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    \textbf{PUZ002-0.ax} Truthtellers and Liars axioms for two types of people

Axioms for two types of people; truthtellers and liars.




$\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{liar}(x))\qquad\mbox{cnf}(\mbox{truthteller\_or\_liar},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{truthteller}(x)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{liar}(x))\qquad\mbox{cnf}(\mbox{not\_both},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{truthteller}(\mbox{truthteller}))\mbox{ and }\mbox{a\_truth}(\mbox{says}(\mbox{truthteller},\mbox{statement})) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{statement})\qquad\mbox{cnf}(\mbox{truthtellers\_tell\_truth},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{liar}(\mbox{liar}))\mbox{ and }\mbox{a\_truth}(\mbox{says}(\mbox{liar},\mbox{statement})) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{statement})\qquad\mbox{cnf}(\mbox{liars\_lie},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{statement})\mbox{ and }\mbox{a\_truth}(\mbox{says}(\mbox{truthteller},\mbox{statement})) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{truthteller}(\mbox{truthteller}))\qquad\mbox{cnf}(\mbox{truths\_are\_told\_by\_truthtellers},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{says}(\mbox{liar},\mbox{statement})) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{statement})\mbox{ or }\mbox{a\_truth}(\mbox{liar}(\mbox{liar})) )\qquad\mbox{cnf}(\mbox{liars\_are\_told\_by\_liars},\mbox{axiom})$

\medskip\textbf{PUZ003-0.ax} Truthtellers and Liars axioms for three types of people

Axioms for three types of people; truthtellers, liars and
 normal people.




$\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ or }\mbox{a\_truth}(\mbox{normal}(x))\qquad\mbox{cnf}(\mbox{person\_is\_one\_type},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{truthteller}(x)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{normal}(x))\qquad\mbox{cnf}(\mbox{not\_truthteller\_and\_normal},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{truthteller}(x)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{liar}(x))\qquad\mbox{cnf}(\mbox{not\_truthteller\_and\_liar},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{liar}(x)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{normal}(x))\qquad\mbox{cnf}(\mbox{not\_liar\_and\_normal},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(\mbox{says}(x,y)) ) \ \Rightarrow \ \mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{truthtellers\_tell\_truth},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(\mbox{says}(x,y)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{liars\_lie},\mbox{axiom})$

$( \mbox{a\_truth}(x)\mbox{ and }\mbox{a\_truth}(\mbox{says}(y,x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{normal}(y)) )\qquad\mbox{cnf}(\mbox{truthtellers\_and\_normal\_tell\_truth},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{says}(y,x)) \ \Rightarrow \ ( \mbox{a\_truth}(x)\mbox{ or }\mbox{a\_truth}(\mbox{liar}(y))\mbox{ or }\mbox{a\_truth}(\mbox{normal}(y)) )\qquad\mbox{cnf}(\mbox{liars\_and\_normal\_lie},\mbox{axiom})$

\medskip\textbf{PUZ001-0.ax} Mars and Venus axioms






$\mbox{from\_mars}(x)\mbox{ or }\mbox{from\_venus}(x)\qquad\mbox{cnf}(\mbox{from\_mars\_or\_venus},\mbox{axiom})$

$\mbox{from\_mars}(x) \ \Rightarrow \ \neg\,\mbox{from\_venus}(x)\qquad\mbox{cnf}(\mbox{not\_from\_mars\_and\_venus},\mbox{axiom})$

$\mbox{male}(x)\mbox{ or }\mbox{female}(x)\qquad\mbox{cnf}(\mbox{male\_or\_female},\mbox{axiom})$

$\mbox{male}(x) \ \Rightarrow \ \neg\,\mbox{female}(x)\qquad\mbox{cnf}(\mbox{not\_male\_and\_female},\mbox{axiom})$

$\mbox{truthteller}(x)\mbox{ or }\mbox{liar}(x)\qquad\mbox{cnf}(\mbox{truthteller\_or\_liar},\mbox{axiom})$

$\mbox{truthteller}(x) \ \Rightarrow \ \neg\,\mbox{liar}(x)\qquad\mbox{cnf}(\mbox{not\_truthteller\_and\_liar},\mbox{axiom})$

$( \mbox{says}(x,y)\mbox{ and }\mbox{a\_truth}(y) ) \ \Rightarrow \ \mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_not},\mbox{axiom})$

$\mbox{says}(x,y) \ \Rightarrow \ y = \mbox{statement\_by}(x)\qquad\mbox{cnf}(\mbox{people\_say\_their\_statements},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{statement\_by}(x)) \ \Rightarrow \ \mbox{truthteller}(x)\qquad\mbox{cnf}(\mbox{true\_statements\_made\_by\_truthtellers},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{statement\_by}(x))\mbox{ or }\mbox{liar}(x)\qquad\mbox{cnf}(\mbox{false\_statements\_made\_by\_liars},\mbox{axiom})$

$( \mbox{from\_venus}(x)\mbox{ and }\mbox{female}(x) ) \ \Rightarrow \ \mbox{truthteller}(x)\qquad\mbox{cnf}(\mbox{venusian\_female\_are\_truthtellers},\mbox{axiom})$

$( \mbox{from\_venus}(x)\mbox{ and }\mbox{male}(x) ) \ \Rightarrow \ \mbox{liar}(x)\qquad\mbox{cnf}(\mbox{venusian\_males\_are\_liars},\mbox{axiom})$

$( \mbox{from\_mars}(x)\mbox{ and }\mbox{male}(x) ) \ \Rightarrow \ \mbox{truthteller}(x)\qquad\mbox{cnf}(\mbox{marsian\_males\_are\_truthtellers},\mbox{axiom})$

$( \mbox{from\_mars}(x)\mbox{ and }\mbox{female}(x) ) \ \Rightarrow \ \mbox{liar}(x)\qquad\mbox{cnf}(\mbox{marsian\_females\_are\_liars},\mbox{axiom})$

$( \mbox{truthteller}(x)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{truthtellers\_make\_true\_statements},\mbox{axiom})$

$( \mbox{liar}(x)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{liars\_make\_false\_statements},\mbox{axiom})$

\medskip\textbf{PUZ045-1.p} Truthtellers and Liars axioms for three types of people

Axioms for three types of people; truthtellers, liars and
 normal people.




$\mbox{include}(\mbox{'Axioms/PUZ003-0.ax'})$

\medskip\textbf{PUZ034-1.003.p} N queens problem

The problem is to place 3 queens on an 3x3 chess board, so that no
 queen can attack another.




$( \mbox{less}(\mbox{low},\mbox{high})\mbox{ and }\mbox{low} + s(n_0){=}\mbox{newLow}\mbox{ and }\mbox{range}(\mbox{newLow},\mbox{high},\mbox{restOfNumbers}) ) \ \Rightarrow \ \mbox{range}(\mbox{low},\mbox{high},\mbox{cons}(\mbox{low},\mbox{restOfNumbers}))\qquad\mbox{cnf}(\mbox{make\_list\_of\_numbers}_1,\mbox{axiom})$

$\mbox{range}(\mbox{same},\mbox{same},\mbox{cons}(\mbox{same},\mbox{empty\_list}))\qquad\mbox{cnf}(\mbox{make\_list\_of\_numbers}_2,\mbox{axiom})$

$\mbox{less}(n_0,s(x))\qquad\mbox{cnf}(\mbox{less}_1,\mbox{axiom})$

$\mbox{less}(x,y) \ \Rightarrow \ \mbox{less}(s(x),s(y))\qquad\mbox{cnf}(\mbox{less}_2,\mbox{axiom})$

$x + n_0{=}x\qquad\mbox{cnf}(\mbox{add}_0,\mbox{axiom})$

$x + y{=}z \ \Rightarrow \ x + s(y){=}s(z)\qquad\mbox{cnf}(\mbox{add},\mbox{axiom})$

$\mbox{select}(\mbox{head},\mbox{cons}(\mbox{head},\mbox{tail}),\mbox{tail})\qquad\mbox{cnf}(\mbox{select}_1,\mbox{axiom})$

$\mbox{select}(\mbox{element},\mbox{tail},\mbox{newTail}) \ \Rightarrow \ \mbox{select}(\mbox{element},\mbox{cons}(\mbox{head},\mbox{tail}),\mbox{cons}(\mbox{head},\mbox{newTail}))\qquad\mbox{cnf}(\mbox{select}_2,\mbox{axiom})$

$\neg\,\mbox{same}(s(x),n_0)\qquad\mbox{cnf}(\mbox{same\_definition}_1,\mbox{axiom})$

$\neg\,\mbox{same}(n_0,s(x))\qquad\mbox{cnf}(\mbox{same\_definition}_2,\mbox{axiom})$

$\mbox{same}(s(x),s(y)) \ \Rightarrow \ \mbox{same}(x,y)\qquad\mbox{cnf}(\mbox{same\_definition}_3,\mbox{axiom})$

$\mbox{attack}(\mbox{queen},\mbox{placedQueens}) \ \Rightarrow \ \mbox{diagonal\_attack}(\mbox{queen},s(n_0),\mbox{placedQueens})\qquad\mbox{cnf}(\mbox{attack},\mbox{axiom})$

$( \mbox{diagonal\_attack}(\mbox{queen},\mbox{queenNumber},\mbox{cons}(\mbox{placedQueen},\mbox{otherPlacedQueens}))\mbox{ and }\mbox{diagonal}_1 + \mbox{queenNumber}{=}\mbox{placedQueen}\mbox{ and }\mbox{placedQueen} + \mbox{queenNumber}{=}\mbox{diagonal}_2\mbox{ and }\mbox{queenNumber} + s(n_0){=}\mbox{nextQueenNumber} ) \ \Rightarrow \ ( \mbox{same}(\mbox{diagonal}_1,\mbox{queen})\mbox{ or }\mbox{same}(\mbox{diagonal}_2,\mbox{queen})\mbox{ or }\mbox{diagonal\_attack}(\mbox{queen},\mbox{nextQueenNumber},\mbox{otherPlacedQueens}) )\qquad\mbox{cnf}(\mbox{check\_diagonals}_1,\mbox{axiom})$

$\neg\,\mbox{diagonal\_attack}(\mbox{queen},\mbox{lastQueen},\mbox{empty\_list})\qquad\mbox{cnf}(\mbox{check\_diagonals}_2,\mbox{axiom})$

$( \mbox{select}(\mbox{aQueen},\mbox{unplacedQueens},\mbox{restOfUnplacedQueens})\mbox{ and }\mbox{do\_queens}(\mbox{restOfUnplacedQueens},\mbox{cons}(\mbox{aQueen},\mbox{safeQueens}),\mbox{placement}) ) \ \Rightarrow \ ( \mbox{do\_queens}(\mbox{unplacedQueens},\mbox{safeQueens},\mbox{placement})\mbox{ or }\mbox{attack}(\mbox{aQueen},\mbox{safeQueens}) )\qquad\mbox{cnf}(\mbox{place\_a\_queen}_1,\mbox{axiom})$

$\mbox{do\_queens}(\mbox{empty\_list},\mbox{placement},\mbox{placement})\qquad\mbox{cnf}(\mbox{place\_a\_queen}_2,\mbox{axiom})$

$( \mbox{numberOfQueens} + s(n_0){=}\mbox{low}\mbox{ and }\mbox{numberOfQueens} + \mbox{numberOfQueens}{=}\mbox{high}\mbox{ and }\mbox{range}(\mbox{low},\mbox{high},\mbox{positions})\mbox{ and }\mbox{do\_queens}(\mbox{positions},\mbox{empty\_list},\mbox{placement}) ) \ \Rightarrow \ \mbox{queens}(\mbox{numberOfQueens},\mbox{placement})\qquad\mbox{cnf}(\mbox{set\_up\_queens},\mbox{axiom})$

$\neg\,\mbox{queens}(s(s(s(n_0))),\mbox{placement})\qquad\mbox{cnf}(\mbox{place\_queens},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ031+2.p} Schubert's Steamroller

Wolves, foxes, birds, caterpillars, and snails are animals, and
 there are some of each of them. Also there are some grains, and
 grains are plants. Every animal either likes to eat all plants
 or all animals much smaller than itself that like to eat some
 plants. Caterpillars and snails are much smaller than birds,
 which are much smaller than foxes, which in turn are much
 smaller than wolves. Wolves do not like to eat foxes or grains,
 while birds like to eat caterpillars but not snails.
 Caterpillars and snails like to eat some plants. Therefore
 there is an animal that likes to eat a grain eating animal.




