) = \mbox{c\_Lambda\_OdB\_OApp}(\mbox{c\_Lambda\_Olift}(\mbox{v\_s},\mbox{v\_k}),\mbox{c\_Lambda\_Olift}(\mbox{v\_t},\mbox{v\_k}))\qquad\mbox{cnf}(\mbox{cls\_lift\_Osimps\_I2\_J}_0,\mbox{axiom})$

$\mbox{c\_InductTermi\_OIT}(\mbox{v\_r}) \ \Rightarrow \ \mbox{c\_InductTermi\_OIT}(\mbox{c\_Lambda\_OdB\_OAbs}(\mbox{v\_r}))\qquad\mbox{cnf}(\mbox{cls\_Lambda}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Osubst}(\mbox{c\_Lambda\_Olift}(\mbox{v\_t},\mbox{v\_k}),\mbox{v\_s},\mbox{v\_k}) = \mbox{v\_t}\qquad\mbox{cnf}(\mbox{cls\_subst\_\_lift}_0,\mbox{axiom})$

$\mbox{c\_Lambda\_Obeta}(\mbox{v\_r},\mbox{v\_s}) \ \Rightarrow \ \mbox{c\_Lambda\_Obeta}(\mbox{c\_Lambda\_Olift}(\mbox{v\_r},\mbox{v\_i}),\mbox{c\_Lambda\_Olift}(\mbox{v\_s},\mbox{v\_i}))\qquad\mbox{cnf}(\mbox{cls\_lift\_\_preserves\_\_beta}_0,\mbox{axiom})$

$\mbox{c\_InductTermi\_OIT}(\mbox{v\_t})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_InductTermi\_OIT}(\mbox{c\_Lambda\_Olift}(\mbox{v\_t},\mbox{v\_i}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL065-1.p} CN-2 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-2 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL165-1.p} A theorem in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{not}(\mbox{or}(\mbox{and}(x,\mbox{or}(x,x)),\mbox{and}(x,x)))\ne \mbox{and}(\mbox{not}(x),\mbox{or}(\mbox{or}(\mbox{not}(x),\mbox{not}(x)),\mbox{and}(\mbox{not}(x),\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL734$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}_3{:}\;( b\to \mbox{\$o} )\to b\to \mbox{\$o},\mbox{xr}_4{:}\;( b\to \mbox{\$o} )\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;( \mbox{xr}_3@ \mbox{xx}@ \mbox{xy}\mbox{ or }\mbox{xr}_4@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;( \mbox{xr}_3@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )\mbox{ or }\mbox{xr}_4@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310B},\mbox{conjecture})$

\medskip\textbf{LCL386-1.p} CN-48 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-48 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(\mbox{not}(y),x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{48},\mbox{negated\_conjecture})$

\medskip\textbf{LCL079-1.p} Transitivity can be derived from Church's system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-18,CN-35,CN-49} by Church. Show
 that transitivity of implies can be derived from the Church
 system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,x),\mbox{implies}(\mbox{not}(x),\mbox{not}(y))))\qquad\mbox{cnf}(\mbox{cn\_49\_reversed},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,b))\qquad\mbox{cnf}(\mbox{a\_implies\_b},\mbox{hypothesis})$

$\mbox{is\_a\_theorem}(\mbox{implies}(b,c))\qquad\mbox{cnf}(\mbox{b\_implies\_c},\mbox{hypothesis})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,c))\qquad\mbox{cnf}(\mbox{prove\_transitivity},\mbox{negated\_conjecture})$

\medskip\textbf{LCL880$\wedge$1.p} Correspondence for axiom 5s






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

$\forall i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},j{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mnot}@ ( \mbox{mbox}@ i@ \mbox{phi} ) )@ ( \mbox{mbox}@ j@ ( \mbox{mnot}@ ( \mbox{mbox}@ i@ \mbox{phi} ) ) ) ) ) )\iff \forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( ( j@ u@ v\mbox{ and }i@ u@ w ) \ \Rightarrow \ ( i@ v@ w ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL273-3.p} Principia Mathematica 4.3






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,q),\mbox{and}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL380-1.p} CN-41 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-41 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(\mbox{not}(x)),y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{41},\mbox{negated\_conjecture})$

\medskip\textbf{LCL444-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL108-1.p} Q-3 depends on the single McCune axiom

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that Q-3 depends on the single McCune axiom.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(\mbox{equivalent}(u,v),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(w,v),\mbox{equivalent}(w,u)),v_6))),\mbox{equivalent}(z,\mbox{equivalent}(\mbox{equivalent}(y,x),v_6))))\qquad\mbox{cnf}(\mbox{lg\_27}_{1690},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(b,a),c)),c))\qquad\mbox{cnf}(\mbox{prove\_q}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL022-1.p} EC-1 depends on YQL

An axiomatisation of the equivalential calculus is {EC-1,
 EC-2} by Lesniewski. Show that EC-1 can be derived from the
 single Lukasiewicz axiom YQL.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(x,z))))\qquad\mbox{cnf}(\mbox{yql},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,a)),\mbox{equivalent}(b,c)))\qquad\mbox{cnf}(\mbox{prove\_ec}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL027-1.p} C0-4 depends on the Church system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-4 can be derived from the
 Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{falsehood}),\mbox{falsehood}),x))\qquad\mbox{cnf}(\mbox{c0}_5,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_6,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{falsehood},a))\qquad\mbox{cnf}(\mbox{prove\_c0}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL603$\wedge$1.p} T and 4 imply reflexivity and transitivity of R in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ a ) ) \ \Rightarrow \ ( \mbox{reflexive}@ r\mbox{ and }\mbox{transitive}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL236-1.p} Principia Mathematica 3.14






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))),\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL923$\wedge$1.p} Modal logic S5






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

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

\medskip\textbf{LCL530+1.p} Prove axiom m3 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_3\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_3,\mbox{conjecture})$

\medskip\textbf{LCL344-3.p} Principia Mathematica 5.54






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{equivalent}(\mbox{and}(p,q),p),\mbox{equivalent}(\mbox{and}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL269-3.p} Principia Mathematica 4.21






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{equivalent}(p,q),\mbox{equivalent}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL072-1.p} CN-3 depends on the Wos system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-3 depends on the Wos system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(y),z)),\mbox{implies}(x,\mbox{implies}(\mbox{implies}(u,z),\mbox{implies}(\mbox{implies}(y,u),z)))))\qquad\mbox{cnf}(\mbox{cn}_{60},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL227-1.p} Principia Mathematica 2.81






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),\mbox{or}(\mbox{not}(r),s))),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(\mbox{or}(p,r)),\mbox{or}(p,s)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL900+1.p} A bounded pocrim with commutative strong disjunction is a hoop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a,b{:}\;a '==>' b '==>' b = b '==>' a '==>' a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17},x_{18}{:}\;x_{17} '+' x_{17} '==>' x_{18} = x_{18} '+' x_{18} '==>' x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL914+1.p} M6S3M9B axiomatization of S5 based on S1-0






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

\medskip\textbf{LCL455+1.p} Prove Principia's r2 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_2\qquad\mbox{fof}(\mbox{principia\_r}_2,\mbox{conjecture})$

\medskip\textbf{LCL240-1.p} Principia Mathematica 3.26






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL116-2.p} MV-50 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-50 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{not}(a),\mbox{implies}(b,\mbox{not}(\mbox{implies}(b,a))))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{50},\mbox{negated\_conjecture})$

\medskip\textbf{LCL136-1.p} A lemma in Wajsberg algebras

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided
 a different axiomatisation. Show that a version of MV-2
 depends on the Wajsberg system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,y) = \mbox{implies}(y,z)\qquad\mbox{cnf}(\mbox{lemma\_antecedent},\mbox{negated\_conjecture})$

$\mbox{implies}(x,z)\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL733$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}{:}\;( b\to \mbox{\$o} )\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;( \exists \mbox{xx}_0{:}\;b{:}\;( \mbox{xx}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xx}_0{:}\;b{:}\;( \mbox{xx}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB}_4,\mbox{conjecture})$

\medskip\textbf{LCL585$\wedge$1.p} Popkorn problem 6






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mdia}@ r@ ( \mbox{mimpl}@ a@ b ) )@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL023-1.p} EC-2 depends on YQL

An axiomatisation of the equivalential calculus is {EC-1,
 EC-2} by Lesniewski. Show that EC-2 can be derived from the
 single Lukasiewicz axiom YQL.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(x,z))))\qquad\mbox{cnf}(\mbox{yql},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(\mbox{equivalent}(a,b),c)))\qquad\mbox{cnf}(\mbox{prove\_ec}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL519+1.p} Prove Principia's r2 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_2\qquad\mbox{fof}(\mbox{principia\_r}_2,\mbox{conjecture})$

\medskip\textbf{LCL499+1.p} Prove Rosser's kn1 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_1\qquad\mbox{fof}(\mbox{rosser\_kn}_1,\mbox{conjecture})$

\medskip\textbf{LCL351-3.p} Principia Mathematica 5.71






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,\mbox{not}(r)),\mbox{equivalent}(\mbox{and}(\mbox{or}(p,q),r),\mbox{and}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL084-3.p} IC-4 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-4 depends on the first Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,x))\qquad\mbox{cnf}(\mbox{ic}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{ic}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_ic}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL147-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{implies}(\mbox{big\_V}(x,y),z)\ne \mbox{big\_hat}(\mbox{implies}(x,z),\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL920$\wedge$1.p} Modal logic M






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

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

\medskip\textbf{LCL451+1.p} Prove Lukasiewicz's cn1 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_1\qquad\mbox{fof}(\mbox{luka\_cn}_1,\mbox{conjecture})$

\medskip\textbf{LCL445-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_excluded\_\_middle}_0,\mbox{axiom})$

\medskip\textbf{LCL267-3.p} Principia Mathematica 4.15






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{implies}(q,\mbox{not}(r))),\mbox{and}(q,\mbox{implies}(r,\mbox{not}(p)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL073-1.p} CN-1 depends on the single Merideth axiom

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-1 depends on the single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(z),\mbox{not}(u))),z),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{cn\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL042-1.p} CN-35 depends on Hilbert's system

Two axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-18,CN-21,CN-35,CN-39,CN-39,
 CN-40,CN-46} by Frege and {CN-3,CN-18,CN-21,CN-22,CN-30,
 CN-54} by Hilbert. Show that CN-35 depends on the simplified
 Hilbert system (without CN-30).




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(y,\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{21},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{22},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),y)))\qquad\mbox{cnf}(\mbox{cn}_{54},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{LCL115-2.p} MV-39 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-39 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{not}(\mbox{implies}(a,b)),\mbox{not}(b))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL134-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,\mbox{truth})\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL437-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_pa},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL458+1.p} Prove Principia's r5 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_5\qquad\mbox{fof}(\mbox{principia\_r}_5,\mbox{conjecture})$

\medskip\textbf{LCL297-3.p} Principia Mathematica 4.63






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{implies}(p,\mbox{not}(q))),\mbox{and}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL055-1.p} CN-37 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-37 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),c),\mbox{implies}(\mbox{not}(a),c)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{37},\mbox{negated\_conjecture})$

\medskip\textbf{LCL047-1.p} CN-18 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-18 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{18},\mbox{negated\_conjecture})$

\medskip\textbf{LCL043-1.p} CN-39 depends on Hilbert's system

Two axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-18,CN-21,CN-35,CN-39,CN-39,
 CN-40,CN-46} by Frege and {CN-3,CN-18,CN-21,CN-22,CN-30,
 CN-54} by Hilbert. Show that CN-39 depends on the simplified
 Hilbert system (without CN-30).




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(y,\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{21},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{22},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),y)))\qquad\mbox{cnf}(\mbox{cn}_{54},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(a)),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL731$\wedge$5.p} TPS problem THM541

Equivalence of global choice at type A (usual way of expressing 
 AC in type theory).




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

$\exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) )\iff \forall \mbox{xs}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{541},\mbox{conjecture})$

\medskip\textbf{LCL678+1.001.p} In S4, formula provable in intuitionistic logic, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\mbox{\$false}\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\mbox{\$false}\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL242-1.p} Principia Mathematica 3.3






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),r)),\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL170-1.p} Principia Mathematica 2.02






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(q),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL009-1.p} EC-5 depends on YQL

An axiomatisation of the equivalential calculus is {EC-4,
 EC-5} by Wajsburg. Show that EC-5 can be derived from the
 single Lukasiewicz axiom YQL.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(x,z))))\qquad\mbox{cnf}(\mbox{yql},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(a,\mbox{equivalent}(b,c))))\qquad\mbox{cnf}(\mbox{prove\_ec}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL529+1.p} Prove axiom m2 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_2\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_2,\mbox{conjecture})$

\medskip\textbf{LCL035-1.p} C0-4 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-4 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{falsehood},a))\qquad\mbox{cnf}(\mbox{prove\_c0}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL882+1.p} An involutive pocrim that does not have unique halving






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' \mbox{'1'} = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} = x_{17} '==>' x_{18}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{17} = x_{19} )\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL116-1.p} MV-50 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that 50 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(a),\mbox{implies}(b,\mbox{not}(\mbox{implies}(b,a)))))\qquad\mbox{cnf}(\mbox{prove\_mv}_{50},\mbox{negated\_conjecture})$

\medskip\textbf{LCL577+1.p} Prove axiom s3 from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{axiom\_s}_3\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_s}_3,\mbox{conjecture})$

\medskip\textbf{LCL109-4.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{big\_V}(\mbox{implies}(x,y),\mbox{implies}(y,x))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL561+1.p} Prove Hilbert's equivalence\_1 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{conjecture})$

\medskip\textbf{LCL564+1.p} Prove axiom K from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom\_K}\qquad\mbox{fof}(\mbox{km5\_axiom\_K},\mbox{conjecture})$

\medskip\textbf{LCL475+1.p} Prove Principia's r1 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_1\qquad\mbox{fof}(\mbox{principia\_r}_1,\mbox{conjecture})$

\medskip\textbf{LCL191-1.p} Principia Mathematica 2.31






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{or}(q,r))),\mbox{or}(\mbox{or}(p,q),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL208-3.p} Principia Mathematica 2.53






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL230-3.p} Principia Mathematica 2.85






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{or}(p,q),\mbox{or}(p,r)),\mbox{or}(p,\mbox{implies}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL904$\wedge$1.p} Axioms for Modal logic S4 under cumulative domains






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

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

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

\medskip\textbf{LCL538+1.p} Prove strict implies modus ponens from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_modus\_ponens\_strict\_implies},\mbox{conjecture})$

\medskip\textbf{LCL180-1.p} Principia Mathematica 2.14






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{not}(p))),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL532+1.p} Prove axiom m5 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_5\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_5,\mbox{conjecture})$

\medskip\textbf{LCL220-1.p} Principia Mathematica 2.69






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),p)),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL414$\wedge$7.p} Peter Andrews Problem THM147






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

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

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

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

$\mbox{not}{:}\;\mbox{mu}\to \mbox{mu}\qquad\mbox{thf}(\mbox{not\_type},\mbox{type})$

$\forall v{:}\;\mbox{\$i},v_1{:}\;\mbox{mu}{:}\;( \mbox{exists\_in\_world}@ ( \mbox{not}@ v_1 )@ v )\qquad\mbox{thf}(\mbox{existence\_of\_not\_ax},\mbox{axiom})$

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

$\forall v{:}\;\mbox{\$i},v_2{:}\;\mbox{mu},v_1{:}\;\mbox{mu}{:}\;( \mbox{exists\_in\_world}@ ( \mbox{implies}@ v_2@ v_1 )@ v )\qquad\mbox{thf}(\mbox{existence\_of\_implies\_ax},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda p{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda q{:}\;\mbox{mu}{:}\;( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ ( \mbox{implies}@ p@ q ) ) ) ) )@ ( \mbox{mor}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ p ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ q ) ) ) ) ) ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda p{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda q{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ ( \mbox{implies}@ p@ ( \mbox{implies}@ q@ p ) ) ) ) ) ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda p{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda q{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda r{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ ( \mbox{implies}@ ( \mbox{implies}@ p@ ( \mbox{implies}@ q@ r ) )@ ( \mbox{implies}@ ( \mbox{implies}@ p@ q )@ ( \mbox{implies}@ p@ r ) ) ) ) ) ) ) ) ) ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda p{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda q{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ ( \mbox{implies}@ ( \mbox{implies}@ ( \mbox{not}@ p )@ ( \mbox{not}@ q ) )@ ( \mbox{implies}@ q@ p ) ) ) ) ) ) ) )@ ( \mbox{mexists\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{a\_truth}@ ( \mbox{implies}@ a@ a ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{thm}_{147},\mbox{conjecture})$

\medskip\textbf{LCL909+1.p} Principia's axiomatization of propositional logic






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

\medskip\textbf{LCL014-1.p} XGF depends on WN

Show that the single Meredith axiom XGF can be derived from
 the single Meredith axiom WN.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(z,\mbox{equivalent}(x,y))))\qquad\mbox{cnf}(\mbox{wn},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(a,c)),\mbox{equivalent}(c,b))))\qquad\mbox{cnf}(\mbox{prove\_xgf},\mbox{negated\_conjecture})$

\medskip\textbf{LCL161-1.p} The 1st Wajsberg algebra axiom, from the alternative axioms






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{xor}(\mbox{truth},\mbox{and\_star}(x,\mbox{xor}(\mbox{truth},y)))\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{implies}(\mbox{truth},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL239-1.p} Principia Mathematica 3.24






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{not}(p))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL056-1.p} CN-39 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-39 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(a)),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL194-1.p} Principia Mathematica 2.37






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(\mbox{or}(q,p)),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL650+1.001.p} In K, black and white polygon with odd number of vertices, size 1

If we have a polygon with n vertices, and all the vertices are
 either black or white, then two adjacent vertices have the same
 colour.




$\neg\,\exists x{:}\;\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }( \neg\,p_8(x)\mbox{ and }\neg\,p_6(x)\mbox{ and }\neg\,p_4(x)\mbox{ and }\neg\,p_2(x) ) ) ) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_5(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,( \neg\,p_3(x)\mbox{ and }\neg\,p_1(x) )\mbox{ or }( p_1(x)\mbox{ and }p_3(x) ) ) ) )\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_4(x) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,( \neg\,p_2(x)\mbox{ and }\neg\,p_3(x) )\mbox{ or }( p_3(x)\mbox{ and }p_2(x) ) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_3(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,( \neg\,p_1(x)\mbox{ and }\neg\,p_2(x) )\mbox{ or }( p_2(x)\mbox{ and }p_1(x) ) ) ) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }( p_4(x)\mbox{ and }p_3(x)\mbox{ and }p_2(x)\mbox{ and }p_1(x) ) ) ) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL368-1.p} CN-23 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-23 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),u),\mbox{implies}(\mbox{implies}(y,\mbox{implies}(x,z)),u)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{23},\mbox{negated\_conjecture})$

\medskip\textbf{LCL403-1.p} CN-69 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-69 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(y),\mbox{not}(\mbox{implies}(x,y)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{69},\mbox{negated\_conjecture})$

\medskip\textbf{LCL430-2.p} Problem about propositional logic






$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{v\_C},\mbox{v\_B},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_ODiff\_\_weaken\_\_left}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{c\_insert}(\mbox{v\_v},\mbox{v\_t},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ohyps\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oinsert\_\_Diff\_\_same}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_c},\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{v\_D},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_c},\mbox{v\_D},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oinsert\_\_Diff\_\_subset2}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_excluded\_\_middle\_\_rule}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_G},\mbox{v\_H},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_F},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL104-1.p} P-1 depends on the 6th McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that P-1 depends on the sixth McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(y,u)),\mbox{equivalent}(z,u)),x)))\qquad\mbox{cnf}(p_4,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(y,x),z)),z))\qquad\mbox{cnf}(q_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(\mbox{equivalent}(e,b),\mbox{equivalent}(\mbox{equivalent}(a,e),c))))\qquad\mbox{cnf}(\mbox{prove\_p}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL681+1.001.p} In S4, in backwards search no way through box and dia, size 1






$\neg\,\exists x{:}\;\neg\,p_1(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL041-1.p} CN-30 depends on the rest of Hilbert's system

An early axiomatisation of Implication/Negation 2 valued
 sentential calculus was {CN-3,CN-18,CN-21,CN-22,CN-30, CN-54}
 by Hilbert. Show, like Lukasiewicz did, that CN-30 depends
 on the rest of this axiomatisation.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(y,\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{21},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{22},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),y)))\qquad\mbox{cnf}(\mbox{cn}_{54},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(a,b)),\mbox{implies}(a,b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{LCL198-3.p} Principia Mathematica 2.42






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(\mbox{not}(p),\mbox{implies}(p,q)),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL583$\wedge$1.p} Popkorn problem 4






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox}@ s@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ a ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL243-3.p} Principia Mathematica 3.31






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(\mbox{and}(p,q),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL040-1.p} CN-21 depends on the rest of Frege's system

The first axiomatisation of Implication/Negation 2 valued
 sentential calculus was {CN-18,CN-21,CN-35,CN-39,CN-39,
 CN-40,CN-46} by Frege. Show, like Lukasiewicz did, that CN-21
 depends on the rest of this axiomatisation.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(x)),x))\qquad\mbox{cnf}(\mbox{cn}_{39},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{not}(\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{cn}_{40},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(y),\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{cn}_{46},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(b,\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{LCL211-1.p} Principia Mathematica 2.56






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(q)),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL578+1.p} Prove axiom m9 from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{axiom\_m}_9\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_9,\mbox{conjecture})$

\medskip\textbf{LCL307-3.p} Principia Mathematica 4.76






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{implies}(p,q),\mbox{implies}(p,r)),\mbox{implies}(p,\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL177-1.p} Principia Mathematica 2.11






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(p,\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL076-2.p} CN-40 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-40 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(x_1)),x_1))\qquad\mbox{cnf}(\mbox{extra\_lemma},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL053-1.p} CN-30 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-30 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(a,b)),\mbox{implies}(a,b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{30},\mbox{negated\_conjecture})$

\medskip\textbf{LCL228-1.p} Principia Mathematica 2.82






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{or}(p,q),r)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{or}(p,\mbox{not}(r)),s)),\mbox{or}(\mbox{or}(p,q),s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL036-1.p} C0-5 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-5 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,\mbox{falsehood}),\mbox{falsehood}),a))\qquad\mbox{cnf}(\mbox{prove\_c0}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL192-3.p} Principia Mathematica 2.32






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(\mbox{or}(p,q),r),\mbox{or}(p,\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL235-3.p} Principia Mathematica 3.13






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{and}(p,q)),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL247-1.p} Principia Mathematica 3.37






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),r)),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{not}(r))))),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL298-3.p} Principia Mathematica 4.51






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{and}(p,q)),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL913+1.p} Axiomatization of S1-0






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

\medskip\textbf{LCL383-1.p} CN-44 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-44 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,\mbox{not}(z))),\mbox{implies}(\mbox{implies}(z,y),\mbox{implies}(x,\mbox{not}(z)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{44},\mbox{negated\_conjecture})$

\medskip\textbf{LCL491+1.p} Prove Hilbert's or\_2 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{conjecture})$

\medskip\textbf{LCL743$\wedge$5.p} TPS problem from AXIOMOFDESCR

Related to the axiom of description.




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

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

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

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

$( \mbox{cF}@ \mbox{cX} ) \ \Rightarrow \ ( \forall y{:}\;a{:}\;( ( \mbox{cF}@ y ) \ \Rightarrow \ \mbox{cX} = y ) \ \Rightarrow \ ( \mbox{cF}@ ( \mbox{cJ}@ \mbox{cF} ) ) )\qquad\mbox{thf}(\mbox{cDESCR\_CHURCH},\mbox{conjecture})$

\medskip\textbf{LCL216-1.p} Principia Mathematica 2.64






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{not}(q))),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL310-3.p} Principia Mathematica 4.79






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(\mbox{implies}(q,p),\mbox{implies}(r,p)),\mbox{implies}(\mbox{and}(q,r),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL878$\wedge$1.p} Correspondence for axiom I






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

$\forall i{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},j{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda \mbox{phi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ i@ \mbox{phi} )@ ( \mbox{mbox}@ j@ \mbox{phi} ) ) ) )\iff \forall u{:}\;\mbox{\$i},v{:}\;\mbox{\$i}{:}\;( ( j@ u@ v ) \ \Rightarrow \ ( i@ u@ v ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL231-1.p} Principia Mathematica 2.86






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(p),r))),\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL359-1.p} CN-08 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-08 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(\mbox{implies}(y,u),\mbox{implies}(z,u)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{08},\mbox{negated\_conjecture})$

\medskip\textbf{LCL697$\wedge$1.p} Propositional intuitionistic logic in HOL

An embedding of propositional intuitionisitc logic in HOL based
 on the McKinsey/Tarski translation of propositional intuitionistic
 logic to modal logic S4.




