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    \textbf{KRS001+1.ax} SZS success ontology node relationships






$\forall s_1,s_2{:}\;( \exists \mbox{ax},c{:}\;( \mbox{status}(\mbox{ax},c,s_1)\mbox{ and }\mbox{status}(\mbox{ax},c,s_2) )\iff \mbox{mighta}(s_1,s_2) )\qquad\mbox{fof}(\mbox{mighta},\mbox{axiom})$

$\forall s_1,s_2{:}\;( \forall \mbox{ax},c{:}\;( \mbox{status}(\mbox{ax},c,s_1) \ \Rightarrow \ \mbox{status}(\mbox{ax},c,s_2) )\iff \mbox{isa}(s_1,s_2) )\qquad\mbox{fof}(\mbox{isa},\mbox{axiom})$

$\forall s_1,s_2{:}\;( \exists \mbox{ax},c{:}\;( \mbox{status}(\mbox{ax},c,s_1)\mbox{ and }\neg\,\mbox{status}(\mbox{ax},c,s_2) )\iff \mbox{nota}(s_1,s_2) )\qquad\mbox{fof}(\mbox{nota},\mbox{axiom})$

$\forall s_1,s_2{:}\;( \forall \mbox{ax},c{:}\;( \mbox{status}(\mbox{ax},c,s_1) \ \Rightarrow \ \neg\,\mbox{status}(\mbox{ax},c,s_2) )\iff \mbox{nevera}(s_1,s_2) )\qquad\mbox{fof}(\mbox{nevera},\mbox{axiom})$

$\forall s_1,s_2{:}\;( \forall \mbox{ax},c{:}\;\mbox{status}(\mbox{ax},c,s_1) <~> \mbox{status}(\mbox{ax},c,s_2)\iff \mbox{xora}(s_1,s_2) )\qquad\mbox{fof}(\mbox{xora},\mbox{axiom})$

$\forall i,f{:}\;\mbox{model}(i,f) <~> \mbox{model}(i,\mbox{not}(f))\qquad\mbox{fof}(\mbox{completeness},\mbox{axiom})$

$\forall i,f{:}\;( \mbox{model}(i,f)\iff \neg\,\mbox{model}(i,\mbox{not}(f)) )\qquad\mbox{fof}(\mbox{not},\mbox{axiom})$

$\exists f{:}\;\forall i{:}\;\mbox{model}(i,f)\qquad\mbox{fof}(\mbox{tautology},\mbox{axiom})$

$\exists f{:}\;( \exists i_1{:}\;\mbox{model}(i_1,f)\mbox{ and }\exists i_2{:}\;\neg\,\mbox{model}(i_2,f) )\qquad\mbox{fof}(\mbox{satisfiable},\mbox{axiom})$

$\exists f{:}\;\forall i{:}\;\neg\,\mbox{model}(i,f)\qquad\mbox{fof}(\mbox{contradiction},\mbox{axiom})$

$\exists \mbox{ax},c{:}\;( \exists i_1{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\mbox{model}(i_1,c) )\mbox{ and }\exists i_2{:}\;( \neg\,\mbox{model}(i_2,\mbox{ax})\mbox{ or }\neg\,\mbox{model}(i_2,c) ) )\qquad\mbox{fof}(\mbox{sat\_non\_taut\_pair},\mbox{axiom})$

$\exists \mbox{ax},c{:}\;( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;( \mbox{model}(i_2,\mbox{ax}) \ \Rightarrow \ \mbox{model}(i_2,c) )\mbox{ and }\exists i_3{:}\;( \neg\,\mbox{model}(i_3,\mbox{ax})\mbox{ and }\mbox{model}(i_3,c) )\mbox{ and }\exists i_4{:}\;\neg\,\mbox{model}(i_4,c) )\qquad\mbox{fof}(\mbox{mixed\_pair},\mbox{axiom})$

$\exists i_1,\mbox{ax},c{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\neg\,\mbox{model}(i_1,c)\mbox{ and }\exists i_2{:}\;\mbox{model}(i_2,c) )\qquad\mbox{fof}(\mbox{non\_thm\_spt},\mbox{axiom})$

\medskip\textbf{KRS001+0.ax} SZS success ontology nodes






$\forall \mbox{ax},c{:}\;( ( \neg\,\exists i_1{:}\;\mbox{model}(i_1,\mbox{ax}) \ \Rightarrow \ \neg\,\exists i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{unp}) )\qquad\mbox{fof}(\mbox{unp},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax}) \ \Rightarrow \ \exists i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{sap}) )\qquad\mbox{fof}(\mbox{sap},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\iff \exists i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{esa}) )\qquad\mbox{fof}(\mbox{esa},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( \exists i_1{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\mbox{model}(i_1,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{sat}) )\qquad\mbox{fof}(\mbox{sat},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( \forall i_1{:}\;( \mbox{model}(i_1,\mbox{ax}) \ \Rightarrow \ \mbox{model}(i_1,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{thm}) )\qquad\mbox{fof}(\mbox{thm},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;( \mbox{model}(i_2,\mbox{ax})\iff \mbox{model}(i_2,c) ) )\iff \mbox{status}(\mbox{ax},c,\mbox{eqv}) )\qquad\mbox{fof}(\mbox{eqv},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{tac}) )\qquad\mbox{fof}(\mbox{tac},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;( \mbox{model}(i_2,\mbox{ax}) \ \Rightarrow \ \mbox{model}(i_2,c) )\mbox{ and }\exists i_3{:}\;( \mbox{model}(i_3,c)\mbox{ and }\neg\,\mbox{model}(i_3,\mbox{ax}) ) )\iff \mbox{status}(\mbox{ax},c,\mbox{wec}) )\qquad\mbox{fof}(\mbox{wec},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\exists i_2{:}\;\neg\,\mbox{model}(i_2,\mbox{ax})\mbox{ and }\forall i_3{:}\;( \mbox{model}(i_3,\mbox{ax})\iff \mbox{model}(i_3,c) ) )\iff \mbox{status}(\mbox{ax},c,\mbox{eth}) )\qquad\mbox{fof}(\mbox{eth},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( \forall i_1{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\mbox{model}(i_1,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{tau}) )\qquad\mbox{fof}(\mbox{tau},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\exists i_2{:}\;\neg\,\mbox{model}(i_2,\mbox{ax})\mbox{ and }\forall i_3{:}\;\mbox{model}(i_3,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{wtc}) )\qquad\mbox{fof}(\mbox{wtc},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;( \mbox{model}(i_2,\mbox{ax}) \ \Rightarrow \ \mbox{model}(i_2,c) )\mbox{ and }\exists i_3{:}\;( \mbox{model}(i_3,c)\mbox{ and }\neg\,\mbox{model}(i_3,\mbox{ax}) )\mbox{ and }\exists i_4{:}\;\neg\,\mbox{model}(i_4,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{wth}) )\qquad\mbox{fof}(\mbox{wth},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( \neg\,\exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\iff \mbox{status}(\mbox{ax},c,\mbox{cax}) )\qquad\mbox{fof}(\mbox{cax},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \neg\,\exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\exists i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{sca}) )\qquad\mbox{fof}(\mbox{sca},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \neg\,\exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;\mbox{model}(i_2,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{tca}) )\qquad\mbox{fof}(\mbox{tca},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \neg\,\exists i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\exists i_2{:}\;\mbox{model}(i_2,c)\mbox{ and }\exists i_3{:}\;\neg\,\mbox{model}(i_3,c) )\iff \mbox{status}(\mbox{ax},c,\mbox{wca}) )\qquad\mbox{fof}(\mbox{wca},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( \exists i_1{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\mbox{model}(i_1,\mbox{not}(c)) )\iff \mbox{status}(\mbox{ax},c,\mbox{csa}) )\qquad\mbox{fof}(\mbox{csa},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \forall i_1{:}\;\mbox{model}(i_1,\mbox{ax})\mbox{ and }\forall i_2{:}\;\mbox{model}(i_2,\mbox{not}(c)) )\iff \mbox{status}(\mbox{ax},c,\mbox{uns}) )\qquad\mbox{fof}(\mbox{uns},\mbox{axiom})$

$\forall \mbox{ax},c{:}\;( ( \exists i_1{:}\;( \mbox{model}(i_1,\mbox{ax})\mbox{ and }\mbox{model}(i_1,c) )\mbox{ and }\exists i_2{:}\;( \mbox{model}(i_2,\mbox{ax})\mbox{ and }\mbox{model}(i_2,\mbox{not}(c)) ) )\iff \mbox{status}(\mbox{ax},c,\mbox{noc}) )\qquad\mbox{fof}(\mbox{noc},\mbox{axiom})$

\medskip\textbf{KRS202+1.p} ESA nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{esa},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_esa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS016-1.p} Paramasivam problem T-Box 5c

c and d exist.




$c(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$d(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$( c(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ s(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( c(x_1)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ r(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$c(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{r1most}(x_1) \ \Rightarrow \ ( c(x_1)\mbox{ or }s(x_1,\mbox{u0r}_1(x_1))\mbox{ or }r(x_1,\mbox{u0r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,\mbox{u0r}_1(x_1))\mbox{ and }s(x_1,\mbox{u0r}_1(x_1)) ) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$( d(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ \neg\,s(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ ( d(x_1)\mbox{ or }s(x_1,\mbox{u2r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ ( d(x_1)\mbox{ or }r(x_1,\mbox{u2r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

\medskip\textbf{KRS277$\wedge$7.p} Database querying






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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{john}@ \mbox{math} )@ ( \mbox{mand}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{teach}@ x@ \mbox{cs} ) )@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{mary}@ \mbox{psych} )@ ( \mbox{teach}@ \mbox{sue}@ \mbox{psych} ) ) ) ) )\qquad\mbox{thf}(\mbox{db},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{teach}@ x@ \mbox{psych} )@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{teach}@ x@ \mbox{cs} ) ) ) ) )\qquad\mbox{thf}(\mbox{query},\mbox{conjecture})$

\medskip\textbf{KRS009-1.p} Paramasivam problem T-Box 3c

e exists.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{p2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{p2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{p2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{p2least}(x_1) \ \Rightarrow \ p(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{p2least}(x_1) \ \Rightarrow \ p(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( p(x_1,x_3)\mbox{ and }p(x_1,x_2) ) \ \Rightarrow \ ( \mbox{p2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{p1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{p1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( \mbox{p1most}(x_1)\mbox{ and }p(x_1,x_3)\mbox{ and }p(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{p1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{p1most}(x_1)\mbox{ or }p(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{p1most}(x_1)\mbox{ or }p(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$a(x_1) \ \Rightarrow \ ( c(x_1)\mbox{ or }d(x_1) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$d(x_1) \ \Rightarrow \ a(x_1)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$c(x_1) \ \Rightarrow \ a(x_1)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$r(x_1,x_2) \ \Rightarrow \ c(x_2)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$r(x_1,x_2) \ \Rightarrow \ t(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$s(x_1,x_2) \ \Rightarrow \ d(x_2)\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$s(x_1,x_2) \ \Rightarrow \ t(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{t3least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$( e(x_1)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ a(x_2)\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$( a(\mbox{u7r}_1(x_1))\mbox{ and }\mbox{t3least}(x_1)\mbox{ and }\mbox{r1most}(x_1)\mbox{ and }\mbox{s1most}(x_1) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$( \mbox{t3least}(x_1)\mbox{ and }\mbox{r1most}(x_1)\mbox{ and }\mbox{s1most}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }t(x_1,\mbox{u7r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u8r}_2(x_1){=}\mbox{u8r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u8r}_3(x_1){=}\mbox{u8r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u8r}_3(x_1){=}\mbox{u8r}_2(x_1)\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u8r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u8r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{axiom})$

$\mbox{t3least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u8r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{axiom})$

$( t(x_1,x_4)\mbox{ and }t(x_1,x_3)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ ( \mbox{t3least}(x_1)\mbox{ or }x_4{=}x_3\mbox{ or }x_4{=}x_2\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{33},\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{34},\mbox{axiom})$

$\mbox{u9r}_2(x_1){=}\mbox{u9r}_1(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{35},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u9r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{36},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u9r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{37},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{38},\mbox{axiom})$

$\mbox{u10r}_2(x_1){=}\mbox{u10r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{39},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u10r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{40},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u10r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{41},\mbox{axiom})$

\medskip\textbf{KRS017-1.p} Paramasivam problem T-Box 7a

a subsumes e.




$e(\mbox{exists})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

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

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u0r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2)\mbox{ and }r(x_3,x_2) ) \ \Rightarrow \ a(x_3)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( a(\mbox{u0r}_2(x_1))\mbox{ and }r(x_1,x_4) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$r(x_1,x_4) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(\mbox{u0r}_2(x_1),\mbox{u0r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$r(x_1,x_4) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u0r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

\medskip\textbf{KRS137+1.p} A variation of equivalentClass-001

This is a variation of equivalentClass-001, showing the use of
 owl:Ontology triples in the premises and conclusions.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cCar}(x)\iff \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iauto})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cAutomobile}(\mbox{iauto})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{icar})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cCar}(\mbox{icar})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cCar}(\mbox{iauto})\mbox{ and }\mbox{cowlThing}(\mbox{iauto})\mbox{ and }\mbox{cowlThing}(\mbox{icar})\mbox{ and }\mbox{cAutomobile}(\mbox{icar})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS173+1.p} A simple infinite loop for implementors to avoid






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cperson}(x)\iff \exists y{:}\;( \mbox{rparent}(x,y)\mbox{ and }\mbox{cperson}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cperson}(\mbox{ifred})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{ifred})\mbox{ and }\mbox{rparent}(\mbox{ifred},\mbox{i2003\_11\_14\_17\_14}_{14998})\mbox{ and }\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{14920})\mbox{ and }\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{14998})\mbox{ and }\mbox{rparent}(\mbox{i2003\_11\_14\_17\_14}_{14998},\mbox{i2003\_11\_14\_17\_14}_{14920})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS144+1.p} An owl:cardinality constraint is simply shorthand

An owl:cardinality constraint is simply shorthand for a pair of
 owl:minCardinality and owl:maxCardinality constraints.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ ( \exists y_0,y_1{:}\;( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }y_0\ne y_1 )\mbox{ and }\forall y_0,y_1,y_2{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }\mbox{rp}(x,y_2) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_1 = y_2 ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \forall y_0,y_1,y_2{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }\mbox{rp}(x,y_2) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_1 = y_2 ) ) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \exists y_0,y_1{:}\;( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }y_0\ne y_1 ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS018+1.p} Nothing can be defined using OWL Lite restrictions






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS035+1.p} DL Test: t5.1 Non-finite model example from paper






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{ca}(a) ) \ \Rightarrow \ \mbox{ca}(b) )\qquad\mbox{fof}(\mbox{ca\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \neg\,\mbox{ca}(x)\mbox{ and }\exists y{:}\;( \mbox{rinvR}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rinvF}(y,z)\mbox{ and }\mbox{ca}(z) ) )\mbox{ and }\exists y{:}\;( \mbox{rinvF}(x,y)\mbox{ and }\mbox{ca}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}(x,y_0)\mbox{ and }\mbox{rf}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{44810})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS131+1.p} The complement of a class can be defined