$\exists a{:}\;\mbox{wolf}(a)\qquad\mbox{fof}(\mbox{wolf\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{fox}(a)\qquad\mbox{fof}(\mbox{fox\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{bird}(a)\qquad\mbox{fof}(\mbox{bird\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{caterpillar}(a)\qquad\mbox{fof}(\mbox{caterpillar\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{snail}(a)\qquad\mbox{fof}(\mbox{snail\_type},\mbox{axiom})$

$\forall x{:}\;( \mbox{wolf}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_1}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{fox}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_2}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{bird}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_3}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{caterpillar}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_4}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{snail}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_4}_2,\mbox{axiom})$

$\exists a{:}\;\mbox{grain}(a)\qquad\mbox{fof}(\mbox{grain\_type},\mbox{axiom})$

$\forall x{:}\;( \mbox{grain}(x) \ \Rightarrow \ \mbox{plant}(x) )\qquad\mbox{fof}(\mbox{pel47\_6}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{animal}(x) \ \Rightarrow \ ( \forall y{:}\;( \mbox{plant}(y) \ \Rightarrow \ \mbox{eats}(x,y) )\mbox{ or }\forall y_1{:}\;( ( \mbox{animal}(y_1)\mbox{ and }\mbox{much\_smaller}(y_1,x)\mbox{ and }\exists z{:}\;( \mbox{plant}(z)\mbox{ and }\mbox{eats}(y_1,z) ) ) \ \Rightarrow \ \mbox{eats}(x,y_1) ) ) )\qquad\mbox{fof}(\mbox{pel47}_7,\mbox{axiom})$

$\forall y,x{:}\;( ( \mbox{bird}(y)\mbox{ and }\mbox{snail}(x) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_8,\mbox{axiom})$

$\forall y,x{:}\;( ( \mbox{bird}(y)\mbox{ and }\mbox{caterpillar}(x) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47\_8a},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{fox}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_9,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{fox}(x)\mbox{ and }\mbox{wolf}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{10},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{wolf}(x)\mbox{ and }\mbox{fox}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{11},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{wolf}(x)\mbox{ and }\mbox{grain}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47\_11a},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{caterpillar}(y) ) \ \Rightarrow \ \mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{12},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{snail}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{13},\mbox{axiom})$

$\forall x{:}\;( \mbox{caterpillar}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{plant}(y)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel47}_{14},\mbox{axiom})$

$\forall x{:}\;( \mbox{snail}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{plant}(y)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel47\_14a},\mbox{axiom})$

$\exists x,y{:}\;( \mbox{animal}(x)\mbox{ and }\mbox{animal}(y)\mbox{ and }\exists z{:}\;( \mbox{grain}(z)\mbox{ and }\mbox{eats}(y,z)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel}_{47},\mbox{conjecture})$

\medskip\textbf{PUZ100$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






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

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_E}_2@ \mbox{xx}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cCKB\_L}_{32000},\mbox{conjecture})$

\medskip\textbf{PUZ047-1.p} Taking the wolf, goat, and cabbage across river






$p(\mbox{south},\mbox{south},\mbox{south},\mbox{south},\mbox{start})\qquad\mbox{cnf}(\mbox{thm100}_1,\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{north},\mbox{south},\mbox{north},a) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{go\_alone}(a))\qquad\mbox{cnf}(\mbox{thm100}_2,\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},a) \ \Rightarrow \ p(\mbox{south},\mbox{north},\mbox{south},\mbox{north},\mbox{go\_alone}(a))\qquad\mbox{cnf}(\mbox{thm100}_3,\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},a) \ \Rightarrow \ p(\mbox{north},\mbox{south},\mbox{north},\mbox{south},\mbox{go\_alone}(a))\qquad\mbox{cnf}(\mbox{thm100}_4,\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{south},\mbox{north},\mbox{south},a) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{go\_alone}(a))\qquad\mbox{cnf}(\mbox{thm100}_5,\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{south},\mbox{south},\mbox{north},a) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{take\_wolf}(a))\qquad\mbox{cnf}(\mbox{thm100}_6,\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},a) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{south},\mbox{north},\mbox{take\_wolf}(a))\qquad\mbox{cnf}(\mbox{thm100}_7,\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},a) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{north},\mbox{south},\mbox{take\_wolf}(a))\qquad\mbox{cnf}(\mbox{thm100}_8,\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{north},\mbox{north},\mbox{south},a) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{take\_wolf}(a))\qquad\mbox{cnf}(\mbox{thm100}_9,\mbox{negated\_conjecture})$

$p(\mbox{south},a,\mbox{south},b,c) \ \Rightarrow \ p(\mbox{north},a,\mbox{north},b,\mbox{take\_goat}(c))\qquad\mbox{cnf}(\mbox{thm100}_{10},\mbox{negated\_conjecture})$

$p(\mbox{north},a,\mbox{north},b,c) \ \Rightarrow \ p(\mbox{south},a,\mbox{south},b,\mbox{take\_goat}(c))\qquad\mbox{cnf}(\mbox{thm100}_{11},\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{north},\mbox{south},\mbox{south},a) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{take\_cabbage}(a))\qquad\mbox{cnf}(\mbox{thm100}_{12},\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},a) \ \Rightarrow \ p(\mbox{south},\mbox{north},\mbox{south},\mbox{south},\mbox{take\_cabbage}(a))\qquad\mbox{cnf}(\mbox{thm100}_{13},\mbox{negated\_conjecture})$

$p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},a) \ \Rightarrow \ p(\mbox{north},\mbox{south},\mbox{north},\mbox{north},\mbox{take\_cabbage}(a))\qquad\mbox{cnf}(\mbox{thm100}_{14},\mbox{negated\_conjecture})$

$p(\mbox{north},\mbox{south},\mbox{north},\mbox{north},a) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{take\_cabbage}(a))\qquad\mbox{cnf}(\mbox{thm100}_{15},\mbox{negated\_conjecture})$

$\neg\,p(\mbox{north},\mbox{north},\mbox{north},\mbox{north},a)\qquad\mbox{cnf}(\mbox{thm100}_{16},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ054-1.p} Take black and white balls from a bag

Start with a bag with 10 white balls and 9 black balls.
 Take out two balls: if they have the same color, put a black
 ball back; if they have a different color, put a white ball back.
 The last ball left cannot be white.




$p(s(s(s(s(s(s(s(s(s(s(n_0)))))))))),s(s(s(s(s(s(s(s(s(n_0))))))))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{axiom})$

$p(s(s(x)),y) \ \Rightarrow \ p(x,s(y))\qquad\mbox{cnf}(\mbox{two\_whites\_out\_one\_black\_in},\mbox{axiom})$

$p(x,s(s(y))) \ \Rightarrow \ p(x,s(y))\qquad\mbox{cnf}(\mbox{two\_blacks\_out\_one\_black\_in},\mbox{axiom})$

$p(s(x),s(y)) \ \Rightarrow \ p(s(x),y)\qquad\mbox{cnf}(\mbox{two\_different\_balls\_out\_one\_white\_in},\mbox{axiom})$

$\neg\,p(s(n_0),n_0)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ067+1.p} Sudoku 8618






$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_4) = n_4\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_5) = n_7\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_7) = n_1\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_2) = n_5\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_4) = n_2\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_3,n_2) = n_3\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_3,n_9) = n_8\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_1) = n_6\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_1\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_4,n_7) = n_7\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_3) = n_4\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_5,n_6) = n_3\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_6,n_8) = n_5\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_7,n_8) = n_3\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_1) = n_1\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_9) = n_4\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ099$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

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

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\neg\,\mbox{cCKB\_E}_2@ c_1@ c_2\qquad\mbox{thf}(\mbox{cCKB\_L}_{38000},\mbox{conjecture})$

\medskip\textbf{PUZ029-1.p} The pigs and balloons puzzle

1) All, who neither dance on tight ropes nor eat penny-buns,
 are old.
 2) Pigs, that are liable to giddiness, are treated with respect.
 3) A wise balloonist takes an umbrella with him.
 4) No one ought to lunch in public who looks ridiculous and eats
 penny-buns.
 5) Young creatures, who go up in balloons, are liable to
 giddiness.
 6) Fat creatures, who look ridiculous, may lunch in public,
 provided that they do not dance on tight ropes.
 7) No wise creatures dance on tight ropes, if liable to
 giddiness.
 8) A pig looks ridiculous, carrying an umbrella.
 9) All, who do not dance on tight ropes, and who are treated
 with respect are fat.
 Show that no wise young pigs go up in balloons.




$\mbox{dances\_on\_tightropes}(x)\mbox{ or }\mbox{eats\_pennybuns}(x)\mbox{ or }\mbox{old}(x)\qquad\mbox{cnf}(\mbox{boring\_old\_people},\mbox{axiom})$

$( \mbox{pig}(x)\mbox{ and }\mbox{liable\_to\_giddiness}(x) ) \ \Rightarrow \ \mbox{treated\_with\_respect}(x)\qquad\mbox{cnf}(\mbox{giddy\_pigs\_reated\_with\_respect},\mbox{axiom})$

$( \mbox{wise}(x)\mbox{ and }\mbox{balloonist}(x) ) \ \Rightarrow \ \mbox{has\_umbrella}(x)\qquad\mbox{cnf}(\mbox{wise\_ballonists\_have\_umbrellas},\mbox{axiom})$

$( \mbox{looks\_ridiculous}(x)\mbox{ and }\mbox{eats\_pennybuns}(x) ) \ \Rightarrow \ \neg\,\mbox{eats\_lunch\_in\_public}(x)\qquad\mbox{cnf}(\mbox{dont\_look\_ridiculous\_eating\_buns\_in\_public},\mbox{axiom})$

$( \mbox{balloonist}(x)\mbox{ and }\mbox{young}(x) ) \ \Rightarrow \ \mbox{liable\_to\_giddiness}(x)\qquad\mbox{cnf}(\mbox{young\_balloonists\_get\_giddy},\mbox{axiom})$

$( \mbox{fat}(x)\mbox{ and }\mbox{looks\_ridiculous}(x) ) \ \Rightarrow \ ( \mbox{dances\_on\_tightropes}(x)\mbox{ or }\mbox{eats\_lunch\_in\_public}(x) )\qquad\mbox{cnf}(\mbox{fat\_ridiculous\_off\_tightrope\_eat\_in\_public},\mbox{axiom})$

$( \mbox{liable\_to\_giddiness}(x)\mbox{ and }\mbox{wise}(x) ) \ \Rightarrow \ \neg\,\mbox{dances\_on\_tightropes}(x)\qquad\mbox{cnf}(\mbox{wise\_giddy\_dont\_dance\_on\_tightrope},\mbox{axiom})$

$( \mbox{pig}(x)\mbox{ and }\mbox{has\_umbrella}(x) ) \ \Rightarrow \ \mbox{looks\_ridiculous}(x)\qquad\mbox{cnf}(\mbox{pigs\_look\_ridiculous\_with\_umbrellas},\mbox{axiom})$

$\mbox{treated\_with\_respect}(x) \ \Rightarrow \ ( \mbox{dances\_on\_tightropes}(x)\mbox{ or }\mbox{fat}(x) )\qquad\mbox{cnf}(\mbox{non\_dancers\_who\_are\_respected\_are\_fat},\mbox{axiom})$

$\mbox{young}(x)\mbox{ or }\mbox{old}(x)\qquad\mbox{cnf}(\mbox{young\_or\_old},\mbox{axiom})$

$\mbox{young}(x) \ \Rightarrow \ \neg\,\mbox{old}(x)\qquad\mbox{cnf}(\mbox{not\_young\_and\_old},\mbox{axiom})$

$\mbox{wise}(\mbox{piggy})\qquad\mbox{cnf}(\mbox{piggy\_is\_wise},\mbox{hypothesis})$

$\mbox{young}(\mbox{piggy})\qquad\mbox{cnf}(\mbox{piggy\_is\_young},\mbox{hypothesis})$

$\mbox{pig}(\mbox{piggy})\qquad\mbox{cnf}(\mbox{piggy\_is\_a\_pig},\mbox{hypothesis})$

$\mbox{balloonist}(\mbox{piggy})\qquad\mbox{cnf}(\mbox{prove\_piggy\_is\_no\_balloonist},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ035-2.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{person}(x) \ \Rightarrow \ ( \mbox{isa}(x,\mbox{knight})\mbox{ or }\mbox{isa}(x,\mbox{knave}) )\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knight}) \ \Rightarrow \ \neg\,\mbox{isa}(x,\mbox{knave})\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knight}) \ \Rightarrow \ \mbox{tell\_the\_truth}(x)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knave}) \ \Rightarrow \ \mbox{lies}(x)\qquad\mbox{cnf}(\mbox{knaves\_make\_false\_statements},\mbox{axiom})$

$\mbox{tell\_the\_truth}(x) \ \Rightarrow \ \neg\,\mbox{lies}(x)\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false},\mbox{axiom})$

$\mbox{person}(x) \ \Rightarrow \ ( \mbox{tell\_the\_truth}(x)\mbox{ or }\mbox{lies}(x) )\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false}_2,\mbox{axiom})$

$( \mbox{response}(\mbox{yes})\mbox{ and }\mbox{tell\_the\_truth}(\mbox{asked}) ) \ \Rightarrow \ ( \mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{true\_one\_is\_a\_knight},\mbox{axiom})$

$( \mbox{response}(\mbox{no})\mbox{ and }\mbox{lies}(\mbox{asked}) ) \ \Rightarrow \ ( \mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{lie\_one\_is\_a\_knight},\mbox{axiom})$

$( \mbox{response}(\mbox{yes})\mbox{ and }\mbox{lies}(\mbox{asked}) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{anyone},\mbox{knight})\qquad\mbox{cnf}(\mbox{knight\_answers}_1,\mbox{axiom})$

$( \mbox{response}(\mbox{no})\mbox{ and }\mbox{tell\_the\_truth}(\mbox{asked}) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{anyone},\mbox{knight})\qquad\mbox{cnf}(\mbox{knight\_answers}_2,\mbox{axiom})$

$( \mbox{tell\_the\_truth}(\mbox{asked})\mbox{ and }\mbox{isa}(\mbox{anyone},\mbox{knight}) ) \ \Rightarrow \ \mbox{response}(\mbox{yes})\qquad\mbox{cnf}(\mbox{knight\_answers}_3,\mbox{axiom})$

$\mbox{tell\_the\_truth}(\mbox{asked}) \ \Rightarrow \ ( \mbox{response}(\mbox{no})\mbox{ or }\mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{truthful\_answer},\mbox{axiom})$

$\mbox{response}(\mbox{yes})\mbox{ or }\mbox{response}(\mbox{no})\qquad\mbox{cnf}(\mbox{two\_answers},\mbox{axiom})$

$\mbox{person}(\mbox{asked})\qquad\mbox{cnf}(\mbox{asked\_person},\mbox{axiom})$

$\mbox{person}(\mbox{other})\qquad\mbox{cnf}(\mbox{other\_person},\mbox{axiom})$

$( \mbox{response}(a)\mbox{ and }\mbox{isa}(\mbox{asked},x) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{other},y)\qquad\mbox{cnf}(\mbox{prove\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ106$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cCKB\_E}_2@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{37000},\mbox{conjecture})$

\medskip\textbf{PUZ041-1.p} Quo vadis 4

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is the true goal from the puzzle.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(s(s(s(o))),s(o)),v_1,v_2,v_3,v_4,h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ128+2.p} Iokaste patricide triangle

Iokaste is a parent of Oedipus. Iokaste is a parent of Polyneikes.
 Oedipus is a parent of Polyneikes. Polyneikes is a parent of 
 Thersandros. Oedipus is a patricide. Thersandros is not a 
 patricide. Therefore, Iokaste is a parent of a patricide that is 
 a parent of somebody who is not a patricide.