$\mbox{include}(\mbox{'Axioms/LCL012\string^0.ax'})$

\medskip\textbf{LCL617$\wedge$1.p} Axiom GL - the Loeb formula






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

\medskip\textbf{LCL176-3.p} Principia Mathematica 2.1 and 2.08






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL071-1.p} CN-2 depends on the Wos system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-2 depends on the Wos system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(y),z)),\mbox{implies}(x,\mbox{implies}(\mbox{implies}(u,z),\mbox{implies}(\mbox{implies}(y,u),z)))))\qquad\mbox{cnf}(\mbox{cn}_{60},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL593$\wedge$1.p} Is axiom T valid in K?






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

\medskip\textbf{LCL286-3.p} Principia Mathematica 4.45






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{and}(p,\mbox{or}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL723$\wedge$1.p} Axioms M and B hold in monomodal logic B






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_b}@ \mbox{phi} )@ \mbox{phi} )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimplies}@ \mbox{phi}@ ( \mbox{mbox\_b}@ ( \mbox{mdia\_b}@ \mbox{phi} ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL021-1.p} XHK depends on XHN

Show that the single Winker axiom XHK can be derived from the
 single Winker axiom XHN.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,x),y))))\qquad\mbox{cnf}(\mbox{xhn},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(a,c),b))))\qquad\mbox{cnf}(\mbox{prove\_xhk},\mbox{negated\_conjecture})$

\medskip\textbf{LCL515+1.p} Prove Lukasiewicz's cn1 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_1\qquad\mbox{fof}(\mbox{luka\_cn}_1,\mbox{conjecture})$

\medskip\textbf{LCL138-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,\mbox{implies}(y,z))\ne \mbox{implies}(y,\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL927-1.p} AxK and AxC in TW+ [AxL,AxTO] + (Resid)






$( p(i(a,b))\mbox{ and }p(a) ) \ \Rightarrow \ p(b)\qquad\mbox{cnf}(\mbox{modus\_ponens},\mbox{axiom})$

$p(i(f(a,b),c)) \ \Rightarrow \ p(i(a,i(b,c)))\qquad\mbox{cnf}(\mbox{resid}_1,\mbox{axiom})$

$p(i(a,i(b,c))) \ \Rightarrow \ p(i(f(a,b),c))\qquad\mbox{cnf}(\mbox{resid}_2,\mbox{axiom})$

$p(i(i(a,b),i(i(b,c),i(a,c))))\qquad\mbox{cnf}(\mbox{axBp},\mbox{axiom})$

$p(i(i(i(x,y),y),i(i(y,x),x)))\qquad\mbox{cnf}(\mbox{axL},\mbox{axiom})$

$p(i(i(i(x,y),i(y,x)),i(y,x)))\qquad\mbox{cnf}(\mbox{axTO},\mbox{axiom})$

$p(i(c_1,i(c_2,c_1))) \ \Rightarrow \ \neg\,p(i(i(a,i(b,c)),i(b,i(a,c))))\qquad\mbox{cnf}(\mbox{axK\_axC},\mbox{negated\_conjecture})$

\medskip\textbf{LCL435-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_pa},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL202-3.p} Principia Mathematica 2.48






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL539+1.p} Prove SSE from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{substitution\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_substitution\_strict\_equiv},\mbox{conjecture})$

\medskip\textbf{LCL716$\wedge$1.p} Axiom implies accessibility relation for weak density






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

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

\medskip\textbf{LCL167-1.p} YRO depends on XHK

Show that the single Meredith axiom YRO can be derived from
 the single Winker axiom XHK.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(x,z),y))))\qquad\mbox{cnf}(\mbox{xhk},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,\mbox{equivalent}(\mbox{equivalent}(c,b),a))))\qquad\mbox{cnf}(\mbox{prove\_yro},\mbox{negated\_conjecture})$

\medskip\textbf{LCL010-1.p} YQL depends on YQF

Show that the single Lukasiewicz axiom YQL can be derived
 from the single Lukasiewicz axiom YQF.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(x,z),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(\mbox{yqf},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(c,b),\mbox{equivalent}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_yql},\mbox{negated\_conjecture})$

\medskip\textbf{LCL070-1.p} CN-1 depends on the Wos system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-1 depends on the Wos system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(y),z)),\mbox{implies}(x,\mbox{implies}(\mbox{implies}(u,z),\mbox{implies}(\mbox{implies}(y,u),z)))))\qquad\mbox{cnf}(\mbox{cn}_{60},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL441-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ofalse\_\_imp}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oimp\_\_false}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_G},\mbox{v\_B},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_tt},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{v\_tt},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{v\_tt},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_tt},\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL411-2.p} Propositional logic deduction axioms






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

\medskip\textbf{LCL295-3.p} Principia Mathematica 4.61






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{implies}(p,q)),\mbox{and}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL197-1.p} Principia Mathematica 2.41






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(q,\mbox{or}(p,q))),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL074-1.p} CN-2 depends on the single Merideth axiom

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-2 depends on the single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(z),\mbox{not}(u))),z),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{cn\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL130-1.p} LG-2 depends on P-4

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on P-4.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u)))),x))\qquad\mbox{cnf}(p_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL899+1.p} A bounded pocrim property

A bounded pocrim with commutative strong disjunction is 
 involutive.




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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a,b{:}\;a '==>' b '==>' b = b '==>' a '==>' a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '==>' \mbox{'1'} '==>' \mbox{'1'} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL695$\wedge$1.p} Propositional intuitionistic logic in HOL

An embedding of propositional intuitionisitc logic in HOL based
 on Goedel's first translation of propositional intuitionistic 
 logic to modal logic S4.




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

\medskip\textbf{LCL119-1.p} WO depends on XGJ

Single axioms for the R calculus are QYF, YQM, WO, all
 by Meredith and XGJ by Winker. Show that WO depends on XGJ.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,\mbox{equivalent}(z,x)),\mbox{equivalent}(y,z))))\qquad\mbox{cnf}(\mbox{xgj},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(c,\mbox{equivalent}(b,a))))\qquad\mbox{cnf}(\mbox{prove\_wo},\mbox{negated\_conjecture})$

\medskip\textbf{LCL735$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xa}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xb}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;\forall \mbox{xw}{:}\;b{:}\;( \mbox{xa}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}\mbox{ or }\mbox{xb}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o},\mbox{xw}{:}\;b{:}\;( \mbox{xa}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}\mbox{ or }\mbox{xb}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw} ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB}_5,\mbox{conjecture})$

\medskip\textbf{LCL218-1.p} Principia Mathematica 2.67






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),q)),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL894+1.p} Weak conjunction is lub in a hoop using horn axioms






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$( c '>=' a\mbox{ and }c '>=' b )\iff c '>=' a '+' a '==>' b\qquad\mbox{fof}(\mbox{goals}_{13},\mbox{conjecture})$

\medskip\textbf{LCL194-3.p} Principia Mathematica 2.37






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{or}(q,p),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL289-3.p} Principia Mathematica 4.53






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{and}(p,\mbox{not}(q))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL482+1.p} Prove Rosser's kn3 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_3\qquad\mbox{fof}(\mbox{rosser\_kn}_3,\mbox{conjecture})$

\medskip\textbf{LCL682+1.001.p} In S4, path through a labyrinth, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{16}(x) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{12}(x) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{14}(x) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{12}(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{15}(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{13}(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{12}(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_{11}(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL159-1.p} The 7th alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{xor}(x,\mbox{xor}(\mbox{truth},y))\ne \mbox{xor}(\mbox{xor}(x,\mbox{truth}),y)\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL693$\wedge$1.p} Prove the Barcan formula in the CS4 translation






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

$\forall a{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cs4\_valid}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_all}@ \lambda i{:}\;\mbox{individuals}{:}\;( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ ( a@ i ) ) ) )@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_all}@ \lambda i{:}\;\mbox{individuals}{:}\;( \mbox{cs4\_atom}@ ( a@ i ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cs4\_barcan},\mbox{conjecture})$

\medskip\textbf{LCL173-1.p} Principia Mathematica 2.05






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(p),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL217-3.p} Principia Mathematica 2.65






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{implies}(p,\mbox{not}(q)),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL444-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_H\_\_MP}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL118-1.p} YQM depends on WO

Single axioms for the R calculus are QYF, YQM, WO, all
 by Meredith and XGJ by Winker. Show that YQM depends on WO.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(z,\mbox{equivalent}(y,x))))\qquad\mbox{cnf}(\mbox{wo},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(c,b),\mbox{equivalent}(c,a))))\qquad\mbox{cnf}(\mbox{prove\_yqm},\mbox{negated\_conjecture})$

\medskip\textbf{LCL552+1.p} Prove Hilbert's implies\_1 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{conjecture})$

\medskip\textbf{LCL302-3.p} Principia Mathematica 4.7






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,q),\mbox{implies}(p,\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL514+1.p} Prove Hilbert's equivalence\_3 axiom from Rosser's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{conjecture})$

\medskip\textbf{LCL425-1.p} BCI+mingle implies Karpenko by condensed detachment

Show that if the mingle formula is added to the logic BCI,
 the Karpenko formula can be derived by condensed detachment.




$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(c,b))))\qquad\mbox{cnf}(b,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(b,\mbox{implies}(a,c))))\qquad\mbox{cnf}(c,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),a),c),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(b,a),a),b),c),c)))\qquad\mbox{cnf}(\mbox{mingle},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(b,b),a)),\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a))))\qquad\mbox{cnf}(\mbox{prove\_karpenko},\mbox{negated\_conjecture})$

\medskip\textbf{LCL887+1.p} An idempotent hoop that is not an involutive pocrim






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '+' a = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '==>' \mbox{'1'} '==>' \mbox{'1'} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL226-1.p} Principia Mathematica 2.8






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(q,r)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(r),s)),\mbox{or}(q,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL896+1.p} Associativity of weak conjunction implies commutativity 






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b,c{:}\;a '+' a '==>' b '+' a '+' a '==>' b '==>' c = a '+' a '==>' b '+' b '==>' c\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall x_{17},x_{18}{:}\;x_{17} '+' x_{17} '==>' x_{18} = x_{18} '+' x_{18} '==>' x_{17}\qquad\mbox{fof}(\mbox{goals}_{13},\mbox{conjecture})$

\medskip\textbf{LCL742$\wedge$5.p} TPS problem from AC-FUNS-THMS






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

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

$\exists \mbox{xj}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ ( g@ \mbox{xx} ) ) ) = f )\qquad\mbox{thf}(\mbox{cTHM588AC\_pme},\mbox{conjecture})$

\medskip\textbf{LCL456+1.p} Prove Principia's r3 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_3\qquad\mbox{fof}(\mbox{principia\_r}_3,\mbox{conjecture})$

\medskip\textbf{LCL555+1.p} Prove Hilbert's and\_1 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{conjecture})$

\medskip\textbf{LCL114-1.p} MV-36 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that 36 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{not}(b),\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_mv}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{LCL595$\wedge$1.p} Is axiom T equivalent to reflexivity of R in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

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

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ a ) )\iff ( \mbox{reflexive}@ r )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL620$\wedge$1.p} A simple theorem of propositional logic






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mand}@ x@ y ) )@ ( \mbox{mor}@ ( \mbox{mnot}@ x )@ ( \mbox{mnot}@ y ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL199-1.p} Principia Mathematica 2.45






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(p,q))),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL274-3.p} Principia Mathematica 4.31






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(p,q),\mbox{or}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL078-1.p} CN-40 depends on CN-18 CN-35 CN-46

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-40 depends on the modified Church system
 {CN-18,CN-35,CN-46}.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,x),\mbox{implies}(\mbox{not}(x),\mbox{not}(y))))\qquad\mbox{cnf}(\mbox{cn}_{46},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL032-1.p} C0-1 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-1 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_c0}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL335-3.p} Principia Mathematica 5.35






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(p,q),\mbox{implies}(q,r)),\mbox{implies}(p,\mbox{equivalent}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL615$\wedge$1.p} Axiom KB






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

\medskip\textbf{LCL238-3.p} Principia Mathematica 3.22






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,q),\mbox{and}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL234-1.p} Principia Mathematica 3.2 and 3.12






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL109-1.p} MV-4 depends on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that MV-4 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(b,a)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL189-3.p} Principia Mathematica 2.26






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{implies}(\mbox{implies}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL632$\wedge$1.p} The muddy forehead puzzle






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

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

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

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

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

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )\qquad\mbox{thf}(\mbox{knowledge\_implies\_truth},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(\mbox{positive\_introspection},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) )@ ( \mbox{mbox}@ r@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{negitive\_introspection},\mbox{axiom})$

$\mbox{ck} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ r@ x@ w ) )\qquad\mbox{thf}(\mbox{common\_knowledge},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfb} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_b},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfc} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfa} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfc} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfa} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfb} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_b},\mbox{axiom})$

$s = ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfa} )@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfa} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfb} )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfb} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfc} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{someone\_knows\_its\_forehead},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mnot}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mnot}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mor}@ \mbox{mfa}@ ( \mbox{mor}@ \mbox{mfb}@ \mbox{mfc} ) ) )@ s ) ) )@ s ) ) )@ s )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL685+1.001.p} In S4, pigeonhole formulae missing a conjunct, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_{201}(x)\mbox{ and }\neg\,p_{101}(x) ) )\mbox{ or }\neg\,p_{201}(x)\mbox{ and }p_{101}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL422-1.p} Prove KA2 from MV1--MV4






$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{not}(b))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(\mbox{implies}(\mbox{implies}(b,c),c))),\mbox{not}(\mbox{implies}(\mbox{implies}(b,c),c))))))\qquad\mbox{cnf}(\mbox{prove\_ka}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL275-3.p} Principia Mathematica 4.32






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(r,\mbox{and}(p,q)),\mbox{and}(p,\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL883+1.p} An involutive pocrim that is not a hoop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' \mbox{'1'} = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17},x_{18}{:}\;x_{17} '+' x_{17} '==>' x_{18} = x_{18} '+' x_{18} '==>' x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL411-1.p} Propositional logic deduction axioms






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

\medskip\textbf{LCL445-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_H\_\_MP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_excluded\_\_middle}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL572+1.p} Prove axiom M from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom\_M}\qquad\mbox{fof}(\mbox{km5\_axiom\_M},\mbox{conjecture})$

\medskip\textbf{LCL264-3.p} Principia Mathematica 4.12






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL296-3.p} Principia Mathematica 4.62






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{not}(q)),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL479+1.p} Prove Principia's r5 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_5\qquad\mbox{fof}(\mbox{principia\_r}_5,\mbox{conjecture})$

\medskip\textbf{LCL916$\wedge$1.p} Multi-Modal Logic






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

\medskip\textbf{LCL190-3.p} Principia Mathematica 2.3






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,\mbox{or}(q,r)),\mbox{or}(p,\mbox{or}(r,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL374-1.p} CN-31 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-31 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{implies}(x,z),u),\mbox{implies}(\mbox{implies}(y,u),u))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{31},\mbox{negated\_conjecture})$

\medskip\textbf{LCL636+1.001.p} In K, the branching formula made provable, size 1

The branching formula plus a negation symbol in front and an
 additional subformula to make the formula provable.




$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }( ( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_2(y)\mbox{ and }\neg\,p_{102}(y)\mbox{ and }p_{101}(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ and }\neg\,p_{102}(y)\mbox{ and }p_{101}(y) ) )\mbox{ or }\neg\,\neg\,p_{101}(x)\mbox{ and }p_{100}(x) )\mbox{ and }( ( ( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ or }\neg\,p_{101}(y) )\mbox{ or }p_2(x) )\mbox{ and }( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y)\mbox{ or }\neg\,p_{101}(y) )\mbox{ or }\neg\,p_2(x) ) )\mbox{ or }\neg\,p_{101}(x) )\mbox{ and }( ( ( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y)\mbox{ or }\neg\,p_{100}(y) )\mbox{ or }p_1(x) )\mbox{ and }( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,p_{100}(y) )\mbox{ or }\neg\,p_1(x) ) )\mbox{ or }\neg\,p_{100}(x) )\mbox{ and }( p_{101}(x)\mbox{ or }\neg\,p_{102}(x) )\mbox{ and }( p_{100}(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL324-3.p} Principia Mathematica 5.16






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{not}(\mbox{and}(\mbox{equivalent}(p,q),\mbox{equivalent}(p,\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL580$\wedge$1.p} Popkorn problem 1






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mbox}@ r@ \mbox{mtrue} )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL013-1.p} UM depends on XGF

Show that the single Meredith axiom UM can be derived from
 the single Meredith axiom XGF.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,\mbox{equivalent}(x,z)),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(\mbox{xgf},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(b,\mbox{equivalent}(c,a))))\qquad\mbox{cnf}(\mbox{prove\_um},\mbox{negated\_conjecture})$

\medskip\textbf{LCL142-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{ordered}(x,y)\qquad\mbox{cnf}(\mbox{antecedent},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered}(\mbox{implies}(x,z),\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL213-1.p} Principia Mathematica 2.61






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(p)),q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL139-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,\mbox{not}(\mbox{truth}))\ne \mbox{not}(x)\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL205-3.p} Principia Mathematica 2.51






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(p,q)),\mbox{implies}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL468+1.p} Prove Hilbert's and\_3 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{conjecture})$

\medskip\textbf{LCL211-3.p} Principia Mathematica 2.56






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(q),\mbox{implies}(\mbox{or}(p,q),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL287-3.p} Principia Mathematica 4.5






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,q),\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL524+1.p} Prove axiom B from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{axiom\_B}\qquad\mbox{fof}(\mbox{km4b\_axiom\_B},\mbox{conjecture})$

\medskip\textbf{LCL740$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}_{12}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xr}_{13}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o},\mbox{xw}_4{:}\;b{:}\;( \mbox{xr}_{12}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_4\mbox{ or }\mbox{xr}_{13}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_4 )\mbox{ or }\neg\,\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;\forall \mbox{xw}_4{:}\;b{:}\;( \mbox{xr}_{12}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_4\mbox{ or }\mbox{xr}_{13}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_4 ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB\_R},\mbox{conjecture})$

\medskip\textbf{LCL568+1.p} Prove axiom 4 from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom}_4\qquad\mbox{fof}(\mbox{km4b\_axiom}_4,\mbox{conjecture})$

\medskip\textbf{LCL493+1.p} Prove Hilbert's equivalence\_1 axiom from Principia's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{conjecture})$

\medskip\textbf{LCL002-1.p} AN-CAMerideth =$>$ AN-1

The Whitehead-Russell axiomatisation of the Disjunction/
 Negation 2 valued sentential calculus is {AN-1,AN-2,AN-3,
 AN-4}. Show that AN-1 can be derived from the Merideth axiom.




$( \mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(x),y)),\mbox{or}(z,\mbox{or}(u,v)))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(u),x)),\mbox{or}(z,\mbox{or}(v,x)))))\qquad\mbox{cnf}(\mbox{an\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(b),c)),\mbox{or}(\mbox{not}(\mbox{or}(a,b)),\mbox{or}(a,c))))\qquad\mbox{cnf}(\mbox{an}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL398-1.p} CN-64 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-64 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(x,x)),y))\qquad\mbox{cnf}(\mbox{prove\_cn}_{64},\mbox{negated\_conjecture})$

\medskip\textbf{LCL121-1.p} LG-1 depends on LG-2

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-1
 depends on LG-2.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u))))))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(c,c)),b))))\qquad\mbox{cnf}(\mbox{prove\_lg}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL376-1.p} CN-33 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-33 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,\mbox{implies}(z,\mbox{implies}(x,u))),\mbox{implies}(z,\mbox{implies}(x,u)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{33},\mbox{negated\_conjecture})$

\medskip\textbf{LCL109-6.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{xor}(\mbox{truth},\mbox{and\_star}(x,\mbox{xor}(\mbox{truth},y)))\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(b,a)),\mbox{implies}(b,a))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL903+1.p} A boolean pocrim is idempotent






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' a '==>' a = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '+' x_{17} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL255-3.p} Principia Mathematica 3.48






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(p,r),\mbox{implies}(q,s)),\mbox{implies}(\mbox{or}(p,q),\mbox{or}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL180-3.p} Principia Mathematica 2.14






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(p)),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL225-3.p} Principia Mathematica 2.77






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL182-3.p} Principia Mathematica 2.16






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{not}(q),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL261-3.p} Principia Mathematica 3.12






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL452+1.p} Prove Lukasiewicz's cn2 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_2\qquad\mbox{fof}(\mbox{luka\_cn}_2,\mbox{conjecture})$

\medskip\textbf{LCL684+1.001.p} In S4, pigeonhole formulae, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_{201}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,p_{201}(x)\mbox{ and }p_{101}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL325-3.p} Principia Mathematica 5.17






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{or}(p,q),\mbox{not}(\mbox{and}(p,q))),\mbox{equivalent}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL439-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_notE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL276-3.p} Principia Mathematica 4.33






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(r,\mbox{or}(p,q)),\mbox{or}(p,\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL579$\wedge$2.p} Leibniz-equality definition means it's an equivalence






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

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

$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ \mbox{cY} ) \ \Rightarrow \ ( p@ \mbox{cX} ) ) \ \Rightarrow \ \forall r{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( r@ \mbox{cX} ) \ \Rightarrow \ ( r@ \mbox{cY} ) )\qquad\mbox{thf}(\mbox{cTHM76A},\mbox{conjecture})$

\medskip\textbf{LCL507+1.p} Prove Hilbert's and\_2 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{conjecture})$

\medskip\textbf{LCL868$\wedge$1.p} Modal logic D45 'includes' modal logic M5






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r ) \ \Rightarrow \ ( \mbox{mserial}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL703$\wedge$1.p} Accessibility relation implies axiom for Euclidianity






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

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

\medskip\textbf{LCL221-3.p} Principia Mathematica 2.73






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{or}(\mbox{or}(p,q),r),\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL624$\wedge$1.p} A simple theorem of K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},p{:}\;\mbox{\$i}\to \mbox{\$o},q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mand}@ ( \mbox{mdia}@ r@ ( \mbox{mbox}@ r@ p ) )@ ( \mbox{mbox}@ r@ ( \mbox{mdia}@ r@ q ) ) )@ ( \mbox{mdia}@ r@ ( \mbox{mdia}@ r@ ( \mbox{mand}@ p@ q ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL531+1.p} Prove axiom m4 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_4\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_4,\mbox{conjecture})$

\medskip\textbf{LCL113-2.p} MV-33 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-33 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(\mbox{not}(y),x))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{33},\mbox{negated\_conjecture})$

\medskip\textbf{LCL554+1.p} Prove Hilbert's implies\_3 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{conjecture})$

\medskip\textbf{LCL164-1.p} The 4th Wajsberg algebra axiom, from the alternative axioms






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{xor}(\mbox{truth},\mbox{and\_star}(x,\mbox{xor}(\mbox{truth},y)))\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL630$\wedge$1.p} The muddy forehead puzzle






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

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

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

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

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

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )\qquad\mbox{thf}(\mbox{knowledge\_implies\_truth},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(\mbox{positive\_introspection},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) )@ ( \mbox{mbox}@ r@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{negitive\_introspection},\mbox{axiom})$

$\mbox{ck} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ r@ x@ w ) )\qquad\mbox{thf}(\mbox{common\_knowledge},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfb} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_b},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfc} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfa} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfc} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfa} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfb} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_b},\mbox{axiom})$

$s = ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfa} )@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfa} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfb} )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfb} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfc} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{someone\_knows\_its\_forehead},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mor}@ \mbox{mfa}@ ( \mbox{mor}@ \mbox{mfb}@ \mbox{mfc} ) ) )@ s ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL633$\wedge$1.p} Goedel's ontological argument on the existence of God






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

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

$\forall x{:}\;a\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( p@ x ) )@ ( p@ \lambda z{:}\;a{:}\;( \mbox{mnot}@ ( x@ z ) ) ) ) )\qquad\mbox{thf}(\mbox{positiveness},\mbox{axiom})$

$g = ( \lambda z{:}\;a,w{:}\;\mbox{\$i}{:}\;\forall x{:}\;a\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimpl}@ ( p@ x )@ ( x@ z )@ w ) )\qquad\mbox{thf}(g,\mbox{definition})$

$e = ( \lambda x{:}\;a\to \mbox{\$i}\to \mbox{\$o},z{:}\;a,p{:}\;\mbox{\$i}{:}\;\forall y{:}\;a\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimpl}@ ( y@ z )@ ( \mbox{mbox}@ r@ \lambda q{:}\;\mbox{\$i}{:}\;\forall w{:}\;a{:}\;( \mbox{mimpl}@ ( x@ w )@ ( y@ w )@ q ) )@ p ) )\qquad\mbox{thf}(e,\mbox{definition})$