The complement of a class can be defined using OWL Lite
 restrictions.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff \exists y{:}\;( \mbox{rq}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotA}(x)\iff \forall y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cnotA}(x)\iff \neg\,\mbox{cA}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS267+1.p} TCA mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{tca},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_tca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS038+1.p} DL Test: t7f.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \mbox{cp}_1(x)\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\mbox{cp}_1(z)\mbox{ and }\forall w{:}\;( \mbox{rinvR}(z,w) \ \Rightarrow \ ( \forall a{:}\;( \mbox{rr}(w,a) \ \Rightarrow \ \mbox{cp}_1(a) )\mbox{ or }\neg\,\mbox{cp}_1(w) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}(x,y)\mbox{ and }\mbox{rf}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{54995})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS021+1.p} Informal semantics for RDF container are not respected by OWL

The informal semantics for RDF container vocabulary, indicated by
 the comment, are not respected by the formal machinery of OWL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{crdfBag}(\mbox{i2003\_11\_14\_17\_14}_{36685})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{rrdf}_1(\mbox{i2003\_11\_14\_17\_14}_{36685},\mbox{i2003\_11\_14\_17\_14}_{36852})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{36852})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS003-1.p} Paramasivam problem T-Box 1c

e and f exist.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$f(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{s2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

\medskip\textbf{KRS204+1.p} EQV nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{eqv},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_eqv\_thm},\mbox{conjecture})$

\medskip\textbf{KRS062+1.p} dc:creator may be declared as an annotation property






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

\medskip\textbf{KRS093+1.p} DL Test: heinsohn1.3

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \mbox{cf}(x)\mbox{ and }\mbox{ce}_3(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \neg\,\mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{07201})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS059+1.p} Abstract syntax restrictions with multiple components

Abstract syntax restrictions with multiple components are in OWL
 DL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{rp}(\mbox{ii},x)\mbox{ and }\mbox{cs}(x) )\qquad\mbox{fof}(\mbox{axiom\_2\_AndLHS},\mbox{axiom})$

$\forall x{:}\;( \mbox{rp}(\mbox{ii},x) \ \Rightarrow \ \mbox{ca}(x) )\qquad\mbox{fof}(\mbox{axiom\_2\_AndRHS},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ii})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS055+1.p} owl:disjointWith edges may be within OWL DL

If the owl:disjointWith edges in the graph form undirected
 complete subgraphs which share URIref nodes but do not share
 blank node then this may be within OWL DL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cE}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cE}(x)\mbox{ and }\mbox{cD}(x)\mbox{ and }\neg\,\mbox{cA}(x)\mbox{ and }\mbox{cD}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cB}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cB}(x)\mbox{ and }\mbox{cC}(x)\mbox{ and }\neg\,\mbox{cA}(x)\mbox{ and }\mbox{cC}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS089+1.p} A test for the interaction of one-of and inverse

A test for the interaction of one-of and inverse using the idea
 of a spy point. Everything is related to the spy via the property
 p and we know that the spy has at most two invP successors, thus
 limiting the cardinality of the domain to being at most 2.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvP}(a,c) ) \ \Rightarrow \ \mbox{rinvP}(b,c) )\qquad\mbox{fof}(\mbox{rinvP\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvP}(c,a) ) \ \Rightarrow \ \mbox{rinvP}(c,b) )\qquad\mbox{fof}(\mbox{rinvP\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \exists y_0,y_1,y_2{:}\;( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1)\mbox{ and }\mbox{rr}(x,y_2)\mbox{ and }y_0\ne y_1\mbox{ and }y_0\ne y_2\mbox{ and }y_1\ne y_2 ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{rp}(x,y)\mbox{ and }y = \mbox{ispy} ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rp}(x,y)\iff \mbox{rinvP}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x_0,x_1,x_2{:}\;( ( \mbox{rinvP}(\mbox{ispy},x_0)\mbox{ and }\mbox{rinvP}(\mbox{ispy},x_1)\mbox{ and }\mbox{rinvP}(\mbox{ispy},x_2) ) \ \Rightarrow \ ( x_0 = x_1\mbox{ or }x_0 = x_2\mbox{ or }x_1 = x_2 ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ispy})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{53168})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS116+1.p} DL Test: t4.1 Dynamic blocking example






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{ca}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\mbox{ca\_Ax}_2(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{caxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x)\iff \forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_7(y) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Ax}_2(x)\iff ( \forall y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_3(y) )\mbox{ and }\forall y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_5(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) )\mbox{ and }\exists y{:}\;( \mbox{rp}(x,y)\mbox{ and }\mbox{cowlThing}(y) )\mbox{ and }\forall y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_4(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_3(x)\iff \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_4(x)\iff \exists y{:}\;( \mbox{rp}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_5(x)\iff \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_6(x)\iff \forall y{:}\;( \mbox{rinvS}(x,y) \ \Rightarrow \ \mbox{caxcomp}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_7(x)\iff \forall y{:}\;( \mbox{rinvP}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_6(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvP}(x,y)\iff \mbox{rp}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvS}(x,y)\iff \mbox{rs}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_{15},\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rp}(x,y)\mbox{ and }\mbox{rp}(y,z) ) \ \Rightarrow \ \mbox{rp}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_{16},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_21}_{33997})\qquad\mbox{fof}(\mbox{axiom}_{17},\mbox{axiom})$

\medskip\textbf{KRS179+1.p} If S1 isa S2 and S1 is nota S3, then S2 is nota S3






$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\forall s_1,s_2,s_3{:}\;( ( \mbox{isa}(s_1,s_2)\mbox{ and }\mbox{isa}(s_2,s_3) ) \ \Rightarrow \ \mbox{isa}(s_1,s_3) )\qquad\mbox{fof}(\mbox{isa\_transitive},\mbox{conjecture})$

\medskip\textbf{KRS100+1.p} DL Test: heinsohn4.1

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ ( \mbox{ce}(y)\mbox{ or }\neg\,\mbox{cd}(y) ) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cd}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\neg\,\mbox{ce}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{32704})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS186+1.p} EQV isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{eqv},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_eqv\_thm},\mbox{conjecture})$

\medskip\textbf{KRS241+1.p} TAC xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{tac},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_tac\_thm},\mbox{conjecture})$

\medskip\textbf{KRS226+1.p} TAU nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{tau},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_tau\_thm},\mbox{conjecture})$

\medskip\textbf{KRS086+1.p} DL Test: t7.3






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff \exists y{:}\;( \mbox{rf}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rinvF}(y,z)\mbox{ and }\exists w{:}\;( \mbox{rf}(z,w)\mbox{ and }\neg\,\mbox{cp}_1(w) ) )\mbox{ and }\mbox{cp}_1(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}(x,y_0)\mbox{ and }\mbox{rf}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{42328})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS248+1.p} SCA xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{sca},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_sca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS166+1.p} Two classes may be different names for the same set of individuals






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cAutomobile}(x) \ \Rightarrow \ \mbox{cCar}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cCar}(x) \ \Rightarrow \ \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cCar}(x)\iff \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS227+1.p} WTC nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{wtc},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_wtc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS189+1.p} ETH isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{eth},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_eth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS265+1.p} CAX mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{cax},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_cax\_thm},\mbox{conjecture})$

\medskip\textbf{KRS245+1.p} WTC xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{wtc},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_wtc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS192+1.p} WTH isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{wth},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_wth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS002-1.p} Paramasivam problem T-Box 1b

e exists.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{s2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ d(x_2)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ c(x_2)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }d(\mbox{u4r}_1(x_1))\mbox{ and }r(x_1,x_3) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$r(x_1,x_3) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

\medskip\textbf{KRS068+1.p} DL Test: fact2.1






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_18}_{23845})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

\medskip\textbf{KRS175+1.p} An inverse to test unionOf-003






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA}(a) ) \ \Rightarrow \ \mbox{cA}(b) )\qquad\mbox{fof}(\mbox{cA\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA\_and\_B}(a) ) \ \Rightarrow \ \mbox{cA\_and\_B}(b) )\qquad\mbox{fof}(\mbox{cA\_and\_B\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cB}(a) ) \ \Rightarrow \ \mbox{cB}(b) )\qquad\mbox{fof}(\mbox{cB\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff x = \mbox{ia} )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA\_and\_B}(x)\iff ( \mbox{cA}(x)\mbox{ or }\mbox{cB}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cB}(x)\iff x = \mbox{ib} )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cA\_and\_B}(x)\iff ( x = \mbox{ib}\mbox{ or }x = \mbox{ia} ) )\mbox{ and }\mbox{cowlThing}(\mbox{ia})\mbox{ and }\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS184+1.p} ESA isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{esa},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_esa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS152+1.p} DL Test: k\_ph ABox test from DL98 systems comparison






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{10}(x)\iff ( \mbox{cC}_2(x)\mbox{ and }\mbox{cC}_4(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{12}(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_{10}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_6(x)\iff ( \neg\,\mbox{cC}_2(x)\mbox{ and }\mbox{cC}_4(x) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_8(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_6(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cTEST}(x)\iff ( \mbox{cC}_{12}(x)\mbox{ and }\neg\,\mbox{cC}_8(x) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cTEST}(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{rR}_1(\mbox{iV}_{21080},x) \ \Rightarrow \ \neg\,\mbox{cC}_6(x) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\neg\,\mbox{cC}_8(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{rR}_1(\mbox{iV}_{21080},\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\neg\,\mbox{cC}_6(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\mbox{cC}_2(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

$\mbox{cC}_4(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{15},\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{21080})\mbox{ and }\mbox{cC}_{12}(\mbox{iV}_{21080})\mbox{ and }\mbox{cC}_{10}(\mbox{iV}_{21081})\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS104+1.p} DL Test: fact1.1

If a, b and c are disjoint, then:
 (a and b) or (b and c) or (c and a)
 is unsatisfiable.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_5(x,y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiablexcomp}(x)\iff ( \mbox{ca\_Cx}_7(x)\mbox{ and }\mbox{ca\_Cx}_8(x)\mbox{ and }\mbox{ca\_Cx}_6(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiablexcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_5(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca}(x) \ \Rightarrow \ \mbox{ca\_Cx}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cb}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_3(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cb}(x) \ \Rightarrow \ \mbox{ccxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cbxcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_3(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ccxcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_1(x)\iff ( \mbox{cbxcomp}(x)\mbox{ and }\mbox{ccxcomp}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_1(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx1xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_6(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_6(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx6xcomp}(x)\iff ( \mbox{ca}(x)\mbox{ and }\mbox{cb}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{15},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx6xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_6(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_{16},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_7(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_7(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_{17},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx7xcomp}(x)\iff ( \mbox{cc}(x)\mbox{ and }\mbox{ca}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{18},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx7xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_7(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{19},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_8(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_8(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{20},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx8xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_8(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_{21},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx8xcomp}(x)\iff ( \mbox{cc}(x)\mbox{ and }\mbox{cb}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{22},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{50869})\qquad\mbox{fof}(\mbox{axiom}_{23},\mbox{axiom})$

\medskip\textbf{KRS135+1.p} This is a typical definition of range from description logic






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rprop}(x,y) \ \Rightarrow \ \mbox{cA}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y{:}\;( \mbox{rprop}(x,y) \ \Rightarrow \ \mbox{cA}(y) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS077+1.p} DL Test: t11.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}(a) ) \ \Rightarrow \ \mbox{cp}(b) )\qquad\mbox{fof}(\mbox{cp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvS}(a,c) ) \ \Rightarrow \ \mbox{rinvS}(b,c) )\qquad\mbox{fof}(\mbox{rinvS\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvS}(c,a) ) \ \Rightarrow \ \mbox{rinvS}(c,b) )\qquad\mbox{fof}(\mbox{rinvS\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(a,c) ) \ \Rightarrow \ \mbox{rs}(b,c) )\qquad\mbox{fof}(\mbox{rs\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(c,a) ) \ \Rightarrow \ \mbox{rs}(c,b) )\qquad\mbox{fof}(\mbox{rs\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \forall y_0,y_1{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\forall z{:}\;( \mbox{rinvS}(y,z) \ \Rightarrow \ \mbox{cp}(z) ) )\mbox{ and }\neg\,\mbox{cp}(x)\mbox{ and }\exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\mbox{cp}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvS}(x,y)\iff \mbox{rs}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{09372})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rs}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

\medskip\textbf{KRS171+1.p} The inverse entailment of test 002 also holds






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasHead}(x,y) \ \Rightarrow \ \mbox{rhasLeader}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasLeader}(x,y) \ \Rightarrow \ \mbox{rhasHead}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y{:}\;( \mbox{rhasLeader}(x,y)\iff \mbox{rhasHead}(x,y) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS252+1.p} UNS xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{uns},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_uns\_thm},\mbox{conjecture})$

\medskip\textbf{KRS124+1.p} DL Test: heinsohn1.2

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{ca\_Ax}_3(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd1xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}_1(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd1xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Ax}_3(x)\iff ( \mbox{cd}(x)\mbox{ and }\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{10903})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

\medskip\textbf{KRS023+1.p} A minimal OWL Lite version of I5.3-005






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{42352})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{42102})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{rp}(\mbox{i2003\_11\_14\_17\_14}_{42102},\mbox{i2003\_11\_14\_17\_14}_{42352})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS025+1.p} Classes can be the object of annotation properties






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{4992})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

\medskip\textbf{KRS211+1.p} CAX nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{cax},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_cax\_thm},\mbox{conjecture})$

\medskip\textbf{KRS136+1.p} Some set theory

The abstract syntax form of the conclusions is:
 EquivalentClasses(restriction( first:p,minCardinality(1)))
 ObjectProperty(first:p). This is trivially true given that
 first:p is an individualvaluedPropertyID.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS163+1.p} Disjoint classes have different members






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA}(a) ) \ \Rightarrow \ \mbox{cA}(b) )\qquad\mbox{fof}(\mbox{cA\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cB}(a) ) \ \Rightarrow \ \mbox{cB}(b) )\qquad\mbox{fof}(\mbox{cB\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cA}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cB}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{cB}(x)\mbox{ and }\mbox{cA}(x)\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{ia})\mbox{ and }\mbox{cowlThing}(\mbox{ib})\mbox{ and }\mbox{ia}\ne \mbox{ib}\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS178+1.p} isa is exclusive of nota, nevera, and xora






$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\forall s_1{:}\;\mbox{isa}(s_1,s_1)\qquad\mbox{fof}(\mbox{isa\_reflexive},\mbox{conjecture})$

\medskip\textbf{KRS273$\wedge$7.p} Querying description logic knowledge bases






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

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

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

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

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

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

$q_2{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q2\_type},\mbox{type})$

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

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

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

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

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

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

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

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

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{mary} )@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{ann} )@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{jane} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{bob} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{john} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{paul} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{bob}@ \mbox{mary} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{bob}@ \mbox{ann} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{john}@ \mbox{paul} )@ ( \mbox{parent}@ \mbox{mary}@ \mbox{jane} ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{abox},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{male}@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{female}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{tbox},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( q_2@ x )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{male}@ x ) )@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{parent}@ x@ y )@ ( \mbox{female}@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{query},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( q_2@ \mbox{john} )@ ( q_2@ \mbox{paul} ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{KRS053+1.p} owl:disjointWith edges may be within OWL DL

If the owl:disjointWith edges in the graph form an undirected
 complete subgraph then this may be within OWL DL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cD}(x)\mbox{ and }\mbox{cC}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cE}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cB}(x)\mbox{ and }\neg\,\mbox{cC}(x)\mbox{ and }\mbox{cE}(x)\mbox{ and }\neg\,\mbox{cC}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cC}(x)\mbox{ and }\mbox{cB}(x)\mbox{ and }\neg\,\mbox{cE}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cE}(x)\mbox{ and }\mbox{cB}(x)\mbox{ and }\neg\,\mbox{cA}(x)\mbox{ and }\mbox{cB}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