$\exists a,b,c,d,e{:}\;( \mbox{parent}(a)\mbox{ and }\mbox{relation}(a,\mbox{of},\mbox{'Oedipus'})\mbox{ and }\mbox{'Iokaste'} = a\mbox{ and }\mbox{parent}(b)\mbox{ and }\mbox{relation}(b,\mbox{of},\mbox{'Polyneikes'})\mbox{ and }\mbox{'Iokaste'} = b\mbox{ and }\mbox{parent}(c)\mbox{ and }\mbox{relation}(c,\mbox{of},\mbox{'Polyneikes'})\mbox{ and }\mbox{'Oedipus'} = c\mbox{ and }\mbox{parent}(d)\mbox{ and }\mbox{relation}(d,\mbox{of},\mbox{'Thersandros'})\mbox{ and }\mbox{'Polyneikes'} = d\mbox{ and }\mbox{patricide}(e)\mbox{ and }\mbox{'Oedipus'} = e\mbox{ and }\neg\,\exists f{:}\;( \mbox{patricide}(f)\mbox{ and }\mbox{'Thersandros'} = f ) )\qquad\mbox{fof}(\mbox{background},\mbox{axiom})$

$\exists a,b,c,d{:}\;( \mbox{parent}(a)\mbox{ and }\mbox{patricide}(b)\mbox{ and }\mbox{parent}(c)\mbox{ and }\mbox{\$true}\mbox{ and }\neg\,\exists e{:}\;( \mbox{patricide}(e)\mbox{ and }d = e )\mbox{ and }\mbox{relation}(c,\mbox{of},d)\mbox{ and }b = c\mbox{ and }\mbox{relation}(a,\mbox{of},b)\mbox{ and }\mbox{'Iokaste'} = a )\qquad\mbox{fof}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{PUZ108$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






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

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_E}_2@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cCKB\_E}_2@ ( s@ \mbox{xx} )@ ( s@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{35000},\mbox{conjecture})$

\medskip\textbf{PUZ027-1.p} Knights and Knaves \#42

There  is an island with exactly three types of people -
 truthtellers who always tell the truth, and liars who always
 lie, and normals who sometimes tell the truth and sometimes
 lie. Liars are said to be of the lowest rank, normals are
 middle rank, and truthtellers of the highest rank. Two people
 A and B on the island make the following statements. A: I am
 of lower rank than B. B: That's not true! What are the ranks
 of A and B, and which of the two statements are true? Answer:
 A is a normal and B is a normal.




$\mbox{include}(\mbox{'Axioms/PUZ003-0.ax'})$

$\mbox{a\_truth}(\mbox{not\_lower}(x,x))\qquad\mbox{cnf}(\mbox{not\_lower\_is\_irreflexive},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{not\_lower}(x,y)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{lower}(x,y))\qquad\mbox{cnf}(\mbox{not\_not\_lower\_and\_lower},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{not\_lower}(x,y))\mbox{ or }\mbox{a\_truth}(\mbox{lower}(x,y))\qquad\mbox{cnf}(\mbox{not\_lower\_or\_lower},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{normal}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y)) )\qquad\mbox{cnf}(\mbox{liars\_lowest},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(x)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{truthteller}(y))\qquad\mbox{cnf}(\mbox{truthtellers\_highest},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{lower}(x,y)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(x))\qquad\mbox{cnf}(\mbox{truthtellers\_lower\_than\_no\_one},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{normal}(x))\mbox{ or }\mbox{a\_truth}(\mbox{liar}(x)) )\qquad\mbox{cnf}(\mbox{normal\_and\_liars\_lower\_than\_truthtellers},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(y)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{liar}(x))\qquad\mbox{cnf}(\mbox{liars\_lower\_than\_normal},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{lower}(x,y)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{liar}(y))\qquad\mbox{cnf}(\mbox{no\_one\_lower\_than\_liars},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{not\_lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{lower}(y,x)) )\qquad\mbox{cnf}(\mbox{not\_lower\_than\_truthteller},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{not\_lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{liar}(y))\mbox{ or }\mbox{a\_truth}(\mbox{lower}(y,x)) )\qquad\mbox{cnf}(\mbox{not\_lower\_than\_liar},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{not\_lower}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{normal}(y))\mbox{ or }\mbox{a\_truth}(\mbox{lower}(y,x)) )\qquad\mbox{cnf}(\mbox{not\_lower\_than\_normal},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{lower}(a,b)))\qquad\mbox{cnf}(\mbox{a\_says\_a\_lower\_than\_b},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(b,\mbox{not\_lower}(a,b)))\qquad\mbox{cnf}(\mbox{b\_says\_a\_not\_lower\_than\_b},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{truthteller}(a))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_and\_b\_truthteller})\qquad\mbox{cnf}(\mbox{a\_and\_b\_truthteller},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{truthteller}(a))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_truthteller\_b\_normal})\qquad\mbox{cnf}(\mbox{a\_truthteller\_b\_normal},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{truthteller}(a))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_truthteller\_b\_liar})\qquad\mbox{cnf}(\mbox{a\_truthteller\_b\_liar},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{normal}(a))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_normal\_b\_truthteller})\qquad\mbox{cnf}(\mbox{a\_normal\_b\_truthteller},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{normal}(a))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_and\_b\_normal})\qquad\mbox{cnf}(\mbox{a\_and\_b\_normal},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{normal}(a))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_normal\_b\_liar})\qquad\mbox{cnf}(\mbox{a\_normal\_b\_liar},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(a))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_liar\_b\_truthteller})\qquad\mbox{cnf}(\mbox{a\_liar\_b\_truthteller},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(a))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_liar\_b\_normal})\qquad\mbox{cnf}(\mbox{a\_liar\_b\_normal},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(a))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(b)) ) \ \Rightarrow \ \mbox{answer}(\mbox{a\_and\_b\_liar})\qquad\mbox{cnf}(\mbox{a\_and\_b\_liar},\mbox{hypothesis})$

$\neg\,\mbox{answer}(\mbox{answer})\qquad\mbox{cnf}(\mbox{prove\_there\_is\_an\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ035-1.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{person}(x) \ \Rightarrow \ ( \mbox{isa}(x,\mbox{knight})\mbox{ or }\mbox{isa}(x,\mbox{knave}) )\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knight}) \ \Rightarrow \ \neg\,\mbox{isa}(x,\mbox{knave})\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knight}) \ \Rightarrow \ \mbox{tell\_the\_truth}(x)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements},\mbox{axiom})$

$\mbox{isa}(x,\mbox{knave}) \ \Rightarrow \ \mbox{lies}(x)\qquad\mbox{cnf}(\mbox{knaves\_make\_false\_statements},\mbox{axiom})$

$\mbox{tell\_the\_truth}(x) \ \Rightarrow \ \neg\,\mbox{lies}(x)\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false},\mbox{axiom})$

$( \mbox{response}(\mbox{yes})\mbox{ and }\mbox{tell\_the\_truth}(\mbox{asked}) ) \ \Rightarrow \ ( \mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{true\_one\_is\_a\_knight},\mbox{axiom})$

$( \mbox{response}(\mbox{no})\mbox{ and }\mbox{lies}(\mbox{asked}) ) \ \Rightarrow \ ( \mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{lie\_one\_is\_a\_knight},\mbox{axiom})$

$( \mbox{response}(\mbox{yes})\mbox{ and }\mbox{lies}(\mbox{asked}) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{anyone},\mbox{knight})\qquad\mbox{cnf}(\mbox{knight\_answers}_1,\mbox{axiom})$

$( \mbox{response}(\mbox{no})\mbox{ and }\mbox{tell\_the\_truth}(\mbox{asked}) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{anyone},\mbox{knight})\qquad\mbox{cnf}(\mbox{knight\_answers}_2,\mbox{axiom})$

$( \mbox{tell\_the\_truth}(\mbox{asked})\mbox{ and }\mbox{isa}(\mbox{anyone},\mbox{knight}) ) \ \Rightarrow \ \mbox{response}(\mbox{yes})\qquad\mbox{cnf}(\mbox{knight\_answers}_3,\mbox{axiom})$

$\mbox{tell\_the\_truth}(\mbox{asked}) \ \Rightarrow \ ( \mbox{response}(\mbox{no})\mbox{ or }\mbox{isa}(\mbox{asked},\mbox{knight})\mbox{ or }\mbox{isa}(\mbox{other},\mbox{knight}) )\qquad\mbox{cnf}(\mbox{truthful\_answer},\mbox{axiom})$

$\mbox{response}(\mbox{yes})\mbox{ or }\mbox{response}(\mbox{no})\qquad\mbox{cnf}(\mbox{two\_answers},\mbox{axiom})$

$\mbox{person}(\mbox{asked})\qquad\mbox{cnf}(\mbox{asked\_person},\mbox{axiom})$

$\mbox{person}(\mbox{other})\qquad\mbox{cnf}(\mbox{other\_person},\mbox{axiom})$

$( \mbox{response}(a)\mbox{ and }\mbox{isa}(\mbox{asked},x) ) \ \Rightarrow \ \neg\,\mbox{isa}(\mbox{other},y)\qquad\mbox{cnf}(\mbox{prove\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ092$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\forall \mbox{aCRES}{:}\;\mbox{\$i},\mbox{mRSACRES}{:}\;\mbox{\$i},\mbox{bARRY}{:}\;\mbox{\$i},\mbox{mRSBARRY}{:}\;\mbox{\$i},\mbox{cOLE}{:}\;\mbox{\$i},\mbox{mRSCOLE}{:}\;\mbox{\$i},\mbox{dIX}{:}\;\mbox{\$i},\mbox{mRSDIX}{:}\;\mbox{\$i},\mbox{eDEN}{:}\;\mbox{\$i},\mbox{mRSEDEN}{:}\;\mbox{\$i},\mbox{hALL}{:}\;\mbox{\$i},\mbox{mRSHALL}{:}\;\mbox{\$i}{:}\;( ( ( ( \mbox{aCRES} = \mbox{mRSACRES}\mbox{ and }\mbox{bARRY} = \mbox{mRSBARRY}\mbox{ and }\mbox{eDEN} = \mbox{mRSHALL} ) \ \Rightarrow \ \mbox{cOLE} = \mbox{mRSDIX} )\mbox{ and }( ( \mbox{aCRES} = \mbox{mRSACRES}\mbox{ and }\mbox{hALL} = \mbox{mRSHALL}\mbox{ and }\mbox{bARRY} = \mbox{mRSCOLE} ) \ \Rightarrow \ \mbox{dIX}\ne \mbox{mRSEDEN} )\mbox{ and }( ( \mbox{cOLE} = \mbox{mRSCOLE}\mbox{ and }\mbox{dIX} = \mbox{mRSDIX}\mbox{ and }\mbox{dIX} = \mbox{cOLE}\mbox{ and }\mbox{aCRES}\ne \mbox{mRSBARRY} ) \ \Rightarrow \ \mbox{eDEN}\ne \mbox{mRSHALL} )\mbox{ and }( ( \mbox{aCRES} = \mbox{mRSACRES}\mbox{ and }\mbox{dIX} = \mbox{mRSDIX}\mbox{ and }\mbox{bARRY}\ne \mbox{mRSCOLE} ) \ \Rightarrow \ \mbox{eDEN} = \mbox{mRSHALL} )\mbox{ and }( ( \mbox{eDEN} = \mbox{mRSEDEN}\mbox{ and }\mbox{hALL} = \mbox{mRSHALL}\mbox{ and }\mbox{cOLE} = \mbox{mRSDIX} ) \ \Rightarrow \ \mbox{aCRES}\ne \mbox{mRSBARRY} )\mbox{ and }( ( \mbox{bARRY} = \mbox{mRSBARRY}\mbox{ and }\mbox{cOLE} = \mbox{mRSCOLE}\mbox{ and }\mbox{cOLE} = \mbox{bARRY}\mbox{ and }\mbox{eDEN}\ne \mbox{mRSHALL} ) \ \Rightarrow \ \mbox{dIX} = \mbox{mRSEDEN} ) ) \ \Rightarrow \ ( \mbox{aCRES}\ne \mbox{mRSACRES}\mbox{ or }\mbox{bARRY}\ne \mbox{mRSBARRY}\mbox{ or }\mbox{cOLE}\ne \mbox{mRSCOLE}\mbox{ or }\mbox{dIX}\ne \mbox{mRSDIX}\mbox{ or }\mbox{eDEN}\ne \mbox{mRSEDEN}\mbox{ or }\mbox{hALL}\ne \mbox{mRSHALL} ) )\qquad\mbox{thf}(\mbox{cSIXFRIENDS\_EASIER},\mbox{conjecture})$

\medskip\textbf{PUZ102$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






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

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_E}_2@ \mbox{xx}@ ( s@ ( s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{33000},\mbox{conjecture})$

\medskip\textbf{PUZ111$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

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

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

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

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

$\mbox{cCKB\_EVEN} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = ( s@ c_1 )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ c_1 ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_EVEN\_def},\mbox{definition})$

$\mbox{cCKB\_ODD} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = c_1\mbox{ or }\mbox{xx} = ( s@ ( s@ c_1 ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_ODD\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_EVEN}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cCKB\_EVEN}@ ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{cCKB\_ODD}@ ( \mbox{cS}@ \mbox{cX} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{10000},\mbox{conjecture})$

\medskip\textbf{PUZ110$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cCKB6\_BLACK}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk} ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{8000},\mbox{conjecture})$

\medskip\textbf{PUZ057-1.p} Show the Hanoi problem is not solvable anymore

Move blocks from pin to pin, until a goal configuration
 is reached. Some rules are removed: one can only move a block
 to a pin next to it; one cannot place the small block on
 the large block.