$\mbox{mvalid}@ \lambda w{:}\;\mbox{\$i}{:}\;\forall z{:}\;a{:}\;( \mbox{mimpl}@ ( g@ z )@ ( e@ g@ z )@ w )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL384-1.p} CN-45 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-45 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(\mbox{not}(z),y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{45},\mbox{negated\_conjecture})$

\medskip\textbf{LCL222-3.p} Principia Mathematica 2.74






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,p),\mbox{implies}(\mbox{or}(\mbox{or}(p,q),r),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL068-1.p} CN-2 depends on the second Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-2 depends on the second Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),z),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),z))))\qquad\mbox{cnf}(\mbox{cn}_{59},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL441-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_a},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

\medskip\textbf{LCL301-3.p} Principia Mathematica 4.67






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{implies}(\mbox{not}(p),\mbox{not}(q))),\mbox{and}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL473+1.p} Prove Hilbert's equivalence\_2 axiom from Lukasiewicz's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{conjecture})$

\medskip\textbf{LCL294-3.p} Principia Mathematica 4.6






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,q),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL867$\wedge$1.p} Modal logic KB5 coincides with K4B






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL185-1.p} Principia Mathematica 2.2






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL017-1.p} YRO depends on PYO

Show that the single Meredith axiom YRO can be derived from
 the single Meredith axiom PYO.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),z),\mbox{equivalent}(y,x)))\qquad\mbox{cnf}(\mbox{pyo},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,\mbox{equivalent}(\mbox{equivalent}(c,b),a))))\qquad\mbox{cnf}(\mbox{prove\_yro},\mbox{negated\_conjecture})$

\medskip\textbf{LCL884+1.p} A bounded hoop that is not an involutive pocrim






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '==>' \mbox{'1'} '==>' \mbox{'1'} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL657+1.001.p} In KT, the branching formula, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL246-1.p} Principia Mathematica 3.35






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{or}(\mbox{not}(p),q))))),q))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL419-1.p} Prove AK1 from MV1--MV4






$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(b),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(b),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(a,b),b)),\mbox{not}(\mbox{implies}(\mbox{implies}(a,c),c))),\mbox{not}(\mbox{implies}(\mbox{implies}(a,c),c))))))\qquad\mbox{cnf}(\mbox{prove\_ak}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL518+1.p} Prove Principia's r1 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_1\qquad\mbox{fof}(\mbox{principia\_r}_1,\mbox{conjecture})$

\medskip\textbf{LCL929-1.p} AxK in TW$\to$ [AxL] + (Resid)






$( p(i(a,b))\mbox{ and }p(a) ) \ \Rightarrow \ p(b)\qquad\mbox{cnf}(\mbox{modus\_ponens},\mbox{axiom})$

$p(i(f(a,b),c)) \ \Rightarrow \ p(i(a,i(b,c)))\qquad\mbox{cnf}(\mbox{resid}_1,\mbox{axiom})$

$p(i(a,i(b,c))) \ \Rightarrow \ p(i(f(a,b),c))\qquad\mbox{cnf}(\mbox{resid}_2,\mbox{axiom})$

$p(i(a,a))\qquad\mbox{cnf}(\mbox{axI},\mbox{axiom})$

$p(i(i(a,b),i(i(b,c),i(a,c))))\qquad\mbox{cnf}(\mbox{axBp},\mbox{axiom})$

$p(i(i(i(a,b),b),i(i(b,a),a)))\qquad\mbox{cnf}(\mbox{axL},\mbox{axiom})$

$\neg\,p(i(a,i(b,a)))\qquad\mbox{cnf}(\mbox{axK},\mbox{negated\_conjecture})$

\medskip\textbf{LCL169-1.p} Principia Mathematica 2.01






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(p))),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL030-1.p} C0-6 depends on the Tarski-Bernays system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-6 can be derived from the
 Tarski-Bernays system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{c0}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{falsehood},x))\qquad\mbox{cnf}(\mbox{c0}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_c0}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LCL618$\wedge$1.p} Axiom GL implies Axiom K4 in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )@ ( \mbox{mbox}@ r@ x ) ) )\qquad\mbox{thf}(\mbox{gl},\mbox{axiom})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(k_4,\mbox{conjecture})$

\medskip\textbf{LCL111-1.p} MV-25 depends on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that MV-25 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(c,b))))\qquad\mbox{cnf}(\mbox{prove\_mv}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{LCL396-1.p} CN-62 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-62 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{not}(x)),y),\mbox{implies}(x,y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{62},\mbox{negated\_conjecture})$

\medskip\textbf{LCL279-3.p} Principia Mathematica 4.38






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{equivalent}(p,r),\mbox{equivalent}(q,s)),\mbox{equivalent}(\mbox{and}(p,q),\mbox{and}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL471+1.p} Prove Hilbert's or\_3 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{conjecture})$

\medskip\textbf{LCL037-1.p} C0-6 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-6 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_c0}_6,\mbox{negated\_conjecture})$

\medskip\textbf{LCL181+1.p} Principia Mathematica 2.15

Judged by [SRM73] to be the 'hardest' of the first 52 theorems
 of [WR27].




$( \neg\,p \ \Rightarrow \ q )\iff ( \neg\,q \ \Rightarrow \ p )\qquad\mbox{fof}(\mbox{pel}_4,\mbox{conjecture})$

\medskip\textbf{LCL545+1.p} Prove axiom m5 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_5\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_5,\mbox{conjecture})$

\medskip\textbf{LCL680+1.001.p} In S4, in backward search find a way through box and dia, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,p_1(x)\mbox{ or }p_1(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL265-3.p} Principia Mathematica 4.13






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{not}(\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL925$\wedge$1.p} Embedding of quantified multimodal logic in simple type theory






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

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

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

\medskip\textbf{LCL277-3.p} Principia Mathematica 4.36






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{and}(p,r),\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL020-1.p} XGK depends on XHK

Show that the single Kalman axiom XGK can be derived from the
 single Winker axiom XHK.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(x,z),y))))\qquad\mbox{cnf}(\mbox{xhk},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(c,a)),\mbox{equivalent}(c,b))))\qquad\mbox{cnf}(\mbox{prove\_xgk},\mbox{negated\_conjecture})$

\medskip\textbf{LCL502+1.p} Prove Hilbert's modus\_tollens axiom from Rosser's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{conjecture})$

\medskip\textbf{LCL477+1.p} Prove Principia's r3 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_3\qquad\mbox{fof}(\mbox{principia\_r}_3,\mbox{conjecture})$

\medskip\textbf{LCL705$\wedge$1.p} Accessibility relation implies axiom for functionality






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

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

\medskip\textbf{LCL044-1.p} CN-40 depends on Hilbert's system

Two axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-18,CN-21,CN-35,CN-39,CN-39,
 CN-40,CN-46} by Frege and {CN-3,CN-18,CN-21,CN-22,CN-30,
 CN-54} by Hilbert. Show that CN-40 depends on the simplified
 Hilbert system (without CN-30).




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(y,\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{21},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{22},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),y)))\qquad\mbox{cnf}(\mbox{cn}_{54},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL085-1.p} IC-5 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-5 depends on the first Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(a,b),b)))\qquad\mbox{cnf}(\mbox{prove\_ic}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL928-1.p} AxTO in BCK$\to$ [AxL] + (Resid)






$( p(i(a,b))\mbox{ and }p(a) ) \ \Rightarrow \ p(b)\qquad\mbox{cnf}(\mbox{modus\_ponens},\mbox{axiom})$

$p(i(f(a,b),c)) \ \Rightarrow \ p(i(a,i(b,c)))\qquad\mbox{cnf}(\mbox{resid}_1,\mbox{axiom})$

$p(i(a,i(b,c))) \ \Rightarrow \ p(i(f(a,b),c))\qquad\mbox{cnf}(\mbox{resid}_2,\mbox{axiom})$

$p(i(i(a,b),i(i(b,c),i(a,c))))\qquad\mbox{cnf}(\mbox{axBp},\mbox{axiom})$

$p(i(i(i(a,b),b),i(i(b,a),a)))\qquad\mbox{cnf}(\mbox{axL},\mbox{axiom})$

$p(i(i(a,i(b,c)),i(b,i(a,c))))\qquad\mbox{cnf}(\mbox{axC},\mbox{axiom})$

$\neg\,p(i(i(i(c_1,c_2),i(c_2,c_1)),i(c_2,c_1)))\qquad\mbox{cnf}(\mbox{axTO},\mbox{negated\_conjecture})$

\medskip\textbf{LCL226-3.p} Principia Mathematica 2.8






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(q,r),\mbox{implies}(\mbox{or}(\mbox{not}(r),s),\mbox{or}(q,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL271-3.p} Principia Mathematica 4.24






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{and}(p,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL619$\wedge$1.p} A simple theorem of K4






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(k_4,\mbox{axiom})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ x@ y ) )@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL199-3.p} Principia Mathematica 2.45






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{or}(p,q)),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL592$\wedge$1.p} Axiom D is valid






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ a@ b ) )@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL918$\wedge$1.p} Modal logic K






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

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

\medskip\textbf{LCL877$\wedge$2.p} Variants of axiom 5






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

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

\medskip\textbf{LCL207-1.p} Principia Mathematica 2.521






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),q))),\mbox{or}(\mbox{not}(q),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL244-1.p} Principia Mathematica 3.33






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{not}(\mbox{or}(\mbox{not}(q),r))))),\mbox{or}(\mbox{not}(p),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL447-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

\medskip\textbf{LCL003-1.p} AN-CAMerideth =$>$ AN-2

The Whitehead-Russell axiomatisation of the Disjunction/
 Negation 2 valued sentential calculus is {AN-1,AN-2,AN-3,
 AN-4}. Show that AN-2 can be derived from the Merideth axiom.




$( \mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(x),y)),\mbox{or}(z,\mbox{or}(u,v)))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(u),x)),\mbox{or}(z,\mbox{or}(v,x)))))\qquad\mbox{cnf}(\mbox{an\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(a,b)),\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{an}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL115-1.p} MV-39 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that 39 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(a,b)),\mbox{not}(b)))\qquad\mbox{cnf}(\mbox{prove\_mv}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL140-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{not}(\mbox{not}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL645+1.001.p} In K, L+ is not provable with instances of L, size 1






$\neg\,\exists x{:}\;\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL193-3.p} Principia Mathematica 2.36






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{or}(p,q),\mbox{or}(r,p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL486+1.p} Prove Hilbert's implies\_3 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{conjecture})$

\medskip\textbf{LCL402-1.p} CN-68 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-68 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(x,\mbox{not}(y))),y))\qquad\mbox{cnf}(\mbox{prove\_cn}_{68},\mbox{negated\_conjecture})$

\medskip\textbf{LCL535+1.p} Prove axiom m9 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_9\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_9,\mbox{conjecture})$

\medskip\textbf{LCL086-1.p} IC-1 depends on the 4th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-1 depends on the fourth Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(t,\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,a))\qquad\mbox{cnf}(\mbox{prove\_ic}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL064-2.p} CN-1 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-1 depends on a simplified Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL699$\wedge$1.p} Accessibility relation implies axiom for reflexivity






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

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

\medskip\textbf{LCL223-1.p} Principia Mathematica 2.75






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{or}(\mbox{not}(q),r))),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL328-3.p} Principia Mathematica 5.23






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{equivalent}(p,q),\mbox{or}(\mbox{and}(p,q),\mbox{and}(\mbox{not}(p),\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL522+1.p} Prove Principia's r5 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_5\qquad\mbox{fof}(\mbox{principia\_r}_5,\mbox{conjecture})$

\medskip\textbf{LCL594$\wedge$1.p} Relation for all propositions making T valid in K

Is there a relation R such that for all modal propositions A,
 axiom T is valid in K




$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ a ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL290-3.p} Principia Mathematica 4.54






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{and}(p,q)),\mbox{not}(\mbox{or}(p,\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL390-1.p} CN-53 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-53 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(\mbox{implies}(x,y),y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{53},\mbox{negated\_conjecture})$

\medskip\textbf{LCL696$\wedge$1.p} Propositional intuitionistic logic in HOL

An embedding of propositional intuitionisitc logic in HOL based
 on Goedel's second translation of propositional intuitionistic
 logic to modal logic S4.




$\mbox{include}(\mbox{'Axioms/LCL011\string^0.ax'})$

\medskip\textbf{LCL688+1.001.p} In S4, formula with T and A4, size 1

T{dia p0/p0} \& box T{~box dia p0/p0} \& A4{dia p0/p0} \&
 box(dia box dia p0 $\to$ (p0 $\to$ box p0)) $\to$ dia box p0 |
 dia box ~p0.




$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_4(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_3(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_3(x) )\mbox{ or }\neg\,p_2(y) )\mbox{ and }p_2(x)\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y) ) ) ) ) )\mbox{ or }( \neg\,p_1(x)\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ and }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) )\mbox{ and }p_1(x) ) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL442-2.p} Problem about propositional logic






$\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_1,\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_pl}_1,\mbox{c\_PropLog\_Othms}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_1,\mbox{v\_tt},\mbox{t\_a}),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_1,\mbox{v\_tt},\mbox{t\_a}),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ or }\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_1,\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_2,\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_pl}_2,\mbox{c\_PropLog\_Othms}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_2,\mbox{v\_tt},\mbox{t\_a}),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_2,\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_2,\mbox{v\_tt},\mbox{t\_a}),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ or }\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_2,\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{v\_pl}_2,\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_1,\mbox{v\_tt},\mbox{t\_a}),\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_2,\mbox{v\_tt},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_1,\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_2,\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{v\_pl}_2,\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_1,\mbox{v\_tt},\mbox{t\_a}),\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_2,\mbox{v\_tt},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pl}_1,\mbox{v\_pl}_2,\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_1,\mbox{v\_tt},\mbox{t\_a}),\mbox{c\_PropLog\_Ohyps}(\mbox{v\_pl}_2,\mbox{v\_tt},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_1,\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_PropLog\_Oeval}(\mbox{v\_tt},\mbox{v\_pl}_2,\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_9,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ofalse\_\_imp}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oimp\_\_false}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_G},\mbox{v\_B},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

\medskip\textbf{LCL426-1.p} Prove the mingle formula by condensed detachment

Show that the mingle axiom can be derived from the three formulas
 given below by condensed detachment.




$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{suffixing},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(a,b),b)))\qquad\mbox{cnf}(\mbox{assertion},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),c),\mbox{implies}(b,a)),c),c))\qquad\mbox{cnf}(\mbox{candidate},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),a),c),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(b,a),a),b),c),c)))\qquad\mbox{cnf}(\mbox{prove\_mingle},\mbox{negated\_conjecture})$

\medskip\textbf{LCL520+1.p} Prove Principia's r3 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_3\qquad\mbox{fof}(\mbox{principia\_r}_3,\mbox{conjecture})$

\medskip\textbf{LCL331-3.p} Principia Mathematica 5.3






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{and}(p,q),r),\mbox{implies}(\mbox{and}(p,q),\mbox{and}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL220-3.p} Principia Mathematica 2.69






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),q),\mbox{implies}(\mbox{implies}(q,p),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL560+1.p} Prove Hilbert's or\_3 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{conjecture})$

\medskip\textbf{LCL377-1.p} CN-34 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-34 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,\mbox{implies}(z,u))),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(y,\mbox{implies}(z,u)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{34},\mbox{negated\_conjecture})$

\medskip\textbf{LCL230+1.p} Principia Mathematica 2.85

Judged by [SRM73] to be the 'hardest' of the
 first 67 theorems of [WR27].




$( ( p\mbox{ or }q ) \ \Rightarrow \ ( p\mbox{ or }r ) ) \ \Rightarrow \ ( p\mbox{ or }( q \ \Rightarrow \ r ) )\qquad\mbox{fof}(\mbox{pel}_5,\mbox{conjecture})$

\medskip\textbf{LCL052-1.p} CN-24 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-24 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{24},\mbox{negated\_conjecture})$

\medskip\textbf{LCL715$\wedge$1.p} Axiom implies accessibility relation for functionality






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

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

\medskip\textbf{LCL063-1.p} CN-CAMerideth depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that the single Meredith axiom depends on the short
 Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{not}(c),\mbox{not}(e))),c),\mbox{falsehood}),\mbox{implies}(\mbox{implies}(\mbox{falsehood},a),\mbox{implies}(e,a))))\qquad\mbox{cnf}(\mbox{prove\_cn\_CAMerideth},\mbox{negated\_conjecture})$

\medskip\textbf{LCL146-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{not}(\mbox{big\_hat}(x,y))\ne \mbox{big\_V}(\mbox{not}(x),\mbox{not}(y))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL245-1.p} Principia Mathematica 3.34






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{not}(\mbox{or}(\mbox{not}(p),q))))),\mbox{or}(\mbox{not}(p),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL901+1.p} An idempotent pocrim property

An idempotent pocrim with commutative strong disjunction is 
 boolean.




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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a,b{:}\;a '==>' b '==>' b = b '==>' a '==>' a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall a{:}\;a '+' a = a\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '==>' \mbox{'1'} '==>' x_{17} '==>' x_{17} = \mbox{'0'}\qquad\mbox{fof}(\mbox{goals}_{15},\mbox{conjecture})$

\medskip\textbf{LCL171-3.p} Principia Mathematica 2.03






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{not}(q)),\mbox{implies}(q,\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL497+1.p} Prove Lukasiewicz's cn2 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_2\qquad\mbox{fof}(\mbox{luka\_cn}_2,\mbox{conjecture})$

\medskip\textbf{LCL105-1.p} LG-2 depends on the 7th McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-2 depends on the seventh McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,y),x))))\qquad\mbox{cnf}(q_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,x),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(q_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(y,x),z)),z))\qquad\mbox{cnf}(q_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(a,c)),\mbox{equivalent}(b,c)),e),e))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL412-1.p} Propositional logic deduction axioms for AND






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

\medskip\textbf{LCL203-3.p} Principia Mathematica 2.49






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL898+1.p} Strong disjunction is commutative in an involutive hoop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' \mbox{'1'} = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$\forall x_{17},x_{18}{:}\;x_{17} '==>' x_{18} '==>' x_{18} = x_{18} '==>' x_{17} '==>' x_{17}\qquad\mbox{fof}(\mbox{goals}_{15},\mbox{conjecture})$

\medskip\textbf{LCL912+1.p} KM4B axiomatization of S5 based on Hilbert's PC






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

\medskip\textbf{LCL557+1.p} Prove Hilbert's and\_3 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{conjecture})$

\medskip\textbf{LCL407-1.p} Wajsberg algebra axioms






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

\medskip\textbf{LCL150-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{implies}(\mbox{big\_hat}(x,y),z)\ne \mbox{big\_V}(\mbox{implies}(x,z),\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL536+1.p} Prove axiom m10 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_{10}\qquad\mbox{fof}(\mbox{s1\_0\_m10\_axiom\_m}_{10},\mbox{conjecture})$

\medskip\textbf{LCL221-1.p} Principia Mathematica 2.73






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{or}(p,q),r)),\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL100-1.p} LG-3 depends on the 2nd McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-3 depends on the second McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(\mbox{equivalent}(u,y),\mbox{equivalent}(\mbox{equivalent}(x,u),z))))\qquad\mbox{cnf}(p_1,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(a,c)),e),\mbox{equivalent}(\mbox{equivalent}(b,c),e)),\mbox{falsehood}),\mbox{falsehood}))\qquad\mbox{cnf}(\mbox{prove\_lg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL028-1.p} C0-CAMerideth depends on the Church system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that the Meredith axiom can be derived
 from the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{falsehood}),\mbox{falsehood}),x))\qquad\mbox{cnf}(\mbox{c0}_5,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_6,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(c,\mbox{falsehood})),e),\mbox{falsehood}),\mbox{implies}(\mbox{implies}(\mbox{falsehood},a),\mbox{implies}(c,a))))\qquad\mbox{cnf}(\mbox{prove\_c0\_CAMerideth},\mbox{negated\_conjecture})$

\medskip\textbf{LCL509+1.p} Prove Hilbert's or\_1 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{conjecture})$

\medskip\textbf{LCL454+1.p} Prove Principia's r1 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_1\qquad\mbox{fof}(\mbox{principia\_r}_1,\mbox{conjecture})$

\medskip\textbf{LCL251-1.p} Principia Mathematica 3.43






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{not}(\mbox{or}(\mbox{not}(p),r))))),\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{or}(\mbox{not}(q),\mbox{not}(r))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL077-1.p} CN-39 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-39 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(a)),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL513+1.p} Prove Hilbert's equivalence\_2 axiom from Rosser's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{conjecture})$

\medskip\textbf{LCL701$\wedge$1.p} Accessibility relation implies axiom for seriality






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

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

\medskip\textbf{LCL590$\wedge$1.p} Popkorn problem 11






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mimpl}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mbox}@ r@ b ) )@ ( \mbox{mimpl}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mdia}@ r@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL461+1.p} Prove Rosser's kn3 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_3\qquad\mbox{fof}(\mbox{rosser\_kn}_3,\mbox{conjecture})$

\medskip\textbf{LCL172-3.p} Principia Mathematica 2.04






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(q,\mbox{implies}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL679+1.001.p} In S4, formula not provable in intuitionistic logic, size 1






$\neg\,\exists x{:}\;\neg\,\mbox{\$false}\mbox{ or }\mbox{\$false}\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL011-1.p} YQF depends on YQJ

Show that the single Lukasiewicz axiom YQF can be derived
 from the single Lukasiewicz axiom YQJ.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,x),\mbox{equivalent}(y,z))))\qquad\mbox{cnf}(\mbox{yqj},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(a,c),\mbox{equivalent}(c,b))))\qquad\mbox{cnf}(\mbox{prove\_yqf},\mbox{negated\_conjecture})$

\medskip\textbf{LCL214-1.p} Principia Mathematica 2.61






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL001-1.p} The Whitehead-Russell system =$>$ the Meredith axiom

The Whitehead-Russell axiomatisation of the Disjunction/
 Negation 2 valued sentential calculus is {AN-1,AN-2,AN-3,
 AN-4}. Show that the Merideth axiom can be derived from the
 Whitehead-Russell axiomatisation.




$( \mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(y),z)),\mbox{or}(\mbox{not}(\mbox{or}(x,y)),\mbox{or}(x,z))))\qquad\mbox{cnf}(\mbox{an}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(x,y)),\mbox{or}(y,x)))\qquad\mbox{cnf}(\mbox{an}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),\mbox{or}(y,x)))\qquad\mbox{cnf}(\mbox{an}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(x,x)),x))\qquad\mbox{cnf}(\mbox{an}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(a),b)),\mbox{or}(c,\mbox{or}(e,\mbox{falsehood})))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(e),a)),\mbox{or}(c,\mbox{or}(\mbox{falsehood},a)))))\qquad\mbox{cnf}(\mbox{prove\_an\_CAMerideth},\mbox{negated\_conjecture})$

\medskip\textbf{LCL741$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}{:}\;( b\to \mbox{\$o} )\to b\to \mbox{\$o},\mbox{xr}_{27}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xr}_{28}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;\forall \mbox{xw}_{11}{:}\;b{:}\;( \mbox{xr}_{27}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_{11}\mbox{ or }\mbox{xr}_{28}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_{11} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o},\mbox{xw}_{11}{:}\;b{:}\;( \mbox{xr}_{27}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_{11}\mbox{ or }\mbox{xr}_{28}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_{11} ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310X\_pme},\mbox{conjecture})$

\medskip\textbf{LCL016-1.p} YRM depends on YRO

Show that the single Meredith axiom YRM can be derived from
 the single Meredith axiom YRO.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(z,\mbox{equivalent}(\mbox{equivalent}(z,y),x))))\qquad\mbox{cnf}(\mbox{yro},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,\mbox{equivalent}(\mbox{equivalent}(b,c),a))))\qquad\mbox{cnf}(\mbox{prove\_yrm},\mbox{negated\_conjecture})$

\medskip\textbf{LCL549+1.p} Prove axiom m10 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_{10}\qquad\mbox{fof}(\mbox{s1\_0\_m10\_axiom\_m}_{10},\mbox{conjecture})$

\medskip\textbf{LCL394-1.p} CN-58 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-58 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),z)),u),\mbox{implies}(\mbox{implies}(\mbox{not}(x),z),u)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{58},\mbox{negated\_conjecture})$

\medskip\textbf{LCL505+1.p} Prove Hilbert's implies\_3 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{conjecture})$

\medskip\textbf{LCL420-1.p} Prove AK2 from MV1--MV4






$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(a,b),b)),\mbox{not}(\mbox{implies}(\mbox{implies}(a,c),c))),\mbox{not}(\mbox{implies}(\mbox{implies}(a,c),c)))),\mbox{implies}(\mbox{implies}(a,\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(b),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(b),\mbox{not}(c)),\mbox{not}(c))))))\qquad\mbox{cnf}(\mbox{prove\_ak}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL443-2.p} Problem about propositional logic






$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL691$\wedge$1.p} Prove reflexivity in the CS4 translation






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cs4\_valid}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ a ) )@ ( \mbox{cs4\_atom}@ a ) ) )\qquad\mbox{thf}(\mbox{cs4\_refl},\mbox{conjecture})$

\medskip\textbf{LCL511+1.p} Prove Hilbert's or\_3 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{conjecture})$

\medskip\textbf{LCL462+1.p} Prove Hilbert's modus\_tollens axiom from Lukasiewicz's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{conjecture})$

\medskip\textbf{LCL166-1.p} UM depends on XHN

Show that the single Meredith axiom UM can be derived from
 the single Winker axiom XHN.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,x),y))))\qquad\mbox{cnf}(\mbox{xhn},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(b,\mbox{equivalent}(c,a))))\qquad\mbox{cnf}(\mbox{prove\_um},\mbox{negated\_conjecture})$

\medskip\textbf{LCL102-1.p} P-1 depends on the 4th McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that P-1 depends on the 4th McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,y),x))))\qquad\mbox{cnf}(q_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,x),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(q_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(\mbox{equivalent}(e,b),\mbox{equivalent}(\mbox{equivalent}(a,e),c))))\qquad\mbox{cnf}(\mbox{prove\_p}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL722$\wedge$1.p} Axiom M holds in monomodal logic M






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_m}@ \mbox{phi} )@ \mbox{phi} )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL309-3.p} Principia Mathematica 4.78






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(\mbox{implies}(p,q),\mbox{implies}(p,r)),\mbox{implies}(p,\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL427-1.p} ORG-D23 is a single axiom for propositional calculus

Show that formula ORG-D23 is a single axiom for propositional
 calculus in terms of the Sheffer stroke by deriving Nicod's
 single axiom.