\medskip\textbf{KRS195+1.p} TCA isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{tca},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_tca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS181+1.p} If S1 isa S2, and S1 nota S3, then S2 nota S3






$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\forall s_1,s_2,s_3{:}\;( ( \mbox{isa}(s_1,s_2)\mbox{ and }\mbox{nota}(s_1,s_3) ) \ \Rightarrow \ \mbox{nota}(s_2,s_3) )\qquad\mbox{fof}(\mbox{nota\_isa\_nota},\mbox{conjecture})$

\medskip\textbf{KRS269+1.p} CSA mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{csa},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_csa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS276$\wedge$7.p} Database querying






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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{john}@ \mbox{math} )@ ( \mbox{mand}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{teach}@ x@ \mbox{cs} ) )@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{mary}@ \mbox{psych} )@ ( \mbox{teach}@ \mbox{sue}@ \mbox{psych} ) ) ) ) )\qquad\mbox{thf}(\mbox{db},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{teach}@ x@ \mbox{cs} ) ) )\qquad\mbox{thf}(\mbox{query},\mbox{conjecture})$

\medskip\textbf{KRS201+1.p} SAP nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{sap},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_sap\_thm},\mbox{conjecture})$

\medskip\textbf{KRS222+1.p} EQV nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{eqv},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_eqv\_thm},\mbox{conjecture})$

\medskip\textbf{KRS056+1.p} owl:disjointWith edges may be within OWL DL

If the owl:disjointWith edges in the graph form undirected
 complete subgraphs which share URIref nodes but do not share
 blank node then this may be within OWL DL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cC}(x)\mbox{ and }\mbox{cD}(x)\mbox{ and }\neg\,\mbox{cC}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cA}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cD}(x)\mbox{ and }\mbox{cB}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cB}(x)\mbox{ and }\mbox{cA}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS246+1.p} WTH xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{wth},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_wth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS019+1.p} The complement of a class can be defined using OWL Lite






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff \exists y{:}\;( \mbox{rq}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotA}(x)\iff \forall y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

\medskip\textbf{KRS257+1.p} SAT mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{sat},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_sat\_thm},\mbox{conjecture})$

\medskip\textbf{KRS253+1.p} NOC xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{noc},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_noc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS105+1.p} DL Test: fact2.1






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Ax}_2(x)\iff \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Ax}_2(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{53634})\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

\medskip\textbf{KRS169+1.p} hasLeader may be stated as the owl:equivalentProperty of hasHead






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iX})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{rhasLeader}(\mbox{iX},\mbox{iY})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iY})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasLeader}(x,y)\iff \mbox{rhasHead}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{iX})\mbox{ and }\mbox{rhasHead}(\mbox{iX},\mbox{iY})\mbox{ and }\mbox{cowlThing}(\mbox{iY})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS272$\wedge$7.p} Generation of abstract instructions: enter a number in a(\#box






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

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

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

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

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

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

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

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

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

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

$\forall v{:}\;\mbox{\$i}{:}\;( \mbox{exists\_in\_world}@ 1@ v )\qquad\mbox{thf}(\mbox{existence\_of\_one\_ax},\mbox{axiom})$

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mexists\_ind}@ \lambda i{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{userid}@ u@ i )@ ( \mbox{string}@ i ) ) ) ) )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mexists\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{entry\_box}@ b )@ ( \mbox{number}@ b@ 1 ) ) ) )\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda s{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda i{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{string}@ i )@ ( \mbox{entry\_box}@ b ) )@ ( \mbox{mexists\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda s_2{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{do}@ s@ a@ s_2 )@ ( \mbox{in}@ i@ b@ s_2 ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ax}_3,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mexists\_ind}@ \lambda i{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda s{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{userid}@ u@ i )@ ( \mbox{mand}@ ( \mbox{entry\_box}@ b )@ ( \mbox{number}@ b@ 1 ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda s_2{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{do}@ s@ a@ s_2 )@ ( \mbox{in}@ i@ b@ s_2 ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{KRS260+1.p} WEC mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{wec},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_wec\_thm},\mbox{conjecture})$

\medskip\textbf{KRS165+1.p} Two classes may be different names for the same set of individuals






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cCar}(x)\iff \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cAutomobile}(x) \ \Rightarrow \ \mbox{cCar}(x) )\mbox{ and }\forall x{:}\;( \mbox{cCar}(x) \ \Rightarrow \ \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS139+1.p} A Lite version of test SymmetricProperty-001






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rpath}(x,y) \ \Rightarrow \ \mbox{rpath}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iAntwerp})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iGhent})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{rpath}(\mbox{iGhent},\mbox{iAntwerp})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{iAntwerp})\mbox{ and }\mbox{rpath}(\mbox{iAntwerp},\mbox{iGhent})\mbox{ and }\mbox{cowlThing}(\mbox{iGhent})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS188+1.p} WEC isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{wec},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_wec\_thm},\mbox{conjecture})$

\medskip\textbf{KRS216+1.p} UNS nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{uns},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_uns\_thm},\mbox{conjecture})$

\medskip\textbf{KRS259+1.p} TAC mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{tac},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_tac\_thm},\mbox{conjecture})$

\medskip\textbf{KRS183+1.p} SAP isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{sap},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_sap\_thm},\mbox{conjecture})$

\medskip\textbf{KRS271+1.p} NOC mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{noc},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_noc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS141+1.p} A simple example






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cr}(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{cc}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cr}(\mbox{ii})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ii})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{rp}(\mbox{ii},\mbox{io})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{io})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cc}(\mbox{io})\mbox{ and }\mbox{cowlThing}(\mbox{io})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS001-1.p} Paramasivam problem T-Box 1a

e exists.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{r2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ ( \mbox{r2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{r1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( c(x_1)\mbox{ and }d(x_1) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

\medskip\textbf{KRS172+1.p} The same property extension means equivalentProperty

If p and q have the same property extension then p
 equivalentProperty q.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rq}(a,c) ) \ \Rightarrow \ \mbox{rq}(b,c) )\qquad\mbox{fof}(\mbox{rq\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rq}(c,a) ) \ \Rightarrow \ \mbox{rq}(c,b) )\qquad\mbox{fof}(\mbox{rq\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \mbox{rq}(x,\mbox{iv}) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \mbox{rp}(x,\mbox{iv}) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rp}(x,y)\mbox{ and }\mbox{rp}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rq}(x,y)\mbox{ and }\mbox{rq}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iv})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y{:}\;( \mbox{rq}(x,y)\iff \mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS061+1.p} User labels in a variety of languages with ruby annotation

This test shows how user labels in a variety of languages can be
 used. Note the use of ruby annotation.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cShakespearePlay}(\mbox{iRomeo\_and\_Juliet})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

\medskip\textbf{KRS142+1.p} An owl:cardinality constraint is simply shorthand

An owl:cardinality constraint is simply shorthand for a pair of
 owl:minCardinality and owl:maxCardinality constraints.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ ( \exists y_0{:}\;\mbox{rp}(x,y_0)\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS167+1.p} Two classes with the same complete description are equivalent






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}_1(a) ) \ \Rightarrow \ \mbox{cc}_1(b) )\qquad\mbox{fof}(\mbox{cc1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}_2(a) ) \ \Rightarrow \ \mbox{cc}_2(b) )\qquad\mbox{fof}(\mbox{cc2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x)\iff ( \exists y_0{:}\;\mbox{rp}(x,y_0)\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_2(x)\iff ( \exists y_0{:}\;\mbox{rp}(x,y_0)\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cc}_1(x)\iff \mbox{cc}_2(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS045+1.p} DL Test: t2.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp2xcomp}(a) ) \ \Rightarrow \ \mbox{cp2xcomp}(b) )\qquad\mbox{fof}(\mbox{cp2xcomp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(a,c) ) \ \Rightarrow \ \mbox{ra\_Px}_1(b,c) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(c,a) ) \ \Rightarrow \ \mbox{ra\_Px}_1(c,b) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(a,c) ) \ \Rightarrow \ \mbox{rf}_1(b,c) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(c,a) ) \ \Rightarrow \ \mbox{rf}_1(c,b) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(a,c) ) \ \Rightarrow \ \mbox{rf}_2(b,c) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(c,a) ) \ \Rightarrow \ \mbox{rf}_2(c,b) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rf}_2(x,y)\mbox{ and }\mbox{cp}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_1(x,y)\mbox{ and }\mbox{cp}_1(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \mbox{cp2xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_2(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp2xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_1(x,y_0)\mbox{ and }\mbox{rf}_1(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_2(x,y_0)\mbox{ and }\mbox{rf}_2(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_16}_{21280})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

\medskip\textbf{KRS213+1.p} TCA nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{tca},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_tca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS039+1.p} DL Test: t8.1






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\forall z{:}\;( \mbox{rinvR}(y,z) \ \Rightarrow \ \forall w{:}\;( \mbox{rr}_1(z,w) \ \Rightarrow \ \mbox{cp}(w) ) ) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\forall z{:}\;( \mbox{rinvR}(y,z) \ \Rightarrow \ \forall w{:}\;( \mbox{rr}_1(z,w) \ \Rightarrow \ \neg\,\mbox{cp}(w) ) ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{58383})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS078+1.p} DL Test: t12.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}(a) ) \ \Rightarrow \ \mbox{cp}(b) )\qquad\mbox{fof}(\mbox{cp\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cq}(a) ) \ \Rightarrow \ \mbox{cq}(b) )\qquad\mbox{fof}(\mbox{cq\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(a,c) ) \ \Rightarrow \ \mbox{rs}(b,c) )\qquad\mbox{fof}(\mbox{rs\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(c,a) ) \ \Rightarrow \ \mbox{rs}(c,b) )\qquad\mbox{fof}(\mbox{rs\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rinvR}(y,z)\mbox{ and }\forall w{:}\;( \mbox{rs}(z,w) \ \Rightarrow \ \mbox{cp}(w) ) )\mbox{ and }\forall z_0,z_1{:}\;( ( \mbox{rinvR}(y,z_0)\mbox{ and }\mbox{rinvR}(y,z_1) ) \ \Rightarrow \ z_0 = z_1 ) )\mbox{ and }\exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\neg\,\mbox{cq}(y)\mbox{ and }\neg\,\mbox{cp}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{13721})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS132+1.p} The union of two classes can be defined

The union of two classes can be defined using OWL Lite
 restrictions, and owl:intersectionOf.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff \exists y{:}\;( \mbox{rq}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cAorB}(x)\iff \exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cB}(x)\iff \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotA}(x)\iff \forall y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotAorB}(x)\iff \forall y{:}\;( \mbox{rs}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotAorB}(x)\iff ( \mbox{cnotB}(x)\mbox{ and }\mbox{cnotA}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotB}(x)\iff \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cAorB}(x)\iff ( \mbox{cB}(x)\mbox{ or }\mbox{cA}(x) ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS200+1.p} UNP nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{unp},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_unp\_thm},\mbox{conjecture})$

\medskip\textbf{KRS197+1.p} CSA isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{csa},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_csa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS223+1.p} TAC nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{tac},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_tac\_thm},\mbox{conjecture})$

\medskip\textbf{KRS085+1.p} DL Test: t7.2






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\mbox{cp}_1(z)\mbox{ and }\forall w{:}\;( \mbox{rinvR}(z,w) \ \Rightarrow \ \neg\,\mbox{cp}_1(w) ) ) )\mbox{ and }\mbox{cp}_1(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}(x,y_0)\mbox{ and }\mbox{rf}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{39537})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS065+1.p} The syntax for using the same restriction twice in OWL Lite

This test shows the syntax for using the same restriction twice
 in OWL Lite.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\exists x{:}\;( \mbox{rop}(\mbox{ia},x)\mbox{ and }\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\exists x{:}\;( \mbox{rop}(\mbox{ib},x)\mbox{ and }\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS082+1.p} DL Test: t4.1 Dynamic blocking example






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rp}(y,z)\mbox{ and }\mbox{cowlThing}(z) )\mbox{ and }\forall z{:}\;( \mbox{rr}(y,z) \ \Rightarrow \ \mbox{cc}(z) )\mbox{ and }\forall z{:}\;( \mbox{rp}(y,z) \ \Rightarrow \ \exists w{:}\;( \mbox{rr}(z,w)\mbox{ and }\mbox{cowlThing}(w) ) )\mbox{ and }\forall z{:}\;( \mbox{rp}(y,z) \ \Rightarrow \ \exists w{:}\;( \mbox{rp}(z,w)\mbox{ and }\mbox{cowlThing}(w) ) )\mbox{ and }\forall z{:}\;( \mbox{rp}(y,z) \ \Rightarrow \ \forall w{:}\;( \mbox{rr}(z,w) \ \Rightarrow \ \mbox{cc}(w) ) )\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\mbox{cowlThing}(z) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{ca}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x)\iff \forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \forall z{:}\;( \mbox{rinvP}(y,z) \ \Rightarrow \ \forall w{:}\;( \mbox{rinvS}(z,w) \ \Rightarrow \ \neg\,\mbox{ca}(w) ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvP}(x,y)\iff \mbox{rp}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvS}(x,y)\iff \mbox{rs}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rp}(x,y)\mbox{ and }\mbox{rp}(y,z) ) \ \Rightarrow \ \mbox{rp}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{28752})\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

\medskip\textbf{KRS067+1.p} DL Test: fact1.1

If a, b and c are disjoint, then:
 (a and b) or (b and c) or (c and a)
 is unsatisfiable.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( ( \mbox{cc}(x)\mbox{ and }\mbox{cb}(x) )\mbox{ or }( \mbox{cb}(x)\mbox{ and }\mbox{ca}(x) )\mbox{ or }( \mbox{cc}(x)\mbox{ and }\mbox{ca}(x) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca}(x) \ \Rightarrow \ \neg\,\mbox{cc}(x)\mbox{ or }\mbox{cb}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cb}(x) \ \Rightarrow \ \neg\,\mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_18}_{1956})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

\medskip\textbf{KRS126+1.p} DL Test: heinsohn1.4

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{cd1xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}_1(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd1xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{23554})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

\medskip\textbf{KRS176+1.p} isa is reflexive






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

\medskip\textbf{KRS206+1.p} WEC nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{wec},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_wec\_thm},\mbox{conjecture})$

\medskip\textbf{KRS228+1.p} WTH nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{wth},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_wth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS235+1.p} NOC nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{noc},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_noc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS129+1.p} An example combinging owl:oneOf and owl:inverseOf