$\mbox{state}(\mbox{nextto}(\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_1)))),\mbox{nextto}(\mbox{pin}(n_2),\mbox{pin}(n_3))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{axiom})$

$\neg\,\mbox{state}(\mbox{nextto}(\mbox{pin}(n_1),\mbox{nextto}(\mbox{pin}(n_2),\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_3)))))))\qquad\mbox{cnf}(\mbox{final\_state},\mbox{negated\_conjecture})$

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

$( \mbox{fits}(x,b)\mbox{ and }\mbox{fits}(x,a) ) \ \Rightarrow \ \mbox{nextto}(\mbox{on}(x,a),b) = \mbox{nextto}(a,\mbox{on}(x,b))\qquad\mbox{cnf}(\mbox{a\_move},\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_1,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_2,\mbox{axiom})$

$\mbox{fits}(\mbox{large},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_3,\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{on}(\mbox{medium},a))\qquad\mbox{cnf}(\mbox{fits}_4,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{on}(\mbox{large},a))\qquad\mbox{cnf}(\mbox{fits}_5,\mbox{axiom})$

\medskip\textbf{PUZ031+3.p} Schubert's Steamroller

Wolves, foxes, birds, caterpillars, and snails are animals, and
 there are some of each of them. Also there are some grains, and
 grains are plants. Every animal either likes to eat all plants
 or all animals much smaller than itself that like to eat some
 plants. Caterpillars and snails are much smaller than birds,
 which are much smaller than foxes, which in turn are much
 smaller than wolves. Wolves do not like to eat foxes or grains,
 while birds like to eat caterpillars but not snails.
 Caterpillars and snails like to eat some plants. Therefore
 there is an animal that likes to eat a grain eating animal.




$\exists a{:}\;\mbox{edible}(a)\qquad\mbox{fof}(\mbox{edible\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{animal}(a)\qquad\mbox{fof}(\mbox{animal\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{animal}(a) \ \Rightarrow \ \mbox{edible}(a) )\qquad\mbox{fof}(\mbox{animal\_is\_edible},\mbox{axiom})$

$\exists a{:}\;\mbox{wolf}(a)\qquad\mbox{fof}(\mbox{wolf\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{wolf}(a) \ \Rightarrow \ \mbox{animal}(a) )\qquad\mbox{fof}(\mbox{wolf\_is\_animal},\mbox{axiom})$

$\exists a{:}\;\mbox{fox}(a)\qquad\mbox{fof}(\mbox{fox\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{fox}(a) \ \Rightarrow \ \mbox{animal}(a) )\qquad\mbox{fof}(\mbox{fox\_is\_animal},\mbox{axiom})$

$\exists a{:}\;\mbox{bird}(a)\qquad\mbox{fof}(\mbox{bird\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{bird}(a) \ \Rightarrow \ \mbox{animal}(a) )\qquad\mbox{fof}(\mbox{bird\_is\_animal},\mbox{axiom})$

$\exists a{:}\;\mbox{caterpillar}(a)\qquad\mbox{fof}(\mbox{caterpillar\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{caterpillar}(a) \ \Rightarrow \ \mbox{animal}(a) )\qquad\mbox{fof}(\mbox{caterpillar\_is\_animal},\mbox{axiom})$

$\exists a{:}\;\mbox{snail}(a)\qquad\mbox{fof}(\mbox{snail\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{snail}(a) \ \Rightarrow \ \mbox{animal}(a) )\qquad\mbox{fof}(\mbox{snail\_is\_animal},\mbox{axiom})$

$\exists a{:}\;\mbox{plant}(a)\qquad\mbox{fof}(\mbox{plant\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{plant}(a) \ \Rightarrow \ \mbox{edible}(a) )\qquad\mbox{fof}(\mbox{plant\_is\_edible},\mbox{axiom})$

$\exists a{:}\;\mbox{grain}(a)\qquad\mbox{fof}(\mbox{grain\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{grain}(a) \ \Rightarrow \ \mbox{plant}(a) )\qquad\mbox{fof}(\mbox{grain\_is\_plant},\mbox{axiom})$

$\forall x{:}\;( \mbox{animal}(x) \ \Rightarrow \ ( \forall y{:}\;( \mbox{plant}(y) \ \Rightarrow \ \mbox{eats}(x,y) )\mbox{ or }\forall y_1{:}\;( \mbox{animal}(y_1) \ \Rightarrow \ ( ( \mbox{much\_smaller}(y_1,x)\mbox{ and }\exists z{:}\;( \mbox{plant}(z)\mbox{ and }\mbox{eats}(y_1,z) ) ) \ \Rightarrow \ \mbox{eats}(x,y_1) ) ) ) )\qquad\mbox{fof}(\mbox{pel47}_7,\mbox{axiom})$

$\forall y,x{:}\;( ( \mbox{bird}(y)\mbox{ and }\mbox{snail}(x) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_8,\mbox{axiom})$

$\forall y,x{:}\;( ( \mbox{bird}(y)\mbox{ and }\mbox{caterpillar}(x) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47\_8a},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{fox}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_9,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{fox}(x)\mbox{ and }\mbox{wolf}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{10},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{wolf}(x)\mbox{ and }\mbox{fox}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{11},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{wolf}(x)\mbox{ and }\mbox{grain}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47\_11a},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{caterpillar}(y) ) \ \Rightarrow \ \mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{12},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{snail}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{13},\mbox{axiom})$

$\forall x{:}\;( \mbox{caterpillar}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{plant}(y)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel47}_{14},\mbox{axiom})$

$\forall x{:}\;( \mbox{snail}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{plant}(y)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel47\_14a},\mbox{axiom})$

$\exists x,y,z{:}\;( \mbox{animal}(x)\mbox{ and }\mbox{animal}(y)\mbox{ and }\mbox{grain}(z)\mbox{ and }\mbox{eats}(y,z)\mbox{ and }\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel}_{47},\mbox{conjecture})$

\medskip\textbf{PUZ139\_1.p} Caramel vanilla coffee helps people stay awake






$\mbox{beverage}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{beverage\_type},\mbox{type})$

$\mbox{syrup}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{syrup\_type},\mbox{type})$

$\mbox{coffee}{:}\;\mbox{beverage}\qquad\mbox{tff}(\mbox{coffee\_type},\mbox{type})$

$\mbox{vanilla\_syrup}{:}\;\mbox{syrup}\qquad\mbox{tff}(\mbox{vanilla\_syrup\_type},\mbox{type})$

$\mbox{caramel\_syrup}{:}\;\mbox{syrup}\qquad\mbox{tff}(\mbox{caramel\_syrup\_type},\mbox{type})$

$\mbox{mixture}{:}\;\forall> \mbox{beverageOrSyrup}{:}\;\mbox{\$tType}{:}\;( \mbox{beverageOrSyrup}\times \mbox{syrup} )\to \mbox{beverageOrSyrup}\qquad\mbox{tff}(\mbox{mixture\_type},\mbox{type})$

$\mbox{help\_people\_stay\_awake}{:}\;\mbox{beverage}\to \mbox{\$o}\qquad\mbox{tff}(\mbox{help\_people\_stay\_awake\_type},\mbox{type})$

$\forall s{:}\;\mbox{syrup}{:}\;\mbox{help\_people\_stay\_awake}(\mbox{mixture}(\mbox{beverage},\mbox{coffee},s))\qquad\mbox{tff}(\mbox{mixture\_of\_coffee\_help\_people\_stay\_awake},\mbox{axiom})$

$\mbox{help\_people\_stay\_awake}(\mbox{mixture}(\mbox{beverage},\mbox{coffee},\mbox{mixture}(\mbox{syrup},\mbox{caramel\_syrup},\mbox{vanilla\_syrup})))\qquad\mbox{tff}(\mbox{caramel\_vanilla\_coffee\_help\_people\_stay\_awake},\mbox{conjecture})$

\medskip\textbf{PUZ006-1.p} Determine sex and race on Mars and Venus

Here's the situation: human observers in this exclusive club
 on Ganymede can't distinguish Martians from Venusians, males
 from females, except for the fact that Venusian women and
 Martian men always tell the truth and Venusian men and
 Martian women always lie.
 Ork says "Bog is from Venus."  Bog says "Ork is from Mars."
 Ork says "Bog is male."  Bog says "Ork is female." Who's
 what?  (sex \& race).




$\mbox{include}(\mbox{'Axioms/PUZ001-0.ax'})$

$\mbox{says}(\mbox{ork},\mbox{bog\_is\_from\_venus})\qquad\mbox{cnf}(\mbox{ork\_says\_bog\_is\_from\_venus},\mbox{hypothesis})$

$\mbox{says}(\mbox{bog},\mbox{ork\_is\_from\_mars})\qquad\mbox{cnf}(\mbox{bog\_says\_ork\_is\_from\_mar},\mbox{hypothesis})$

$\mbox{says}(\mbox{ork},\mbox{bog\_is\_male})\qquad\mbox{cnf}(\mbox{ork\_says\_bog\_is\_male},\mbox{hypothesis})$

$\mbox{says}(\mbox{bog},\mbox{ork\_is\_female})\qquad\mbox{cnf}(\mbox{bog\_says\_ork\_is\_female},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{bog\_is\_from\_venus}) \ \Rightarrow \ \mbox{from\_venus}(\mbox{bog})\qquad\mbox{cnf}(\mbox{bog\_is\_from\_venus}_1,\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{ork\_is\_from\_mars}) \ \Rightarrow \ \mbox{from\_mars}(\mbox{ork})\qquad\mbox{cnf}(\mbox{ork\_is\_from\_mars}_1,\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{bog\_is\_male}) \ \Rightarrow \ \mbox{male}(\mbox{bog})\qquad\mbox{cnf}(\mbox{bog\_is\_male}_1,\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{ork\_is\_female}) \ \Rightarrow \ \mbox{female}(\mbox{ork})\qquad\mbox{cnf}(\mbox{ork\_is\_female}_1,\mbox{hypothesis})$

$\mbox{from\_venus}(\mbox{bog}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{bog\_is\_from\_venus})\qquad\mbox{cnf}(\mbox{bog\_is\_from\_venus}_2,\mbox{hypothesis})$

$\mbox{from\_mars}(\mbox{ork}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{ork\_is\_from\_mars})\qquad\mbox{cnf}(\mbox{ork\_is\_from\_mars}_2,\mbox{hypothesis})$

$\mbox{male}(\mbox{bog}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{bog\_is\_male})\qquad\mbox{cnf}(\mbox{bog\_is\_male}_2,\mbox{hypothesis})$

$\mbox{female}(\mbox{ork}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{ork\_is\_female})\qquad\mbox{cnf}(\mbox{ork\_is\_female}_2,\mbox{hypothesis})$

$\neg\,\mbox{female}(\mbox{bog})\qquad\mbox{cnf}(\mbox{prove\_bog\_is\_female},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ020-1.p} A knights \& knaves problem, if he's a knight, so is she






$\mbox{person}(x) \ \Rightarrow \ ( \mbox{knight}(x)\mbox{ or }\mbox{knave}(x) )\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{knight}(x) ) \ \Rightarrow \ \neg\,\mbox{knave}(x)\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{says}(x,y)\mbox{ and }\mbox{a\_truth}(y) ) \ \Rightarrow \ \mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false},\mbox{axiom})$

$\mbox{says}(x,y) \ \Rightarrow \ x\ne y\qquad\mbox{cnf}(\mbox{people\_do\_not\_equal\_their\_statements}_1,\mbox{axiom})$

$\mbox{says}(x,y) \ \Rightarrow \ y = \mbox{statement\_by}(x)\qquad\mbox{cnf}(\mbox{peoples\_statements},\mbox{axiom})$

$\mbox{person}(x) \ \Rightarrow \ x\ne \mbox{statement\_by}(y)\qquad\mbox{cnf}(\mbox{people\_do\_not\_equal\_their\_statement}_2,\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{a\_truth}(\mbox{statement\_by}(x)) ) \ \Rightarrow \ \mbox{knight}(x)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements},\mbox{axiom})$

$\mbox{person}(x) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{statement\_by}(x))\mbox{ or }\mbox{knave}(x) )\qquad\mbox{cnf}(\mbox{knaves\_make\_false\_statements},\mbox{axiom})$

$( \mbox{knight}(x)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{knights\_say\_the\_truth},\mbox{axiom})$

$( \mbox{knave}(x)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(y)\qquad\mbox{cnf}(\mbox{knaves\_do\_not\_say\_the\_truth},\mbox{axiom})$

$\mbox{person}(\mbox{husband})\qquad\mbox{cnf}(\mbox{husband},\mbox{hypothesis})$

$\mbox{person}(\mbox{wife})\qquad\mbox{cnf}(\mbox{wife},\mbox{hypothesis})$

$\mbox{husband}\ne \mbox{wife}\qquad\mbox{cnf}(\mbox{husband\_not\_wife},\mbox{hypothesis})$

$\mbox{says}(\mbox{husband},\mbox{statement\_by}(\mbox{husband}))\qquad\mbox{cnf}(\mbox{husband\_makes\_statements},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{statement\_by}(\mbox{husband}))\mbox{ and }\mbox{knight}(\mbox{husband}) ) \ \Rightarrow \ \mbox{knight}(\mbox{wife})\qquad\mbox{cnf}(\mbox{truthful\_knight\_husband\_means\_knight\_wife},\mbox{hypothesis})$

$\mbox{knight}(\mbox{husband}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{statement\_by}(\mbox{husband}))\qquad\mbox{cnf}(\mbox{knight\_husband\_makes\_true\_statements},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{statement\_by}(\mbox{husband}))\mbox{ or }\mbox{knight}(\mbox{wife})\qquad\mbox{cnf}(\mbox{knight\_wife\_or\_truthful\_husband},\mbox{hypothesis})$

$\mbox{knight}(\mbox{wife}) \ \Rightarrow \ \mbox{a\_truth}(\mbox{statement\_by}(\mbox{husband}))\qquad\mbox{cnf}(\mbox{knight\_wife\_means\_truthful\_husband},\mbox{hypothesis})$

$\neg\,\mbox{knight}(\mbox{husband})\qquad\mbox{cnf}(\mbox{prove\_knight\_husband},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ083$\wedge$1.p} Peter the untruthful

Peter says that everything he says is false. Show that not
 everything Peter says is true.




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

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

$\mbox{says}@ \mbox{peter}@ \forall x{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ \mbox{peter}@ x ) \ \Rightarrow \ \neg\,x )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\neg\,\forall x{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ \mbox{peter}@ x ) \ \Rightarrow \ x )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{PUZ011-1.p} An ocean that borders on an African and an Asian country

There is a database of assertions about various countries and
 oceans and their relationships. Find which ocean borders
 on African and Asian countries.