$( \mbox{is\_a\_theorem}(\mbox{detach}(a,\mbox{detach}(b,c)))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(c)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{detach}(\mbox{detach}(a,\mbox{detach}(b,c)),\mbox{detach}(\mbox{detach}(a,\mbox{detach}(b,c)),\mbox{detach}(\mbox{detach}(d,c),\mbox{detach}(\mbox{detach}(c,d),\mbox{detach}(a,d))))))\qquad\mbox{cnf}(\mbox{org\_d}_{23},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{detach}(\mbox{detach}(a,\mbox{detach}(b,c)),\mbox{detach}(\mbox{detach}(e,\mbox{detach}(e,e)),\mbox{detach}(\mbox{detach}(f,b),\mbox{detach}(\mbox{detach}(a,f),\mbox{detach}(a,f))))))\qquad\mbox{cnf}(\mbox{prove\_nicod},\mbox{negated\_conjecture})$

\medskip\textbf{LCL732$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

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

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

$\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}_0{:}\;b{:}\;( \exists \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx}_0 ) \ \Rightarrow \ ( p@ y ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xx}_0{:}\;\mbox{\$i}{:}\;( p@ \mbox{xx}_0 ) \ \Rightarrow \ ( p@ y ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB}_3,\mbox{conjecture})$

\medskip\textbf{LCL255-1.p} Principia Mathematica 3.48






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),r)),\mbox{not}(\mbox{or}(\mbox{not}(q),s))))),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL446-2.p} Problem about propositional logic






$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

\medskip\textbf{LCL239-3.p} Principia Mathematica 3.24






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{not}(\mbox{and}(p,\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL137-1.p} A lemma in Wajsberg algebras

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided
 a different axiomatisation. Show that MV-3 depends on the
 Wajsberg system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z)))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL363-1.p} CN-12 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-12 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(\mbox{not}(y),y),y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{12},\mbox{negated\_conjecture})$

\medskip\textbf{LCL316-3.p} Principia Mathematica 4.85






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{implies}(r,p),\mbox{implies}(r,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL865$\wedge$1.p} Modal logic S5(=M5) coincides with DB5






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mserial}@ r\mbox{ and }\mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL718$\wedge$1.p} Axiom implies accessibility relation for weak directedness






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

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

\medskip\textbf{LCL327-3.p} Principia Mathematica 5.21






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{not}(p),\mbox{not}(q)),\mbox{equivalent}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL720$\wedge$1.p} Distribution axiom holds in monomodal logic K






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

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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ ( \mbox{mimplies}@ \mbox{phi}@ \mbox{psi} ) )@ ( \mbox{mimplies}@ ( \mbox{mbox\_k}@ \mbox{phi} )@ ( \mbox{mbox\_k}@ \mbox{psi} ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL242-3.p} Principia Mathematica 3.3






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{and}(p,q),r),\mbox{implies}(p,\mbox{implies}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL890+1.p} Halving is unique in a hoop, rule for a/2 $>$= x






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} '==>' x_{18} '>=' x_{17}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{19} '>=' x_{17} )\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL712$\wedge$1.p} Axiom implies accessibility relation for transitivity






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

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

\medskip\textbf{LCL599$\wedge$1.p} Is axiom 4 equivalent to symmetry?






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\iff ( \mbox{symmetric}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL160-1.p} The 8th alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{and\_star}(\mbox{xor}(\mbox{and\_star}(\mbox{xor}(\mbox{truth},x),y),\mbox{truth}),y)\ne \mbox{and\_star}(\mbox{xor}(\mbox{and\_star}(\mbox{xor}(\mbox{truth},y),x),\mbox{truth}),x)\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL214-3.p} Principia Mathematica 2.61






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{implies}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL893+1.p} In a coop, x/2 = x implies x = 0






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall a{:}\;h(a) = h(a) '==>' a\qquad\mbox{fof}(\mbox{sos}_{14},\mbox{axiom})$

$\forall x_{17}{:}\;( h(x_{17}) = x_{17} \ \Rightarrow \ x_{17} = \mbox{'0'} )\qquad\mbox{fof}(\mbox{goals}_{15},\mbox{conjecture})$

\medskip\textbf{LCL472+1.p} Prove Hilbert's equivalence\_1 axiom from Lukasiewicz's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{conjecture})$

\medskip\textbf{LCL210-3.p} Principia Mathematica 2.55






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(p),\mbox{implies}(\mbox{or}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL317-3.p} Principia Mathematica 4.86






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{equivalent}(p,r),\mbox{equivalent}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL450+1.p} Congruence of equiv lemmas, to admit substitution of equivalents






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_ponens},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{axiom})$

$\forall x{:}\;\mbox{is\_a\_theorem}(\mbox{equiv}(x,x))\mbox{ and }\forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{not}(x),\mbox{not}(y))) )\mbox{ and }\forall x_1,x_2,y_1,y_2{:}\;( ( \mbox{is\_a\_theorem}(\mbox{equiv}(x_1,x_2))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{equiv}(y_1,y_2)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{and}(x_1,y_1),\mbox{and}(x_2,y_2))) )\mbox{ and }\forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) )\qquad\mbox{fof}(\mbox{equiv\_congruence},\mbox{conjecture})$

\medskip\textbf{LCL730$\wedge$5.p} TPS problem X5310

Related to the axiom of choice.




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

$\forall \mbox{xr}{:}\;( b\to \mbox{\$o} )\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;( \mbox{xr}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX}_{5310},\mbox{conjecture})$

\medskip\textbf{LCL496+1.p} Prove Lukasiewicz's cn1 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_1\qquad\mbox{fof}(\mbox{luka\_cn}_1,\mbox{conjecture})$

\medskip\textbf{LCL644+1.001.p} In K, H2 $\to$ L, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x) )\mbox{ or }p_1(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y)\mbox{ or }\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) )\mbox{ and }p_1(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }p_1(x) )\mbox{ or }p_2(y)\mbox{ or }\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) )\mbox{ and }p_1(y) )\mbox{ and }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x) )\mbox{ or }p_1(y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_1(x)\mbox{ and }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) )\mbox{ and }p_1(x) ) )\mbox{ and }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x) )\mbox{ or }p_1(y)\mbox{ or }p_2(y)\mbox{ or }\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) )\mbox{ and }p_1(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) )\mbox{ and }p_1(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) )\mbox{ and }p_1(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL579$\wedge$1.p} Leibniz-equality definition means it's an equivalence






$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) \ \Rightarrow \ \forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ x ) ) )\qquad\mbox{thf}(a,\mbox{conjecture})$

\medskip\textbf{LCL186-3.p} Principia Mathematica 2.21






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(p),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL143-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{ordered}(x,y)\qquad\mbox{cnf}(\mbox{antecedent},\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered}(\mbox{implies}(z,x),\mbox{implies}(z,y))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL908+1.p} Lukasiewicz's axiomatization of propositional logic






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

\medskip\textbf{LCL690$\wedge$1.p} Prove K in the CS4 translation






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cs4\_valid}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_atom}@ a )@ ( \mbox{cs4\_atom}@ b ) ) )@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ a ) )@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{cs4\_k},\mbox{conjecture})$

\medskip\textbf{LCL423-1.p} Luka-23 is a single axiom

Show that the formula Luka-23 is a single axiom for two-valued
 logic by deriving the Lukasiewicz 3-basis.




$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(c),\mbox{not}(d)),e),c)),\mbox{implies}(f,\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(d,a)))))\qquad\mbox{cnf}(\mbox{luka}_{23},\mbox{axiom})$

$( \mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a)) ) \ \Rightarrow \ \neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_luka\_3\_basis},\mbox{negated\_conjecture})$

\medskip\textbf{LCL179-1.p} Principia Mathematica 2.13






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(p,\mbox{not}(\mbox{not}(\mbox{not}(p)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL243-1.p} Principia Mathematica 3.31






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL459+1.p} Prove Rosser's kn1 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_1\qquad\mbox{fof}(\mbox{rosser\_kn}_1,\mbox{conjecture})$

\medskip\textbf{LCL597$\wedge$1.p} Relation and proposition for 4 in K

Is there a relation R and a modal proposition A for which axiom
 4 is valid in K?




$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\exists r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL521+1.p} Prove Principia's r4 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_4\qquad\mbox{fof}(\mbox{principia\_r}_4,\mbox{conjecture})$

\medskip\textbf{LCL440-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ofalse\_\_imp}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oimp\_\_false}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_G},\mbox{v\_B},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_Un2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL917$\wedge$1.p} Translating constructive S4 (CS4) to bimodal classical S4 (BS4)






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

\medskip\textbf{LCL106-1.p} Q-2 depends on Q-1, Q-4

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that Q-2 depends on Q-1 and Q-4.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(z,y),x))))\qquad\mbox{cnf}(q_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)))\qquad\mbox{cnf}(q_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(c,a),\mbox{equivalent}(c,b))))\qquad\mbox{cnf}(\mbox{prove\_q}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL623$\wedge$1.p} The Loeb formula is a theorem in GL






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )@ ( \mbox{mbox}@ r@ x ) ) )\qquad\mbox{thf}(\mbox{gl},\mbox{axiom})$

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

\medskip\textbf{LCL174-3.p} Principia Mathematica 2.06






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL238-1.p} Principia Mathematica 3.22






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),\mbox{not}(\mbox{or}(\mbox{not}(q),\mbox{not}(p)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL148-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{implies}(x,\mbox{big\_hat}(y,z))\ne \mbox{big\_hat}(\mbox{implies}(x,y),\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL213-3.p} Principia Mathematica 2.61






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{implies}(\mbox{not}(p),q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL109-3.p} MV-4 depends on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that MV-4 depends
 on the Meredith system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,x) = \mbox{truth}\qquad\mbox{cnf}(\mbox{lemma}_1,\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{implies}(y,x) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{lemma}_2,\mbox{axiom})$

$\mbox{implies}(x,\mbox{truth}) = \mbox{truth}\qquad\mbox{cnf}(\mbox{lemma}_3,\mbox{axiom})$

$\mbox{implies}(x,\mbox{implies}(y,x)) = \mbox{truth}\qquad\mbox{cnf}(\mbox{lemma}_4,\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{implies}(y,z) \ \Rightarrow \ \mbox{implies}(x,z) = \mbox{truth}\qquad\mbox{cnf}(\mbox{lemma}_5,\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(z,y))) = \mbox{truth}\qquad\mbox{cnf}(\mbox{lemma}_6,\mbox{axiom})$

$\mbox{implies}(x,\mbox{implies}(y,z)) = \mbox{implies}(y,\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{lemma}_7,\mbox{axiom})$

$\mbox{implies}(x,\mbox{not}(\mbox{truth})) = \mbox{not}(x)\qquad\mbox{cnf}(\mbox{lemma}_8,\mbox{axiom})$

$\mbox{not}(\mbox{not}(x)) = x\qquad\mbox{cnf}(\mbox{lemma}_9,\mbox{axiom})$

$\mbox{implies}(\mbox{not}(x),\mbox{not}(y)) = \mbox{implies}(y,x)\qquad\mbox{cnf}(\mbox{lemma}_{10},\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(b,a)),\mbox{implies}(b,a))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL873$\wedge$1.p} Commutativity implies orthogonality in 2-D modal logic S5






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

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

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

$\mbox{mreflexive}@ \mbox{ra}\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\mbox{mreflexive}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{ra}\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\mbox{meuclidean}@ \mbox{ra}\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\mbox{meuclidean}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_6,\mbox{axiom})$

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

\medskip\textbf{LCL410-1.p} Alternative Wajsberg algebra definitions






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

\medskip\textbf{LCL860$\wedge$1.p} Modal logic S5(=M5) coincides with M4B5






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mreflexive}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL687+1.001.p} In S4, formula not provable in S5 embedding, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_6(y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_6(y) ) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL562+1.p} Prove Hilbert's equivalence\_2 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{conjecture})$

\medskip\textbf{LCL465+1.p} Prove Hilbert's implies\_3 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{conjecture})$

\medskip\textbf{LCL083-2.p} IC-3 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-3 depends on the first Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,x))\qquad\mbox{cnf}(\mbox{ic}_1,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_ic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL080-2.p} The 1st Lukasiewicz axiom depends on Tarski-Bernays system

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and Lukasiewicz
 axioms. Show that the 1st Lukasiewicz axiom depends on the
 Tarski-Bernays system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,x))\qquad\mbox{cnf}(\mbox{ic}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{ic}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{ic}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{ic}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),c),\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(e,a))))\qquad\mbox{cnf}(\mbox{prove\_ic\_JLukasiewicz},\mbox{negated\_conjecture})$

\medskip\textbf{LCL724$\wedge$1.p} Axioms M and 4 hold in monomodal logic S4






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ \mbox{phi} )@ \mbox{phi} )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ \mbox{phi} )@ ( \mbox{mbox\_s}_4@ ( \mbox{mbox\_s}_4@ \mbox{phi} ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL407-2.p} Alternative Wajsberg algebra axioms






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

\medskip\textbf{LCL596$\wedge$1.p} Is axiom 4 valid in K?






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

\medskip\textbf{LCL270-3.p} Principia Mathematica 4.22






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{equivalent}(p,q),\mbox{equivalent}(q,r)),\mbox{equivalent}(p,r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL707$\wedge$1.p} Accessibility relation implies axiom for weak connectedness






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

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

\medskip\textbf{LCL926-1.p} IO in TW+ [AxL,AxTO]






$( p(i(a,b))\mbox{ and }p(a) ) \ \Rightarrow \ p(b)\qquad\mbox{cnf}(\mbox{modus\_ponens},\mbox{axiom})$

$p(i(i(a,b),i(i(b,c),i(a,c))))\qquad\mbox{cnf}(\mbox{axBp},\mbox{axiom})$

$p(i(i(i(x,y),y),i(i(y,x),x)))\qquad\mbox{cnf}(\mbox{axL},\mbox{axiom})$

$p(i(i(i(x,y),i(y,x)),i(y,x)))\qquad\mbox{cnf}(\mbox{axTO},\mbox{axiom})$

$p(i(x,\mbox{or}(x,y)))\qquad\mbox{cnf}(\mbox{axorI}_1,\mbox{axiom})$

$p(i(y,\mbox{or}(x,y)))\qquad\mbox{cnf}(\mbox{axorI}_2,\mbox{axiom})$

$\neg\,p(i(i(i(x,y),y),\mbox{or}(x,y)))\qquad\mbox{cnf}(\mbox{io},\mbox{negated\_conjecture})$

\medskip\textbf{LCL227-3.p} Principia Mathematica 2.81






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,\mbox{implies}(r,s)),\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{or}(p,r),\mbox{or}(p,s)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL259-3.p} Principia Mathematica 2.43






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{implies}(p,q)),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL413-1.p} Propositional logic deduction axioms for EQUIVALENT






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

\medskip\textbf{LCL547+1.p} Prove axiom s3 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_s}_3\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_s}_3,\mbox{conjecture})$

\medskip\textbf{LCL576+1.p} Prove axiom m6 from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{axiom\_m}_6\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_6,\mbox{conjecture})$

\medskip\textbf{LCL110-1.p} MV-24 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that MV24 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(a)),a))\qquad\mbox{cnf}(\mbox{prove\_mv}_{24},\mbox{negated\_conjecture})$

\medskip\textbf{LCL905-1.p} Alternative Wajsberg algebra






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

\medskip\textbf{LCL095-1.p} IC-5 depends on the 5th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-5 depends on the fifth Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(t,\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(a,b),b)))\qquad\mbox{cnf}(\mbox{prove\_ic}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL076-1.p} CN-40 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-40 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL498+1.p} Prove Lukasiewicz's cn3 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_3\qquad\mbox{fof}(\mbox{luka\_cn}_3,\mbox{conjecture})$

\medskip\textbf{LCL260-3.p} Principia Mathematica 2.63






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{or}(\mbox{not}(p),q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL120-1.p} XGJ depends on QYF

Single axioms for the R calculus are QYF, YQM, WO, all
 by Meredith and XGJ by Winker. Show that XGJ depends on QYF.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(z,y)))\qquad\mbox{cnf}(\mbox{qyf},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(c,a)),\mbox{equivalent}(b,c))))\qquad\mbox{cnf}(\mbox{prove\_xgj},\mbox{negated\_conjecture})$

\medskip\textbf{LCL313-3.p} Principia Mathematica 4.82






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{implies}(p,q),\mbox{implies}(p,\mbox{not}(q))),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL364-1.p} CN-13 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-13 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(z,\mbox{implies}(\mbox{implies}(y,x),x))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{13},\mbox{negated\_conjecture})$

\medskip\textbf{LCL217-1.p} Principia Mathematica 2.65






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL638+1.001.p} In K, D \& A4 \& B{~p0/p0} $\to$ T, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ and }\neg\,p_1(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) )\mbox{ and }p_1(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\mbox{\$false} ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL332-3.p} Principia Mathematica 5.31






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(r,\mbox{implies}(p,q)),\mbox{implies}(p,\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL889+1.p} Halving is unique in a hoop, rule for x $>$= a/2






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} '>=' x_{17} '==>' x_{18}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{17} '>=' x_{19} )\qquad\mbox{fof}(\mbox{goals}_{13},\mbox{conjecture})$

\medskip\textbf{LCL278-3.p} Principia Mathematica 4.37






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{or}(p,r),\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL189-1.p} Principia Mathematica 2.26 and 2.27






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL392-1.p} CN-56 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-56 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),z),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{56},\mbox{negated\_conjecture})$

\medskip\textbf{LCL133-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,y) = \mbox{implies}(y,x)\qquad\mbox{cnf}(\mbox{lemma\_antecedent},\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL866$\wedge$1.p} Modal logic KB5 coincides with K4B5






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL205-1.p} Principia Mathematica 2.51






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),q))),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL527+1.p} Prove adjunction from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{adjunction}\qquad\mbox{fof}(\mbox{s1\_0\_adjunction},\mbox{conjecture})$

\medskip\textbf{LCL076-3.p} CN-40 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-40 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$( \mbox{is\_a\_theorem}(\mbox{implies}(x_1,x_2))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{implies}(x_2,x_3)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{implies}(x_1,x_3))\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL318-3.p} Principia Mathematica 4.87






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{and}(p,q),r),\mbox{equivalent}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(q,\mbox{equivalent}(\mbox{implies}(p,r),\mbox{implies}(\mbox{and}(q,p),r))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL196-3.p} Principia Mathematica 2.4






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,\mbox{or}(p,q)),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL258-3.p} Principia Mathematica 2.27






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{implies}(\mbox{implies}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL064-1.p} CN-1 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-X depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL203-1.p} Principia Mathematica 2.49






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(p,q))),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL432-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_p},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ocompleteness\_\_0}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{v\_q},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_PropLog\_Osat}(\mbox{v\_H},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Osat\_\_imp}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_H\_\_MP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL457+1.p} Prove Principia's r4 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_4\qquad\mbox{fof}(\mbox{principia\_r}_4,\mbox{conjecture})$

\medskip\textbf{LCL864$\wedge$1.p} Modal logic S5(=M5) coincides with D4B5






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mserial}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL548+1.p} Prove axiom m9 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_9\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_9,\mbox{conjecture})$

\medskip\textbf{LCL280-3.p} Principia Mathematica 4.39






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{equivalent}(p,r),\mbox{equivalent}(q,s)),\mbox{equivalent}(\mbox{or}(p,r),\mbox{or}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL399-1.p} CN-65 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-65 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(\mbox{implies}(y,y))),x))\qquad\mbox{cnf}(\mbox{prove\_cn}_{65},\mbox{negated\_conjecture})$

\medskip\textbf{LCL706$\wedge$1.p} Accessibility relation implies axiom for weak density






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

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

\medskip\textbf{LCL219-3.p} Principia Mathematica 2.68






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),q),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL528+1.p} Prove axiom m1 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_1\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_1,\mbox{conjecture})$

\medskip\textbf{LCL550+1.p} Prove Hilbert's modus ponens rule from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_ponens},\mbox{conjecture})$

\medskip\textbf{LCL516+1.p} Prove Lukasiewicz's cn2 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_2\qquad\mbox{fof}(\mbox{luka\_cn}_2,\mbox{conjecture})$

\medskip\textbf{LCL235-1.p} Principia Mathematica 3.13






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))))),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL293-3.p} Principia Mathematica 4.57






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{and}(\mbox{not}(p),\mbox{not}(q))),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL215-3.p} Principia Mathematica 2.621






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{or}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL251-3.p} Principia Mathematica 3.43






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(p,q),\mbox{implies}(p,r)),\mbox{implies}(p,\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL558+1.p} Prove Hilbert's or\_1 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{conjecture})$

\medskip\textbf{LCL400-1.p} CN-66 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-66 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(x,y)),x))\qquad\mbox{cnf}(\mbox{prove\_cn}_{66},\mbox{negated\_conjecture})$

\medskip\textbf{LCL713$\wedge$1.p} Axiom implies accessibility relation for Euclidianity






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

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

\medskip\textbf{LCL256-1.p} A formula that can be derived from the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that not(not(p $\to$ p)) can be derived from the short
 Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{not}(\mbox{not}(\mbox{implies}(p,p))))\qquad\mbox{cnf}(\mbox{prove\_not\_not\_implies},\mbox{negated\_conjecture})$

\medskip\textbf{LCL606$\wedge$1.p} LAMBDA$\wedge$mm\_1 validates the Barcan formula axioms






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mall}@ \lambda x{:}\;\mbox{individuals}{:}\;( \mbox{mbox}@ r@ ( a@ x ) ) )@ ( \mbox{mbox}@ r@ ( \mbox{mall}@ \lambda x{:}\;\mbox{individuals}{:}\;( a@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL354+1.p} Independence of an Axiom for Temporal Intervals

Shows that the 5th axiom of temporal intervals is not dependant
 on the first three by building a model of the first three and
 the negation of the 5th.