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEUCountry}(a) ) \ \Rightarrow \ \mbox{cEUCountry}(b) )\qquad\mbox{fof}(\mbox{cEUCountry\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEuroMP}(a) ) \ \Rightarrow \ \mbox{cEuroMP}(b) )\qquad\mbox{fof}(\mbox{cEuroMP\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEuropeanCountry}(a) ) \ \Rightarrow \ \mbox{cEuropeanCountry}(b) )\qquad\mbox{fof}(\mbox{cEuropeanCountry\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cPerson}(a) ) \ \Rightarrow \ \mbox{cPerson}(b) )\qquad\mbox{fof}(\mbox{cPerson\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rhasEuroMP}(a,c) ) \ \Rightarrow \ \mbox{rhasEuroMP}(b,c) )\qquad\mbox{fof}(\mbox{rhasEuroMP\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rhasEuroMP}(c,a) ) \ \Rightarrow \ \mbox{rhasEuroMP}(c,b) )\qquad\mbox{fof}(\mbox{rhasEuroMP\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{risEuroMPFrom}(a,c) ) \ \Rightarrow \ \mbox{risEuroMPFrom}(b,c) )\qquad\mbox{fof}(\mbox{risEuroMPFrom\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{risEuroMPFrom}(c,a) ) \ \Rightarrow \ \mbox{risEuroMPFrom}(c,b) )\qquad\mbox{fof}(\mbox{risEuroMPFrom\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cEUCountry}(x)\iff ( x = \mbox{iBE}\mbox{ or }x = \mbox{iFR}\mbox{ or }x = \mbox{iES}\mbox{ or }x = \mbox{iUK}\mbox{ or }x = \mbox{iNL}\mbox{ or }x = \mbox{iPT} ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cEuroMP}(x)\iff \exists y{:}\;( \mbox{risEuroMPFrom}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasEuroMP}(x,y) \ \Rightarrow \ \mbox{cEUCountry}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{risEuroMPFrom}(x,y)\iff \mbox{rhasEuroMP}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iBE})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iES})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iFR})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{cPerson}(\mbox{iKinnock})\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iNL})\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iPT})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iUK})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{rhasEuroMP}(\mbox{iUK},\mbox{iKinnock})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cEuroMP}(\mbox{iKinnock})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS006-1.p} Paramasivam problem T-Box 2b

Inconsistent concept definition with disjoint concepts.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{r1most}(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$r(\mbox{exist},\mbox{u4r}_1(\mbox{exist}))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$d(\mbox{u4r}_2(\mbox{exist}))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

$c(\mbox{u4r}_1(\mbox{exist}))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{s2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ d(\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ c(\mbox{u4r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,x_2)\mbox{ and }c(x_2)\mbox{ and }r(x_1,x_3)\mbox{ and }d(x_3) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$( \mbox{r1most}(x_1)\mbox{ and }r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$\mbox{u5r}_2(x_1){=}\mbox{u5r}_1(x_1) \ \Rightarrow \ \mbox{r1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u5r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

$\mbox{r1most}(x_1)\mbox{ or }r(x_1,\mbox{u5r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{axiom})$

\medskip\textbf{KRS069+1.p} DL Test: fact3.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(a,c) ) \ \Rightarrow \ \mbox{rf}_1(b,c) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(c,a) ) \ \Rightarrow \ \mbox{rf}_1(c,b) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(a,c) ) \ \Rightarrow \ \mbox{rf}_2(b,c) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(c,a) ) \ \Rightarrow \ \mbox{rf}_2(c,b) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_3(a,c) ) \ \Rightarrow \ \mbox{rf}_3(b,c) )\qquad\mbox{fof}(\mbox{rf3\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_3(c,a) ) \ \Rightarrow \ \mbox{rf}_3(c,b) )\qquad\mbox{fof}(\mbox{rf3\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rf}_1(x,y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_2(x,y)\mbox{ and }\neg\,\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_3(x,y)\mbox{ and }\mbox{cp}_2(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_1(x,y)\mbox{ and }\mbox{rf}_1(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_2(x,y)\mbox{ and }\mbox{rf}_2(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_3(x,y)\mbox{ and }\mbox{rf}_3(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_18}_{2750})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}_3(x,y) \ \Rightarrow \ \mbox{rf}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}_3(x,y) \ \Rightarrow \ \mbox{rf}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS199+1.p} NOC isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{noc},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_noc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS099+1.p} DL Test: heinsohn3c.1

Tbox tests from [HK+94]




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{ca}(a) ) \ \Rightarrow \ \mbox{ca}(b) )\qquad\mbox{fof}(\mbox{ca\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rtt}(a,c) ) \ \Rightarrow \ \mbox{rtt}(b,c) )\qquad\mbox{fof}(\mbox{rtt\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rtt}(c,a) ) \ \Rightarrow \ \mbox{rtt}(c,b) )\qquad\mbox{fof}(\mbox{rtt\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y_0,y_1,y_2{:}\;( \mbox{rtt}(x,y_0)\mbox{ and }\mbox{rtt}(x,y_1)\mbox{ and }\mbox{rtt}(x,y_2)\mbox{ and }y_0\ne y_1\mbox{ and }y_0\ne y_2\mbox{ and }y_1\ne y_2 )\mbox{ and }\forall y{:}\;( \mbox{rtt}(x,y) \ \Rightarrow \ \mbox{ca}(y) )\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rtt}(x,y_0)\mbox{ and }\mbox{rtt}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 )\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rtt}(x,y_0)\mbox{ and }\mbox{rtt}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca}(x) \ \Rightarrow \ ( \mbox{cd}(x)\mbox{ or }\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{29215})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

\medskip\textbf{KRS190+1.p} TAU isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{tau},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_tau\_thm},\mbox{conjecture})$

\medskip\textbf{KRS140+1.p} Test illustrating extensional semantics of owl:TransitiveProperty






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rsymProp}(a,c) ) \ \Rightarrow \ \mbox{rsymProp}(b,c) )\qquad\mbox{fof}(\mbox{rsymProp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rsymProp}(c,a) ) \ \Rightarrow \ \mbox{rsymProp}(c,b) )\qquad\mbox{fof}(\mbox{rsymProp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rsymProp}(x,y) \ \Rightarrow \ ( y = \mbox{ia}\mbox{ or }y = \mbox{ib} ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rsymProp}(x,y) \ \Rightarrow \ \mbox{rsymProp}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{rsymProp}(\mbox{ia},\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{rsymProp}(\mbox{ib},\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y,z{:}\;( ( \mbox{rsymProp}(x,y)\mbox{ and }\mbox{rsymProp}(y,z) ) \ \Rightarrow \ \mbox{rsymProp}(x,z) )\mbox{ and }\exists x{:}\;( \mbox{rsymProp}(\mbox{ia},x)\mbox{ and }\mbox{cowlThing}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS194+1.p} SCA isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{sca},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_sca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS244+1.p} TAU xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{tau},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_tau\_thm},\mbox{conjecture})$

\medskip\textbf{KRS027+1.p} An example of use






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

\medskip\textbf{KRS026+1.p} The extension of OWL Thing may be a singleton in OWL DL






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\iff x = \mbox{is} )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{is})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS087+1.p} DL Test: t7f.2






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \mbox{cp}_1(x)\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\forall w{:}\;( \mbox{rinvR}(z,w) \ \Rightarrow \ \neg\,\mbox{cp}_1(w) )\mbox{ and }\mbox{cp}_1(z) ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}(x,y)\mbox{ and }\mbox{rf}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{46763})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS229+1.p} CAX nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{cax},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_cax\_thm},\mbox{conjecture})$

\medskip\textbf{KRS013-1.p} Paramasivam problem T-Box 4b

f subsumes e.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

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

$c(x_1) \ \Rightarrow \ \mbox{t2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( t(x_1,x_3)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ ( \mbox{t2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{t1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{t1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( \mbox{t1most}(x_1)\mbox{ and }t(x_1,x_3)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{t1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{t1most}(x_1)\mbox{ or }t(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{t1most}(x_1)\mbox{ or }t(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_3) ) \ \Rightarrow \ d(x_3)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2)\mbox{ and }\mbox{s2least}(x_2) ) \ \Rightarrow \ c(x_2)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }d(\mbox{u4r}_2(x_1)) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$c(\mbox{u4r}_1(x_1)) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$d(\mbox{u4r}_2(x_1)) \ \Rightarrow \ ( e(x_1)\mbox{ or }\mbox{s2least}(\mbox{u4r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$e(x_1)\mbox{ or }\mbox{s2least}(\mbox{u4r}_1(x_1))\mbox{ or }r(x_1,\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$d(\mbox{u4r}_2(x_1)) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1))\mbox{ or }r(x_1,\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u5r}_2(x_1){=}\mbox{u5r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u5r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u5r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

$( f(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ \mbox{s1most}(x_2)\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{axiom})$

$\mbox{s1most}(\mbox{u6r}_1(x_1)) \ \Rightarrow \ f(x_1)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{axiom})$

$f(x_1)\mbox{ or }r(x_1,\mbox{u6r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{axiom})$

$\mbox{u7r}_2(x_1){=}\mbox{u7r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u7r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{32},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u7r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{33},\mbox{axiom})$

\medskip\textbf{KRS221+1.p} SAT nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{sat},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_sat\_thm},\mbox{conjecture})$

\medskip\textbf{KRS261+1.p} ETH mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{eth},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_eth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS083+1.p} DL Test: t6.1 Double blocking example






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \exists z{:}\;( \mbox{rinvF}(y,z)\mbox{ and }\mbox{cd}(z) ) )\mbox{ and }\neg\,\mbox{cc}(x)\mbox{ and }\exists y{:}\;( \mbox{rinvF}(x,y)\mbox{ and }\mbox{cd}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff ( \exists y{:}\;( \mbox{rf}(x,y)\mbox{ and }\neg\,\mbox{cc}(y) )\mbox{ and }\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}(x,y_0)\mbox{ and }\mbox{rf}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{32337})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

\medskip\textbf{KRS134+1.p} This is a typical definition of range from description logic






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rprop}(x,y) \ \Rightarrow \ \mbox{cA}(y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rprop}(x,y) \ \Rightarrow \ \mbox{cA}(y) ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS160+1.p} DL Test: k\_ph ABox test from DL98 systems comparison






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{10}(x)\iff ( \mbox{cC}_4(x)\mbox{ and }\mbox{cC}_2(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{12}(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_{10}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_2(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC2xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_6(x)\iff ( \mbox{cC2xcomp}(x)\mbox{ and }\mbox{cC}_4(x) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_6(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_4(x,y) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC6xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_4(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_8(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_8(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_6(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{cC8xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\forall x{:}\;( \mbox{cTEST}(x)\iff ( \mbox{cC8xcomp}(x)\mbox{ and }\mbox{cC}_{12}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{cTEST}(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\mbox{cC8xcomp}(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{21080})\qquad\mbox{fof}(\mbox{axiom}_{15},\mbox{axiom})$

$\forall x{:}\;( \mbox{rR}_1(\mbox{iV}_{21080},x) \ \Rightarrow \ \mbox{cC6xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_{16},\mbox{axiom})$

$\mbox{rR}_1(\mbox{iV}_{21080},\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{17},\mbox{axiom})$

$\mbox{cC}_4(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{18},\mbox{axiom})$

$\mbox{cC6xcomp}(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{19},\mbox{axiom})$

$\mbox{cC}_2(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{20},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{axiom}_{21},\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{21080})\mbox{ and }\mbox{cC}_{12}(\mbox{iV}_{21080})\mbox{ and }\mbox{cC}_{10}(\mbox{iV}_{21081})\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{21081})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS130+1.p} owl:Nothing can be defined using OWL Lite restrictions

A class like owl:Nothing can be defined using OWL Lite
 restrictions.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cNothing}(x)\iff \mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS237+1.p} SAP xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{sap},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_sap\_thm},\mbox{conjecture})$

\medskip\textbf{KRS003\_1.p} Paramasivam problem T-Box 1c

e and f exist.




$\mbox{unreal}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{unreal\_type},\mbox{type})$

$\mbox{real}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{real\_type},\mbox{type})$

$\mbox{u1r}_1{:}\;\mbox{unreal}\to \mbox{real}\qquad\mbox{tff}(\mbox{u1r1\_type},\mbox{type})$

$\mbox{u1r}_2{:}\;\mbox{unreal}\to \mbox{real}\qquad\mbox{tff}(\mbox{u1r2\_type},\mbox{type})$

$\mbox{u3r}_1{:}\;\mbox{unreal}\to \mbox{real}\qquad\mbox{tff}(\mbox{u3r1\_type},\mbox{type})$

$\mbox{u3r}_2{:}\;\mbox{unreal}\to \mbox{real}\qquad\mbox{tff}(\mbox{u3r2\_type},\mbox{type})$

$\mbox{exist}{:}\;\mbox{unreal}\qquad\mbox{tff}(\mbox{exist\_type},\mbox{type})$

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

$d{:}\;\mbox{unreal}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{d\_type},\mbox{type})$

$e{:}\;\mbox{unreal}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{e\_type},\mbox{type})$

$\mbox{s1most}{:}\;\mbox{unreal}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{s1most\_type},\mbox{type})$

$s{:}\;( \mbox{unreal}\times \mbox{real} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{s\_type},\mbox{type})$

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

$={:}\;( \mbox{real}\times \mbox{real} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{equalish\_type},\mbox{type})$

$\mbox{s2least}{:}\;\mbox{unreal}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{s2least\_type},\mbox{type})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( c(x_1) \ \Rightarrow \ \mbox{s2least}(x_1) )\qquad\mbox{tff}(\mbox{clause}_3,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s2least}(x_1) \ \Rightarrow \ c(x_1) )\qquad\mbox{tff}(\mbox{clause}_4,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;\neg\,\mbox{s2least}(x_1)\mbox{ and }\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{tff}(\mbox{clause}_5,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_1(x_1)) )\qquad\mbox{tff}(\mbox{clause}_6,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_2(x_1)) )\qquad\mbox{tff}(\mbox{clause}_7,\mbox{axiom})$

$\forall x_2{:}\;\mbox{real},x_3{:}\;\mbox{real},x_1{:}\;\mbox{unreal}{:}\;( ( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 ) )\qquad\mbox{tff}(\mbox{clause}_8,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1) )\qquad\mbox{tff}(\mbox{clause}_9,\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s1most}(x_1) \ \Rightarrow \ d(x_1) )\qquad\mbox{tff}(\mbox{clause}_{10},\mbox{axiom})$

$\forall x_2{:}\;\mbox{real},x_3{:}\;\mbox{real},x_1{:}\;\mbox{unreal}{:}\;( ( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2 )\qquad\mbox{tff}(\mbox{clause}_{11},\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1) )\qquad\mbox{tff}(\mbox{clause}_{12},\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1)) )\qquad\mbox{tff}(\mbox{clause}_{13},\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( \mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1)) )\qquad\mbox{tff}(\mbox{clause}_{14},\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( e(x_1) \ \Rightarrow \ c(x_1) )\qquad\mbox{tff}(\mbox{clause}_{15},\mbox{axiom})$

$\forall x_1{:}\;\mbox{unreal}{:}\;( f(x_1) \ \Rightarrow \ d(x_1) )\qquad\mbox{tff}(\mbox{clause}_{16},\mbox{axiom})$

$\neg\,e(\mbox{exist})\mbox{ or }\neg\,f(\mbox{exist})\qquad\mbox{tff}(\mbox{clause\_1\_\_clause}_2,\mbox{conjecture})$

\medskip\textbf{KRS037+1.p} DL Test: t7.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\forall w{:}\;( \mbox{rinvR}(z,w) \ \Rightarrow \ ( \neg\,\mbox{cp}_1(w)\mbox{ or }\forall a{:}\;( \mbox{rr}(w,a) \ \Rightarrow \ \mbox{cp}_1(a) ) ) )\mbox{ and }\mbox{cp}_1(z) ) )\mbox{ and }\mbox{cp}_1(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}(x,y_0)\mbox{ and }\mbox{rf}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{51999})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS125+1.p} DL Test: heinsohn1.3

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \mbox{ce}_3(x)\mbox{ and }\mbox{cf}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd1xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}_1(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd1xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{17947})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

\medskip\textbf{KRS255+1.p} SAP mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{sap},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_sap\_thm},\mbox{conjecture})$