$\mbox{ocean}(\mbox{atlantic})\qquad\mbox{cnf}(\mbox{atlantic},\mbox{hypothesis})$

$\mbox{ocean}(\mbox{indian})\qquad\mbox{cnf}(\mbox{indian},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{brazil})\qquad\mbox{cnf}(\mbox{atlantic\_brazil},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{uruguay})\qquad\mbox{cnf}(\mbox{atlantic\_uruguay},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{venesuela})\qquad\mbox{cnf}(\mbox{atlantic\_venesuela},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{zaire})\qquad\mbox{cnf}(\mbox{atlantic\_zaire},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{nigeria})\qquad\mbox{cnf}(\mbox{atlantic\_nigeria},\mbox{hypothesis})$

$\mbox{borders}(\mbox{atlantic},\mbox{angola})\qquad\mbox{cnf}(\mbox{atlantic\_angola},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{india})\qquad\mbox{cnf}(\mbox{indian\_india},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{pakistan})\qquad\mbox{cnf}(\mbox{indian\_pakistan},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{iran})\qquad\mbox{cnf}(\mbox{indian\_iran},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{somalia})\qquad\mbox{cnf}(\mbox{indian\_somalia},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{kenya})\qquad\mbox{cnf}(\mbox{indian\_kenya},\mbox{hypothesis})$

$\mbox{borders}(\mbox{indian},\mbox{tanzania})\qquad\mbox{cnf}(\mbox{indian\_tanzania},\mbox{hypothesis})$

$\mbox{south\_american}(\mbox{brazil})\qquad\mbox{cnf}(\mbox{brazil},\mbox{hypothesis})$

$\mbox{south\_american}(\mbox{uruguay})\qquad\mbox{cnf}(\mbox{uruguay},\mbox{hypothesis})$

$\mbox{south\_american}(\mbox{venesuela})\qquad\mbox{cnf}(\mbox{venezuela},\mbox{hypothesis})$

$\mbox{african}(\mbox{zaire})\qquad\mbox{cnf}(\mbox{zaire},\mbox{hypothesis})$

$\mbox{african}(\mbox{nigeria})\qquad\mbox{cnf}(\mbox{nigeria},\mbox{hypothesis})$

$\mbox{african}(\mbox{angola})\qquad\mbox{cnf}(\mbox{angola},\mbox{hypothesis})$

$\mbox{african}(\mbox{somalia})\qquad\mbox{cnf}(\mbox{somalia},\mbox{hypothesis})$

$\mbox{african}(\mbox{kenya})\qquad\mbox{cnf}(\mbox{kenya},\mbox{hypothesis})$

$\mbox{african}(\mbox{tanzania})\qquad\mbox{cnf}(\mbox{tanzania},\mbox{hypothesis})$

$\mbox{asian}(\mbox{india})\qquad\mbox{cnf}(\mbox{india},\mbox{hypothesis})$

$\mbox{asian}(\mbox{pakistan})\qquad\mbox{cnf}(\mbox{pakistan},\mbox{hypothesis})$

$\mbox{asian}(\mbox{iran})\qquad\mbox{cnf}(\mbox{iran},\mbox{hypothesis})$

$( \mbox{ocean}(\mbox{ocean})\mbox{ and }\mbox{borders}(\mbox{ocean},\mbox{african})\mbox{ and }\mbox{african}(\mbox{african})\mbox{ and }\mbox{borders}(\mbox{ocean},\mbox{asian}) ) \ \Rightarrow \ \neg\,\mbox{asian}(\mbox{asian})\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_common\_ocean},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ035-7.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(p,\mbox{knight})) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ and }\mbox{reply}(p,q,\mbox{yes}) ) \ \Rightarrow \ \mbox{truth}(q)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_1,\mbox{axiom})$

$( \mbox{truth}(q)\mbox{ and }\mbox{reply}(p,q,\mbox{yes}) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_2,\mbox{axiom})$

$( \mbox{truth}(q)\mbox{ and }\mbox{truth}(\mbox{isa}(p,\mbox{knight})) ) \ \Rightarrow \ \mbox{reply}(p,q,\mbox{yes})\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_3,\mbox{axiom})$

$\mbox{truth}(q)\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{reply}(p,q,\mbox{yes})\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_4,\mbox{axiom})$

$\mbox{reply}(p,q,\mbox{yes})\mbox{ or }\mbox{reply}(p,q,\mbox{no})\qquad\mbox{cnf}(\mbox{yes\_or\_no},\mbox{axiom})$

$\mbox{reply}(p,q,\mbox{yes}) \ \Rightarrow \ \neg\,\mbox{reply}(p,q,\mbox{no})\qquad\mbox{cnf}(\mbox{not\_yes\_and\_no},\mbox{axiom})$

$\mbox{truth}(c)\mbox{ or }\mbox{truth}(\mbox{not}(c))\qquad\mbox{cnf}(\mbox{true}_1,\mbox{axiom})$

$\mbox{truth}(c) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(c))\qquad\mbox{cnf}(\mbox{true}_2,\mbox{axiom})$

$\mbox{truth}(\mbox{or}(a,b)) \ \Rightarrow \ ( \mbox{truth}(a)\mbox{ or }\mbox{truth}(b) )\qquad\mbox{cnf}(\mbox{or}_1,\mbox{axiom})$

$\mbox{truth}(a) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_2,\mbox{axiom})$

$\mbox{truth}(b) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_3,\mbox{axiom})$

$( \mbox{reply}(\mbox{asked},\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight})),r)\mbox{ and }\mbox{truth}(\mbox{isa}(\mbox{asked},x)) ) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(\mbox{other},y))\qquad\mbox{cnf}(\mbox{prove\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ071+1.p} Thursday's Sudoku

45  8
 71
 2
 16   3
 5   4 2
 
 2     8
 1 3




$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_1) = n_4\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_2) = n_5\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_5) = n_8\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_8) = n_7\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_9) = n_1\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_3,n_2) = n_2\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_4,n_3) = n_1\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_4) = n_6\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_8) = n_3\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_5\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_5) = n_4\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_5,n_7) = n_2\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_7,n_1) = n_2\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_7) = n_8\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_8,n_4) = n_1\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_6) = n_3\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ001-1.p} Dreadbury Mansion

Someone who lives in Dreadbury Mansion killed Aunt Agatha.
 Agatha, the butler, and Charles live in Dreadbury Mansion,
 and are the only people who live therein. A killer always
 hates his victim, and is never richer than his victim.
 Charles hates no one that Aunt Agatha hates. Agatha hates
 everyone except the butler. The butler hates everyone not
 richer than Aunt Agatha. The butler hates everyone Aunt
 Agatha hates. No one hates everyone. Agatha is not the
 butler. Therefore : Agatha killed herself.




$\mbox{lives}(\mbox{agatha})\qquad\mbox{cnf}(\mbox{agatha},\mbox{hypothesis})$

$\mbox{lives}(\mbox{butler})\qquad\mbox{cnf}(\mbox{butler},\mbox{hypothesis})$

$\mbox{lives}(\mbox{charles})\qquad\mbox{cnf}(\mbox{charles},\mbox{hypothesis})$

$\mbox{killed}(x,y) \ \Rightarrow \ \neg\,\mbox{richer}(x,y)\qquad\mbox{cnf}(\mbox{poorer\_killer},\mbox{hypothesis})$

$\mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \neg\,\mbox{hates}(\mbox{charles},x)\qquad\mbox{cnf}(\mbox{different\_hates},\mbox{hypothesis})$

$( \mbox{hates}(x,\mbox{agatha})\mbox{ and }\mbox{hates}(x,\mbox{butler}) ) \ \Rightarrow \ \neg\,\mbox{hates}(x,\mbox{charles})\qquad\mbox{cnf}(\mbox{no\_one\_hates\_everyone},\mbox{hypothesis})$

$\mbox{hates}(\mbox{agatha},\mbox{agatha})\qquad\mbox{cnf}(\mbox{agatha\_hates\_agatha},\mbox{hypothesis})$

$\mbox{hates}(\mbox{agatha},\mbox{charles})\qquad\mbox{cnf}(\mbox{agatha\_hates\_charles},\mbox{hypothesis})$

$\mbox{killed}(x,y) \ \Rightarrow \ \mbox{hates}(x,y)\qquad\mbox{cnf}(\mbox{killer\_hates\_victim},\mbox{hypothesis})$

$\mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \mbox{hates}(\mbox{butler},x)\qquad\mbox{cnf}(\mbox{same\_hates},\mbox{hypothesis})$

$\mbox{lives}(x) \ \Rightarrow \ ( \mbox{richer}(x,\mbox{agatha})\mbox{ or }\mbox{hates}(\mbox{butler},x) )\qquad\mbox{cnf}(\mbox{butler\_hates\_poor},\mbox{hypothesis})$

$\mbox{killed}(\mbox{butler},\mbox{agatha})\mbox{ or }\mbox{killed}(\mbox{charles},\mbox{agatha})\qquad\mbox{cnf}(\mbox{prove\_neither\_charles\_nor\_butler\_did\_it},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ118$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

$c_5{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c5\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_H}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( g@ \mbox{xu}@ \mbox{xv} )\ne c_5 )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{31000},\mbox{conjecture})$

\medskip\textbf{PUZ065+1.p} Sudoku 13273






$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_2) = n_1\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_9) = n_9\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_2,n_4) = n_3\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_7) = n_8\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_3,n_7) = n_6\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_1\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_4,n_6) = n_2\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_7) = n_4\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_7\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_5,n_3) = n_3\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_6,n_1) = n_5\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_7,n_1) = n_8\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_6\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_8,n_5) = n_4\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_8,n_8) = n_2\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_9,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_8) = n_5\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ030-1.p} Salt and Mustard Problem






$\mbox{both}(x) \ \Rightarrow \ \mbox{salt}(x)\qquad\mbox{cnf}(\mbox{both}_1,\mbox{axiom})$

$\mbox{both}(x) \ \Rightarrow \ \mbox{mustard}(x)\qquad\mbox{cnf}(\mbox{both}_2,\mbox{axiom})$

$( \mbox{salt}(x)\mbox{ and }\mbox{mustard}(x) ) \ \Rightarrow \ \mbox{both}(x)\qquad\mbox{cnf}(\mbox{both}_3,\mbox{axiom})$

$\mbox{oneof}(x) \ \Rightarrow \ ( \mbox{salt}(x)\mbox{ or }\mbox{mustard}(x) )\qquad\mbox{cnf}(\mbox{oneof}_1,\mbox{axiom})$

$\mbox{oneof}(x) \ \Rightarrow \ \neg\,\mbox{both}(x)\qquad\mbox{cnf}(\mbox{oneof}_2,\mbox{axiom})$

$\mbox{oneof}(x) \ \Rightarrow \ \neg\,\mbox{neither}(x)\qquad\mbox{cnf}(\mbox{oneof}_3,\mbox{axiom})$

$\mbox{both}(x)\mbox{ or }\mbox{neither}(x)\mbox{ or }\mbox{oneof}(x)\qquad\mbox{cnf}(\mbox{one\_condition\_holds}_1,\mbox{axiom})$

$( \mbox{oneof}(x)\mbox{ and }\mbox{salt}(x) ) \ \Rightarrow \ \neg\,\mbox{mustard}(x)\qquad\mbox{cnf}(\mbox{oneof}_4,\mbox{axiom})$

$\mbox{both}(x) \ \Rightarrow \ \neg\,\mbox{neither}(x)\qquad\mbox{cnf}(\mbox{neither}_1,\mbox{axiom})$

$\mbox{neither}(x) \ \Rightarrow \ \neg\,\mbox{salt}(x)\qquad\mbox{cnf}(\mbox{neither}_2,\mbox{axiom})$

$\mbox{neither}(x) \ \Rightarrow \ \neg\,\mbox{mustard}(x)\qquad\mbox{cnf}(\mbox{neither}_3,\mbox{axiom})$

$\mbox{salt}(x)\mbox{ or }\mbox{mustard}(x)\mbox{ or }\mbox{neither}(x)\qquad\mbox{cnf}(\mbox{neither}_4,\mbox{axiom})$

$\mbox{salt}(\mbox{barry}) \ \Rightarrow \ ( \mbox{oneof}(\mbox{cole})\mbox{ or }\mbox{oneof}(\mbox{lang}) )\qquad\mbox{cnf}(\mbox{rule1}_1,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{cole}) \ \Rightarrow \ \mbox{salt}(\mbox{barry})\qquad\mbox{cnf}(\mbox{rule1}_2,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{lang}) \ \Rightarrow \ \mbox{salt}(\mbox{barry})\qquad\mbox{cnf}(\mbox{rule1}_3,\mbox{hypothesis})$

$\mbox{mustard}(\mbox{barry}) \ \Rightarrow \ ( \mbox{neither}(\mbox{dix})\mbox{ or }\mbox{both}(\mbox{mill}) )\qquad\mbox{cnf}(\mbox{rule2}_1,\mbox{hypothesis})$

$\mbox{neither}(\mbox{dix}) \ \Rightarrow \ \mbox{mustard}(\mbox{barry})\qquad\mbox{cnf}(\mbox{rule2}_2,\mbox{hypothesis})$

$\mbox{both}(\mbox{mill}) \ \Rightarrow \ \mbox{mustard}(\mbox{barry})\qquad\mbox{cnf}(\mbox{rule2}_3,\mbox{hypothesis})$

$\mbox{salt}(\mbox{cole}) \ \Rightarrow \ ( \mbox{oneof}(\mbox{barry})\mbox{ or }\mbox{neither}(\mbox{mill}) )\qquad\mbox{cnf}(\mbox{rule3}_1,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{barry}) \ \Rightarrow \ \mbox{salt}(\mbox{cole})\qquad\mbox{cnf}(\mbox{rule3}_2,\mbox{hypothesis})$

$\mbox{neither}(\mbox{mill}) \ \Rightarrow \ \mbox{salt}(\mbox{cole})\qquad\mbox{cnf}(\mbox{rule3}_3,\mbox{hypothesis})$

$\mbox{mustard}(\mbox{cole}) \ \Rightarrow \ ( \mbox{both}(\mbox{dix})\mbox{ or }\mbox{both}(\mbox{lang}) )\qquad\mbox{cnf}(\mbox{rule4}_1,\mbox{hypothesis})$

$\mbox{both}(\mbox{dix}) \ \Rightarrow \ \mbox{mustard}(\mbox{cole})\qquad\mbox{cnf}(\mbox{rule4}_2,\mbox{hypothesis})$