$\forall p,q,r,s{:}\;( ( \mbox{meets}(p,q)\mbox{ and }\mbox{meets}(p,s)\mbox{ and }\mbox{meets}(r,q) ) \ \Rightarrow \ \mbox{meets}(r,s) )\qquad\mbox{fof}(m_1,\mbox{axiom})$

$\forall p,q,r,s{:}\;( ( \mbox{meets}(p,q)\mbox{ and }\mbox{meets}(r,s) ) \ \Rightarrow \ \mbox{meets}(p,s) <~> \exists t{:}\;( \mbox{meets}(p,t)\mbox{ and }\mbox{meets}(t,s) ) <~> \exists t{:}\;( \mbox{meets}(r,t)\mbox{ and }\mbox{meets}(t,q) ) )\qquad\mbox{fof}(m_2,\mbox{axiom})$

$\forall p{:}\;\exists q,r{:}\;( \mbox{meets}(q,p)\mbox{ and }\mbox{meets}(p,r) )\qquad\mbox{fof}(m_3,\mbox{axiom})$

$\neg\,\forall p,q{:}\;( \mbox{meets}(p,q) \ \Rightarrow \ \exists r,s,t{:}\;( \mbox{meets}(r,p)\mbox{ and }\mbox{meets}(q,s)\mbox{ and }\mbox{meets}(r,t)\mbox{ and }\mbox{meets}(t,s) ) )\qquad\mbox{fof}(\mbox{not\_m}_5,\mbox{axiom})$

\medskip\textbf{LCL094-1.p} IC-5 depends on the 4th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-5 depends on the fourth Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(t,\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(a,b),b)))\qquad\mbox{cnf}(\mbox{prove\_ic}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL129-1.p} LG-2 depends on S-4

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on S-4.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(u,z)),\mbox{equivalent}(y,u))))\qquad\mbox{cnf}(s_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL739$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\forall \mbox{xr}_{24}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xr}_{23}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;\forall \mbox{xw}_6{:}\;b{:}\;( \mbox{xr}_{23}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_6\mbox{ or }\mbox{xr}_{24}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_6 ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o},\mbox{xw}_6{:}\;b{:}\;( \mbox{xr}_{23}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_6\mbox{ or }\mbox{xr}_{24}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_6 ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB}_2,\mbox{conjecture})$

\medskip\textbf{LCL356-1.p} CN-05 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-05 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(u,y),\mbox{implies}(x,\mbox{implies}(u,z)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{05},\mbox{negated\_conjecture})$

\medskip\textbf{LCL598$\wedge$1.p} Is axiom 4 equivalent to irreflexivity?






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\iff ( \mbox{irreflexive}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL389-1.p} CN-52 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-52 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,\mbox{not}(z))),\mbox{implies}(x,\mbox{implies}(z,\mbox{not}(y)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{52},\mbox{negated\_conjecture})$

\medskip\textbf{LCL029-1.p} C0-5 depends on the Tarski-Bernays system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-5 can be derived from the
 Tarski-Bernays system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{c0}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{falsehood},x))\qquad\mbox{cnf}(\mbox{c0}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,\mbox{falsehood}),\mbox{falsehood}),a))\qquad\mbox{cnf}(\mbox{prove\_c0}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL729$\wedge$5.p} TPS problem THM560

AC1(A) equiv AC3(OA,A) from [RR93]. Note that AC3 usually refers
 to 'relations' where here we are using it for relations on OA x A.




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

$\forall \mbox{xr}{:}\;( a\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\exists \mbox{xg}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xy}{:}\;a{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ ( \mbox{xg}@ \mbox{xx} ) ) )\iff \forall \mbox{xs}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{560},\mbox{conjecture})$

\medskip\textbf{LCL057-1.p} CN-40 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-40 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{40},\mbox{negated\_conjecture})$

\medskip\textbf{LCL080-1.p} The 1st Lukasiewicz axiom depends on Tarski-Bernays system

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that the 1st Lukasiewicz axiom depends on the
 Tarski-Bernays system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{ic}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{ic}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{ic}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),c),\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(e,a))))\qquad\mbox{cnf}(\mbox{prove\_ic\_JLukasiewicz},\mbox{negated\_conjecture})$

\medskip\textbf{LCL571+1.p} Prove necessitation from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{necessitation}\qquad\mbox{fof}(\mbox{km5\_necessitation},\mbox{conjecture})$

\medskip\textbf{LCL186-1.p} Principia Mathematica 2.21






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(p)),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL393-1.p} CN-57 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-57 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(\mbox{implies}(x,z),\mbox{implies}(\mbox{implies}(z,y),y))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{57},\mbox{negated\_conjecture})$

\medskip\textbf{LCL329-3.p} Principia Mathematica 5.24






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{or}(\mbox{and}(p,q),\mbox{and}(\mbox{not}(p),\mbox{not}(q)))),\mbox{or}(\mbox{and}(p,\mbox{not}(q)),\mbox{and}(q,\mbox{not}(p)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL436-2.p} Problem about propositional logic






$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

\medskip\textbf{LCL157-1.p} The 5th alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{and\_star}(x,\mbox{falsehood})\ne \mbox{falsehood}\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL183-1.p} Principia Mathematica 2.17






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(q)),\mbox{not}(p))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL433-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_p},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ocompleteness\_\_0}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{v\_q},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_PropLog\_Osat}(\mbox{v\_H},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Osat\_\_imp}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_H\_\_MP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_F},\mbox{c\_Finite\_\_Set\_OFinites},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{v\_xa},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_xa},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{v\_F},\mbox{v\_U},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_U},\mbox{c\_PropLog\_Othms}(\mbox{v\_F},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL466+1.p} Prove Hilbert's and\_1 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{conjecture})$

\medskip\textbf{LCL435-2.p} Problem about propositional logic






$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_pa},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

\medskip\textbf{LCL369-1.p} CN-25 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-25 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(x,u),\mbox{implies}(\mbox{implies}(u,y),z))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{LCL249-1.p} Principia Mathematica 3.41






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),r)),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL183-3.p} Principia Mathematica 2.17






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(q),\mbox{not}(p)),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL252-3.p} Principia Mathematica 3.44






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(q,p),\mbox{implies}(r,p)),\mbox{implies}(\mbox{or}(q,r),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL175-1.p} Principia Mathematica 2.07






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(p,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL888+1.p} Halving is unique in a hoop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} = x_{17} '==>' x_{18}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{17} = x_{19} )\qquad\mbox{fof}(\mbox{goals}_{13},\mbox{conjecture})$

\medskip\textbf{LCL501+1.p} Prove Rosser's kn3 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_3\qquad\mbox{fof}(\mbox{rosser\_kn}_3,\mbox{conjecture})$

\medskip\textbf{LCL370-1.p} CN-26 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-26 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{26},\mbox{negated\_conjecture})$

\medskip\textbf{LCL191-3.p} Principia Mathematica 2.31






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,\mbox{or}(q,r)),\mbox{or}(\mbox{or}(p,q),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL478+1.p} Prove Principia's r4 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$r_4\qquad\mbox{fof}(\mbox{principia\_r}_4,\mbox{conjecture})$

\medskip\textbf{LCL463+1.p} Prove Hilbert's implies\_1 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{conjecture})$

\medskip\textbf{LCL871$\wedge$1.p} The converse Barcan formula is valid in quantified modal logic K






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox}@ r@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( p@ x ) ) )@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox}@ r@ ( p@ x ) ) ) )\qquad\mbox{thf}(\mbox{ex2b},\mbox{conjecture})$

\medskip\textbf{LCL886+1.p} An involutive pocrim that is not idempotent






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' \mbox{'1'} = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '+' x_{17} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL587$\wedge$1.p} Popkorn problem 8






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ b )@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL414-1.p} Peter Andrews Problem THM147






$( \mbox{a\_truth}(a)\mbox{ and }\mbox{a\_truth}(\mbox{implies}(a,b)) ) \ \Rightarrow \ \mbox{a\_truth}(b)\qquad\mbox{cnf}(\mbox{thm147}_1,\mbox{negated\_conjecture})$

$\mbox{a\_truth}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{thm147}_2,\mbox{negated\_conjecture})$

$\mbox{a\_truth}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{thm147}_3,\mbox{negated\_conjecture})$

$\mbox{a\_truth}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{thm147}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{a\_truth}(\mbox{implies}(\mbox{sk}_1,\mbox{sk}_1))\qquad\mbox{cnf}(\mbox{thm147}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL207-3.p} Principia Mathematica 2.521






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(p,q)),\mbox{implies}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL081-1.p} IC-1 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-1 depends on the first Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,a))\qquad\mbox{cnf}(\mbox{prove\_ic}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL201-3.p} Principia Mathematica 2.47






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL863$\wedge$1.p} Modal logic S5(=M5) coincides with D4B






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mserial}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL725$\wedge$1.p} Axioms M, 4, and B hold in monomodal logic S5






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ \mbox{phi} )@ \mbox{phi} )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_5@ \mbox{phi} )@ ( \mbox{mbox\_s}_5@ ( \mbox{mbox\_s}_5@ \mbox{phi} ) ) )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimplies}@ \mbox{phi}@ ( \mbox{mbox\_s}_5@ ( \mbox{mdia\_s}_5@ \mbox{phi} ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL124-1.p} LG-5 depends on LG-2

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-5
 depends on LG-2.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u))))))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,\mbox{equivalent}(c,\mbox{equivalent}(e,f)))),\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(f,c)),\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(f,e)),f))))\qquad\mbox{cnf}(\mbox{prove\_lg}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL209-3.p} Principia Mathematica 2.54






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(p),q),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL450+2.p} Congruence of equiv lemmas, to admit substitution of equivalents






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_ponens},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{axiom})$

$\mbox{implies}_3\qquad\mbox{fof}(\mbox{hilbert\_implies}_3,\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{axiom})$

$\forall x{:}\;\mbox{is\_a\_theorem}(\mbox{equiv}(x,x))\mbox{ and }\forall x,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{not}(x),\mbox{not}(y))) )\mbox{ and }\forall x_1,x_2,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x_1,x_2)) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{and}(x_1,y),\mbox{and}(x_2,y))) )\mbox{ and }\forall x_1,x_2,y{:}\;( \mbox{is\_a\_theorem}(\mbox{equiv}(x_1,x_2)) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{equiv}(\mbox{and}(y,x_1),\mbox{and}(y,x_2))) )\mbox{ and }\forall x,y{:}\;( ( \mbox{is\_a\_theorem}(x)\mbox{ and }\mbox{is\_a\_theorem}(\mbox{equiv}(x,y)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) )\qquad\mbox{fof}(\mbox{equiv\_congruence},\mbox{conjecture})$

\medskip\textbf{LCL033-1.p} C0-2 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-2 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_c0}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL224-1.p} Principia Mathematica 2.76






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{or}(\mbox{not}(q),r))),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL038-1.p} C0-1 depends on a single axiom

An axiomatisation for the Implication/Falsehood 2 valued
 sentential calculus is {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays. Show that C0-1 can be derived from the first
 Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_c0}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL181-2.p} Principia Mathematica 2.15

Judged by [SRM73] to be the 'hardest' of the first 52 theorems
 of [WR27].




$p\mbox{ or }q\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$q \ \Rightarrow \ \neg\,p\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\neg\,q\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\neg\,p\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL655+1.001.p} In KT, A5 not provable with instances of A4, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_1(y)\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL131-1.p} LG-2 depends on S-6

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on S-6.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(u,z)),\mbox{equivalent}(y,u))),x))\qquad\mbox{cnf}(s_6,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL114-2.p} MV-36 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-36 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(y),\mbox{not}(x)))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{LCL218-3.p} Principia Mathematica 2.67






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{or}(p,q),q),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL483+1.p} Prove Hilbert's modus\_tollens axiom from Principia's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{conjecture})$

\medskip\textbf{LCL464+1.p} Prove Hilbert's implies\_2 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{conjecture})$

\medskip\textbf{LCL178-1.p} Principia Mathematica 2.12






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL702$\wedge$1.p} Accessibility relation implies axiom for transitivity






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

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

\medskip\textbf{LCL111-2.p} MV-25 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-25 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(z,y)))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{25},\mbox{negated\_conjecture})$

\medskip\textbf{LCL714$\wedge$1.p} Axiom implies accessibility relation for partial functionality






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

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

\medskip\textbf{LCL046-1.p} CN-16 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-16 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{16},\mbox{negated\_conjecture})$

\medskip\textbf{LCL230$\wedge$4.p} Principia Mathematica 2.85






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

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

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

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

$\mbox{ivalid}@ ( \mbox{iimplies}@ ( \mbox{iimplies}@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ q ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iatom}@ r ) ) )@ ( \mbox{ior}@ ( \mbox{iatom}@ p )@ ( \mbox{iimplies}@ ( \mbox{iatom}@ q )@ ( \mbox{iatom}@ r ) ) ) )\qquad\mbox{thf}(\mbox{pel}_5,\mbox{conjecture})$

\medskip\textbf{LCL126-1.p} Q-2 depends on the 2nd McCune system

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that Q-2
 depends on the second McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(q_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(x,z),\mbox{equivalent}(y,z))))\qquad\mbox{cnf}(q_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,b)),\mbox{equivalent}(a,c)))\qquad\mbox{cnf}(\mbox{prove\_q}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL340-3.p} Principia Mathematica 5.44






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{equivalent}(\mbox{implies}(p,r),\mbox{implies}(p,\mbox{and}(q,r)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL600$\wedge$1.p} Is axiom 4 equivalent to transitivity of R in K?






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\iff ( \mbox{transitive}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL436-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL151-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{big\_V}(\mbox{big\_hat}(x,y),z)\ne \mbox{big\_hat}(\mbox{big\_V}(x,z),\mbox{big\_V}(y,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL311-3.p} Principia Mathematica 4.8






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{not}(p)),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL195-3.p} Principia Mathematica 2.38






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{or}(q,p),\mbox{or}(r,p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL915+1.p} M10 axiomatization of S5 based on S1-0






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

\medskip\textbf{LCL474+1.p} Prove Hilbert's equivalence\_3 axiom from Lukasiewicz's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{conjecture})$

\medskip\textbf{LCL588$\wedge$1.p} Popkorn problem 9






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mimpl}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mbox}@ r@ b ) )@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL254-3.p} Principia Mathematica 3.47






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(p,r),\mbox{implies}(q,s)),\mbox{implies}(\mbox{and}(p,q),\mbox{and}(r,s))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL187-3.p} Principia Mathematica 2.24






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{implies}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL556+1.p} Prove Hilbert's and\_2 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{conjecture})$

\medskip\textbf{LCL446-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{LCL639+1.001.p} In K, A5 not provable with instances of D, A4, and T, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_1(y)\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_1(y)\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ and }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\mbox{\$false} ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL589$\wedge$1.p} Popkorn problem 10






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mimpl}@ ( \mbox{mdia}@ r@ a )@ ( \mbox{mbox}@ r@ b ) )@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL178-3.p} Principia Mathematica 2.12






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{not}(\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL303-3.p} Principia Mathematica 4.71






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,q),\mbox{equivalent}(q,\mbox{or}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL323-3.p} Principia Mathematica 5.14






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{implies}(p,q),\mbox{implies}(q,r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL388-1.p} CN-51 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-51 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(x,\mbox{implies}(\mbox{not}(z),\mbox{not}(y)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{51},\mbox{negated\_conjecture})$

\medskip\textbf{LCL525+1.p} Prove strict implies modus ponens from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_modus\_ponens\_strict\_implies},\mbox{conjecture})$

\medskip\textbf{LCL391-1.p} CN-55 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-55 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(x,\mbox{not}(y)),\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{55},\mbox{negated\_conjecture})$

\medskip\textbf{LCL245-3.p} Principia Mathematica 3.34






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(\mbox{implies}(q,r),\mbox{implies}(p,q)),\mbox{implies}(p,r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL353-3.p} Principia Mathematica 5.75






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(\mbox{and}(\mbox{implies}(r,\mbox{not}(q)),p),\mbox{or}(q,r)),\mbox{equivalent}(\mbox{and}(p,\mbox{not}(q)),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL281-3.p} Principia Mathematica 4.4






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{or}(q,r)),\mbox{or}(\mbox{and}(p,q),\mbox{and}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL089-1.p} IC-4 depends on the 4th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-4 depends on the fourth Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(t,\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_ic}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL176-1.p} Principia Mathematica 2.1 and 2.08






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL113-1.p} MV-33 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that 33 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),b),\mbox{implies}(\mbox{not}(b),a)))\qquad\mbox{cnf}(\mbox{prove\_mv}_{33},\mbox{negated\_conjecture})$

\medskip\textbf{LCL162-1.p} The 2nd Wajsberg algebra axiom, from the alternative axioms






$\mbox{include}(\mbox{'Axioms/LCL002-0.ax'})$

$\mbox{xor}(x,y) = \mbox{xor}(y,x)\qquad\mbox{cnf}(\mbox{xor\_commutativity},\mbox{axiom})$

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

$\mbox{and\_star}(x,y) = \mbox{and\_star}(y,x)\qquad\mbox{cnf}(\mbox{and\_star\_commutativity},\mbox{axiom})$

$\mbox{not}(\mbox{truth}) = \mbox{falsehood}\qquad\mbox{cnf}(\mbox{false\_definition},\mbox{axiom})$

$\mbox{implies}(x,y) = \mbox{xor}(\mbox{truth},\mbox{and\_star}(x,\mbox{xor}(\mbox{truth},y)))\qquad\mbox{cnf}(\mbox{implies\_definition},\mbox{axiom})$

$\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z)))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL361-1.p} CN-10 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-10 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x),\mbox{implies}(\mbox{implies}(y,x),x))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{10},\mbox{negated\_conjecture})$

\medskip\textbf{LCL416-1.p} Prove reflexivity from formula XCB by condensed detachment






$( \mbox{is\_a\_theorem}(\mbox{equivalent}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,b)),c)))\qquad\mbox{cnf}(\mbox{xcb},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,a))\qquad\mbox{cnf}(\mbox{prove\_reflexivity},\mbox{negated\_conjecture})$

\medskip\textbf{LCL366-1.p} CN-15 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-15 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{15},\mbox{negated\_conjecture})$

\medskip\textbf{LCL185-3.p} Principia Mathematica 2.2






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL608$\wedge$1.p} LAMBDA$\wedge$mm\_1 validates the modus ponens rule






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ a\mbox{ and }\mbox{mvalid}@ ( \mbox{mimpl}@ a@ b ) ) \ \Rightarrow \ ( \mbox{mvalid}@ b ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL365-1.p} CN-14 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-14 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(y,z),z)),u),\mbox{implies}(\mbox{implies}(\mbox{not}(z),y),u)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{14},\mbox{negated\_conjecture})$

\medskip\textbf{LCL031-1.p} C0-CAMerideth depends on the Tarski-Bernays system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that the single Meredith axiom can
 be derived from the Tarski-Bernays system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),x),x))\qquad\mbox{cnf}(\mbox{c0}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{falsehood},x))\qquad\mbox{cnf}(\mbox{c0}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(c,\mbox{falsehood})),e),\mbox{falsehood}),\mbox{implies}(\mbox{implies}(\mbox{falsehood},a),\mbox{implies}(c,a))))\qquad\mbox{cnf}(\mbox{prove\_c0\_CAMerideth},\mbox{negated\_conjecture})$

\medskip\textbf{LCL672+1.001.p} In S4, A5{box p0/p0} \& box A5{~p0/p0} $\to$ A5, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) )\mbox{ and }p_1(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) ) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\mbox{\$false} ) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL586$\wedge$1.p} Popkorn problem 7






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mdia}@ r@ a ) )@ ( \mbox{mbox}@ r@ ( \mbox{mimpl}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL357-1.p} CN-06 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-06 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{implies}(x,z),u),\mbox{implies}(\mbox{implies}(y,z),u))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{06},\mbox{negated\_conjecture})$

\medskip\textbf{LCL355-1.p} CN-04 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-04 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,y)),u),\mbox{implies}(\mbox{implies}(z,x),u)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{04},\mbox{negated\_conjecture})$

\medskip\textbf{LCL096-1.p} LG-1 depends on LG-2, LG-3, LG-4

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-1 depends on a part of the Kalman system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),u),\mbox{equivalent}(\mbox{equivalent}(y,z),u)),v),v))\qquad\mbox{cnf}(\mbox{lg}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),u),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,v),z),\mbox{equivalent}(\mbox{equivalent}(y,v),u))))\qquad\mbox{cnf}(\mbox{lg}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,b),a)),c),c))\qquad\mbox{cnf}(\mbox{prove\_lg}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL637+1.001.p} In K, the branching formula, size 1






$\neg\,\exists x{:}\;( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }( ( \neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,p_2(y)\mbox{ and }\neg\,p_{102}(y)\mbox{ and }p_{101}(y) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ and }\neg\,p_{102}(y)\mbox{ and }p_{101}(y) ) )\mbox{ or }\neg\,\neg\,p_{101}(x)\mbox{ and }p_{100}(x) )\mbox{ and }( ( ( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_2(y)\mbox{ or }\neg\,p_{101}(y) )\mbox{ or }p_2(x) )\mbox{ and }( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y)\mbox{ or }\neg\,p_{101}(y) )\mbox{ or }\neg\,p_2(x) ) )\mbox{ or }\neg\,p_{101}(x) )\mbox{ and }( ( ( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y)\mbox{ or }\neg\,p_{100}(y) )\mbox{ or }p_1(x) )\mbox{ and }( \forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y)\mbox{ or }\neg\,p_{100}(y) )\mbox{ or }\neg\,p_1(x) ) )\mbox{ or }\neg\,p_{100}(x) )\mbox{ and }( p_{101}(x)\mbox{ or }\neg\,p_{102}(x) )\mbox{ and }( p_{100}(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL656+1.001.p} In KT, the branching formula made provable, size 1

The branching formula plus a negation symbol in front and an
 additional subformula to make the formula provable.




$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL184-1.p} Principia Mathematica 2.18






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(p)),p)),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL045-1.p} CN-46 depends on Hilbert's system

Two axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-18,CN-21,CN-35,CN-39,CN-39,
 CN-40,CN-46} by Frege and {CN-3,CN-18,CN-21,CN-22,CN-30,
 CN-54} by Hilbert. Show that CN-46 depends on the simplified
 Hilbert system (without CN-30).