\medskip\textbf{KRS096+1.p} DL Test: heinsohn2.2

Tbox tests from [HK+94]




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cc}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cd}(y) )\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{18265})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS251+1.p} CSA xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{csa},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_csa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS177+1.p} isa is transitive






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

\medskip\textbf{KRS101+1.p} DL Test: heinsohn4.2

Tbox tests from [HK+94]




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(a,c) ) \ \Rightarrow \ \mbox{rs}(b,c) )\qquad\mbox{fof}(\mbox{rs\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(c,a) ) \ \Rightarrow \ \mbox{rs}(c,b) )\qquad\mbox{fof}(\mbox{rs\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cd}(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ ( \neg\,\exists z_0,z_1{:}\;( \mbox{rs}(y,z_0)\mbox{ and }\mbox{rs}(y,z_1)\mbox{ and }z_0\ne z_1 )\mbox{ or }\mbox{cc}(y) ) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\neg\,\forall z_0,z_1{:}\;( ( \mbox{rs}(y,z_0)\mbox{ and }\mbox{rs}(y,z_1) ) \ \Rightarrow \ z_0 = z_1 ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{36582})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS270+1.p} UNS mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{uns},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_uns\_thm},\mbox{conjecture})$

\medskip\textbf{KRS230+1.p} SCA nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{sca},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_sca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS057+1.p} A possible mapping of the EquivalentClasses axiom

A possible mapping of the EquivalentClasses axiom, which is
 connected but without a Hamiltonian path.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cB}(a) ) \ \Rightarrow \ \mbox{cB}(b) )\qquad\mbox{fof}(\mbox{cB\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cC}(a) ) \ \Rightarrow \ \mbox{cC}(b) )\qquad\mbox{fof}(\mbox{cC\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cD}(a) ) \ \Rightarrow \ \mbox{cD}(b) )\qquad\mbox{fof}(\mbox{cD\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iA})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( ( \mbox{cB}(x)\iff \mbox{cC}(x) )\mbox{ and }( \mbox{cB}(x)\iff x = \mbox{iA} )\mbox{ and }( \mbox{cB}(x)\iff \neg\,\mbox{cD}(x) )\mbox{ and }( \mbox{cC}(x)\iff x = \mbox{iA} )\mbox{ and }( \mbox{cC}(x)\iff \neg\,\mbox{cD}(x) )\mbox{ and }( x = \mbox{iA}\iff \neg\,\mbox{cD}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS088+1.p} DL Test: t7f.3






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff \exists y{:}\;( \mbox{rf}(x,y)\mbox{ and }\forall z{:}\;( \mbox{rinvF}(y,z) \ \Rightarrow \ \exists w{:}\;( \mbox{rf}(z,w)\mbox{ and }\neg\,\mbox{cp}_1(w) ) )\mbox{ and }\mbox{cp}_1(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}(x,y)\mbox{ and }\mbox{rf}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{49673})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS010-1.p} Paramasivam problem T-Box 3d

f subsumes e.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

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

$a(x_1) \ \Rightarrow \ ( c(x_1)\mbox{ or }d(x_1) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ a(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$c(x_1) \ \Rightarrow \ a(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_4)\mbox{ and }r(x_1,x_3)\mbox{ and }c(x_4)\mbox{ and }c(x_3) ) \ \Rightarrow \ x_4{=}x_3\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{r3least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ a(x_2)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$( a(\mbox{u1r}_1(x_1))\mbox{ and }\mbox{r3least}(x_1)\mbox{ and }\mbox{u1r}_3(x_1){=}\mbox{u1r}_2(x_1) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( a(\mbox{u1r}_1(x_1))\mbox{ and }\mbox{r3least}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }c(\mbox{u1r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( a(\mbox{u1r}_1(x_1))\mbox{ and }\mbox{r3least}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }c(\mbox{u1r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$( a(\mbox{u1r}_1(x_1))\mbox{ and }\mbox{r3least}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$( a(\mbox{u1r}_1(x_1))\mbox{ and }\mbox{r3least}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$( \mbox{r3least}(x_1)\mbox{ and }\mbox{u1r}_3(x_1){=}\mbox{u1r}_2(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\mbox{ or }c(\mbox{u1r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\mbox{ or }c(\mbox{u1r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\mbox{ or }r(x_1,\mbox{u1r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\mbox{ or }r(x_1,\mbox{u1r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u2r}_2(x_1){=}\mbox{u2r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u2r}_3(x_1){=}\mbox{u2r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ \neg\,\mbox{u2r}_3(x_1){=}\mbox{u2r}_2(x_1)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u2r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u2r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$\mbox{r3least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u2r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$( r(x_1,x_4)\mbox{ and }r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ ( \mbox{r3least}(x_1)\mbox{ or }x_4{=}x_3\mbox{ or }x_4{=}x_2\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ d(\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ d(\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{27},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ \neg\,\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{28},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ r(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{29},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ r(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{30},\mbox{axiom})$

$( r(x_1,x_3)\mbox{ and }r(x_1,x_2)\mbox{ and }d(x_3)\mbox{ and }d(x_2) ) \ \Rightarrow \ ( f(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{31},\mbox{axiom})$

\medskip\textbf{KRS207+1.p} ETH nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{eth},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_eth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS180+1.p} isa is incompatible with nota, nevera, and xora






$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\forall s_1,s_2{:}\;\exists \mbox{ax},c{:}\;( \mbox{status}(\mbox{ax},c,s_1) \ \Rightarrow \ \neg\,\mbox{isa}(s_1,s_2)\mbox{ and }( \mbox{nota}(s_1,s_2)\mbox{ or }\mbox{nevera}(s_1,s_2)\mbox{ or }\mbox{xora}(s_1,s_2) ) )\qquad\mbox{fof}(\mbox{isa\_exclusive},\mbox{conjecture})$

\medskip\textbf{KRS032+1.p} DL Test: t3.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}(a) ) \ \Rightarrow \ \mbox{cp}(b) )\qquad\mbox{fof}(\mbox{cp\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_3(a) ) \ \Rightarrow \ \mbox{cp}_3(b) )\qquad\mbox{fof}(\mbox{cp3\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_4(a) ) \ \Rightarrow \ \mbox{cp}_4(b) )\qquad\mbox{fof}(\mbox{cp4\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_5(a) ) \ \Rightarrow \ \mbox{cp}_5(b) )\qquad\mbox{fof}(\mbox{cp5\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_3(y) )\mbox{ and }\forall y_0,y_1,y_2,y_3{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1)\mbox{ and }\mbox{rr}(x,y_2)\mbox{ and }\mbox{rr}(x,y_3) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_0 = y_3\mbox{ or }y_1 = y_2\mbox{ or }y_1 = y_3\mbox{ or }y_2 = y_3 ) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_3(y)\mbox{ and }\mbox{cp}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_2(y)\mbox{ and }\mbox{cp}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}(y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_1(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_5(x)\mbox{ or }\mbox{cp}_2(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_2(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_3(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_4(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{33836})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

\medskip\textbf{KRS231+1.p} TCA nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{tca},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_tca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS020+1.p} The union of two classes can be defined using OWL Lite

The union of two classes can be defined using OWL Lite
 restrictions, and owl:intersectionOf.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff \exists y{:}\;( \mbox{rq}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cAorB}(x)\iff \exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cB}(x)\iff \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \neg\,\exists y{:}\;\mbox{rp}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cNothing}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotA}(x)\iff \forall y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotAorB}(x)\iff \forall y{:}\;( \mbox{rs}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotAorB}(x)\iff ( \mbox{cnotB}(x)\mbox{ and }\mbox{cnotA}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cnotB}(x)\iff \forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cNothing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

\medskip\textbf{KRS240+1.p} EQV xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{eqv},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_eqv\_thm},\mbox{conjecture})$

\medskip\textbf{KRS220+1.p} ESA nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{esa},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_esa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS233+1.p} CSA nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{csa},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_csa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS187+1.p} TAC isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{tac},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_tac\_thm},\mbox{conjecture})$

\medskip\textbf{KRS275$\wedge$7.p} Database querying






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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{john}@ \mbox{math} )@ ( \mbox{mand}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{teach}@ x@ \mbox{cs} ) )@ ( \mbox{mand}@ ( \mbox{teach}@ \mbox{mary}@ \mbox{psych} )@ ( \mbox{teach}@ \mbox{sue}@ \mbox{psych} ) ) ) ) )\qquad\mbox{thf}(\mbox{db},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mexists\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{teach}@ \mbox{john}@ x ) ) )\qquad\mbox{thf}(\mbox{query},\mbox{conjecture})$

\medskip\textbf{KRS266+1.p} SCA mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{sca},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_sca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS050+1.p} DL Test: t8.1






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{ca\_Vx}_4(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{ca\_Vx}_5(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cpxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_2(x)\iff \forall y{:}\;( \mbox{rr}_1(x,y) \ \Rightarrow \ \mbox{cp}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_3(x)\iff \forall y{:}\;( \mbox{rr}_1(x,y) \ \Rightarrow \ \mbox{cpxcomp}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_4(x)\iff \forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_2(y) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_5(x)\iff \forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \mbox{ca\_Vx}_3(y) ) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_16}_{39209})\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

\medskip\textbf{KRS217+1.p} NOC nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{noc},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_noc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS215+1.p} CSA nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{csa},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_csa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS258+1.p} EQV mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{eqv},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_eqv\_thm},\mbox{conjecture})$

\medskip\textbf{KRS060+1.p} Description cannot be expressed as a multicomponent restriction

This description cannot be expressed as a multicomponent
 restriction in the abstract syntax.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{rp}(\mbox{ii},x) \ \Rightarrow \ \mbox{ca}(x) )\qquad\mbox{fof}(\mbox{axiom\_2\_AndLHS},\mbox{axiom})$

$\exists x{:}\;( \mbox{rq}(\mbox{ii},x)\mbox{ and }\mbox{cs}(x) )\qquad\mbox{fof}(\mbox{axiom\_2\_AndRHS},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ii})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip\textbf{KRS084+1.p} DL Test: t6f.1 Double blocking example






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rinvF}(x,y)\mbox{ and }\mbox{cd}(y) )\mbox{ and }\forall y{:}\;( \mbox{rinvR}(x,y) \ \Rightarrow \ \exists z{:}\;( \mbox{rinvF}(y,z)\mbox{ and }\mbox{cd}(z) ) )\mbox{ and }\neg\,\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff ( \exists y{:}\;( \mbox{rf}(x,y)\mbox{ and }\neg\,\mbox{cc}(y) )\mbox{ and }\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}(x,y)\mbox{ and }\mbox{rf}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{35232})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

\medskip\textbf{KRS263+1.p} WTC mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{wtc},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_wtc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS249+1.p} TCA xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{tca},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_tca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS092+1.p} DL Test: heinsohn1.2

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ ( \mbox{cd}(y)\mbox{ and }\mbox{cc}(y) ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \neg\,\mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{04172})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS254+1.p} UNP mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{unp},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_unp\_thm},\mbox{conjecture})$

\medskip\textbf{KRS224+1.p} WEC nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{wec},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_wec\_thm},\mbox{conjecture})$

\medskip\textbf{KRS264+1.p} WTH mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{wth},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_wth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS029+1.p} DL Test: t1.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_3(a) ) \ \Rightarrow \ \mbox{cp}_3(b) )\qquad\mbox{fof}(\mbox{cp3\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_4(a) ) \ \Rightarrow \ \mbox{cp}_4(b) )\qquad\mbox{fof}(\mbox{cp4\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_5(a) ) \ \Rightarrow \ \mbox{cp}_5(b) )\qquad\mbox{fof}(\mbox{cp5\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff \exists y{:}\;( \mbox{rinvR}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\mbox{cp}_1(z) )\mbox{ and }\forall z_0,z_1{:}\;( ( \mbox{rr}(y,z_0)\mbox{ and }\mbox{rr}(y,z_1) ) \ \Rightarrow \ z_0 = z_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_2(x)\mbox{ or }\mbox{cp}_5(x)\mbox{ or }\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_3(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_2(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x)\mbox{ or }\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_3(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_3(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x)\mbox{ or }\mbox{cp}_4(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_4(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{22537})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS247+1.p} CAX xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{cax},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_cax\_thm},\mbox{conjecture})$

\medskip\textbf{KRS242+1.p} WEC xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{wec},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_wec\_thm},\mbox{conjecture})$

\medskip\textbf{KRS012-1.p} Paramasivam problem T-Box 4a

f subsumes c.




$c(\mbox{exists})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

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

$( c(x_1)\mbox{ and }r(x_1,x_3)\mbox{ and }d(x_3) ) \ \Rightarrow \ e(x_3)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( c(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ d(x_2)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$( d(\mbox{u0r}_1(x_1))\mbox{ and }e(\mbox{u0r}_2(x_1)) ) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$d(\mbox{u0r}_1(x_1)) \ \Rightarrow \ ( c(x_1)\mbox{ or }d(\mbox{u0r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$d(\mbox{u0r}_1(x_1)) \ \Rightarrow \ ( c(x_1)\mbox{ or }r(x_1,\mbox{u0r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$e(\mbox{u0r}_2(x_1)) \ \Rightarrow \ ( c(x_1)\mbox{ or }r(x_1,\mbox{u0r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$c(x_1)\mbox{ or }r(x_1,\mbox{u0r}_1(x_1))\mbox{ or }d(\mbox{u0r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$c(x_1)\mbox{ or }r(x_1,\mbox{u0r}_1(x_1))\mbox{ or }r(x_1,\mbox{u0r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( f(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ e(x_2)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$e(\mbox{u1r}_1(x_1)) \ \Rightarrow \ f(x_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$f(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

\medskip\textbf{KRS170+1.p} Deduction from hasLeader

A reasoner can also deduce that hasLeader is a subProperty of
 hasHead and hasHead is a subProperty of hasLeader.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasLeader}(x,y)\iff \mbox{rhasHead}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y{:}\;( \mbox{rhasLeader}(x,y) \ \Rightarrow \ \mbox{rhasHead}(x,y) )\mbox{ and }\forall x,y{:}\;( \mbox{rhasHead}(x,y) \ \Rightarrow \ \mbox{rhasLeader}(x,y) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS143+1.p} An owl:cardinality constraint is simply shorthand

An owl:cardinality constraint is simply shorthand for a pair of
 owl:minCardinality and owl:maxCardinality constraints.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \exists y_0{:}\;\mbox{rp}(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ ( \exists y_0{:}\;\mbox{rp}(x,y_0)\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS036+1.p} DL Test: t5f.1 Non-finite model example from paper