$\mbox{both}(\mbox{lang}) \ \Rightarrow \ \mbox{mustard}(\mbox{cole})\qquad\mbox{cnf}(\mbox{rule4}_3,\mbox{hypothesis})$

$\mbox{salt}(\mbox{dix}) \ \Rightarrow \ ( \mbox{neither}(\mbox{barry})\mbox{ or }\mbox{both}(\mbox{cole}) )\qquad\mbox{cnf}(\mbox{rule5}_1,\mbox{hypothesis})$

$\mbox{neither}(\mbox{barry}) \ \Rightarrow \ \mbox{salt}(\mbox{dix})\qquad\mbox{cnf}(\mbox{rule5}_2,\mbox{hypothesis})$

$\mbox{both}(\mbox{cole}) \ \Rightarrow \ \mbox{salt}(\mbox{dix})\qquad\mbox{cnf}(\mbox{rule5}_3,\mbox{hypothesis})$

$\mbox{mustard}(\mbox{dix}) \ \Rightarrow \ ( \mbox{neither}(\mbox{lang})\mbox{ or }\mbox{neither}(\mbox{mill}) )\qquad\mbox{cnf}(\mbox{rule6}_1,\mbox{hypothesis})$

$\mbox{neither}(\mbox{lang}) \ \Rightarrow \ \mbox{mustard}(\mbox{dix})\qquad\mbox{cnf}(\mbox{rule6}_2,\mbox{hypothesis})$

$\mbox{neither}(\mbox{mill}) \ \Rightarrow \ \mbox{mustard}(\mbox{dix})\qquad\mbox{cnf}(\mbox{rule6}_3,\mbox{hypothesis})$

$\mbox{salt}(\mbox{lang}) \ \Rightarrow \ ( \mbox{oneof}(\mbox{barry})\mbox{ or }\mbox{oneof}(\mbox{dix}) )\qquad\mbox{cnf}(\mbox{rule7}_1,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{barry}) \ \Rightarrow \ \mbox{salt}(\mbox{lang})\qquad\mbox{cnf}(\mbox{rule7}_2,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{dix}) \ \Rightarrow \ \mbox{salt}(\mbox{lang})\qquad\mbox{cnf}(\mbox{rule7}_3,\mbox{hypothesis})$

$\mbox{mustard}(\mbox{lang}) \ \Rightarrow \ ( \mbox{neither}(\mbox{cole})\mbox{ or }\mbox{neither}(\mbox{mill}) )\qquad\mbox{cnf}(\mbox{rule8}_1,\mbox{hypothesis})$

$\mbox{neither}(\mbox{cole}) \ \Rightarrow \ \mbox{mustard}(\mbox{lang})\qquad\mbox{cnf}(\mbox{rule8}_2,\mbox{hypothesis})$

$\mbox{neither}(\mbox{mill}) \ \Rightarrow \ \mbox{mustard}(\mbox{lang})\qquad\mbox{cnf}(\mbox{rule8}_3,\mbox{hypothesis})$

$\mbox{salt}(\mbox{mill}) \ \Rightarrow \ ( \mbox{both}(\mbox{barry})\mbox{ or }\mbox{both}(\mbox{lang}) )\qquad\mbox{cnf}(\mbox{rule9}_1,\mbox{hypothesis})$

$\mbox{both}(\mbox{barry}) \ \Rightarrow \ \mbox{salt}(\mbox{mill})\qquad\mbox{cnf}(\mbox{rule9}_2,\mbox{hypothesis})$

$\mbox{both}(\mbox{lang}) \ \Rightarrow \ \mbox{mustard}(\mbox{mill})\qquad\mbox{cnf}(\mbox{rule9}_3,\mbox{hypothesis})$

$\mbox{mustard}(\mbox{mill}) \ \Rightarrow \ ( \mbox{oneof}(\mbox{cole})\mbox{ or }\mbox{oneof}(\mbox{dix}) )\qquad\mbox{cnf}(\mbox{rule10}_1,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{cole}) \ \Rightarrow \ \mbox{mustard}(\mbox{mill})\qquad\mbox{cnf}(\mbox{rule10}_2,\mbox{hypothesis})$

$\mbox{oneof}(\mbox{dix}) \ \Rightarrow \ \mbox{mustard}(\mbox{mill})\qquad\mbox{cnf}(\mbox{rule10}_3,\mbox{hypothesis})$

$( \mbox{neither}(\mbox{cole})\mbox{ and }\mbox{neither}(\mbox{dix})\mbox{ and }\mbox{both}(\mbox{barry})\mbox{ and }\mbox{oneof}(\mbox{lang})\mbox{ and }\mbox{salt}(\mbox{mill})\mbox{ and }\mbox{mustard}(\mbox{lang}) ) \ \Rightarrow \ \neg\,\mbox{oneof}(\mbox{mill})\qquad\mbox{cnf}(\mbox{prove\_who\_takes\_what},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ008-1.p} Missionaries and Cannibals

There are 3 missionaries, 3 cannibals, and a boat on the west
 bank of a river. All wish to cross, but the boat holds
 at most 2 people. If the cannibals ever outnumber the
 missionaries on either bank of the river the outnumbered
 missionaries will be eaten. Can they all safely cross the
 river?  If so, how? (The boat cannot cross empty.)




$\mbox{achievable}(\mbox{west}(m(x),c(s(y))),\mbox{boatonwest},\mbox{east}(m(z),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(z),c(s(w))))\qquad\mbox{cnf}(\mbox{cannibal\_west\_to\_east},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(z),c(s(w)))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(s(y))),\mbox{boatonwest},\mbox{east}(m(z),c(w)))\qquad\mbox{cnf}(\mbox{cannibal\_east\_to\_west},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(s(s(y)))),\mbox{boatonwest},\mbox{east}(m(z),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(z),c(s(s(w)))))\qquad\mbox{cnf}(\mbox{two\_cannibals\_west\_to\_east},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(z),c(s(s(w))))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(s(s(y)))),\mbox{boatonwest},\mbox{east}(m(z),c(w)))\qquad\mbox{cnf}(\mbox{two\_cannibals\_east\_to\_west},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(x)),c(y)),\mbox{boatonwest},\mbox{east}(m(z),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(z)),c(w)))\qquad\mbox{cnf}(\mbox{missionary\_west\_to\_east},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(z)),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(s(x)),c(y)),\mbox{boatonwest},\mbox{east}(m(z),c(w)))\qquad\mbox{cnf}(\mbox{missionary\_east\_to\_west},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(s(x))),c(y)),\mbox{boatonwest},\mbox{east}(m(z),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(s(z))),c(w)))\qquad\mbox{cnf}(\mbox{two\_missionaries\_west\_to\_east},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(s(z))),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(s(s(x))),c(y)),\mbox{boatonwest},\mbox{east}(m(z),c(w)))\qquad\mbox{cnf}(\mbox{two\_missionaries\_east\_to\_west},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(x)),c(s(y))),\mbox{boatonwest},\mbox{east}(m(z),c(w))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(z)),c(s(w))))\qquad\mbox{cnf}(\mbox{missionary\_and\_cannibal\_west\_to\_east},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(x),c(y)),\mbox{boatoneast},\mbox{east}(m(s(z)),c(s(w)))) \ \Rightarrow \ \mbox{achievable}(\mbox{west}(m(s(x)),c(s(y))),\mbox{boatonwest},\mbox{east}(m(z),c(w)))\qquad\mbox{cnf}(\mbox{missionary\_and\_cannibal\_east\_to\_west},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(x)),c(s(s(x)))),y,\mbox{east}(z,w))\qquad\mbox{cnf}(\mbox{extra\_cannibal\_meal\_on\_west\_bank},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(x)),c(s(s(s(x))))),y,\mbox{east}(z,w))\qquad\mbox{cnf}(\mbox{two\_extra\_cannibals\_meal\_on\_west\_bank},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(x,y),z,\mbox{east}(m(s(w)),c(s(s(w)))))\qquad\mbox{cnf}(\mbox{extra\_cannibal\_meal\_on\_east\_bank},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(x,y),z,\mbox{east}(m(s(w)),c(s(s(s(w))))))\qquad\mbox{cnf}(\mbox{two\_extra\_cannibals\_meal\_on\_east\_bank},\mbox{axiom})$

$\mbox{achievable}(\mbox{west}(m(s(s(s(n_0)))),c(s(s(s(n_0))))),\mbox{boatonwest},\mbox{east}(m(n_0),c(n_0)))\qquad\mbox{cnf}(\mbox{start\_on\_west\_bank},\mbox{hypothesis})$

$\neg\,\mbox{achievable}(\mbox{west}(m(n_0),c(n_0)),x,\mbox{east}(m(s(s(s(n_0)))),c(s(s(s(n_0))))))\qquad\mbox{cnf}(\mbox{prove\_can\_get\_to\_east\_bank},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ084$\wedge$1.p} The friends puzzle - reflexivity for Peter's wife

(i) Peter is a friend of John, so if Peter knows that John knows
 something then John knows that Peter knows the same thing. 
 (ii) Peter is married, so if Peter's wife knows something, then
 Peter knows the same thing. John and Peter have an appointment,
 let us consider the following situation: (a) Peter knows the time
 of their appointment. (b) Peter also knows that John knows the
 place of their appointment. Moreover, (c) Peter's wife knows that
 if Peter knows the time of their appointment, then John knows
 that too (since John and Peter are friends). Finally, (d) Peter
 knows that if John knows the place and the time of their
 appointment, then John knows that he has an appointment. From
 this situation we want to prove (e) that each of the two friends
 knows that the other one knows that he has an appointment.




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

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

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

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

$\mbox{mreflexive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{refl\_peter},\mbox{axiom})$

$\mbox{mreflexive}@ \mbox{john}\qquad\mbox{thf}(\mbox{refl\_john},\mbox{axiom})$

$\mbox{mreflexive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{refl\_wife\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{trans\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{john}\qquad\mbox{thf}(\mbox{trans\_john},\mbox{axiom})$

$\mbox{mtransitive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{trans\_wife\_peter},\mbox{axiom})$

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

\medskip\textbf{PUZ059-1.p} Quo vadis 7 - an unreachable state

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is a simple goal for testing.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(s(s(s(o))),o),v_3(s(o),s(s(s(o)))),v_4(s(s(s(s(o)))),s(s(s(o)))),h(s(s(s(s(o)))),s(s(o))),s_1(s(s(o)),o),s_2(s(s(o)),s(s(o))),s_3(o,s(s(s(o)))),s_4(s(s(s(s(o)))),s(o)),e_1(s(s(o)),s(o)),e_2(s(s(s(o))),s(o)))\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ120$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx}\mbox{ and }\mbox{cCKB6\_NUM}@ \mbox{xy}\mbox{ and }( s@ ( s@ ( s@ \mbox{xx} ) ) ) = ( s@ ( s@ ( s@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{26000},\mbox{conjecture})$

\medskip\textbf{PUZ066+1.p} Sudoku 19262






$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_2) = n_6\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_4) = n_2\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_7) = n_5\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_2) = n_1\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_3) = n_3\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_2,n_5) = n_6\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_4,n_2) = n_7\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_8) = n_3\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_9) = n_1\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_8\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_4) = n_5\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_6,n_4) = n_4\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_7,n_5) = n_1\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_8) = n_6\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_8,n_1) = n_9\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_7) = n_2\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_1) = n_4\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ061+1.p} Food problems






$\forall \mbox{peanuts},\mbox{john},\mbox{bill},\mbox{sue},\mbox{apples},\mbox{chicken}{:}\;( ( \forall x,y{:}\;( ( \mbox{alive}(y)\mbox{ and }\mbox{eats}(y,\mbox{chicken}) ) \ \Rightarrow \ \mbox{likes}(y,x) )\mbox{ and }\forall x{:}\;( \mbox{food}(x) \ \Rightarrow \ \mbox{likes}(\mbox{john},x) )\mbox{ and }\mbox{food}(\mbox{apples})\mbox{ and }\mbox{food}(\mbox{chicken})\mbox{ and }\forall x{:}\;( \exists y{:}\;( \mbox{eats}(y,x)\mbox{ and }\mbox{not\_killed\_by}(y,x) ) \ \Rightarrow \ \mbox{food}(x) )\mbox{ and }\mbox{eats}(\mbox{bill},\mbox{peanuts})\mbox{ and }\mbox{alive}(\mbox{bill})\mbox{ and }\forall x{:}\;( \mbox{eats}(\mbox{bill},x) \ \Rightarrow \ \mbox{eats}(\mbox{sue},x) )\mbox{ and }\forall y{:}\;( \mbox{alive}(y) \ \Rightarrow \ \forall x{:}\;\mbox{not\_killed\_by}(y,x) ) ) \ \Rightarrow \ \mbox{likes}(\mbox{john},\mbox{peanuts}) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{PUZ140$\wedge$1.p} A mixture of coffee and syrup that is hot






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

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

$\mbox{coffee}{:}\;\mbox{beverage}\qquad\mbox{thf}(\mbox{coffee\_type},\mbox{type})$

$\mbox{heat}{:}\;\mbox{beverage}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{heat\_type},\mbox{type})$

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

$\mbox{mix}{:}\;\mbox{beverage}\to \mbox{syrup}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{mix\_type},\mbox{type})$

$\mbox{hot\_mixture}{:}\;\mbox{beverage}\to \mbox{syrup}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{hot\_mixture\_type},\mbox{type})$

$\mbox{cold\_mixture}{:}\;\mbox{beverage}\to \mbox{syrup}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{cold\_mixture\_type},\mbox{type})$

$\mbox{hot\_mixture} = ( \lambda b{:}\;\mbox{beverage},s{:}\;\mbox{syrup}{:}\;( \mbox{heat}@ ( \mbox{mix}@ b@ s ) ) )\qquad\mbox{thf}(\mbox{hot\_mixture\_definition},\mbox{definition})$

$\mbox{cold\_mixture} = ( \lambda b{:}\;\mbox{beverage},s{:}\;\mbox{syrup}{:}\;( \mbox{mix}@ b@ s ) )\qquad\mbox{thf}(\mbox{cold\_mixture\_definition},\mbox{definition})$

$\forall b{:}\;\mbox{beverage}{:}\;( \mbox{hot}@ ( \mbox{heat}@ b ) )\qquad\mbox{thf}(\mbox{its\_hot},\mbox{axiom})$

$\exists \mbox{mixture}{:}\;\mbox{beverage}\to \mbox{syrup}\to \mbox{beverage}{:}\;\forall s{:}\;\mbox{syrup}{:}\;\exists b{:}\;\mbox{beverage}{:}\;( ( \mbox{mixture}@ \mbox{coffee}@ s ) = b\mbox{ and }\mbox{hot}@ b )\qquad\mbox{thf}(\mbox{hot\_coffee},\mbox{conjecture})$

\medskip\textbf{PUZ002-1.p} The Animals Puzzle

1)  The only animals in this house are cats.
 2)  Every animal is suitable for a pet, that loves to gaze at
 the moon.
 3)  When I detest an animal, I avoid it.
 4)  No animals are carnivorous, unless they prowl at night.
 5)  No cat fails to kill mice.
 6)  No animals ever take to me, except what are in this house.
 7)  Kangaroos are not suitable for pets.
 8)  None but carnivora kill mice.
 9)  I detest animals that do not take to me.
 10) Animals that prowl at night always love to gaze at the moon.
 The problem is to prove that "I always avoid a kangaroo".