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(y,\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{21},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{22},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),y)))\qquad\mbox{cnf}(\mbox{cn}_{54},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{not}(b),\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{46},\mbox{negated\_conjecture})$

\medskip\textbf{LCL489+1.p} Prove Hilbert's and\_3 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{conjecture})$

\medskip\textbf{LCL248-1.p} Principia Mathematica 3.4






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL448+1.p} Redundant axiom in Principia axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{op\_implies\_or}\qquad\mbox{fof}(\mbox{principia\_op\_implies\_or},\mbox{axiom})$

$\mbox{op\_and}\qquad\mbox{fof}(\mbox{principia\_op\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{principia\_op\_equiv},\mbox{axiom})$

$\mbox{modus\_ponens}\qquad\mbox{fof}(\mbox{principia\_modus\_ponens},\mbox{axiom})$

$r_1\qquad\mbox{fof}(\mbox{principia\_r}_1,\mbox{axiom})$

$r_2\qquad\mbox{fof}(\mbox{principia\_r}_2,\mbox{axiom})$

$r_3\qquad\mbox{fof}(\mbox{principia\_r}_3,\mbox{axiom})$

$r_5\qquad\mbox{fof}(\mbox{principia\_r}_5,\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$r_4\qquad\mbox{fof}(\mbox{principia\_r}_4,\mbox{conjecture})$

\medskip\textbf{LCL921$\wedge$1.p} Modal logic B






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

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

\medskip\textbf{LCL404-1.p} CN-70 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-70 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,\mbox{not}(\mbox{implies}(x,\mbox{not}(y))))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{70},\mbox{negated\_conjecture})$

\medskip\textbf{LCL145-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{not}(\mbox{big\_V}(x,y))\ne \mbox{big\_hat}(\mbox{not}(x),\mbox{not}(y))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL284-3.p} Principia Mathematica 4.43






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{and}(\mbox{or}(p,q),\mbox{or}(p,\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL372-1.p} CN-28 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-28 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),z),\mbox{implies}(\mbox{implies}(\mbox{implies}(y,u),x),z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{28},\mbox{negated\_conjecture})$

\medskip\textbf{LCL891+1.p} Halving is unique in a hoop, rule for a/2 $>$= x






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} '==>' x_{18} '>=' x_{17}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{19} '>=' x_{17} )\qquad\mbox{fof}(\mbox{goals}_{13},\mbox{conjecture})$

\medskip\textbf{LCL429-2.p} Problem about propositional logic






$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{v\_C},\mbox{v\_B},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_ODiff\_\_weaken\_\_left}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{c\_minus}(\mbox{v\_t},\mbox{c\_insert}(\mbox{v\_v},\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{c\_emptyset},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ohyps\_\_Diff}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oinsert\_\_Diff\_\_same}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_minus}(\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_c},\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{v\_D},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_insert}(\mbox{v\_a},\mbox{c\_minus}(\mbox{v\_B},\mbox{c\_insert}(\mbox{v\_c},\mbox{v\_D},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oinsert\_\_Diff\_\_subset2}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_excluded\_\_middle\_\_rule}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_G},\mbox{v\_H},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_F},\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_insert}(\mbox{c\_PropLog\_Opl\_Ovar}(\mbox{v\_v},\mbox{t\_a}),\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL154-1.p} The 2nd alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{xor}(x,\mbox{falsehood})\ne x\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL268-3.p} Principia Mathematica 4.2






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL308-3.p} Principia Mathematica 4.77






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{implies}(q,p),\mbox{implies}(r,p)),\mbox{implies}(\mbox{or}(q,r),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL168-1.p} XEH is not a single axiom for the R-calculus

To show that XEH is not a single axiom, attempt to derive
 from it any one of YQM, WO, XGJ or QYF, which are known
 single axioms.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,\mbox{equivalent}(\mbox{equivalent}(y,z),x)),z)))\qquad\mbox{cnf}(\mbox{xeh},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(a,c)),\mbox{equivalent}(c,b)))\qquad\mbox{cnf}(\mbox{try\_prove\_qyf},\mbox{negated\_conjecture})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(c,b),\mbox{equivalent}(c,a))))\qquad\mbox{cnf}(\mbox{try\_prove\_yqm},\mbox{negated\_conjecture})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(c,\mbox{equivalent}(b,a))))\qquad\mbox{cnf}(\mbox{try\_prove\_wo},\mbox{negated\_conjecture})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(c,a)),\mbox{equivalent}(b,c))))\qquad\mbox{cnf}(\mbox{try\_prove\_xgj},\mbox{negated\_conjecture})$

\medskip\textbf{LCL409-1.p} Wajsberg algebra AND and OR definitions






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

\medskip\textbf{LCL352-3.p} Principia Mathematica 5.74






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{equivalent}(q,r)),\mbox{equivalent}(\mbox{implies}(p,q),\mbox{implies}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL241-3.p} Principia Mathematica 3.27






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,q),q))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL153-1.p} The 1st alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{not}(x)\ne \mbox{xor}(x,\mbox{truth})\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL440-2.p} Problem about propositional logic






$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

\medskip\textbf{LCL008-1.p} EC-4 depends on YQL

An axiomatisation of the equivalential calculus is {EC-4,
 EC-5} by Wajsburg. Show that EC-4 can be derived from the
 single Lukasiewicz axiom YQL.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(z,y),\mbox{equivalent}(x,z))))\qquad\mbox{cnf}(\mbox{yql},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_ec}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL087-1.p} IC-2 depends on the 4th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-2 depends on the fourth Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(t,\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_ic}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL187-1.p} Principia Mathematica 2.24






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(\mbox{not}(p)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL181-1.p} Principia Mathematica 2.15






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(p)),q)),\mbox{or}(\mbox{not}(\mbox{not}(q)),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL574+1.p} Prove axiom 4 from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom}_4\qquad\mbox{fof}(\mbox{km4b\_axiom}_4,\mbox{conjecture})$

\medskip\textbf{LCL494+1.p} Prove Hilbert's equivalence\_2 axiom from Principia's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_2\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_2,\mbox{conjecture})$

\medskip\textbf{LCL417-2.p} XCB is a single axiom for the equivalential calculus

Show that formula XCB is a single axiom for the equivalential
 calculus by deriving the pair of axioms {symmetry, transitivity}.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,b)),c)))\qquad\mbox{cnf}(\mbox{xcb},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(b,a))) \ \Rightarrow \ \neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_symmetry\_and\_transitivity},\mbox{negated\_conjecture})$

\medskip\textbf{LCL212-3.p} Principia Mathematica 2.6






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(p),\mbox{implies}(q,\mbox{implies}(\mbox{implies}(p,q),q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL149-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{implies}(x,\mbox{big\_V}(y,z))\ne \mbox{big\_V}(\mbox{implies}(x,y),\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL066-1.p} CN-3 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-3 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL299-3.p} Principia Mathematica 4.65






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{implies}(\mbox{not}(p),q)),\mbox{and}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL869$\wedge$1.p} Modal logic M5 'includes' modal logic D45






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mserial}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{meuclidean}@ r ) \ \Rightarrow \ ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL026-1.p} C0-3 depends on the Church system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-3 can be derived from the
 Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{falsehood}),\mbox{falsehood}),x))\qquad\mbox{cnf}(\mbox{c0}_5,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_6,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_c0}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL048-1.p} CN-19 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-19 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),c),\mbox{implies}(b,c)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{19},\mbox{negated\_conjecture})$

\medskip\textbf{LCL005-1.p} AN-CAMerideth =$>$ AN-4

The Whitehead-Russell axiomatisation of the Disjunction/
 Negation 2 valued sentential calculus is {AN-1,AN-2,AN-3,
 AN-4}. Show that AN-4 can be derived from the Merideth axiom.




$( \mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(x),y)),\mbox{or}(z,\mbox{or}(u,v)))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(u),x)),\mbox{or}(z,\mbox{or}(v,x)))))\qquad\mbox{cnf}(\mbox{an\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(a,a)),a))\qquad\mbox{cnf}(\mbox{an}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL004-1.p} AN-CAMerideth =$>$ AN-3

The Whitehead-Russell axiomatisation of the Disjunction/
 Negation 2 valued sentential calculus is {AN-1,AN-2,AN-3,
 AN-4}. Show that AN-3 can be derived from the Merideth axiom.




$( \mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(x),y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(x),y)),\mbox{or}(z,\mbox{or}(u,v)))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(u),x)),\mbox{or}(z,\mbox{or}(v,x)))))\qquad\mbox{cnf}(\mbox{an\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{or}(\mbox{not}(a),\mbox{or}(b,a)))\qquad\mbox{cnf}(\mbox{an}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL135-1.p} A lemma in Wajsberg algebras

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided
 a different axiomatisation. Show that MV-1 depends on the
 Wajsberg system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,\mbox{implies}(y,x))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL054-1.p} CN-35 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-35 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{35},\mbox{negated\_conjecture})$

\medskip\textbf{LCL540+1.p} Prove adjunction from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{adjunction}\qquad\mbox{fof}(\mbox{s1\_0\_adjunction},\mbox{conjecture})$

\medskip\textbf{LCL208-1.p} Principia Mathematica 2.53






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(\mbox{not}(\mbox{not}(p)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL418-1.p} Is formula YQE a single axiom for the right group calculus?






$( \mbox{is\_a\_theorem}(\mbox{equivalent}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(a,c),\mbox{equivalent}(b,c))))\qquad\mbox{cnf}(\mbox{yqe},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,d),\mbox{equivalent}(c,d))))))\qquad\mbox{cnf}(\mbox{prove\_l}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL346-3.p} Principia Mathematica 5.6






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{and}(p,\mbox{not}(q)),r),\mbox{implies}(p,\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL443-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL304-3.p} Principia Mathematica 4.72






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,q),\mbox{equivalent}(q,\mbox{or}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL694$\wedge$1.p} Prove the converse Barcan formula in the CS4 translation






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL009\string^0.ax'})$

$\forall a{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{cs4\_valid}@ ( \mbox{cs4\_impl}@ ( \mbox{cs4\_box}@ ( \mbox{cs4\_all}@ \lambda i{:}\;\mbox{individuals}{:}\;( \mbox{cs4\_atom}@ ( a@ i ) ) ) )@ ( \mbox{cs4\_all}@ \lambda i{:}\;\mbox{individuals}{:}\;( \mbox{cs4\_box}@ ( \mbox{cs4\_atom}@ ( a@ i ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cs4\_cbarcan},\mbox{conjecture})$

\medskip\textbf{LCL367-1.p} CN-17 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-17 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{17},\mbox{negated\_conjecture})$

\medskip\textbf{LCL263-3.p} Principia Mathematica 4.11






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL169-3.p} Principia Mathematica 2.01






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,\mbox{not}(p)),\mbox{not}(p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL125-1.p} LG-2 depends on the 1st McCune system

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on the first McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(z,y)),\mbox{equivalent}(x,z)))\qquad\mbox{cnf}(q_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(q_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL225-1.p} Principia Mathematica 2.77






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(p),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL481+1.p} Prove Rosser's kn2 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_2\qquad\mbox{fof}(\mbox{rosser\_kn}_2,\mbox{conjecture})$

\medskip\textbf{LCL553+1.p} Prove Hilbert's implies\_2 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{conjecture})$

\medskip\textbf{LCL484+1.p} Prove Hilbert's implies\_1 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{conjecture})$

\medskip\textbf{LCL602$\wedge$1.p} T and 4 equivalent to reflexivity and transitivity of R in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ a ) )\iff ( \mbox{reflexive}@ r\mbox{ and }\mbox{transitive}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL128-1.p} LG-2 depends on S-3

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on S-3.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(u,z)),\mbox{equivalent}(y,u)))))\qquad\mbox{cnf}(s_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL222-1.p} Principia Mathematica 2.74






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),p)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{or}(p,q),r)),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL060-1.p} CN-54 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-54 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(\mbox{not}(a),b),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{54},\mbox{negated\_conjecture})$

\medskip\textbf{LCL487+1.p} Prove Hilbert's and\_1 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{conjecture})$

\medskip\textbf{LCL897+1.p} Weak conjunction is associative in a hoop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '==>' a = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'0'} = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;\mbox{'0'} '==>' a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

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

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$a '+' a '==>' b '+' a '+' a '==>' b '==>' c = a '+' a '==>' b '+' b '==>' c\qquad\mbox{fof}(\mbox{goals}_{09},\mbox{conjecture})$

\medskip\textbf{LCL190-1.p} Principia Mathematica 2.3






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{or}(q,r))),\mbox{or}(p,\mbox{or}(r,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL567+1.p} Prove axiom 5 from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom}_5\qquad\mbox{fof}(\mbox{km5\_axiom}_5,\mbox{conjecture})$

\medskip\textbf{LCL626$\wedge$1.p} Loeb axiom is valid in this frame

In a frame that is transitive and upwards well-founded, the Loeb
 axiom is valid.




$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{transitive}@ r\mbox{ and }\mbox{upwards\_well\_founded}@ r )\qquad\mbox{thf}(\mbox{upwf\_trans},\mbox{axiom})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(k_4,\mbox{conjecture})$

\medskip\textbf{LCL099-1.p} LG-5 depends on the 1st McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-5 depends on the first McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),u),\mbox{equivalent}(\mbox{equivalent}(y,z),u)),v),v))\qquad\mbox{cnf}(\mbox{lg}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,a),c)),\mbox{equivalent}(\mbox{equivalent}(e,a),\mbox{falsehood})),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),e),c),\mbox{falsehood})))\qquad\mbox{cnf}(\mbox{prove\_lg}_5,\mbox{negated\_conjecture})$

\medskip\textbf{LCL510+1.p} Prove Hilbert's or\_2 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{conjecture})$

\medskip\textbf{LCL654+1.001.p} In KT, A5{box p0/p0} \& box A5{~p0/p0} $\to$ A4, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_1(y) ) )\mbox{ and }p_1(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\mbox{ and }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) ) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) ) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\mbox{\$false} ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) ) )\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL206-1.p} Principia Mathematica 2.52






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),q))),\mbox{or}(\mbox{not}(\mbox{not}(p)),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL158-1.p} The 6th alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{and\_star}(\mbox{xor}(\mbox{truth},x),x)\ne \mbox{falsehood}\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL566+1.p} Prove axiom M from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom\_M}\qquad\mbox{fof}(\mbox{km5\_axiom\_M},\mbox{conjecture})$

\medskip\textbf{LCL546+1.p} Prove axiom m6 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_6\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_6,\mbox{conjecture})$

\medskip\textbf{LCL007-1.p} EC-2 depends on the Wajsberg system

Two axiomatisations of the equivalential calculus are
 {EC-1,EC-2} by Lesniewski, and {EC-4,EC-5} by Wajsburg. Show
 that EC-2 can be derived from the Wajsburg system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(y,x)))\qquad\mbox{cnf}(\mbox{ec}_4,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(x,\mbox{equivalent}(y,z))))\qquad\mbox{cnf}(\mbox{ec}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(\mbox{equivalent}(a,b),c)))\qquad\mbox{cnf}(\mbox{prove\_ec}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL342-3.p} Principia Mathematica 5.501






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{equivalent}(q,\mbox{equivalent}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL861$\wedge$1.p} Modal logic S5(=M5) coincides with M45






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mreflexive}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL467+1.p} Prove Hilbert's and\_2 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_2\qquad\mbox{fof}(\mbox{hilbert\_and}_2,\mbox{conjecture})$

\medskip\textbf{LCL551+1.p} Prove Hilbert's modus\_tollens axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{modus\_tollens}\qquad\mbox{fof}(\mbox{hilbert\_modus\_tollens},\mbox{conjecture})$

\medskip\textbf{LCL563+1.p} Prove Hilbert's equivalence\_3 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{conjecture})$

\medskip\textbf{LCL408-1.p} Wajsberg algebra lattice structure definitions






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

\medskip\textbf{LCL333-3.p} Principia Mathematica 5.32






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{equivalent}(q,r)),\mbox{equivalent}(\mbox{and}(p,q),\mbox{and}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL172-1.p} Principia Mathematica 2.04






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))),\mbox{or}(\mbox{not}(q),\mbox{or}(\mbox{not}(p),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL885+1.p} An involutive pocrim that is not a coop






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' \mbox{'1'} = a\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;\exists x_{18}{:}\;x_{18} = x_{18} '==>' x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL337-3.p} Principia Mathematica 5.4






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{implies}(p,q)),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL604$\wedge$1.p} T and 4 implied by reflexivity and transitivity of R in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) )\mbox{ and }\mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ a ) ) \ \Leftarrow \ ( \mbox{reflexive}@ r\mbox{ and }\mbox{transitive}@ r ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL704$\wedge$1.p} Accessibility relation implies axiom for partial functionality






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

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

\medskip\textbf{LCL144-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{ordered}(x,\mbox{implies}(y,z))\mbox{ or }\mbox{ordered}(y,\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{antecedent},\mbox{negated\_conjecture})$

$\mbox{ordered}(x,\mbox{implies}(y,z)) \ \Rightarrow \ \neg\,\mbox{ordered}(y,\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL196-1.p} Principia Mathematica 2.4






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(p,\mbox{or}(p,q))),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL250-3.p} Principia Mathematica 3.42






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{and}(p,q),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL288-3.p} Principia Mathematica 4.52






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{not}(q)),\mbox{or}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL112-2.p} MV-29 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-29 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,\mbox{not}(\mbox{not}(x)))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{LCL875-1.p} The Rezus formula






$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(c,a),\mbox{implies}(c,b))))\qquad\mbox{cnf}(b,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(b,\mbox{implies}(a,c))))\qquad\mbox{cnf}(c,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,a))\qquad\mbox{cnf}(i,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(u,\mbox{implies}(u,v)),\mbox{implies}(u,v)))\qquad\mbox{cnf}(w,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(\mbox{implies}(y,y),\mbox{implies}(\mbox{implies}(z,z),\mbox{implies}(\mbox{implies}(u,u),\mbox{implies}(\mbox{implies}(v,v),\mbox{implies}(x,w))))),w)),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(v_6,v_7),\mbox{implies}(\mbox{implies}(v_7,v_8),\mbox{implies}(v_6,v_8))),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(v_9,\mbox{implies}(v_9,v_{10})),\mbox{implies}(v_9,v_{10})),\mbox{implies}(\mbox{implies}(\mbox{implies}(v_{11},v_{12}),\mbox{implies}(\mbox{implies}(v_{12},v_{13}),\mbox{implies}(v_{11},v_{13}))),v_{14})),v_{14}),v_{15})),v_{15}),\mbox{implies}(\mbox{implies}(v_{16},\mbox{implies}(\mbox{implies}(\mbox{implies}(v_{17},v_{17}),\mbox{implies}(\mbox{implies}(v_{18},v_{18}),\mbox{implies}(\mbox{implies}(v_{19},v_{19}),\mbox{implies}(\mbox{implies}(v_{20},v_{20}),\mbox{implies}(v_{16},v_{21}))))),v_{21})),v_{22}))),v_{22}))\qquad\mbox{cnf}(\mbox{prove\_rezus},\mbox{negated\_conjecture})$

\medskip\textbf{LCL090-1.p} IC-1 depends on the 5th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-1 depends on the fifth Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(t,\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,a))\qquad\mbox{cnf}(\mbox{prove\_ic}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL728$\wedge$5.p} TPS problem THM532

AC1 =$>$ AC3 from [RR93].




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

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

$\forall \mbox{xs}{:}\;( b\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xy}{:}\;b{:}\;( x@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall x{:}\;b\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) ) \ \Rightarrow \ \forall \mbox{xr}{:}\;a\to b\to \mbox{\$o}{:}\;\exists \mbox{xg}{:}\;a\to b{:}\;\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xy}{:}\;b{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ ( \mbox{xg}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{532},\mbox{conjecture})$

\medskip\textbf{LCL347-3.p} Principia Mathematica 5.61






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{or}(p,q),\mbox{not}(q)),\mbox{and}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL662+1.001.p} In KT, in backward search find a way through box and dia, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,p_1(x)\mbox{ or }p_1(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL206-3.p} Principia Mathematica 2.52






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(p,q)),\mbox{implies}(\mbox{not}(p),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL006-1.p} EC-1 depends on the Wajsberg system

Two axiomatisations of the equivalential calculus are
 {EC-1,EC-2} by Lesniewski, and {EC-4,EC-5} by Wajsburg. Show
 that EC-1 can be derived from the Wajsburg system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(y,x)))\qquad\mbox{cnf}(\mbox{ec}_4,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(x,\mbox{equivalent}(y,z))))\qquad\mbox{cnf}(\mbox{ec}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,a)),\mbox{equivalent}(b,c)))\qquad\mbox{cnf}(\mbox{prove\_ec}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL109-2.p} MV-4 depends on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided
 a different axiomatisation. Show that MV-4 depends on the
 Wajsberg system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(b,a)),\mbox{implies}(b,a))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL424-1.p} Mer-21 is a single axiom for two-valued logic

Show that the formula Mer-21 is a single axiom for two-valued
 logic by deriving the Lukasiewicz 3-basis.




$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{not}(c),\mbox{not}(d))),c),e),\mbox{implies}(\mbox{implies}(e,a),\mbox{implies}(d,a))))\qquad\mbox{cnf}(\mbox{mer}_{21},\mbox{axiom})$

$( \mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),a),a)) ) \ \Rightarrow \ \neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_luka\_3\_basis},\mbox{negated\_conjecture})$

\medskip\textbf{LCL246-3.p} Principia Mathematica 3.35






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,\mbox{implies}(p,q)),q))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL084-2.p} IC-4 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-4 depends on the first Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,x))\qquad\mbox{cnf}(\mbox{ic}_1,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_ic}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL291-3.p} Principia Mathematica 4.55






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{not}(\mbox{not}(\mbox{and}(p,q))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL892+1.p} Halving is unique in a hoop, rule for a/2 $>$= x






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17},x_{18},x_{19}{:}\;( ( x_{17} '==>' x_{18} '>=' x_{17}\mbox{ and }x_{19} = x_{19} '==>' x_{18} ) \ \Rightarrow \ x_{19} '>=' x_{17} )\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL195-1.p} Principia Mathematica 2.38






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(\mbox{or}(q,p)),\mbox{or}(r,p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL717$\wedge$1.p} Axiom implies accessibility relation for weak connectedness






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

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

\medskip\textbf{LCL666+1.001.p} In KT, pigeonhole formulae, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }p_{101}(y) )\mbox{ or }\neg\,p_{201}(x)\mbox{ and }p_{101}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL859$\wedge$1.p} Modal logic S5(=M5) coincides with MB5 






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mreflexive}@ r\mbox{ and }\mbox{msymmetric}@ r\mbox{ and }\mbox{meuclidean}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL123-1.p} LG-4 depends on LG-2

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-4
 depends on LG-2.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u))))))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(e,\mbox{equivalent}(b,f))),\mbox{equivalent}(a,\mbox{equivalent}(e,\mbox{equivalent}(c,f)))))\qquad\mbox{cnf}(\mbox{prove\_lg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL565+1.p} Prove necessitation from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{necessitation}\qquad\mbox{fof}(\mbox{km5\_necessitation},\mbox{conjecture})$

\medskip\textbf{LCL075-1.p} CN-3 depends on the single Merideth axiom

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-3 depends on the single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{not}(z),\mbox{not}(u))),z),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{cn\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL127-1.p} LG-2 depends on S-2

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-2
 depends on S-2.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u)))))\qquad\mbox{cnf}(s_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,c),\mbox{equivalent}(\mbox{equivalent}(b,e),\mbox{equivalent}(c,e))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL919$\wedge$1.p} Modal logic D






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

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

\medskip\textbf{LCL156-1.p} The 4th alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{and\_star}(x,\mbox{truth})\ne x\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL314-3.p} Principia Mathematica 4.83






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{implies}(p,q),\mbox{implies}(\mbox{not}(p),q)),q))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL877$\wedge$1.p} Variants of axiom 5






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

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

\medskip\textbf{LCL512+1.p} Prove Hilbert's equivalence\_1 axiom from Rosser's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_1\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_1,\mbox{conjecture})$

\medskip\textbf{LCL612$\wedge$1.p} Modus Ponens holds in K






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mvalid}@ x\mbox{ and }\mbox{mvalid}@ ( \mbox{mimpl}@ x@ y ) ) \ \Rightarrow \ ( \mbox{mvalid}@ y ) )\qquad\mbox{thf}(\mbox{modus\_ponens},\mbox{conjecture})$

\medskip\textbf{LCL490+1.p} Prove Hilbert's or\_1 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_1\qquad\mbox{fof}(\mbox{hilbert\_or}_1,\mbox{conjecture})$

\medskip\textbf{LCL698$\wedge$1.p} Embedding of quantified multimodal logic in simple type theory






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

\medskip\textbf{LCL421-1.p} Prove KA1 from MV1--MV4






$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(\mbox{implies}(\mbox{implies}(b,c),c))),\mbox{not}(\mbox{implies}(\mbox{implies}(b,c),c)))),\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{not}(b))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(c)),\mbox{not}(c)))),\mbox{not}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(c)),\mbox{not}(c))))))\qquad\mbox{cnf}(\mbox{prove\_ka}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL385-1.p} CN-47 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-47 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{not}(y)),\mbox{implies}(y,\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{47},\mbox{negated\_conjecture})$

\medskip\textbf{LCL726$\wedge$5.p} TPS problem THM534

AC1 =$>$ AC17 from [RR93].