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{ca}(a) ) \ \Rightarrow \ \mbox{ca}(b) )\qquad\mbox{fof}(\mbox{ca\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(a,c) ) \ \Rightarrow \ \mbox{rf}(b,c) )\qquad\mbox{fof}(\mbox{rf\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}(c,a) ) \ \Rightarrow \ \mbox{rf}(c,b) )\qquad\mbox{fof}(\mbox{rf\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(a,c) ) \ \Rightarrow \ \mbox{rinvF}(b,c) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvF}(c,a) ) \ \Rightarrow \ \mbox{rinvF}(c,b) )\qquad\mbox{fof}(\mbox{rinvF\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \neg\,\mbox{ca}(x)\mbox{ and }\exists y{:}\;( \mbox{rinvR}(x,y)\mbox{ and }\exists z{:}\;( \mbox{rinvF}(y,z)\mbox{ and }\mbox{ca}(z) ) )\mbox{ and }\exists y{:}\;( \mbox{rinvF}(x,y)\mbox{ and }\mbox{ca}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}(x,y)\mbox{ and }\mbox{rf}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvF}(x,y)\iff \mbox{rf}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rr}(x,y)\mbox{ and }\mbox{rr}(y,z) ) \ \Rightarrow \ \mbox{rr}(x,z) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{47436})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS113+1.p} DL Test: t11.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{ca\_Vx}_2(a) ) \ \Rightarrow \ \mbox{ca\_Vx}_2(b) )\qquad\mbox{fof}(\mbox{ca\_Vx2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}(a) ) \ \Rightarrow \ \mbox{cp}(b) )\qquad\mbox{fof}(\mbox{cp\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cpxcomp}(a) ) \ \Rightarrow \ \mbox{cpxcomp}(b) )\qquad\mbox{fof}(\mbox{cpxcomp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(a,c) ) \ \Rightarrow \ \mbox{ra\_Px}_1(b,c) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(c,a) ) \ \Rightarrow \ \mbox{ra\_Px}_1(c,b) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvS}(a,c) ) \ \Rightarrow \ \mbox{rinvS}(b,c) )\qquad\mbox{fof}(\mbox{rinvS\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvS}(c,a) ) \ \Rightarrow \ \mbox{rinvS}(c,b) )\qquad\mbox{fof}(\mbox{rinvS\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(a,c) ) \ \Rightarrow \ \mbox{rs}(b,c) )\qquad\mbox{fof}(\mbox{rs\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rs}(c,a) ) \ \Rightarrow \ \mbox{rs}(c,b) )\qquad\mbox{fof}(\mbox{rs\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \forall y_0,y_1{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{ca\_Vx}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rs}(x,y)\mbox{ and }\mbox{cp}(y) )\mbox{ and }\mbox{cpxcomp}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cpxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Vx}_2(x)\iff \forall y{:}\;( \mbox{rinvS}(x,y) \ \Rightarrow \ \mbox{cp}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvS}(x,y)\iff \mbox{rs}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_21}_{22376})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rs}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS064+1.p} Something of type owl:Nothing

The triple asserts something of type owl:Nothing, however that is
 the empty class.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlNothing}(\mbox{i2003\_11\_14\_17\_18}_{08754})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

\medskip\textbf{KRS090+1.p} A pattern comes up a lot in more complex ontologies

This kind of pattern comes up a lot in more complex ontologies.
 Failure to cope with this kind of pattern is one of the reasons
 that many reasoners have been unable to cope with such ontologies.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_1(x) \ \Rightarrow \ ( ( \mbox{cB}_5(x)\mbox{ or }\mbox{cA}_5(x) )\mbox{ and }( \mbox{cB}_{13}(x)\mbox{ or }\mbox{cA}_{13}(x) )\mbox{ and }( \mbox{cA}_1(x)\mbox{ or }\mbox{cB}_1(x) )\mbox{ and }( \mbox{cA}_{27}(x)\mbox{ or }\mbox{cB}_{27}(x) )\mbox{ and }( \mbox{cA}_4(x)\mbox{ or }\mbox{cB}_4(x) )\mbox{ and }( \mbox{cB}_{12}(x)\mbox{ or }\mbox{cA}_{12}(x) )\mbox{ and }( \mbox{cB}_{30}(x)\mbox{ or }\mbox{cA}_{30}(x) )\mbox{ and }( \mbox{cA}_{20}(x)\mbox{ or }\mbox{cB}_{20}(x) )\mbox{ and }( \mbox{cB}_{14}(x)\mbox{ or }\mbox{cA}_{14}(x) )\mbox{ and }( \mbox{cB}_{28}(x)\mbox{ or }\mbox{cA}_{28}(x) )\mbox{ and }( \mbox{cA}_3(x)\mbox{ or }\mbox{cB}_3(x) )\mbox{ and }( \mbox{cB}_7(x)\mbox{ or }\mbox{cA}_7(x) )\mbox{ and }( \mbox{cB}_{21}(x)\mbox{ or }\mbox{cA}_{21}(x) )\mbox{ and }( \mbox{cB}_{22}(x)\mbox{ or }\mbox{cA}_{22}(x) )\mbox{ and }( \mbox{cB}_{17}(x)\mbox{ or }\mbox{cA}_{17}(x) )\mbox{ and }( \mbox{cB}_{11}(x)\mbox{ or }\mbox{cA}_{11}(x) )\mbox{ and }( \mbox{cB}_{19}(x)\mbox{ or }\mbox{cA}_{19}(x) )\mbox{ and }( \mbox{cA}_8(x)\mbox{ or }\mbox{cB}_8(x) )\mbox{ and }( \mbox{cA}_{26}(x)\mbox{ or }\mbox{cB}_{26}(x) )\mbox{ and }( \mbox{cA}_{25}(x)\mbox{ or }\mbox{cB}_{25}(x) )\mbox{ and }( \mbox{cA}_{29}(x)\mbox{ or }\mbox{cB}_{29}(x) )\mbox{ and }( \mbox{cB}_{23}(x)\mbox{ or }\mbox{cA}_{23}(x) )\mbox{ and }( \mbox{cB}_{18}(x)\mbox{ or }\mbox{cA}_{18}(x) )\mbox{ and }( \mbox{cB}_{10}(x)\mbox{ or }\mbox{cA}_{10}(x) )\mbox{ and }( \mbox{cB}_2(x)\mbox{ or }\mbox{cA}_2(x) )\mbox{ and }( \mbox{cA}_{16}(x)\mbox{ or }\mbox{cB}_{16}(x) )\mbox{ and }( \mbox{cB}_0(x)\mbox{ or }\mbox{cA}_0(x) )\mbox{ and }( \mbox{cB}_{31}(x)\mbox{ or }\mbox{cA}_{31}(x) )\mbox{ and }( \mbox{cB}_9(x)\mbox{ or }\mbox{cA}_9(x) )\mbox{ and }( \mbox{cB}_6(x)\mbox{ or }\mbox{cA}_6(x) )\mbox{ and }( \mbox{cB}_{24}(x)\mbox{ or }\mbox{cA}_{24}(x) )\mbox{ and }( \mbox{cB}_{15}(x)\mbox{ or }\mbox{cA}_{15}(x) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_2(x) \ \Rightarrow \ ( ( \neg\,\mbox{cB}(x)\mbox{ or }\mbox{cA}(x) )\mbox{ and }( \mbox{cB}(x)\mbox{ or }\mbox{cA}(x) ) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_3(x) \ \Rightarrow \ ( ( \neg\,\mbox{cB}(x)\mbox{ or }\neg\,\mbox{cA}(x) )\mbox{ and }( \mbox{cB}(x)\mbox{ or }\neg\,\mbox{cA}(x) ) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_4(x) \ \Rightarrow \ \exists y{:}\;( \mbox{rR}(x,y)\mbox{ and }\mbox{cC}_2(y) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_5(x) \ \Rightarrow \ \forall y{:}\;( \mbox{rR}(x,y) \ \Rightarrow \ \mbox{cC}_3(y) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cTEST}(x) \ \Rightarrow \ ( \mbox{cC}_4(x)\mbox{ and }\mbox{cC}_1(x)\mbox{ and }\mbox{cC}_5(x) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cTEST}(\mbox{i2003\_11\_14\_17\_19}_{57994})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS210+1.p} WTH nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{wth},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_wth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS238+1.p} ESA xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{esa},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_esa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS198+1.p} UNS isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{uns},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_uns\_thm},\mbox{conjecture})$

\medskip\textbf{KRS193+1.p} CAX isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{cax},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_cax\_thm},\mbox{conjecture})$

\medskip\textbf{KRS209+1.p} WTC nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{wtc},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_wtc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS243+1.p} ETH xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{eth},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_eth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS091+1.p} DL Test: heinsohn1.1

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \mbox{cd}(x)\mbox{ and }\mbox{cc}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \neg\,\mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{00819})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS072+1.p} DL Test: t1.3






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_3(a) ) \ \Rightarrow \ \mbox{cp}_3(b) )\qquad\mbox{fof}(\mbox{cp3\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_4(a) ) \ \Rightarrow \ \mbox{cp}_4(b) )\qquad\mbox{fof}(\mbox{cp4\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_5(a) ) \ \Rightarrow \ \mbox{cp}_5(b) )\qquad\mbox{fof}(\mbox{cp5\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rinvR}(x,y)\mbox{ and }\forall z_0,z_1{:}\;( ( \mbox{rr}(y,z_0)\mbox{ and }\mbox{rr}(y,z_1) ) \ \Rightarrow \ z_0 = z_1 )\mbox{ and }\exists z{:}\;( \mbox{rr}(y,z)\mbox{ and }\mbox{cp}_1(z) ) )\mbox{ and }\mbox{cp}_2(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_2(x)\mbox{ or }\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_2(x) \ \Rightarrow \ \neg\,\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_3(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_4(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_18}_{50190})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS236+1.p} UNP xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{unp},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_unp\_thm},\mbox{conjecture})$

\medskip\textbf{KRS150+1.p} DL Test: k\_lin ABox test from DL98 systems comparison






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{10}(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\neg\,\mbox{cC}_2(y) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{12}(x)\iff ( \neg\,\mbox{cC}_2(x)\mbox{ and }\neg\,\mbox{cC}_{10}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{14}(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_{12}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{16}(x)\iff ( \mbox{cC}_{14}(x)\mbox{ and }\mbox{cC}_8(x) ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_{18}(x)\iff ( \mbox{cTOP}(x)\mbox{ and }\mbox{cC}_{16}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_4(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\neg\,\mbox{cC}_2(y) ) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_6(x)\iff ( \neg\,\mbox{cC}_2(x)\mbox{ and }\neg\,\mbox{cC}_4(x) ) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cC}_8(x)\iff \exists y{:}\;( \mbox{rR}_1(x,y)\mbox{ and }\mbox{cC}_6(y) ) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cTEST}(x)\iff ( \mbox{cC}_{18}(x)\mbox{ and }\mbox{cTOP}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{cTOP}(\mbox{iV}_{16560})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{cTEST}(\mbox{iV}_{16560})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{16560})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\mbox{rR}_1(\mbox{iV}_{16560},\mbox{iV}_{16562})\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

$\mbox{rR}_1(\mbox{iV}_{16560},\mbox{iV}_{16561})\qquad\mbox{fof}(\mbox{axiom}_{15},\mbox{axiom})$

$\neg\,\mbox{cC}_4(\mbox{iV}_{16561})\qquad\mbox{fof}(\mbox{axiom}_{16},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{16561})\qquad\mbox{fof}(\mbox{axiom}_{17},\mbox{axiom})$

$\forall x{:}\;( \mbox{rR}_1(\mbox{iV}_{16561},x) \ \Rightarrow \ \mbox{cC}_2(x) )\qquad\mbox{fof}(\mbox{axiom}_{18},\mbox{axiom})$

$\neg\,\mbox{cC}_2(\mbox{iV}_{16561})\qquad\mbox{fof}(\mbox{axiom}_{19},\mbox{axiom})$

$\neg\,\mbox{cC}_{10}(\mbox{iV}_{16562})\qquad\mbox{fof}(\mbox{axiom}_{20},\mbox{axiom})$

$\neg\,\mbox{cC}_2(\mbox{iV}_{16562})\qquad\mbox{fof}(\mbox{axiom}_{21},\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iV}_{16562})\qquad\mbox{fof}(\mbox{axiom}_{22},\mbox{axiom})$

$\forall x{:}\;( \mbox{rR}_1(\mbox{iV}_{16562},x) \ \Rightarrow \ \mbox{cC}_2(x) )\qquad\mbox{fof}(\mbox{axiom}_{23},\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cC}_{18}(\mbox{iV}_{16560})\mbox{ and }\mbox{cC}_{16}(\mbox{iV}_{16560})\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{16560})\mbox{ and }\mbox{cC}_{14}(\mbox{iV}_{16560})\mbox{ and }\mbox{cC}_8(\mbox{iV}_{16560})\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{16561})\mbox{ and }\mbox{cC}_6(\mbox{iV}_{16561})\mbox{ and }\mbox{cowlThing}(\mbox{iV}_{16562})\mbox{ and }\mbox{cC}_{12}(\mbox{iV}_{16562})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS095+1.p} DL Test: heinsohn2.1

Tbox tests from [HK+94]




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y_0,y_1{:}\;( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1)\mbox{ and }y_0\ne y_1 )\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{14253})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS063+1.p} An example combining owl:oneOf and owl:inverseOf






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEUCountry}(a) ) \ \Rightarrow \ \mbox{cEUCountry}(b) )\qquad\mbox{fof}(\mbox{cEUCountry\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEuroMP}(a) ) \ \Rightarrow \ \mbox{cEuroMP}(b) )\qquad\mbox{fof}(\mbox{cEuroMP\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cEuropeanCountry}(a) ) \ \Rightarrow \ \mbox{cEuropeanCountry}(b) )\qquad\mbox{fof}(\mbox{cEuropeanCountry\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cPerson}(a) ) \ \Rightarrow \ \mbox{cPerson}(b) )\qquad\mbox{fof}(\mbox{cPerson\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rhasEuroMP}(a,c) ) \ \Rightarrow \ \mbox{rhasEuroMP}(b,c) )\qquad\mbox{fof}(\mbox{rhasEuroMP\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rhasEuroMP}(c,a) ) \ \Rightarrow \ \mbox{rhasEuroMP}(c,b) )\qquad\mbox{fof}(\mbox{rhasEuroMP\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{risEuroMPFrom}(a,c) ) \ \Rightarrow \ \mbox{risEuroMPFrom}(b,c) )\qquad\mbox{fof}(\mbox{risEuroMPFrom\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{risEuroMPFrom}(c,a) ) \ \Rightarrow \ \mbox{risEuroMPFrom}(c,b) )\qquad\mbox{fof}(\mbox{risEuroMPFrom\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cEUCountry}(x)\iff ( x = \mbox{iPT}\mbox{ or }x = \mbox{iBE}\mbox{ or }x = \mbox{iNL}\mbox{ or }x = \mbox{iES}\mbox{ or }x = \mbox{iFR}\mbox{ or }x = \mbox{iUK} ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cEuroMP}(x)\iff \exists y{:}\;( \mbox{risEuroMPFrom}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rhasEuroMP}(x,y) \ \Rightarrow \ \mbox{cEUCountry}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{risEuroMPFrom}(x,y)\iff \mbox{rhasEuroMP}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iBE})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iES})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iFR})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{cPerson}(\mbox{iKinnock})\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\neg\,\mbox{cEuroMP}(\mbox{iKinnock})\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iNL})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iPT})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

$\mbox{cEuropeanCountry}(\mbox{iUK})\qquad\mbox{fof}(\mbox{axiom}_{13},\mbox{axiom})$

$\mbox{rhasEuroMP}(\mbox{iUK},\mbox{iKinnock})\qquad\mbox{fof}(\mbox{axiom}_{14},\mbox{axiom})$

\medskip\textbf{KRS079+1.p} DL Test: t2.2






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(a,c) ) \ \Rightarrow \ \mbox{rf}_1(b,c) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(c,a) ) \ \Rightarrow \ \mbox{rf}_1(c,b) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(a,c) ) \ \Rightarrow \ \mbox{rf}_2(b,c) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(c,a) ) \ \Rightarrow \ \mbox{rf}_2(c,b) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rf}_2(x,y)\mbox{ and }\mbox{cp}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_1(x,y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cowlThing}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_2(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_1(x,y_0)\mbox{ and }\mbox{rf}_1(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_2(x,y_0)\mbox{ and }\mbox{rf}_2(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_19}_{17492})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS162+1.p} Entailment for three natural numbers