$\mbox{in\_house}(\mbox{cat}) \ \Rightarrow \ \mbox{cat}(\mbox{cat})\qquad\mbox{cnf}(\mbox{only\_cats\_in\_house},\mbox{axiom})$

$\mbox{gazer}(\mbox{gazer}) \ \Rightarrow \ \mbox{suitable\_pet}(\mbox{gazer})\qquad\mbox{cnf}(\mbox{gazers\_are\_suitable\_pets},\mbox{axiom})$

$\mbox{detested}(\mbox{detested}) \ \Rightarrow \ \mbox{avoided}(\mbox{detested})\qquad\mbox{cnf}(\mbox{avoid\_detested},\mbox{axiom})$

$\mbox{carnivore}(\mbox{carnivore}) \ \Rightarrow \ \mbox{prowler}(\mbox{carnivore})\qquad\mbox{cnf}(\mbox{carnivores\_are\_prowlers},\mbox{axiom})$

$\mbox{cat}(\mbox{cat}) \ \Rightarrow \ \mbox{mouse\_killer}(\mbox{cat})\qquad\mbox{cnf}(\mbox{cats\_are\_mice\_killers},\mbox{axiom})$

$\mbox{takes\_to\_me}(\mbox{taken\_animal}) \ \Rightarrow \ \mbox{in\_house}(\mbox{taken\_animal})\qquad\mbox{cnf}(\mbox{in\_house\_if\_takes\_to\_me},\mbox{axiom})$

$\mbox{kangaroo}(\mbox{kangaroo}) \ \Rightarrow \ \neg\,\mbox{suitable\_pet}(\mbox{kangaroo})\qquad\mbox{cnf}(\mbox{kangaroos\_are\_not\_pets},\mbox{axiom})$

$\mbox{mouse\_killer}(\mbox{killer}) \ \Rightarrow \ \mbox{carnivore}(\mbox{killer})\qquad\mbox{cnf}(\mbox{mouse\_killers\_are\_carnivores},\mbox{axiom})$

$\mbox{takes\_to\_me}(\mbox{animal})\mbox{ or }\mbox{detested}(\mbox{animal})\qquad\mbox{cnf}(\mbox{takes\_to\_me\_or\_detested},\mbox{axiom})$

$\mbox{prowler}(\mbox{prowler}) \ \Rightarrow \ \mbox{gazer}(\mbox{prowler})\qquad\mbox{cnf}(\mbox{prowlers\_are\_gazers},\mbox{axiom})$

$\mbox{kangaroo}(\mbox{the\_kangaroo})\qquad\mbox{cnf}(\mbox{kangaroo\_is\_a\_kangaroo},\mbox{axiom})$

$\neg\,\mbox{avoided}(\mbox{the\_kangaroo})\qquad\mbox{cnf}(\mbox{avoid\_kangaroo},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ049-1.p} Quo vadis 6 - intermediate to final

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is the true goal from the puzzle.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(s(s(o)),s(s(o))),v_1(o,o),v_2(o,s(o)),v_3(o,s(s(o))),v_4(o,s(s(s(o)))),h(s(s(s(s(o)))),o),s_1(s(s(o)),o),s_2(s(s(o)),s(o)),s_3(s(s(o)),o),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(o))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{intermediate\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(s(s(s(o))),s(o)),v_1,v_2,v_3,v_4,h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ064-1.p} Problem about mutilated chessboard problem






$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

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

$\mbox{c\_in}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{c\_Suc}(\mbox{v\_j}),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{c\_Mutil\_Odomino},\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Odomino\_Ohoriz}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{c\_Suc}(\mbox{v\_i}),\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{c\_Mutil\_Odomino},\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Odomino\_Overtl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}) = \mbox{c\_emptyset} ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_OUn}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_emptyset},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_Oempty}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{t\_a}) = \mbox{c\_union}(\mbox{v\_A},\mbox{c\_union}(\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_assoc}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{v\_u},\mbox{t\_a}) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_inter}(\mbox{v\_t},\mbox{v\_u},\mbox{t\_a}) = \mbox{c\_emptyset}\mbox{ and }\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{v\_t},\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{PUZ035-3.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{person}(p) \ \Rightarrow \ ( \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knave})) )\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(p,\mbox{knight})) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(\mbox{isa}(p,\mbox{knight})) ) \ \Rightarrow \ \mbox{truth}(s)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_1,\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(s) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_2,\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(\mbox{isa}(p,\mbox{knave})) ) \ \Rightarrow \ \mbox{truth}(\mbox{not}(s))\qquad\mbox{cnf}(\mbox{knaves\_make\_false\_statements}_1,\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(\mbox{not}(s)) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{knaves\_make\_false\_statements}_2,\mbox{axiom})$

$\mbox{says}(\mbox{asked},\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight})))\mbox{ or }\mbox{says}(\mbox{asked},\mbox{not}(\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight}))))\qquad\mbox{cnf}(\mbox{one\_is\_knight\_answer},\mbox{axiom})$

$\mbox{says}(\mbox{anyone},s) \ \Rightarrow \ ( \mbox{truth}(s)\mbox{ or }\mbox{truth}(\mbox{not}(s)) )\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false},\mbox{axiom})$

$\mbox{says}(\mbox{anyone},\mbox{not}(s)) \ \Rightarrow \ ( \mbox{truth}(s)\mbox{ or }\mbox{truth}(\mbox{not}(s)) )\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false}_2,\mbox{axiom})$

$\mbox{truth}(s) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(s))\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false}_3,\mbox{axiom})$

$\mbox{truth}(\mbox{or}(s_1,s_2)) \ \Rightarrow \ ( \mbox{truth}(s_1)\mbox{ or }\mbox{truth}(s_2) )\qquad\mbox{cnf}(\mbox{or\_def},\mbox{axiom})$

$\mbox{truth}(s_1) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(\mbox{or}(s_1,\mbox{anything})))\qquad\mbox{cnf}(\mbox{not}_1,\mbox{axiom})$

$\mbox{truth}(s_2) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(\mbox{or}(\mbox{anything},s_2)))\qquad\mbox{cnf}(\mbox{not}_2,\mbox{axiom})$

$\mbox{person}(\mbox{asked})\qquad\mbox{cnf}(\mbox{asked\_person},\mbox{axiom})$

$\mbox{person}(\mbox{other})\qquad\mbox{cnf}(\mbox{other\_person},\mbox{axiom})$

$( \mbox{truth}(\mbox{isa}(\mbox{asked},p_1))\mbox{ and }\mbox{truth}(\mbox{isa}(\mbox{other},p_2)) ) \ \Rightarrow \ \neg\,\mbox{says}(\mbox{asked},s)\qquad\mbox{cnf}(\mbox{prove\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ072-1.p} Friday's Sudoku

1      9
 3  8
 6
 124
 7 3
 5
 8  6
 4  2
 7   5




$\mbox{include}(\mbox{'Axioms/PUZ005-0.ax'})$

$\mbox{el}(s(n_0),s(s(n_0)),s(n_0))\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(s(s(s(s(s(n_0))))))))))\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(n_0)))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(n_0))))),s(n_0))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(s(n_0)))))),s(s(n_0)))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(n_0),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(n_0))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(n_0)))))),s(n_0),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(n_0),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(n_0)))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(n_0))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(n_0)))\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

\medskip\textbf{PUZ107$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

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

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

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

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\mbox{cCKB\_FIN} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{cCKB\_INF}@ \mbox{xk} )\qquad\mbox{thf}(\mbox{cCKB\_FIN\_def},\mbox{definition})$

$\mbox{cCKB\_FIN}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xu} = c_1\mbox{ and }\mbox{xv} = c_1 )\qquad\mbox{thf}(\mbox{cL}_{3000},\mbox{conjecture})$

\medskip\textbf{PUZ068+2.p} Monday's Sudoku

47 1
 5 2
 3      8
 6   1 7
 4  3
 5
 7   3
 1
 4




$\mbox{include}(\mbox{'Axioms/PUZ006+0.ax'})$

$p(n_1,n_4,n_4)\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$p(n_1,n_5,n_7)\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$p(n_1,n_7,n_1)\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$p(n_2,n_2,n_5)\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$p(n_2,n_4,n_2)\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$p(n_3,n_2,n_3)\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$p(n_3,n_9,n_8)\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$p(n_4,n_1,n_6)\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$p(n_4,n_5,n_1)\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$p(n_4,n_7,n_7)\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$p(n_5,n_3,n_4)\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$p(n_5,n_6,n_3)\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$p(n_6,n_8,n_5)\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$p(n_7,n_4,n_7)\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$p(n_7,n_8,n_3)\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$p(n_8,n_1,n_1)\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$p(n_9,n_9,n_4)\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ035-6.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(p,\mbox{knight})) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ and }\mbox{says}(p,s) ) \ \Rightarrow \ \mbox{truth}(s)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_1,\mbox{axiom})$

$( \mbox{truth}(s)\mbox{ and }\mbox{says}(p,s) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_2,\mbox{axiom})$

$\mbox{truth}(\mbox{or}(a,b)) \ \Rightarrow \ ( \mbox{truth}(a)\mbox{ or }\mbox{truth}(b) )\qquad\mbox{cnf}(\mbox{or}_1,\mbox{axiom})$

$\mbox{truth}(a) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_2,\mbox{axiom})$

$\mbox{truth}(b) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_3,\mbox{axiom})$

$\mbox{truth}(c)\mbox{ or }\mbox{truth}(\mbox{not}(c))\qquad\mbox{cnf}(\mbox{not}_1,\mbox{axiom})$

$\mbox{truth}(c) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(c))\qquad\mbox{cnf}(\mbox{not}_2,\mbox{axiom})$

$\mbox{says}(\mbox{asked},\mbox{not}(\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight}))))\qquad\mbox{cnf}(\mbox{says\_yes},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(\mbox{asked},x)) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(\mbox{other},y))\qquad\mbox{cnf}(\mbox{query},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ056-1.p} Tower of Hanoi






$\mbox{state}(\mbox{nextto}(\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_1)))),\mbox{nextto}(\mbox{pin}(n_2),\mbox{pin}(n_3))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{axiom})$

$\neg\,\mbox{state}(\mbox{nextto}(\mbox{pin}(n_1),\mbox{nextto}(\mbox{pin}(n_2),\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_3)))))))\qquad\mbox{cnf}(\mbox{final\_state},\mbox{negated\_conjecture})$

$\mbox{nextto}(x,y) = \mbox{nextto}(y,x)\qquad\mbox{cnf}(\mbox{nextto\_commutative},\mbox{axiom})$

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

$( \mbox{fits}(x,b)\mbox{ and }\mbox{fits}(x,a) ) \ \Rightarrow \ \mbox{nextto}(\mbox{on}(x,a),b) = \mbox{nextto}(a,\mbox{on}(x,b))\qquad\mbox{cnf}(\mbox{a\_move},\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_1,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_2,\mbox{axiom})$

$\mbox{fits}(\mbox{large},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_3,\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{on}(\mbox{medium},a))\qquad\mbox{cnf}(\mbox{fits}_4,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{on}(\mbox{large},a))\qquad\mbox{cnf}(\mbox{fits}_5,\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{on}(\mbox{large},a))\qquad\mbox{cnf}(\mbox{fits}_6,\mbox{axiom})$

\medskip\textbf{PUZ081$\wedge$1.p} 1 of http://philosophy.hku.hk/think/logic/knight.php

A very special island is inhabited only by knights and knaves.
 Knights always tell the truth, and knaves always lie. You meet two
 inhabitants: Zoey and Mel. Zoey tells you that Mel is a knave. Mel
 says, `Neither Zoey nor I are knaves'. Who is a knight and who is
 a knave?




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

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

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

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

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

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

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

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{is\_a}@ x@ \mbox{islander} ) \ \Rightarrow \ ( \mbox{is\_a}@ x@ \mbox{knight}\mbox{ or }\mbox{is\_a}@ x@ \mbox{knave} ) )\qquad\mbox{thf}(\mbox{kk\_6}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{is\_a}@ x@ \mbox{knight} ) \ \Rightarrow \ \forall a{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ x@ a ) \ \Rightarrow \ a ) )\qquad\mbox{thf}(\mbox{kk\_6}_2,\mbox{axiom})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{is\_a}@ x@ \mbox{knave} ) \ \Rightarrow \ \forall a{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ x@ a ) \ \Rightarrow \ \neg\,a ) )\qquad\mbox{thf}(\mbox{kk\_6}_3,\mbox{axiom})$

$\mbox{is\_a}@ \mbox{zoey}@ \mbox{islander}\mbox{ and }\mbox{is\_a}@ \mbox{mel}@ \mbox{islander}\qquad\mbox{thf}(\mbox{kk\_6}_4,\mbox{axiom})$

$\mbox{says}@ \mbox{zoey}@ ( \mbox{is\_a}@ \mbox{mel}@ \mbox{knave} )\qquad\mbox{thf}(\mbox{kk\_6}_5,\mbox{axiom})$

$\mbox{says}@ \mbox{mel}@ \neg\,\mbox{is\_a}@ \mbox{zoey}@ \mbox{knave}\mbox{ or }\mbox{is\_a}@ \mbox{mel}@ \mbox{knave}\qquad\mbox{thf}(\mbox{kk\_6}_6,\mbox{axiom})$

$\exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( y = \mbox{knight} <~> y = \mbox{knave} )\mbox{ and }( z = \mbox{knight} <~> z = \mbox{knave} )\mbox{ and }\mbox{is\_a}@ \mbox{mel}@ y\mbox{ and }\mbox{is\_a}@ \mbox{zoey}@ z )\qquad\mbox{thf}(\mbox{query},\mbox{conjecture})$

\medskip\textbf{PUZ023-1.p} Knights and Knaves \#27

There is an island with exactly two types of people :
 truthtellers who always tell the truth and liars who always
 lie. There are a group of three people, A, B, and C on the
 island. A stranger passes by and asks A, "How many
 truthtellers are among you ?" A answers indistinctly. So the
 stranger asks B, "what did A say?". B replies "A said that
 there is exactly one truthteller among us." Then C says,
 "Don't believe B; he is lying!" What are B and C. Answer:
 B is a liar and C is a truth-teller.