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

$\forall \mbox{xs}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) ) \ \Rightarrow \ \forall \mbox{xg}{:}\;( ( a\to \mbox{\$o} )\to a )\to a\to \mbox{\$o}{:}\;( \forall \mbox{xh}{:}\;( a\to \mbox{\$o} )\to a{:}\;\exists \mbox{xu}{:}\;a{:}\;( \mbox{xg}@ \mbox{xh}@ \mbox{xu} ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;( \mbox{xg}@ \mbox{xf}@ ( \mbox{xf}@ ( \mbox{xg}@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{534},\mbox{conjecture})$

\medskip\textbf{LCL631$\wedge$1.p} The muddy forehead puzzle






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

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

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

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

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

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ x ) )\qquad\mbox{thf}(\mbox{knowledge\_implies\_truth},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(\mbox{positive\_introspection},\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) )@ ( \mbox{mbox}@ r@ ( \mbox{mnot}@ ( \mbox{mbox}@ r@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{negitive\_introspection},\mbox{axiom})$

$\mbox{ck} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},w{:}\;\mbox{\$i}{:}\;\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mbox}@ r@ x@ w ) )\qquad\mbox{thf}(\mbox{common\_knowledge},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfb} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_b},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfc} )@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_a\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfa} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfc} )@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) )\qquad\mbox{thf}(\mbox{what\_b\_knows\_about\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfa} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfa} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{ck}@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfb} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfb} ) ) ) )\qquad\mbox{thf}(\mbox{what\_c\_knows\_about\_b},\mbox{axiom})$

$s = ( \mbox{mor}@ ( \mbox{mbox}@ a@ \mbox{mfa} )@ ( \mbox{mor}@ ( \mbox{mbox}@ a@ ( \mbox{mnot}@ \mbox{mfa} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ \mbox{mfb} )@ ( \mbox{mor}@ ( \mbox{mbox}@ b@ ( \mbox{mnot}@ \mbox{mfb} ) )@ ( \mbox{mor}@ ( \mbox{mbox}@ c@ \mbox{mfc} )@ ( \mbox{mbox}@ c@ ( \mbox{mnot}@ \mbox{mfc} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{someone\_knows\_its\_forehead},\mbox{definition})$

$\mbox{mvalid}@ ( \mbox{mnot}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mnot}@ ( \mbox{mimpl}@ ( \mbox{ck}@ ( \mbox{mor}@ \mbox{mfa}@ ( \mbox{mor}@ \mbox{mfb}@ \mbox{mfc} ) ) )@ s ) ) )@ s ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL321-3.p} Principia Mathematica 5.12






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{implies}(p,q),\mbox{implies}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL266-3.p} Principia Mathematica 4.14






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{implies}(q,r)),\mbox{and}(p,\mbox{implies}(\mbox{not}(r),\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL092-1.p} IC-3 depends on the 5th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-3 depends on the fifth Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(t,\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_ic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL122-1.p} LG-3 depends on LG-2

Kalman's axiomatisation of the right group calculus
 is {LG-1,LG-2,LG-3,LG-4,LG-5}. McCune has shown that LG-2
 is a single axiom. Other axiomatisations by McCune are
 {Q-2,Q-3}, {Q-3,Q-4}, S-2, S-3, S-4, P-4, S-6. Show that LG-3
 depends on LG-2.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(\mbox{equivalent}(y,u),\mbox{equivalent}(z,u))))))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(b,\mbox{equivalent}(c,e)),\mbox{equivalent}(b,\mbox{equivalent}(\mbox{equivalent}(c,f),\mbox{equivalent}(e,f)))))))\qquad\mbox{cnf}(\mbox{prove\_lg}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL319-3.p} Principia Mathematica 5.1






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,q),\mbox{equivalent}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL249-3.p} Principia Mathematica 3.41






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,r),\mbox{implies}(\mbox{and}(p,q),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL417-1.p} XCB is a single axiom for the equivalential calculus

Show that formula XCB is a single axiom for the equivalential
 calculus by deriving the single axiom WN by condensed detachment.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(a,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(c,b)),c)))\qquad\mbox{cnf}(\mbox{xcb},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),\mbox{equivalent}(c,\mbox{equivalent}(a,b))))\qquad\mbox{cnf}(\mbox{prove\_wn},\mbox{negated\_conjecture})$

\medskip\textbf{LCL542+1.p} Prove axiom m2 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_2\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_2,\mbox{conjecture})$

\medskip\textbf{LCL541+1.p} Prove axiom m1 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_1\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_1,\mbox{conjecture})$

\medskip\textbf{LCL248-3.p} Principia Mathematica 3.4






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,q),\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL348-3.p} Principia Mathematica 5.62






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(\mbox{and}(p,q),\mbox{not}(q)),\mbox{or}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL434-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{v\_pa},\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_pa},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

\medskip\textbf{LCL155-1.p} The 3rd alternative Wajsberg algebra axiom






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL002-1.ax'})$

$\mbox{xor}(x,x)\ne \mbox{falsehood}\qquad\mbox{cnf}(\mbox{prove\_alternative\_wajsberg\_axiom},\mbox{negated\_conjecture})$

\medskip\textbf{LCL508+1.p} Prove Hilbert's and\_3 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_3\qquad\mbox{fof}(\mbox{hilbert\_and}_3,\mbox{conjecture})$

\medskip\textbf{LCL202-1.p} Principia Mathematica 2.48






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(p,q))),\mbox{or}(p,\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL229-1.p} Principia Mathematica 2.83






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),r))),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(r),s))),\mbox{or}(\mbox{not}(p),\mbox{or}(\mbox{not}(q),s)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL874$\wedge$1.p} Inclusion statement in a 2-D logic of knowledge and belief

Suppose we want to work with a 2-dimensional logic combining a 
 modality box\_k of knowledge with a modality box\_b of belief. 
 Moreover, suppose we model box\_k as an S5 modality and box\_b as 
 an D45 modality and let us furthermore add two axioms 
 characterizing their relationship. We may then want to check 
 whether or not box\_k and box\_b coincide, i.e., whether box\_k 
 includes box\_b




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

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

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

$\mbox{mreflexive}@ \mbox{rk}\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\mbox{mserial}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{rk}\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_4,\mbox{axiom})$

$\mbox{meuclidean}@ \mbox{rk}\qquad\mbox{thf}(\mbox{ax}_5,\mbox{axiom})$

$\mbox{meuclidean}@ \mbox{rb}\qquad\mbox{thf}(\mbox{ax}_6,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ \mbox{rk}@ a )@ ( \mbox{mbox}@ \mbox{rb}@ a ) ) )\qquad\mbox{thf}(\mbox{ax}_7,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ \mbox{rb}@ a )@ ( \mbox{mbox}@ \mbox{rb}@ ( \mbox{mbox}@ \mbox{rk}@ a ) ) ) )\qquad\mbox{thf}(\mbox{ax}_8,\mbox{axiom})$

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

\medskip\textbf{LCL181-3.p} Principia Mathematica 2.15






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(p),q),\mbox{implies}(\mbox{not}(q),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL922$\wedge$1.p} Modal logic S4






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

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

\medskip\textbf{LCL902+1.p} A boolean pocrim is involutive






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '>=' a\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( x_0 '>=' x_1\mbox{ and }x_1 '>=' x_2 ) \ \Rightarrow \ x_0 '>=' x_2 )\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall x_3,x_4{:}\;( ( x_3 '>=' x_4\mbox{ and }x_4 '>=' x_3 ) \ \Rightarrow \ x_3 = x_4 )\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

$\forall x_5,x_6,x_7{:}\;( x_5 '+' x_6 '>=' x_7\iff x_6 '>=' x_5 '==>' x_7 )\qquad\mbox{fof}(\mbox{sos}_{07},\mbox{axiom})$

$\forall a{:}\;a '>=' \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$\forall x_8,x_9,x_{10}{:}\;( x_8 '>=' x_9 \ \Rightarrow \ x_8 '+' x_{10} '>=' x_9 '+' x_{10} )\qquad\mbox{fof}(\mbox{sos}_{09},\mbox{axiom})$

$\forall x_{11},x_{12},x_{13}{:}\;( x_{11} '>=' x_{12} \ \Rightarrow \ x_{12} '==>' x_{13} '>=' x_{11} '==>' x_{13} )\qquad\mbox{fof}(\mbox{sos}_{10},\mbox{axiom})$

$\forall x_{14},x_{15},x_{16}{:}\;( x_{14} '>=' x_{15} \ \Rightarrow \ x_{16} '==>' x_{14} '>=' x_{16} '==>' x_{15} )\qquad\mbox{fof}(\mbox{sos}_{11},\mbox{axiom})$

$\forall a{:}\;a '+' \mbox{'1'} = \mbox{'1'}\qquad\mbox{fof}(\mbox{sos}_{12},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'1'} '==>' a '==>' a = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{13},\mbox{axiom})$

$\forall x_{17}{:}\;x_{17} '==>' \mbox{'1'} '==>' \mbox{'1'} = x_{17}\qquad\mbox{fof}(\mbox{goals}_{14},\mbox{conjecture})$

\medskip\textbf{LCL375-1.p} CN-32 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-32 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(x,u),\mbox{implies}(\mbox{implies}(u,z),z))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{32},\mbox{negated\_conjecture})$

\medskip\textbf{LCL371-1.p} CN-27 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-27 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{27},\mbox{negated\_conjecture})$

\medskip\textbf{LCL197-3.p} Principia Mathematica 2.41






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(q,\mbox{or}(p,q)),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL709$\wedge$1.p} Axiom implies accessibility relation for reflexivity






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

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

\medskip\textbf{LCL173-3.p} Principia Mathematica 2.05






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(q,r),\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL077-2.p} CN-39 depends on the Church system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-39 depends on the Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$( \mbox{is\_a\_theorem}(\mbox{implies}(x_1,x_2))\mbox{ and }\mbox{is\_a\_theorem}(\mbox{implies}(x_2,x_3)) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{implies}(x_1,x_3))\qquad\mbox{cnf}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(\mbox{not}(a)),a))\qquad\mbox{cnf}(\mbox{prove\_cn}_{39},\mbox{negated\_conjecture})$

\medskip\textbf{LCL253-3.p} Principia Mathematica 3.45






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(\mbox{and}(p,r),\mbox{and}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL201-1.p} Principia Mathematica 2.47






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(p,q))),\mbox{or}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL523+1.p} Prove axiom 4 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{axiom}_4\qquad\mbox{fof}(\mbox{km4b\_axiom}_4,\mbox{conjecture})$

\medskip\textbf{LCL061-1.p} CN-59 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-59 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),c),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(\mbox{implies}(a,b),c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{59},\mbox{negated\_conjecture})$

\medskip\textbf{LCL358-1.p} CN-07 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-07 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(y,z),u)),\mbox{implies}(\mbox{implies}(y,v),\mbox{implies}(x,\mbox{implies}(\mbox{implies}(v,z),u)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{07},\mbox{negated\_conjecture})$

\medskip\textbf{LCL069-1.p} CN-3 depends on the second Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-3 depends on the second Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(y,z)))\qquad\mbox{cnf}(\mbox{cn}_{19},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{not}(x),z)))\qquad\mbox{cnf}(\mbox{cn}_{37},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),z),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(\mbox{implies}(x,y),z))))\qquad\mbox{cnf}(\mbox{cn}_{59},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(a),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL480+1.p} Prove Rosser's kn1 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_1\qquad\mbox{fof}(\mbox{rosser\_kn}_1,\mbox{conjecture})$

\medskip\textbf{LCL320-3.p} Principia Mathematica 5.11






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{implies}(p,q),\mbox{implies}(\mbox{not}(p),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL236-3.p} Principia Mathematica 3.14






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)),\mbox{not}(\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL570+1.p} Prove axiom K from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom\_K}\qquad\mbox{fof}(\mbox{km5\_axiom\_K},\mbox{conjecture})$

\medskip\textbf{LCL224-3.p} Principia Mathematica 2.76






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,\mbox{implies}(q,r)),\mbox{implies}(\mbox{or}(p,q),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL447-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL059-1.p} CN-49 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-49 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(a),\mbox{not}(b)),\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{49},\mbox{negated\_conjecture})$

\medskip\textbf{LCL719$\wedge$1.p} Necessitation rule holds in monomodal logic K






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

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

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

$( \mbox{mvalid}@ \mbox{phi} ) \ \Rightarrow \ ( \mbox{mvalid}@ ( \mbox{mbox\_k}@ \mbox{phi} ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL397-1.p} CN-63 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-63 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{63},\mbox{negated\_conjecture})$

\medskip\textbf{LCL012-1.p} YQJ depends on UM

Show that the single Lukasiewicz axiom YQJ can be derived
 from the single Meredith axiom UM.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(y,\mbox{equivalent}(z,x))))\qquad\mbox{cnf}(\mbox{um},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(\mbox{equivalent}(c,a),\mbox{equivalent}(b,c))))\qquad\mbox{cnf}(\mbox{prove\_yqj},\mbox{negated\_conjecture})$

\medskip\textbf{LCL034-1.p} C0-3 depends on the Merideth axiom

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-3 can be derived from the
 single Meredith axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(z,\mbox{falsehood})),u),v),\mbox{implies}(\mbox{implies}(v,x),\mbox{implies}(z,x))))\qquad\mbox{cnf}(\mbox{c0\_CAMerideth},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_c0}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL083-1.p} IC-3 depends on the 1st Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms. Show that IC-3 depends on the first Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),z),\mbox{implies}(\mbox{implies}(z,x),\mbox{implies}(u,x))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_ic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL230-1.p} Principia Mathematica 2.85






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(p,r))),\mbox{or}(p,\mbox{or}(\mbox{not}(q),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL584$\wedge$1.p} Popkorn problem 5






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

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

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

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

$\mbox{mvalid}@ ( \mbox{miff}@ ( \mbox{mbox}@ r@ ( \mbox{mand}@ a@ b ) )@ ( \mbox{mand}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ b ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL611$\wedge$1.p} LAMBDA$\wedge$mm\_1 validates the converse Barcan formula






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{individuals}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ ( \mbox{mall}@ \lambda x{:}\;\mbox{individuals}{:}\;( a@ x ) ) )@ ( \mbox{mall}@ \lambda x{:}\;\mbox{individuals}{:}\;( \mbox{mbox}@ r@ ( a@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL601$\wedge$1.p} Axiom 4 for all R means all R are valid

If axiom 4 is valid for all relations R then all relations R are
 transitive.




$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ a )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ a ) ) ) )\iff \forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{transitive}@ r )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL025-1.p} C0-1 depends on the Church system

Axiomatisations for the Implication/Falsehood 2 valued
 sentential calculus are {C0-1,C0-2,C0-3,C0-4}
 by Tarski-Bernays, {C0-2,C0-5,C0-6} by Church, and the single
 Meredith axioms. Show that C0-1 can be derived from the
 Church system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{c0}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{falsehood}),\mbox{falsehood}),x))\qquad\mbox{cnf}(\mbox{c0}_5,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{c0}_6,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_c0}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL338-3.p} Principia Mathematica 5.41






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(q,r)),\mbox{implies}(p,\mbox{implies}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL379-1.p} CN-38 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-38 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{not}(x)),\mbox{not}(x)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{38},\mbox{negated\_conjecture})$

\medskip\textbf{LCL406-1.p} Generate LTL structures of size 4






$x \leq x\qquad\mbox{cnf}(\mbox{reflexivity\_of\_less\_or\_equal},\mbox{axiom})$

$( x \leq y\mbox{ and }y \leq z ) \ \Rightarrow \ x \leq z\qquad\mbox{cnf}(\mbox{transitivity\_of\_less\_or\_equal},\mbox{axiom})$

$x \leq y\mbox{ or }y \leq x\qquad\mbox{cnf}(\mbox{completeness\_of\_less\_or\_equal},\mbox{axiom})$

$\mbox{successor}(x) = y \ \Rightarrow \ x \leq y\qquad\mbox{cnf}(\mbox{predecessor\_less\_or\_equal},\mbox{axiom})$

$x \leq y \ \Rightarrow \ ( y = x\mbox{ or }y = \mbox{successor}(x)\mbox{ or }y = \mbox{successor}(\mbox{successor}(x))\mbox{ or }y = \mbox{successor}(\mbox{successor}(\mbox{successor}(x))) )\qquad\mbox{cnf}(\mbox{four\_step\_successor},\mbox{axiom})$

\medskip\textbf{LCL339-3.p} Principia Mathematica 5.42






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{implies}(p,\mbox{implies}(q,r)),\mbox{implies}(p,\mbox{implies}(q,\mbox{and}(p,r)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL930$\wedge$1.p} Embedding of second order modal logic S5 with universal access






$\mbox{include}(\mbox{'Axioms/LCL017\string^0.ax'})$

\medskip\textbf{LCL182-1.p} Principia Mathematica 2.16






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{not}(q)),\mbox{not}(p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL686+1.001.p} In S4, formula provable in S5 embedding, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_3(y)\mbox{ or }\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\mbox{\$false} )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,( p_2(y)\mbox{ and }\neg\,p_1(y) )\mbox{ or }( \neg\,p_2(y)\mbox{ and }p_1(y) ) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_3(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( \neg\,p_1(y)\mbox{ and }\neg\,p_2(y) )\mbox{ or }( p_2(y)\mbox{ and }p_1(y) ) ) ) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL438-2.p} Problem about propositional logic






$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_pa},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

\medskip\textbf{LCL039-1.p} A theorem from Morgan

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-18,CN-35,CN-49} by Church. This
 can be extended to the  modal logic T by the addition
 of three axioms for the modal operators. This problem proves
 a simple result of T.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{18},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,z)),\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_{35},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{cn}_{49},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessary}(\mbox{implies}(x,y)),\mbox{implies}(\mbox{necessary}(x),\mbox{necessary}(y))))\qquad\mbox{cnf}(\mbox{necessitation}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessary}(x),x))\qquad\mbox{cnf}(\mbox{necessitation}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(x) \ \Rightarrow \ \mbox{is\_a\_theorem}(\mbox{necessary}(x))\qquad\mbox{cnf}(\mbox{axiom\_of\_necessitation},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{necessary}(a),\mbox{not}(\mbox{necessary}(\mbox{not}(a)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL907+1.p} Hilbert's axiomatization of propositional logic






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

\medskip\textbf{LCL231-3.p} Principia Mathematica 2.86






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(p,r)),\mbox{implies}(p,\mbox{implies}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL240-3.p} Principia Mathematica 3.26






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{and}(p,q),p))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL470+1.p} Prove Hilbert's or\_2 axiom from Lukasiewicz's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+3.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{conjecture})$

\medskip\textbf{LCL500+1.p} Prove Rosser's kn2 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_2\qquad\mbox{fof}(\mbox{rosser\_kn}_2,\mbox{conjecture})$

\medskip\textbf{LCL112-1.p} MV-29 depnds on the Merideth system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Show that 29 depends
 on the Meredith system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{not}(\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_mv}_{29},\mbox{negated\_conjecture})$

\medskip\textbf{LCL711$\wedge$1.p} Axiom implies accessibility relation for seriality






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

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

\medskip\textbf{LCL434-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_p},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OK}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_q},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_r},\mbox{t\_a}),\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OS}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_pa},\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_pa},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL192-1.p} Principia Mathematica 2.32 and 2.33






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{or}(p,q),r)),\mbox{or}(p,\mbox{or}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL103-1.p} LG-2 depends on the 5th McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-2 depends on the 5th McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),z),\mbox{equivalent}(\mbox{equivalent}(u,y),\mbox{equivalent}(\mbox{equivalent}(x,u),z))))\qquad\mbox{cnf}(p_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(\mbox{equivalent}(y,x),z)),z))\qquad\mbox{cnf}(q_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),\mbox{equivalent}(a,c)),\mbox{equivalent}(b,c)),e),e))\qquad\mbox{cnf}(\mbox{prove\_lg}_2,\mbox{negated\_conjecture})$

\medskip\textbf{LCL237-3.p} Principia Mathematica 3.21






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(q,\mbox{implies}(p,\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL533+1.p} Prove axiom m6 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_6\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_m}_6,\mbox{conjecture})$

\medskip\textbf{LCL591$\wedge$1.p} Axiom N is valid






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

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

\medskip\textbf{LCL537+1.p} Prove axiom 5 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{axiom}_5\qquad\mbox{fof}(\mbox{km5\_axiom}_5,\mbox{conjecture})$

\medskip\textbf{LCL349-3.p} Principia Mathematica 5.63






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(p,q),\mbox{or}(p,\mbox{and}(\mbox{not}(p),q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL646+1.001.p} In K, path through a labyrinth, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_6(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_4(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_5(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_3(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_1(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL517+1.p} Prove Lukasiewicz's cn3 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_3\qquad\mbox{fof}(\mbox{luka\_cn}_3,\mbox{conjecture})$

\medskip\textbf{LCL534+1.p} Prove axiom s3 from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_s}_3\qquad\mbox{fof}(\mbox{s1\_0\_m6s3m9b\_axiom\_s}_3,\mbox{conjecture})$

\medskip\textbf{LCL649+1.001.p} In K, pigeonhole formulae missing a conjunct, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }\neg\,p_{101}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }p_{101}(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL664+1.001.p} In KT, path through a labyrinth, size 1






$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{16}(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{12}(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{14}(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{12}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{15}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{13}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{12}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_{11}(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL252-1.p} Principia Mathematica 3.44






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),p)),\mbox{not}(\mbox{or}(\mbox{not}(r),p))))),\mbox{or}(\mbox{not}(\mbox{or}(q,r)),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL727$\wedge$5.p} TPS problem THM533

AC3 =$>$ AC1 from [RR93].




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

$\forall \mbox{xr}{:}\;( a\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;\exists \mbox{xg}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xy}{:}\;a{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xr}@ \mbox{xx}@ ( \mbox{xg}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xs}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{xs}@ x ) \ \Rightarrow \ ( x@ ( \mbox{xf}@ x ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{533},\mbox{conjecture})$

\medskip\textbf{LCL215-1.p} Principia Mathematica 2.62 and 2.63






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL431-2.p} Problem about propositional logic






$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_p},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_emptyset},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_emptyset},\mbox{v\_p},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_minus}(\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_U},\mbox{t\_a}),\mbox{c\_PropLog\_Ohyps}(\mbox{v\_p},\mbox{v\_t}_0,\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Ocompleteness\_\_0\_\_lemma\_\_dest}_0,\mbox{axiom})$

$\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_Set\_ODiff\_\_cancel}_0,\mbox{axiom})$

\medskip\textbf{LCL322-3.p} Principia Mathematica 5.13






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{implies}(p,q),\mbox{implies}(q,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL097-1.p} LG-4 depends on LG-2, LG-3

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that LG-4 depends on a part of the Kalman system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),u),\mbox{equivalent}(\mbox{equivalent}(y,z),u)),v),v))\qquad\mbox{cnf}(\mbox{lg}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),e),\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{falsehood}),c),\mbox{equivalent}(\mbox{equivalent}(b,\mbox{falsehood}),e))))\qquad\mbox{cnf}(\mbox{prove\_lg}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL170-3.p} Principia Mathematica 2.02






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(q,\mbox{implies}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL141-1.p} A lemma in Wajsberg algebras

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided
 a different axiomatisation. Show that MV-5 depends on the
 Wajsberg system.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{not}(x),\mbox{not}(y))\ne \mbox{implies}(y,x)\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL428-1.p} Prove the Harris/Rezus axiom from MV1-MV3 and MV5






$( \mbox{is\_a\_theorem}(\mbox{implies}(a,b))\mbox{ and }\mbox{is\_a\_theorem}(a) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(b)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(b,a)))\qquad\mbox{cnf}(\mbox{mv}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{mv}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),b),\mbox{implies}(\mbox{implies}(b,a),a)))\qquad\mbox{cnf}(\mbox{mv}_3,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(y,x)),\mbox{implies}(y,x)))\qquad\mbox{cnf}(\mbox{mv}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(y,x)),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(z,u),\mbox{implies}(\mbox{implies}(v,z),\mbox{implies}(v,u))),\mbox{implies}(\mbox{implies}(w,\mbox{implies}(v_6,w)),v_7)),v_7),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(v_8,v_9),v_9),\mbox{implies}(\mbox{implies}(v_9,v_8),v_8)),v_{10})),v_{10}),\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{implies}(v_{11},v_{12}),\mbox{implies}(v_{12},v_{11})),\mbox{implies}(v_{12},v_{11})),v_{13})),v_{13}),\mbox{implies}(\mbox{implies}(v_{14},\mbox{implies}(v_{15},v_{14})),v_{16}))),v_{16}))\qquad\mbox{cnf}(\mbox{prove\_hr},\mbox{negated\_conjecture})$

\medskip\textbf{LCL109-5.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{ordered}(x,y) \ \Rightarrow \ \mbox{ordered}(\mbox{implies}(x,z),\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{lemma}_1,\mbox{axiom})$

$\mbox{ordered}(x,y) \ \Rightarrow \ \mbox{ordered}(\mbox{implies}(z,x),\mbox{implies}(z,y))\qquad\mbox{cnf}(\mbox{lemma}_2,\mbox{axiom})$

$\mbox{ordered}(x,\mbox{implies}(y,z)) \ \Rightarrow \ \mbox{ordered}(y,\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{lemma\_3}_1,\mbox{axiom})$

$\mbox{ordered}(y,\mbox{implies}(x,z)) \ \Rightarrow \ \mbox{ordered}(x,\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{lemma\_3}_2,\mbox{axiom})$

$\mbox{not}(\mbox{big\_V}(x,y)) = \mbox{big\_hat}(\mbox{not}(x),\mbox{not}(y))\qquad\mbox{cnf}(\mbox{lemma}_4,\mbox{axiom})$

$\mbox{not}(\mbox{big\_hat}(x,y)) = \mbox{big\_V}(\mbox{not}(x),\mbox{not}(y))\qquad\mbox{cnf}(\mbox{lemma}_5,\mbox{axiom})$

$\mbox{implies}(\mbox{big\_V}(x,y),z) = \mbox{big\_hat}(\mbox{implies}(x,z),\mbox{implies}(y,z))\qquad\mbox{cnf}(\mbox{lemma}_6,\mbox{axiom})$

$\mbox{implies}(x,\mbox{big\_hat}(y,z)) = \mbox{big\_hat}(\mbox{implies}(x,y),\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{lemma}_7,\mbox{axiom})$

$\mbox{big\_V}(\mbox{implies}(x,y),\mbox{implies}(y,x))\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL049-1.p} CN-20 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-20 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(a,\mbox{implies}(\mbox{implies}(a,b),b)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{20},\mbox{negated\_conjecture})$

\medskip\textbf{LCL862$\wedge$1.p} Modal logic S5(=M5) coincides with M4B






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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{mreflexive}@ r\mbox{ and }\mbox{meuclidean}@ r )\iff ( \mbox{mreflexive}@ r\mbox{ and }\mbox{mtransitive}@ r\mbox{ and }\mbox{msymmetric}@ r ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{LCL345-3.p} Principia Mathematica 5.55






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{equivalent}(\mbox{or}(p,q),p),\mbox{equivalent}(\mbox{or}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL663+1.001.p} In KT, in backwards search no way through box and dia, size 1