This entailment can be replicated for any three natural numbers
 i, j, k such that i+j $>$= k. In this example, they are chosen as
 2, 3 and 5.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA}(a) ) \ \Rightarrow \ \mbox{cA}(b) )\qquad\mbox{fof}(\mbox{cA\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cB}(a) ) \ \Rightarrow \ \mbox{cB}(b) )\qquad\mbox{fof}(\mbox{cB\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rq}(a,c) ) \ \Rightarrow \ \mbox{rq}(b,c) )\qquad\mbox{fof}(\mbox{rq\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rq}(c,a) ) \ \Rightarrow \ \mbox{rq}(c,b) )\qquad\mbox{fof}(\mbox{rq\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{cA}(y) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{cB}(y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{cB}(x)\mbox{ and }\mbox{cA}(x)\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rq}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rp}(x,y) \ \Rightarrow \ \mbox{rr}(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( ( \exists y_0,y_1{:}\;( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }y_0\ne y_1 )\mbox{ and }\exists y_0,y_1,y_2{:}\;( \mbox{rq}(x,y_0)\mbox{ and }\mbox{rq}(x,y_1)\mbox{ and }\mbox{rq}(x,y_2)\mbox{ and }y_0\ne y_1\mbox{ and }y_0\ne y_2\mbox{ and }y_1\ne y_2 ) ) \ \Rightarrow \ \exists y_0,y_1,y_2,y_3,y_4{:}\;( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1)\mbox{ and }\mbox{rr}(x,y_2)\mbox{ and }\mbox{rr}(x,y_3)\mbox{ and }\mbox{rr}(x,y_4)\mbox{ and }y_0\ne y_1\mbox{ and }y_0\ne y_2\mbox{ and }y_0\ne y_3\mbox{ and }y_0\ne y_4\mbox{ and }y_1\ne y_2\mbox{ and }y_1\ne y_3\mbox{ and }y_1\ne y_4\mbox{ and }y_2\ne y_3\mbox{ and }y_2\ne y_4\mbox{ and }y_3\ne y_4 ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS138+1.p} Extensional semantics of owl:SymmetricProperty

Test illustrating extensional semantics of owl:SymmetricProperty.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA}(a) ) \ \Rightarrow \ \mbox{cA}(b) )\qquad\mbox{fof}(\mbox{cA\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{requalityOnA}(a,c) ) \ \Rightarrow \ \mbox{requalityOnA}(b,c) )\qquad\mbox{fof}(\mbox{requalityOnA\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{requalityOnA}(c,a) ) \ \Rightarrow \ \mbox{requalityOnA}(c,b) )\qquad\mbox{fof}(\mbox{requalityOnA\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff ( x = \mbox{ib}\mbox{ or }x = \mbox{ia} ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{requalityOnA}(y,x)\mbox{ and }\mbox{requalityOnA}(z,x) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{requalityOnA}(x,y) \ \Rightarrow \ \mbox{cA}(y) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{requalityOnA}(\mbox{ia},\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{requalityOnA}(\mbox{ib},\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ic})\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x,y{:}\;( \mbox{requalityOnA}(x,y) \ \Rightarrow \ ( x = \mbox{ia}\mbox{ or }x = \mbox{ib}\mbox{ or }x = \mbox{ic} ) )\mbox{ and }\forall x,y{:}\;( \mbox{requalityOnA}(x,y) \ \Rightarrow \ \mbox{requalityOnA}(y,x) )\mbox{ and }\mbox{cowlThing}(\mbox{ia})\mbox{ and }\mbox{requalityOnA}(\mbox{ia},\mbox{ia})\mbox{ and }\mbox{cowlThing}(\mbox{ib})\mbox{ and }\mbox{cowlThing}(\mbox{ic})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS205+1.p} TAC nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{tac},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_tac\_thm},\mbox{conjecture})$

\medskip\textbf{KRS225+1.p} ETH nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{eth},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_eth\_thm},\mbox{conjecture})$

\medskip\textbf{KRS203+1.p} SAT nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{sat},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_sat\_thm},\mbox{conjecture})$

\medskip\textbf{KRS274$\wedge$7.p} Querying description logic knowledge bases






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

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

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

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

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

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

$q_3{:}\;\mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{q3\_type},\mbox{type})$

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

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

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

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

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

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

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

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

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

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{mary} )@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{ann} )@ ( \mbox{mand}@ ( \mbox{female}@ \mbox{jane} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{bob} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{john} )@ ( \mbox{mand}@ ( \mbox{male}@ \mbox{paul} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{bob}@ \mbox{mary} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{bob}@ \mbox{ann} )@ ( \mbox{mand}@ ( \mbox{parent}@ \mbox{john}@ \mbox{paul} )@ ( \mbox{parent}@ \mbox{mary}@ \mbox{jane} ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{abox},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{male}@ x ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{female}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{tbox},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( q_3@ x )@ ( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{parent}@ y@ x ) )@ ( \mbox{mforall\_ind}@ \lambda z{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{parent}@ y@ z ) )@ ( \mbox{qmltpeq}@ z@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{query},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( q_3@ \mbox{jane} )@ ( q_3@ \mbox{paul} ) )\qquad\mbox{thf}(\mbox{con},\mbox{conjecture})$

\medskip\textbf{KRS214+1.p} WCA nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{wca},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_wca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS182+1.p} UNP isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{unp},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_unp\_thm},\mbox{conjecture})$

\medskip\textbf{KRS031+1.p} DL Test: t2.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cSatisfiable}(a) ) \ \Rightarrow \ \mbox{cSatisfiable}(b) )\qquad\mbox{fof}(\mbox{cSatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(a,c) ) \ \Rightarrow \ \mbox{rf}_1(b,c) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(c,a) ) \ \Rightarrow \ \mbox{rf}_1(c,b) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(a,c) ) \ \Rightarrow \ \mbox{rf}_2(b,c) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(c,a) ) \ \Rightarrow \ \mbox{rf}_2(c,b) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cSatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rf}_1(x,y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_2(x,y)\mbox{ and }\mbox{cp}_2(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_2(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_2(x,y_0)\mbox{ and }\mbox{rf}_2(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x) \ \Rightarrow \ \forall y_0,y_1{:}\;( ( \mbox{rf}_1(x,y_0)\mbox{ and }\mbox{rf}_1(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cSatisfiable}(\mbox{i2003\_11\_14\_17\_15}_{2938})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{rf}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS196+1.p} WCA isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{wca},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_wca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS007-1.p} Paramasivam problem T-Box 3a

f subsumes e.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

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

$c(x_1) \ \Rightarrow \ \mbox{s2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$\mbox{s2least}(x_1) \ \Rightarrow \ s(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$\mbox{s1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$( \mbox{s1most}(x_1)\mbox{ and }s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{s1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{s1most}(x_1)\mbox{ or }s(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ d(\mbox{u4r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ c(\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$( r(x_1,x_2)\mbox{ and }r(x_1,x_3)\mbox{ and }c(x_3)\mbox{ and }r(x_1,x_4)\mbox{ and }d(x_4) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$f(x_1) \ \Rightarrow \ \mbox{r2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ f(x_1)\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u6r}_2(x_1){=}\mbox{u6r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u6r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$\mbox{r2least}(x_1) \ \Rightarrow \ r(x_1,\mbox{u6r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$( r(x_1,x_3)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ ( \mbox{r2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

\medskip\textbf{KRS015-1.p} Paramasivam problem T-Box 5b

e exists.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ \mbox{t2least}(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ c(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ \neg\,\mbox{u1r}_2(x_1){=}\mbox{u1r}_1(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u1r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$\mbox{t2least}(x_1) \ \Rightarrow \ t(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( t(x_1,x_3)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ ( \mbox{t2least}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \mbox{t1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{t1most}(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( \mbox{t1most}(x_1)\mbox{ and }t(x_1,x_3)\mbox{ and }t(x_1,x_2) ) \ \Rightarrow \ x_3{=}x_2\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{u3r}_2(x_1){=}\mbox{u3r}_1(x_1) \ \Rightarrow \ \mbox{t1most}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$\mbox{t1most}(x_1)\mbox{ or }t(x_1,\mbox{u3r}_1(x_1))\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{t1most}(x_1)\mbox{ or }t(x_1,\mbox{u3r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u4r}_4(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_4) ) \ \Rightarrow \ s(x_1,x_4)\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( e(x_1)\mbox{ and }s(x_1,x_4) ) \ \Rightarrow \ r(x_1,x_4)\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$( e(x_1)\mbox{ and }s(x_1,x_3) ) \ \Rightarrow \ d(x_3)\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$( e(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ c(x_2)\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }d(\mbox{u4r}_2(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }s(x_1,\mbox{u4r}_3(x_1))\mbox{ or }r(x_1,\mbox{u4r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{19},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }d(\mbox{u4r}_2(x_1))\mbox{ and }r(x_1,\mbox{u4r}_3(x_1))\mbox{ and }s(x_1,\mbox{u4r}_3(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_{20},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }s(x_1,\mbox{u4r}_2(x_1))\mbox{ or }s(x_1,\mbox{u4r}_3(x_1))\mbox{ or }r(x_1,\mbox{u4r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{21},\mbox{axiom})$

$( c(\mbox{u4r}_1(x_1))\mbox{ and }r(x_1,\mbox{u4r}_3(x_1))\mbox{ and }s(x_1,\mbox{u4r}_3(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }s(x_1,\mbox{u4r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{22},\mbox{axiom})$

$( d(\mbox{u4r}_2(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1))\mbox{ or }s(x_1,\mbox{u4r}_3(x_1))\mbox{ or }r(x_1,\mbox{u4r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{23},\mbox{axiom})$

$( d(\mbox{u4r}_2(x_1))\mbox{ and }r(x_1,\mbox{u4r}_3(x_1))\mbox{ and }s(x_1,\mbox{u4r}_3(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{24},\mbox{axiom})$

$r(x_1,x_5) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1))\mbox{ or }s(x_1,\mbox{u4r}_2(x_1))\mbox{ or }s(x_1,\mbox{u4r}_3(x_1))\mbox{ or }r(x_1,\mbox{u4r}_3(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{25},\mbox{axiom})$

$( r(x_1,\mbox{u4r}_3(x_1))\mbox{ and }s(x_1,\mbox{u4r}_3(x_1))\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }r(x_1,\mbox{u4r}_1(x_1))\mbox{ or }s(x_1,\mbox{u4r}_2(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{26},\mbox{axiom})$

\medskip\textbf{KRS022+1.p} Informal semantics for RDF container are not respected by OWL

The informal semantics indicated by comments concerning user
 defined classes are not respected by the formal machinery of OWL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cBag}(\mbox{i2003\_11\_14\_17\_14}_{39627})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$r_1(\mbox{i2003\_11\_14\_17\_14}_{39627},\mbox{i2003\_11\_14\_17\_14}_{39661})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{39661})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS268+1.p} WCA mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{wca},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_wca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS106+1.p} DL Test: fact3.1






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp1xcomp}(a) ) \ \Rightarrow \ \mbox{cp1xcomp}(b) )\qquad\mbox{fof}(\mbox{cp1xcomp\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(a,c) ) \ \Rightarrow \ \mbox{ra\_Px}_1(b,c) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(c,a) ) \ \Rightarrow \ \mbox{ra\_Px}_1(c,b) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(a,c) ) \ \Rightarrow \ \mbox{rf}_1(b,c) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_1(c,a) ) \ \Rightarrow \ \mbox{rf}_1(c,b) )\qquad\mbox{fof}(\mbox{rf1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(a,c) ) \ \Rightarrow \ \mbox{rf}_2(b,c) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_2(c,a) ) \ \Rightarrow \ \mbox{rf}_2(c,b) )\qquad\mbox{fof}(\mbox{rf2\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_3(a,c) ) \ \Rightarrow \ \mbox{rf}_3(b,c) )\qquad\mbox{fof}(\mbox{rf3\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rf}_3(c,a) ) \ \Rightarrow \ \mbox{rf}_3(c,b) )\qquad\mbox{fof}(\mbox{rf3\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rf}_3(x,y)\mbox{ and }\mbox{cp}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_1(x,y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rf}_2(x,y)\mbox{ and }\mbox{cp1xcomp}(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp1xcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_1(x,y)\mbox{ and }\mbox{rf}_1(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_2(x,y)\mbox{ and }\mbox{rf}_2(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y,z{:}\;( ( \mbox{rf}_3(x,y)\mbox{ and }\mbox{rf}_3(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{57644})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}_3(x,y) \ \Rightarrow \ \mbox{rf}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rf}_3(x,y) \ \Rightarrow \ \mbox{rf}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

\medskip\textbf{KRS145+1.p} An owl:cardinality constraint is simply shorthand

An owl:cardinality constraint is simply shorthand for a pair of
 owl:minCardinality and owl:maxCardinality constraints.




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(a,c) ) \ \Rightarrow \ \mbox{rp}(b,c) )\qquad\mbox{fof}(\mbox{rp\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rp}(c,a) ) \ \Rightarrow \ \mbox{rp}(c,b) )\qquad\mbox{fof}(\mbox{rp\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ ( \forall y_0,y_1,y_2{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }\mbox{rp}(x,y_2) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_1 = y_2 ) )\mbox{ and }\exists y_0,y_1{:}\;( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }y_0\ne y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ ( \exists y_0,y_1{:}\;( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }y_0\ne y_1 )\mbox{ and }\forall y_0,y_1,y_2{:}\;( ( \mbox{rp}(x,y_0)\mbox{ and }\mbox{rp}(x,y_1)\mbox{ and }\mbox{rp}(x,y_2) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_1 = y_2 ) ) ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS123+1.p} DL Test: heinsohn1.1

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \mbox{cc}(x)\mbox{ and }\mbox{cd}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd1xcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}_1(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd1xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{02803})\qquad\mbox{fof}(\mbox{axiom}_{12},\mbox{axiom})$

\medskip\textbf{KRS218+1.p} UNP nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{unp},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_unp\_thm},\mbox{conjecture})$

\medskip\textbf{KRS219+1.p} SAP nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{sap},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_sap\_thm},\mbox{conjecture})$

\medskip\textbf{KRS071+1.p} DL Test: t1.2






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_1(a) ) \ \Rightarrow \ \mbox{cp}_1(b) )\qquad\mbox{fof}(\mbox{cp1\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_2(a) ) \ \Rightarrow \ \mbox{cp}_2(b) )\qquad\mbox{fof}(\mbox{cp2\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_3(a) ) \ \Rightarrow \ \mbox{cp}_3(b) )\qquad\mbox{fof}(\mbox{cp3\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_4(a) ) \ \Rightarrow \ \mbox{cp}_4(b) )\qquad\mbox{fof}(\mbox{cp4\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cp}_5(a) ) \ \Rightarrow \ \mbox{cp}_5(b) )\qquad\mbox{fof}(\mbox{cp5\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(a,c) ) \ \Rightarrow \ \mbox{rinvR}(b,c) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rinvR}(c,a) ) \ \Rightarrow \ \mbox{rinvR}(c,b) )\qquad\mbox{fof}(\mbox{rinvR\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_1(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_2(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cp}_3(y) )\mbox{ and }\forall y_0,y_1,y_2{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1)\mbox{ and }\mbox{rr}(x,y_2) ) \ \Rightarrow \ ( y_0 = y_1\mbox{ or }y_0 = y_2\mbox{ or }y_1 = y_2 ) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_1(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_2(x)\mbox{ or }\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_2(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_3(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_3(x) \ \Rightarrow \ \neg\,\mbox{cp}_4(x)\mbox{ or }\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{cp}_4(x) \ \Rightarrow \ \neg\,\mbox{cp}_5(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{rinvR}(x,y)\iff \mbox{rr}(y,x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_18}_{39380})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS212+1.p} SCA nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{sca},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_sca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS191+1.p} WTC isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{wtc},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_wtc\_thm},\mbox{conjecture})$