$\neg\,\exists x{:}\;\neg\,p_1(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL648+1.001.p} In K, pigeonhole formulae, size 1






$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }p_{101}(y) )\mbox{ or }\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }\neg\,p_{201}(y)\mbox{ and }p_{101}(y) )\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL569+1.p} Prove axiom m10 from the S1-0M6S3M9B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+5.ax'})$

$\mbox{axiom\_m}_{10}\qquad\mbox{fof}(\mbox{s1\_0\_m10\_axiom\_m}_{10},\mbox{conjecture})$

\medskip\textbf{LCL234-3.p} Principia Mathematica 3.2






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{implies}(q,\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL460+1.p} Prove Rosser's kn2 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{rosser\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{rosser\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{rosser\_op\_equiv},\mbox{axiom})$

$\mbox{kn}_2\qquad\mbox{fof}(\mbox{rosser\_kn}_2,\mbox{conjecture})$

\medskip\textbf{LCL495+1.p} Prove Hilbert's equivalence\_3 axiom from Principia's system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{equivalence}_3\qquad\mbox{fof}(\mbox{hilbert\_equivalence}_3,\mbox{conjecture})$

\medskip\textbf{LCL378-1.p} CN-36 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-36 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{not}(x),\mbox{implies}(x,y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{36},\mbox{negated\_conjecture})$

\medskip\textbf{LCL621$\wedge$1.p} A simple theorem of K4






$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mbox}@ r@ x )@ ( \mbox{mbox}@ r@ ( \mbox{mbox}@ r@ x ) ) ) )\qquad\mbox{thf}(k_4,\mbox{axiom})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mvalid}@ ( \mbox{mimpl}@ ( \mbox{mand}@ ( \mbox{mbox}@ r@ ( \mbox{mdia}@ r@ x ) )@ ( \mbox{mbox}@ r@ y ) )@ ( \mbox{mbox}@ r@ ( \mbox{mdia}@ r@ ( \mbox{mand}@ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{LCL906-1.p} Lattice theory (equality) axioms






$\mbox{include}(\mbox{'Axioms/LAT001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/LAT001-3.ax'})$

\medskip\textbf{LCL062-1.p} CN-60 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-60 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(\mbox{not}(b),c)),\mbox{implies}(a,\mbox{implies}(\mbox{implies}(e,c),\mbox{implies}(\mbox{implies}(b,e),c)))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{60},\mbox{negated\_conjecture})$

\medskip\textbf{LCL381-1.p} CN-42 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-42 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(\mbox{not}(x)),y),z),\mbox{implies}(\mbox{implies}(x,y),z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{42},\mbox{negated\_conjecture})$

\medskip\textbf{LCL292-3.p} Principia Mathematica 4.56






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(\mbox{not}(p),\mbox{not}(q)),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL710$\wedge$1.p} Axiom implies accessibility relation for symmetry






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

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

\medskip\textbf{LCL019-1.p} PYM depends on XGK

Show that the single Meredith axiom PYM can be derived from
 the single Kalman axiom XGK.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,\mbox{equivalent}(z,x)),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(\mbox{xgk},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),b),\mbox{equivalent}(c,a)))\qquad\mbox{cnf}(\mbox{prove\_pym},\mbox{negated\_conjecture})$

\medskip\textbf{LCL559+1.p} Prove Hilbert's or\_2 axiom from the S1-0 system






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{or}_2\qquad\mbox{fof}(\mbox{hilbert\_or}_2,\mbox{conjecture})$

\medskip\textbf{LCL058-1.p} CN-46 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-46 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{not}(b),\mbox{not}(a))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{46},\mbox{negated\_conjecture})$

\medskip\textbf{LCL895+1.p} Weak conjunction is lub in a hoop using equational axioms






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

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

$\forall a{:}\;a '+' \mbox{'0'} = a\qquad\mbox{fof}(\mbox{sos}_{03},\mbox{axiom})$

$\forall a{:}\;a '==>' a = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{04},\mbox{axiom})$

$\forall a{:}\;a '==>' \mbox{'0'} = \mbox{'0'}\qquad\mbox{fof}(\mbox{sos}_{05},\mbox{axiom})$

$\forall a{:}\;\mbox{'0'} '==>' a = a\qquad\mbox{fof}(\mbox{sos}_{06},\mbox{axiom})$

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

$\forall a,b{:}\;a '+' a '==>' b = b '+' b '==>' a\qquad\mbox{fof}(\mbox{sos}_{08},\mbox{axiom})$

$( c '==>' a = \mbox{'0'}\mbox{ and }c '==>' b = \mbox{'0'} )\iff c '==>' a '+' a '==>' b = \mbox{'0'}\qquad\mbox{fof}(\mbox{goals}_{09},\mbox{conjecture})$

\medskip\textbf{LCL219-1.p} Principia Mathematica 2.68






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),q)),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL674+1.001.p} In S4, the branching formula made provable, size 1

The branching formula plus a negation symbol in front and an
 additional subformula to make the formula provable.




$\forall x{:}\;r_1(x,x)\qquad\mbox{fof}(\mbox{reflexivity},\mbox{axiom})$

$\forall x,y,z{:}\;( ( r_1(x,y)\mbox{ and }r_1(y,z) ) \ \Rightarrow \ r_1(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{axiom})$

$\neg\,\exists x{:}\;\neg\,\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }p_2(y) )\mbox{ or }\neg\,\forall y{:}\;( \neg\,r_1(x,y)\mbox{ or }( ( ( \neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) )\mbox{ and }\neg\,\forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ and }\neg\,p_{102}(x)\mbox{ and }p_{101}(x) ) )\mbox{ or }\neg\,\neg\,p_{101}(y)\mbox{ and }p_{100}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }p_2(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_2(x)\mbox{ or }\neg\,p_{101}(x) )\mbox{ or }\neg\,p_2(y) ) )\mbox{ or }\neg\,p_{101}(y) )\mbox{ and }( ( ( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }\neg\,p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }p_1(y) )\mbox{ and }( \forall x{:}\;( \neg\,r_1(y,x)\mbox{ or }p_1(x)\mbox{ or }\neg\,p_{100}(x) )\mbox{ or }\neg\,p_1(y) ) )\mbox{ or }\neg\,p_{100}(y) )\mbox{ and }( p_{101}(y)\mbox{ or }\neg\,p_{102}(y) )\mbox{ and }( p_{100}(y)\mbox{ or }\neg\,p_{101}(y) ) ) )\mbox{ and }\neg\,p_{101}(x)\mbox{ and }p_{100}(x)\qquad\mbox{fof}(\mbox{main},\mbox{conjecture})$

\medskip\textbf{LCL437-2.p} Problem about propositional logic






$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_pa},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_pa},\mbox{t\_a}),\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{v\_p},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_ODN}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_right}_0,\mbox{axiom})$

\medskip\textbf{LCL625$\wedge$1.p} GL/K4 axiom is valid in this frame

In a frame that is transitive and upwards well-founded, the GL/K4
 axiom is valid.




$\mbox{include}(\mbox{'Axioms/LCL008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{transitive}@ r\mbox{ and }\mbox{upwards\_well\_founded}@ r )\qquad\mbox{thf}(\mbox{upwf\_trans},\mbox{axiom})$

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

\medskip\textbf{LCL306-3.p} Principia Mathematica 4.74






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{not}(p),\mbox{equivalent}(q,\mbox{or}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL247-3.p} Principia Mathematica 3.37






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{implies}(\mbox{and}(p,q),r),\mbox{implies}(\mbox{and}(p,\mbox{not}(r)),\mbox{not}(q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL573+1.p} Prove axiom 5 from the S1-0M10 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+6.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{substitution\_of\_equivalents}\qquad\mbox{fof}(\mbox{substitution\_of\_equivalents},\mbox{axiom})$

$\mbox{axiom}_5\qquad\mbox{fof}(\mbox{km5\_axiom}_5,\mbox{conjecture})$

\medskip\textbf{LCL101-1.p} P-1 depends on the 3rd McCune system

Axiomatisations of the left group calculus are {LG-1,
 LG-2,LG-3,LG-4,LG-5} by Kalman, {LG-2,LG-3}, {LG-2,P-1},
 {LG-2,P-4}, {LG-2,Q-1,Q-2}, {P-1,Q-3}, {P-4,Q-3}, {Q-1,
 Q-2,Q-3}, {Q-1,Q-3,Q-4}, {LG-27-1690} all by McCune. Show
 that P-1 depends on the 3rd McCune system.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,y),\mbox{equivalent}(x,z)),\mbox{equivalent}(y,z)),u),u))\qquad\mbox{cnf}(\mbox{lg}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(y,z),\mbox{equivalent}(y,u)),\mbox{equivalent}(z,u)),x)))\qquad\mbox{cnf}(p_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,b),c),\mbox{equivalent}(\mbox{equivalent}(e,b),\mbox{equivalent}(\mbox{equivalent}(a,e),c))))\qquad\mbox{cnf}(\mbox{prove\_p}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL503+1.p} Prove Hilbert's implies\_1 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_1\qquad\mbox{fof}(\mbox{hilbert\_implies}_1,\mbox{conjecture})$

\medskip\textbf{LCL395-1.p} CN-61 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-61 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(z,y),\mbox{implies}(\mbox{implies}(\mbox{not}(x),z),y))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{61},\mbox{negated\_conjecture})$

\medskip\textbf{LCL204-1.p} Principia Mathematica 2.5






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),q))),\mbox{or}(\mbox{not}(\mbox{not}(p)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL152-1.p} A theorem in the lattice structure of Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL001-1.ax'})$

$\mbox{implies}(\mbox{big\_hat}(x,y),z)\ne \mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(x,z))\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_theorem},\mbox{negated\_conjecture})$

\medskip\textbf{LCL175-3.p} Principia Mathematica 2.07






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{or}(p,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL433-2.p} Problem about propositional logic






$\mbox{c\_PropLog\_Osat}(\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{v\_xa},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_xa},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_F},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{v\_F},\mbox{v\_U},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_U},\mbox{c\_PropLog\_Othms}(\mbox{v\_F},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_PropLog\_Osat}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{v\_q},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_PropLog\_Osat}(\mbox{v\_H},\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Osat\_\_imp}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OH}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_OMP}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

\medskip\textbf{LCL193-1.p} Principia Mathematica 2.36






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),\mbox{or}(r,p))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL272-3.p} Principia Mathematica 4.25






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{or}(p,p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL315-3.p} Principia Mathematica 4.84






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{equivalent}(p,q),\mbox{equivalent}(\mbox{implies}(p,r),\mbox{implies}(q,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL721$\wedge$1.p} Axiom D holds in monomodal logic D






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

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

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

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

\medskip\textbf{LCL093-1.p} IC-4 depends on the 5th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-4 depends on the fifth Lukasiewicz axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(t,\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_5,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,b),\mbox{implies}(\mbox{implies}(b,c),\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_ic}_4,\mbox{negated\_conjecture})$

\medskip\textbf{LCL305-3.p} Principia Mathematica 4.73






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(q,\mbox{equivalent}(p,\mbox{and}(p,q))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL382-1.p} CN-43 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-43 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,\mbox{not}(x)),\mbox{not}(x))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{43},\mbox{negated\_conjecture})$

\medskip\textbf{LCL492+1.p} Prove Hilbert's or\_3 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{or}_3\qquad\mbox{fof}(\mbox{hilbert\_or}_3,\mbox{conjecture})$

\medskip\textbf{LCL506+1.p} Prove Hilbert's and\_1 axiom from Rosser's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+5.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{and}_1\qquad\mbox{fof}(\mbox{hilbert\_and}_1,\mbox{conjecture})$

\medskip\textbf{LCL341-3.p} Principia Mathematica 5.5






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(p,\mbox{equivalent}(\mbox{implies}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL708$\wedge$1.p} Accessibility relation implies axiom for weak directedness






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

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

\medskip\textbf{LCL334-3.p} Principia Mathematica 5.33






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{and}(p,\mbox{implies}(q,r)),\mbox{and}(p,\mbox{implies}(\mbox{and}(p,q),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL050-1.p} CN-21 depends on the Lukasiewicz system

Axiomatisations of the Implication/Negation 2 valued
 sentential calculus are {CN-1,CN-2,CN-3} by Lukasiewicz,
 {CN-18,CN-21,CN-35,CN-39,CN-39,CN-40,CN-46} by Frege,
 {CN-3,CN-18,CN-21,CN-22,CN-30,CN-54} by Hilbert, {CN-18,
 CN-35,CN-49} by Church, {CN-19,CN-37,CN-59} by Lukasiewicz,
 {CN-19,CN-37,CN-60} by Wos, and the single Meredith axiom.
 Show that CN-21 depends on the short Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(a,\mbox{implies}(b,c)),\mbox{implies}(b,\mbox{implies}(a,c))))\qquad\mbox{cnf}(\mbox{prove\_cn}_{21},\mbox{negated\_conjecture})$

\medskip\textbf{LCL188-3.p} Principia Mathematica 2.25






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(p,\mbox{implies}(\mbox{or}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL924$\wedge$1.p} Region Connection Calculus






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

$\mbox{include}(\mbox{'Axioms/LCL014\string^0.ax'})$

\medskip\textbf{LCL283-3.p} Principia Mathematica 4.42






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(p,\mbox{or}(\mbox{and}(p,q),\mbox{and}(p,\mbox{not}(q)))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL526+1.p} Prove SSE from KM5 axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+2.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{substitution\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_substitution\_strict\_equiv},\mbox{conjecture})$

\medskip\textbf{LCL132-1.p} A lemma in Wajsberg algebras






$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(x,x)\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_wajsberg\_lemma},\mbox{negated\_conjecture})$

\medskip\textbf{LCL209-1.p} Principia Mathematica 2.54






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(p)),q)),\mbox{or}(p,q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL870$\wedge$1.p} The Barcan formula is valid in quantified modal logic K






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

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

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

$\mbox{mvalid}@ ( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox}@ r@ ( p@ x ) ) )@ ( \mbox{mbox}@ r@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( p@ x ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{conjecture})$

\medskip\textbf{LCL223-3.p} Principia Mathematica 2.75






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{or}(p,\mbox{implies}(q,r)),\mbox{or}(p,r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL738$\wedge$5.p} TPS problem from AC-THMS

Related to the axiom of choice.




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

$\exists \mbox{xr}_{24}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o},\mbox{xr}_{23}{:}\;( b\to \mbox{\$o} )\to b\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;\exists \mbox{xy}{:}\;b{:}\;\forall \mbox{xw}_6{:}\;b{:}\;( \mbox{xr}_{23}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_6\mbox{ or }\mbox{xr}_{24}@ \mbox{xx}@ \mbox{xy}@ \mbox{xw}_6 ) \ \Rightarrow \ \exists \mbox{xf}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xx}{:}\;b\to \mbox{\$o},\mbox{xw}_6{:}\;b{:}\;( \mbox{xr}_{23}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_6\mbox{ or }\mbox{xr}_{24}@ \mbox{xx}@ ( \mbox{xf}@ \mbox{xx} )@ \mbox{xw}_6 ) ) \ \Rightarrow \ \exists \mbox{xj}{:}\;( b\to \mbox{\$o} )\to b{:}\;\forall \mbox{xp}{:}\;b\to \mbox{\$o}{:}\;( \exists \mbox{xz}{:}\;b{:}\;( \mbox{xp}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{xj}@ \mbox{xp} ) ) )\qquad\mbox{thf}(\mbox{cX5310\_SUB},\mbox{conjecture})$

\medskip\textbf{LCL188-1.p} Principia Mathematica 2.25






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(p,\mbox{or}(\mbox{not}(\mbox{or}(p,q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL088-1.p} IC-3 depends on the 4th Lukasiewicz axiom

Axiomatisations of the Implicational propositional calculus
 are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz
 axioms.Show that IC-3 depends on the fourth Lukasiewicz
 axiom.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(p,q),\mbox{implies}(r,s)),\mbox{implies}(t,\mbox{implies}(\mbox{implies}(s,p),\mbox{implies}(r,p)))))\qquad\mbox{cnf}(\mbox{ic\_JLukasiewicz}_4,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(a,b),a),a))\qquad\mbox{cnf}(\mbox{prove\_ic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{LCL544+1.p} Prove axiom m4 from KM4B axiomatization of S5






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL007+3.ax'})$

$\mbox{op\_possibly}\qquad\mbox{fof}(\mbox{s1\_0\_op\_possibly},\mbox{axiom})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{s1\_0\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_implies},\mbox{axiom})$

$\mbox{op\_strict\_implies}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_equiv},\mbox{axiom})$

$\mbox{op\_strict\_equiv}\qquad\mbox{fof}(\mbox{s1\_0\_op\_strict\_equiv},\mbox{axiom})$

$\mbox{axiom\_m}_4\qquad\mbox{fof}(\mbox{s1\_0\_axiom\_m}_4,\mbox{conjecture})$

\medskip\textbf{LCL237-1.p} Principia Mathematica 3.21






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(q),\mbox{or}(\mbox{not}(p),\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL210-1.p} Principia Mathematica 2.55






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{not}(p)),\mbox{or}(\mbox{not}(\mbox{or}(p,q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL216-3.p} Principia Mathematica 2.64






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{implies}(\mbox{or}(p,q),\mbox{implies}(\mbox{or}(p,\mbox{not}(q)),p)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL362-1.p} CN-11 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-11 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,\mbox{implies}(\mbox{implies}(\mbox{not}(y),y),y)),\mbox{implies}(\mbox{implies}(\mbox{not}(y),y),y)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{11},\mbox{negated\_conjecture})$

\medskip\textbf{LCL485+1.p} Prove Hilbert's implies\_2 axiom from Principia's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+4.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{hilbert\_op\_or},\mbox{axiom})$

$\mbox{op\_implies\_and}\qquad\mbox{fof}(\mbox{hilbert\_op\_implies\_and},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{hilbert\_op\_equiv},\mbox{axiom})$

$\mbox{implies}_2\qquad\mbox{fof}(\mbox{hilbert\_implies}_2,\mbox{conjecture})$

\medskip\textbf{LCL387-1.p} CN-50 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-50 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),z),\mbox{implies}(\mbox{implies}(\mbox{not}(y),x),z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{50},\mbox{negated\_conjecture})$

\medskip\textbf{LCL360-1.p} CN-09 depends on the Lukasiewicz system

An axiomatisation of the Implication/Negation 2 valued
 sentential calculus is {CN-1,CN-2,CN-3} by Lukasiewicz.
 Show that CN-09 depends on the Lukasiewicz system.




$( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{cnf}(\mbox{cn}_1,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),x),x))\qquad\mbox{cnf}(\mbox{cn}_2,\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(\mbox{not}(x),y)))\qquad\mbox{cnf}(\mbox{cn}_3,\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),y),z),\mbox{implies}(x,z)))\qquad\mbox{cnf}(\mbox{prove\_cn}_{09},\mbox{negated\_conjecture})$

\medskip\textbf{LCL024-1.p} PYO depends on XGK

Show that Kalman's shortest single axiom for the
 equivalential calculus, XGK, can be derived from the Meredith
 single axiom PYO.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(x,\mbox{equivalent}(\mbox{equivalent}(y,\mbox{equivalent}(z,x)),\mbox{equivalent}(z,y))))\qquad\mbox{cnf}(\mbox{xgk},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),c),\mbox{equivalent}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_pyo},\mbox{negated\_conjecture})$

\medskip\textbf{LCL453+1.p} Prove Lukasiewicz's cn3 axiom from Hilbert's axiomatization






$\mbox{include}(\mbox{'Axioms/LCL006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL006+2.ax'})$

$\mbox{op\_or}\qquad\mbox{fof}(\mbox{luka\_op\_or},\mbox{axiom})$

$\mbox{op\_implies}\qquad\mbox{fof}(\mbox{luka\_op\_implies},\mbox{axiom})$

$\mbox{op\_equiv}\qquad\mbox{fof}(\mbox{luka\_op\_equiv},\mbox{axiom})$

$\mbox{cn}_3\qquad\mbox{fof}(\mbox{luka\_cn}_3,\mbox{conjecture})$

\medskip\textbf{LCL250-1.p} Principia Mathematica 3.42






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(q)))),r)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL212-1.p} Principia Mathematica 2.6






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(\mbox{not}(p)),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL253-1.p} Principia Mathematica 3.45






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{not}(\mbox{or}(\mbox{not}(p),\mbox{not}(r)))),\mbox{not}(\mbox{or}(\mbox{not}(q),\mbox{not}(r))))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL174-1.p} Principia Mathematica 2.06






$\mbox{include}(\mbox{'Axioms/LCL003-0.ax'})$

$\neg\,\mbox{theorem}(\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(p),q)),\mbox{or}(\mbox{not}(\mbox{or}(\mbox{not}(q),r)),\mbox{or}(\mbox{not}(p),r))))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL330-3.p} Principia Mathematica 5.25






$\mbox{include}(\mbox{'Axioms/LCL004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-1.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL004-2.ax'})$

$\neg\,\mbox{theorem}(\mbox{equivalent}(\mbox{or}(p,q),\mbox{implies}(\mbox{implies}(p,q),q)))\qquad\mbox{cnf}(\mbox{prove\_this},\mbox{negated\_conjecture})$

\medskip\textbf{LCL439-1.p} Problem about propositional logic






$\mbox{include}(\mbox{'Axioms/LCL005-0.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_p},\mbox{v\_H},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Odeduction}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_q},\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Othms\_\_notE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{v\_G},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_p},\mbox{c\_PropLog\_Othms}(\mbox{c\_insert}(\mbox{v\_a},\mbox{v\_G},\mbox{tc\_PropLog\_Opl}(\mbox{t\_a})),\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_PropLog\_Oweaken\_\_left\_\_insert}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{c\_PropLog\_Opl\_Ofalse},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_PropLog\_Opl\_Oop\_A\_N}_{62}(\mbox{v\_p},\mbox{v\_q},\mbox{t\_a}),\mbox{c\_PropLog\_Othms}(\mbox{v\_H},\mbox{t\_a}),\mbox{tc\_PropLog\_Opl}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{LCL018-1.p} PYO depends on PYM

Show that the single Meredith axiom PYO can be derived from
 the single Meredith axiom PYM.




$( \mbox{is\_a\_theorem}(\mbox{equivalent}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y)\qquad\mbox{cnf}(\mbox{condensed\_detachment},\mbox{axiom})$

$\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(x,\mbox{equivalent}(y,z)),y),\mbox{equivalent}(z,x)))\qquad\mbox{cnf}(\mbox{pym},\mbox{axiom})$

$\neg\,\mbox{is\_a\_theorem}(\mbox{equivalent}(\mbox{equivalent}(\mbox{equivalent}(a,\mbox{equivalent}(b,c)),c),\mbox{equivalent}(b,a)))\qquad\mbox{cnf}(\mbox{prove\_pyo},\mbox{negated\_conjecture})$

\medskip\textbf{LCL876+1.p} Prove Mv5 from MV1--MV4






$\forall y,x{:}\;( ( \mbox{is\_a\_theorem}(\mbox{implies}(x,y))\mbox{ and }\mbox{is\_a\_theorem}(x) ) \ \Rightarrow \ \mbox{is\_a\_theorem}(y) )\qquad\mbox{fof}(\mbox{cd},\mbox{axiom})$

$\forall y,x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(x,\mbox{implies}(y,x)))\qquad\mbox{fof}(\mbox{mv}_1,\mbox{axiom})$

$\forall z,y,x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(\mbox{implies}(y,z),\mbox{implies}(x,z))))\qquad\mbox{fof}(\mbox{mv}_2,\mbox{axiom})$

$\forall y,x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),y),\mbox{implies}(\mbox{implies}(y,x),x)))\qquad\mbox{fof}(\mbox{mv}_3,\mbox{axiom})$

$\forall y,x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{not}(x),\mbox{not}(y)),\mbox{implies}(y,x)))\qquad\mbox{fof}(\mbox{mv}_4,\mbox{axiom})$

$\forall y,x{:}\;\mbox{is\_a\_theorem}(\mbox{implies}(\mbox{implies}(\mbox{implies}(x,y),\mbox{implies}(y,x)),\mbox{implies}(y,x)))\qquad\mbox{fof}(\mbox{mv}_5,\mbox{conjecture})$

\medskip\textbf{LCL110-2.p} MV-24 depends on the Meredith system

An axiomatisation of the many valued sentential calculus
 is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented
 an equality axiomatisation. Show that MV-24 depends on the
 Wajsberg axiomatisation.




$\mbox{include}(\mbox{'Axioms/LCL001-0.ax'})$

$\mbox{implies}(\mbox{not}(\mbox{not}(x)),x)\ne \mbox{truth}\qquad\mbox{cnf}(\mbox{prove\_mv}_{24},\mbox{negated\_conjecture})$

\medskip
\end{document}