\medskip\textbf{KRS256+1.p} ESA mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{esa},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_esa\_thm},\mbox{conjecture})$

\medskip\textbf{KRS168+1.p} De Morgan's law






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( ( \neg\,\mbox{cB}(x)\mbox{ and }\neg\,\mbox{cA}(x) )\iff \neg\,\mbox{cA}(x)\mbox{ or }\mbox{cB}(x) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS058+1.p} A simple test for infinite loops in imports processing code






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

\medskip\textbf{KRS127+1.p} DL Test: heinsohn2.2

Tbox tests from [HK+94]




$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cUnsatisfiable}(a) ) \ \Rightarrow \ \mbox{cUnsatisfiable}(b) )\qquad\mbox{fof}(\mbox{cUnsatisfiable\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cc}(a) ) \ \Rightarrow \ \mbox{cc}(b) )\qquad\mbox{fof}(\mbox{cc\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cd}(a) ) \ \Rightarrow \ \mbox{cd}(b) )\qquad\mbox{fof}(\mbox{cd\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cdxcomp}(a) ) \ \Rightarrow \ \mbox{cdxcomp}(b) )\qquad\mbox{fof}(\mbox{cdxcomp\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(a,c) ) \ \Rightarrow \ \mbox{ra\_Px}_1(b,c) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{ra\_Px}_1(c,a) ) \ \Rightarrow \ \mbox{ra\_Px}_1(c,b) )\qquad\mbox{fof}(\mbox{ra\_Px1\_substitution}_2,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(a,c) ) \ \Rightarrow \ \mbox{rr}(b,c) )\qquad\mbox{fof}(\mbox{rr\_substitution}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a = b\mbox{ and }\mbox{rr}(c,a) ) \ \Rightarrow \ \mbox{rr}(c,b) )\qquad\mbox{fof}(\mbox{rr\_substitution}_2,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cc}(y) )\mbox{ and }\exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cd}(y) )\mbox{ and }\forall y_0,y_1{:}\;( ( \mbox{rr}(x,y_0)\mbox{ and }\mbox{rr}(x,y_1) ) \ \Rightarrow \ y_0 = y_1 ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{27794})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

\medskip\textbf{KRS185+1.p} SAT isa THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{isa}(\mbox{sat},\mbox{thm})\qquad\mbox{fof}(\mbox{isa\_sat\_thm},\mbox{conjecture})$

\medskip\textbf{KRS094+1.p} DL Test: heinsohn1.4

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff \mbox{cc}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \neg\,\mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \neg\,\mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}_1(x) \ \Rightarrow \ \mbox{cd}_1(x) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}_3(x) \ \Rightarrow \ \mbox{cc}(x) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cf}(x) \ \Rightarrow \ \mbox{cd}(x) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_20}_{11330})\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

\medskip\textbf{KRS239+1.p} SAT xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{sat},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_sat\_thm},\mbox{conjecture})$

\medskip\textbf{KRS128+1.p} DL Test: heinsohn4.1

Tbox tests from [HK+94]




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cUnsatisfiable}(x)\iff ( \exists y{:}\;( \mbox{rr}(x,y)\mbox{ and }\mbox{cexcomp}(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{cd}(y) )\mbox{ and }\forall y{:}\;( \mbox{rr}(x,y) \ \Rightarrow \ \mbox{ca\_Cx}_4(y) ) ) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cc}(x) \ \Rightarrow \ \mbox{cdxcomp}(x) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cd}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_2(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\forall x{:}\;( \mbox{cdxcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_2(x,y) )\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{ce}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_1(x,y) )\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cexcomp}(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_1(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_7,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx}_4(x)\iff \exists y_0{:}\;\mbox{ra\_Px}_4(x,y_0) )\qquad\mbox{fof}(\mbox{axiom}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx4xcomp}(x)\iff \neg\,\exists y{:}\;\mbox{ra\_Px}_4(x,y) )\qquad\mbox{fof}(\mbox{axiom}_9,\mbox{axiom})$

$\forall x{:}\;( \mbox{ca\_Cx4xcomp}(x)\iff ( \mbox{cd}(x)\mbox{ and }\mbox{cexcomp}(x) ) )\qquad\mbox{fof}(\mbox{axiom}_{10},\mbox{axiom})$

$\mbox{cUnsatisfiable}(\mbox{i2003\_11\_14\_17\_22}_{31584})\qquad\mbox{fof}(\mbox{axiom}_{11},\mbox{axiom})$

\medskip\textbf{KRS066+1.p} The extension of OWL Thing may not be emtpy in OWL Lite






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\iff \mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

\medskip\textbf{KRS234+1.p} UNS nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{uns},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_uns\_thm},\mbox{conjecture})$

\medskip\textbf{KRS004-1.p} Paramasivam problem T-Box 1d

c exists.




$c(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$c(x_1) \ \Rightarrow \ d(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$c(x_1) \ \Rightarrow \ \neg\,d(x_1)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$d(x_1) \ \Rightarrow \ \neg\,c(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

\medskip\textbf{KRS005-1.p} Paramasivam problem T-Box 2a

Inconsistent concept definition; e exists.




$e(\mbox{exist})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( e(x_1)\mbox{ and }r(x_1,x_5) ) \ \Rightarrow \ x_5{=}\mbox{u0r}_4(x_1)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u0r}_4(x_1))\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \neg\,\mbox{u0r}_3(x_1){=}\mbox{u0r}_2(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u0r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ r(x_1,\mbox{u0r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( r(x_1,x_3)\mbox{ and }r(x_1,x_2)\mbox{ and }r(x_1,x_4)\mbox{ and }\mbox{u0r}_1(x_4,x_1){=}x_4 ) \ \Rightarrow \ ( e(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( r(x_1,x_3)\mbox{ and }r(x_1,x_2)\mbox{ and }r(x_1,x_4) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }x_3{=}x_2\mbox{ or }r(x_1,\mbox{u0r}_1(x_4,x_1)) )\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

\medskip\textbf{KRS024+1.p} An OWL Lite version of I5.3-007






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{i2003\_11\_14\_17\_14}_{46558})\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{xsd\_string}(\mbox{xsd\_string}_0)\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{rdp}(\mbox{i2003\_11\_14\_17\_14}_{46558},\mbox{xsd\_string}_0)\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

\medskip\textbf{KRS014-1.p} Paramasivam problem T-Box 5a

e exists.




$e(\mbox{exists})\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$( e(x_1)\mbox{ and }r(x_1,x_2) ) \ \Rightarrow \ s(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$( e(x_1)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ r(x_1,x_2)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{s2exact}(x_1)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$e(x_1) \ \Rightarrow \ \mbox{r1exact}(x_1)\qquad\mbox{cnf}(\mbox{clause}_5,\mbox{axiom})$

$( \mbox{r1exact}(x_1)\mbox{ and }\mbox{s2exact}(x_1) ) \ \Rightarrow \ ( e(x_1)\mbox{ or }s(x_1,\mbox{u0r}_1(x_1))\mbox{ or }r(x_1,\mbox{u0r}_1(x_1)) )\qquad\mbox{cnf}(\mbox{clause}_6,\mbox{axiom})$

$( \mbox{r1exact}(x_1)\mbox{ and }\mbox{s2exact}(x_1)\mbox{ and }r(x_1,\mbox{u0r}_1(x_1))\mbox{ and }s(x_1,\mbox{u0r}_1(x_1)) ) \ \Rightarrow \ e(x_1)\qquad\mbox{cnf}(\mbox{clause}_7,\mbox{axiom})$

$( \mbox{r1exact}(x_1)\mbox{ and }r(x_1,x_3) ) \ \Rightarrow \ x_3{=}\mbox{u1r}_2(x_1)\qquad\mbox{cnf}(\mbox{clause}_8,\mbox{axiom})$

$\mbox{r1exact}(x_1) \ \Rightarrow \ r(x_1,\mbox{u1r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_9,\mbox{axiom})$

$( r(x_1,x_2)\mbox{ and }\mbox{u1r}_1(x_2,x_1){=}x_2 ) \ \Rightarrow \ \mbox{r1exact}(x_1)\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$r(x_1,x_2) \ \Rightarrow \ ( \mbox{r1exact}(x_1)\mbox{ or }r(x_1,\mbox{u1r}_1(x_2,x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{11},\mbox{axiom})$

$( \mbox{s2exact}(x_1)\mbox{ and }s(x_1,x_4) ) \ \Rightarrow \ ( x_4{=}\mbox{u2r}_3(x_1)\mbox{ or }x_4{=}\mbox{u2r}_2(x_1) )\qquad\mbox{cnf}(\mbox{clause}_{12},\mbox{axiom})$

$\mbox{s2exact}(x_1) \ \Rightarrow \ \neg\,\mbox{u2r}_3(x_1){=}\mbox{u2r}_2(x_1)\qquad\mbox{cnf}(\mbox{clause}_{13},\mbox{axiom})$

$\mbox{s2exact}(x_1) \ \Rightarrow \ s(x_1,\mbox{u2r}_2(x_1))\qquad\mbox{cnf}(\mbox{clause}_{14},\mbox{axiom})$

$\mbox{s2exact}(x_1) \ \Rightarrow \ s(x_1,\mbox{u2r}_3(x_1))\qquad\mbox{cnf}(\mbox{clause}_{15},\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2)\mbox{ and }\mbox{u2r}_1(x_3,x_2,x_1){=}x_2 ) \ \Rightarrow \ ( \mbox{s2exact}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{16},\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2)\mbox{ and }\mbox{u2r}_1(x_3,x_2,x_1){=}x_3 ) \ \Rightarrow \ ( \mbox{s2exact}(x_1)\mbox{ or }x_3{=}x_2 )\qquad\mbox{cnf}(\mbox{clause}_{17},\mbox{axiom})$

$( s(x_1,x_3)\mbox{ and }s(x_1,x_2) ) \ \Rightarrow \ ( \mbox{s2exact}(x_1)\mbox{ or }x_3{=}x_2\mbox{ or }s(x_1,\mbox{u2r}_1(x_3,x_2,x_1)) )\qquad\mbox{cnf}(\mbox{clause}_{18},\mbox{axiom})$

\medskip\textbf{KRS208+1.p} TAU nota THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nota}(\mbox{tau},\mbox{thm})\qquad\mbox{fof}(\mbox{nota\_tau\_thm},\mbox{conjecture})$

\medskip\textbf{KRS262+1.p} TAU mighta THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{mighta}(\mbox{tau},\mbox{thm})\qquad\mbox{fof}(\mbox{mighta\_tau\_thm},\mbox{conjecture})$

\medskip\textbf{KRS174+1.p} Sets with appropriate extensions are related by unionOf






$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA}(a) ) \ \Rightarrow \ \mbox{cA}(b) )\qquad\mbox{fof}(\mbox{cA\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cA\_and\_B}(a) ) \ \Rightarrow \ \mbox{cA\_and\_B}(b) )\qquad\mbox{fof}(\mbox{cA\_and\_B\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cB}(a) ) \ \Rightarrow \ \mbox{cB}(b) )\qquad\mbox{fof}(\mbox{cB\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlNothing}(a) ) \ \Rightarrow \ \mbox{cowlNothing}(b) )\qquad\mbox{fof}(\mbox{cowlNothing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{cowlThing}(a) ) \ \Rightarrow \ \mbox{cowlThing}(b) )\qquad\mbox{fof}(\mbox{cowlThing\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_integer}(a) ) \ \Rightarrow \ \mbox{xsd\_integer}(b) )\qquad\mbox{fof}(\mbox{xsd\_integer\_substitution}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a = b\mbox{ and }\mbox{xsd\_string}(a) ) \ \Rightarrow \ \mbox{xsd\_string}(b) )\qquad\mbox{fof}(\mbox{xsd\_string\_substitution}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA}(x)\iff x = \mbox{ia} )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{cA\_and\_B}(x)\iff ( x = \mbox{ib}\mbox{ or }x = \mbox{ia} ) )\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{cB}(x)\iff x = \mbox{ib} )\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ia})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{ib})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\forall x{:}\;( \mbox{cA\_and\_B}(x)\iff ( \mbox{cB}(x)\mbox{ or }\mbox{cA}(x) ) )\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS164+1.p} Two classes may have the same class extension






$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{cCar}(x)\iff \mbox{cAutomobile}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\mbox{cAutomobile}(\mbox{iauto})\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{iauto})\qquad\mbox{fof}(\mbox{axiom}_4,\mbox{axiom})$

$\mbox{cCar}(\mbox{icar})\qquad\mbox{fof}(\mbox{axiom}_5,\mbox{axiom})$

$\mbox{cowlThing}(\mbox{icar})\qquad\mbox{fof}(\mbox{axiom}_6,\mbox{axiom})$

$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\mbox{ and }\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\mbox{ and }\mbox{cCar}(\mbox{iauto})\mbox{ and }\mbox{cowlThing}(\mbox{iauto})\mbox{ and }\mbox{cAutomobile}(\mbox{icar})\mbox{ and }\mbox{cowlThing}(\mbox{icar})\qquad\mbox{fof}(\mbox{the\_axiom},\mbox{conjecture})$

\medskip\textbf{KRS250+1.p} WCA xora THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{xora}(\mbox{wca},\mbox{thm})\qquad\mbox{fof}(\mbox{xora\_wca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS232+1.p} WCA nevera THM






$\mbox{include}(\mbox{'Axioms/KRS001+0.ax'})$

$\mbox{include}(\mbox{'Axioms/KRS001+1.ax'})$

$\mbox{nevera}(\mbox{wca},\mbox{thm})\qquad\mbox{fof}(\mbox{nevera\_wca\_thm},\mbox{conjecture})$

\medskip\textbf{KRS054+1.p} owl:disjointWith edges may be within OWL DL

If the owl:disjointWith edges in the graph form unconnected
 undirected complete subgraphs then this may be within OWL DL.




$\forall x{:}\;( \mbox{cowlThing}(x)\mbox{ and }\neg\,\mbox{cowlNothing}(x) )\qquad\mbox{fof}(\mbox{axiom}_0,\mbox{axiom})$

$\forall x{:}\;( \mbox{xsd\_string}(x)\iff \neg\,\mbox{xsd\_integer}(x) )\qquad\mbox{fof}(\mbox{axiom}_1,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{cE}(x)\mbox{ and }\mbox{cD}(x)\mbox{ and }\neg\,\mbox{cE}(x)\mbox{ and }\mbox{cA}(x)\mbox{ and }\neg\,\mbox{cD}(x)\mbox{ and }\mbox{cA}(x) )\qquad\mbox{fof}(\mbox{axiom}_2,\mbox{axiom})$

$\forall x{:}\;\neg\,\mbox{cB}(x)\mbox{ and }\mbox{cC}(x)\qquad\mbox{fof}(\mbox{axiom}_3,\mbox{axiom})$

\medskip
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