$\mbox{include}(\mbox{'Axioms/PUZ002-0.ax'})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{one\_truthteller}) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{one\_is\_the\_truthteller},\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{one\_truthteller})\qquad\mbox{cnf}(\mbox{two\_truthtellers}_1,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(z)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{one\_truthteller})\qquad\mbox{cnf}(\mbox{two\_truthtellers}_2,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(z)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{one\_truthteller})\qquad\mbox{cnf}(\mbox{two\_truthtellers}_3,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{identify\_one\_truthteller}_1,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{identify\_one\_truthteller}_2,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(z)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(x)) )\qquad\mbox{cnf}(\mbox{identify\_one\_truthteller}_3,\mbox{axiom})$

$\mbox{people}(a,b,c)\qquad\mbox{cnf}(\mbox{a\_b\_and\_c\_are\_people},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{garbage}))\qquad\mbox{cnf}(\mbox{a\_says\_garbage},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(b,\mbox{says}(a,\mbox{one\_truthteller})))\qquad\mbox{cnf}(\mbox{b\_says\_a\_says\_one\_truthteller},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(c,\mbox{liar}(b)))\qquad\mbox{cnf}(\mbox{c\_says\_b\_lies},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(b))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(c)) ) \ \Rightarrow \ \mbox{an\_answer}(\mbox{b\_and\_c\_liars})\qquad\mbox{cnf}(\mbox{b\_and\_c\_liars},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(b))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(c)) ) \ \Rightarrow \ \mbox{an\_answer}(\mbox{b\_liar\_and\_c\_truthteller})\qquad\mbox{cnf}(\mbox{b\_liar\_and\_c\_truthteller},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{truthteller}(b))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(c)) ) \ \Rightarrow \ \mbox{an\_answer}(\mbox{b\_truthteller\_and\_c\_liar})\qquad\mbox{cnf}(\mbox{b\_truthteller\_and\_c\_liar},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{truthteller}(b))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(c)) ) \ \Rightarrow \ \mbox{an\_answer}(\mbox{b\_and\_c\_truthtellers})\qquad\mbox{cnf}(\mbox{b\_and\_c\_truthtellers},\mbox{hypothesis})$

$\neg\,\mbox{an\_answer}(x)\qquad\mbox{cnf}(\mbox{prove\_there\_is\_an\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ015-1.p} Checkerboard and Dominoes : Opposing corners removed

There is a checker board whose upper left and lower right
 squares have been removed. There is a box of dominoes that
 are one square by two squares in size. Can you exactly cover
 the checker board with dominoes?




$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(\mbox{not\_covered},\mbox{not\_covered},y_3,y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(\mbox{covered},\mbox{covered},y_3,y_4,y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_1\_and}_2,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,\mbox{not\_covered},\mbox{not\_covered},y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,\mbox{covered},\mbox{covered},y_4,y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_2\_and}_3,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,\mbox{not\_covered},\mbox{not\_covered},y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,\mbox{covered},\mbox{covered},y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_3\_and}_4,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,\mbox{not\_covered},\mbox{not\_covered},y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,\mbox{covered},\mbox{covered},y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_4\_and}_5,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,\mbox{not\_covered},\mbox{not\_covered},y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,\mbox{covered},\mbox{covered},y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_5\_and}_6,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,\mbox{not\_covered},\mbox{not\_covered},y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,\mbox{covered},\mbox{covered},y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_6\_and}_7,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,\mbox{not\_covered},\mbox{not\_covered})) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,\mbox{covered},\mbox{covered}))\qquad\mbox{cnf}(\mbox{cover\_columns\_7\_and}_8,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(\mbox{successor}(x)),\mbox{squares}(y_1',y_2',y_3',y_4',y_5',y_6',y_7',y_8'))\qquad\mbox{cnf}(\mbox{place\_vertical\_pieces},\mbox{axiom})$

$\mbox{successor}(n_1) = n_2\qquad\mbox{cnf}(\mbox{successor\_of\_1\_is}_2,\mbox{axiom})$

$\mbox{successor}(n_2) = n_3\qquad\mbox{cnf}(\mbox{successor\_of\_2\_is}_3,\mbox{axiom})$

$\mbox{successor}(n_3) = n_4\qquad\mbox{cnf}(\mbox{successor\_of\_3\_is}_4,\mbox{axiom})$

$\mbox{successor}(n_4) = n_5\qquad\mbox{cnf}(\mbox{successor\_of\_4\_is}_5,\mbox{axiom})$

$\mbox{successor}(n_5) = n_6\qquad\mbox{cnf}(\mbox{successor\_of\_5\_is}_6,\mbox{axiom})$

$\mbox{successor}(n_6) = n_7\qquad\mbox{cnf}(\mbox{successor\_of\_6\_is}_7,\mbox{axiom})$

$\mbox{successor}(n_7) = n_8\qquad\mbox{cnf}(\mbox{successor\_of\_7\_is}_8,\mbox{axiom})$

$\mbox{successor}(n_8) = n_9\qquad\mbox{cnf}(\mbox{successor\_of\_8\_is}_9,\mbox{axiom})$

$\mbox{covered}' = \mbox{not\_covered}\qquad\mbox{cnf}(\mbox{complement\_of\_covered\_is\_not\_covered},\mbox{axiom})$

$\mbox{not\_covered}' = \mbox{covered}\qquad\mbox{cnf}(\mbox{complement\_of\_not\_covered\_is\_covered},\mbox{axiom})$

$\mbox{removed}' = \mbox{not\_covered}\qquad\mbox{cnf}(\mbox{complement\_of\_r\_is\_not\_covered},\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(n_1),\mbox{squares}(\mbox{removed},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered}))\qquad\mbox{cnf}(\mbox{first\_row\_initially},\mbox{hypothesis})$

$\neg\,\mbox{achievable}(\mbox{row}(n_8),\mbox{squares}(\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{not\_covered}))\qquad\mbox{cnf}(\mbox{try\_prove\_row\_8\_can\_be\_covered},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ028-1.p} People at a party

We can always choose 3 persons who are either familiar with
 each other or not familiar with each other, from 6 persons
 who meet at a party.




$\mbox{person}(1)\qquad\mbox{cnf}(\mbox{person}_1,\mbox{axiom})$

$\mbox{person}(\mbox{two})\qquad\mbox{cnf}(\mbox{person}_2,\mbox{axiom})$

$\mbox{person}(\mbox{three})\qquad\mbox{cnf}(\mbox{person}_3,\mbox{axiom})$

$\mbox{person}(\mbox{four})\qquad\mbox{cnf}(\mbox{person}_4,\mbox{axiom})$

$\mbox{person}(\mbox{five})\qquad\mbox{cnf}(\mbox{person}_5,\mbox{axiom})$

$\mbox{person}(\mbox{six})\qquad\mbox{cnf}(\mbox{person}_6,\mbox{axiom})$

$\mbox{after}(1,\mbox{two})\qquad\mbox{cnf}(\mbox{order}_1,\mbox{axiom})$

$\mbox{after}(\mbox{two},\mbox{three})\qquad\mbox{cnf}(\mbox{order}_2,\mbox{axiom})$

$\mbox{after}(\mbox{three},\mbox{four})\qquad\mbox{cnf}(\mbox{order}_3,\mbox{axiom})$

$\mbox{after}(\mbox{four},\mbox{five})\qquad\mbox{cnf}(\mbox{order}_4,\mbox{axiom})$

$\mbox{after}(\mbox{five},\mbox{six})\qquad\mbox{cnf}(\mbox{order}_5,\mbox{axiom})$

$( \mbox{after}(\mbox{large},\mbox{medium})\mbox{ and }\mbox{after}(\mbox{medium},\mbox{small}) ) \ \Rightarrow \ \mbox{after}(\mbox{large},\mbox{small})\qquad\mbox{cnf}(\mbox{transitivity\_of\_order},\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{person}(y)\mbox{ and }\mbox{after}(x,y) ) \ \Rightarrow \ ( \mbox{familiar}(x,y)\mbox{ or }\mbox{not\_familiar}(x,y) )\qquad\mbox{cnf}(\mbox{familiar\_or\_not},\mbox{axiom})$

$( \mbox{familiar}(x_1,x_2)\mbox{ and }\mbox{familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_familiar},\mbox{negated\_conjecture})$

$( \mbox{not\_familiar}(x_1,x_2)\mbox{ and }\mbox{not\_familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{not\_familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_not\_familiar},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ104$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cCKB6\_NUM}@ ( s@ ( s@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{4000},\mbox{conjecture})$

\medskip\textbf{PUZ133+2.p} N queens problem has the variable symmetry property






$\mbox{queens\_p} \ \Rightarrow \ \forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n) ) \ \Rightarrow \ ( p(i)\ne p(j)\mbox{ and }p(i) + i\ne p(j) + j\mbox{ and }-p(i)\ne -p(j) ) )\qquad\mbox{fof}(\mbox{queens\_p},\mbox{axiom})$

$\forall i{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n) ) \ \Rightarrow \ \mbox{perm}(i) = -s(n) )\qquad\mbox{fof}(\mbox{permutation},\mbox{axiom})$

$\forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n)\mbox{ and }( \mbox{le}(s(i),j)\iff \mbox{le}(s(\mbox{perm}(j)),\mbox{perm}(i)) ) ) \ \Rightarrow \ ( q(i)\ne q(j)\mbox{ and }q(i) + i\ne q(j) + j\mbox{ and }-q(i)\ne -q(j) ) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_q},\mbox{axiom})$

$( \mbox{queens\_p}\mbox{ and }\forall i{:}\;q(i) = p(\mbox{perm}(i)) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_sym},\mbox{conjecture})$

$\forall i{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n) ) \ \Rightarrow \ ( \mbox{le}(s(n_0),\mbox{perm}(i))\mbox{ and }\mbox{le}(\mbox{perm}(i),n) ) )\qquad\mbox{fof}(\mbox{permutation\_range},\mbox{axiom})$

$\forall j,i{:}\;-i = -\mbox{perm}(j)\qquad\mbox{fof}(\mbox{permutation\_another\_one},\mbox{axiom})$

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

$\forall x{:}\;\mbox{le}(x,s(x))\qquad\mbox{fof}(\mbox{succ\_le},\mbox{axiom})$

$\forall i,j,k,l{:}\;( i + j = k + l\iff -i = -l )\qquad\mbox{fof}(\mbox{plus}_1,\mbox{axiom})$

$\forall i,j,k,l{:}\;( -i = -k\iff -i = -j )\qquad\mbox{fof}(\mbox{minus}_1,\mbox{axiom})$

\medskip\textbf{PUZ136$\wedge$1.p} Under two assumptions there are at least two individuals.






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

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

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

$\mbox{parent}@ \mbox{kronus}@ \mbox{zeus}\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

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

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

$\neg\,\mbox{parent}@ \mbox{sutekh}@ \mbox{horus}\qquad\mbox{thf}(\mbox{ax}_2,\mbox{axiom})$

$\exists p{:}\;\mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( p@ x\mbox{ and }\neg\,p@ y )\qquad\mbox{thf}(\mbox{hotwo},\mbox{conjecture})$

\medskip\textbf{PUZ001-3.p} Dreadbury Mansion

Someone who lives in Dreadbury Mansion killed Aunt Agatha.
 Agatha, the butler, and Charles live in Dreadbury Mansion,
 and are the only people who live therein. A killer always
 hates his victim, and is never richer than his victim.
 Charles hates no one that Aunt Agatha hates. Agatha hates
 everyone except the butler. The butler hates everyone not
 richer than Aunt Agatha. The butler hates everyone Aunt
 Agatha hates. No one hates everyone. Agatha is not the
 butler. Therefore : Agatha killed herself.




$\mbox{lives}(\mbox{agatha})\qquad\mbox{cnf}(\mbox{agatha},\mbox{axiom})$

$\mbox{lives}(\mbox{butler})\qquad\mbox{cnf}(\mbox{butler},\mbox{axiom})$

$\mbox{lives}(\mbox{charles})\qquad\mbox{cnf}(\mbox{charles},\mbox{axiom})$

$\mbox{killed}(x,y) \ \Rightarrow \ \neg\,\mbox{richer}(x,y)\qquad\mbox{cnf}(\mbox{poorer\_killer},\mbox{axiom})$

$\mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \neg\,\mbox{hates}(\mbox{charles},x)\qquad\mbox{cnf}(\mbox{different\_hates},\mbox{axiom})$

$( \mbox{hates}(x,\mbox{agatha})\mbox{ and }\mbox{hates}(x,\mbox{butler}) ) \ \Rightarrow \ \neg\,\mbox{hates}(x,\mbox{charles})\qquad\mbox{cnf}(\mbox{no\_one\_hates\_everyone},\mbox{axiom})$

$\mbox{hates}(\mbox{agatha},\mbox{agatha})\qquad\mbox{cnf}(\mbox{agatha\_hates\_agatha},\mbox{axiom})$

$\mbox{hates}(\mbox{agatha},\mbox{charles})\qquad\mbox{cnf}(\mbox{agatha\_hates\_charles},\mbox{axiom})$

$\mbox{killed}(x,y) \ \Rightarrow \ \mbox{hates}(x,y)\qquad\mbox{cnf}(\mbox{killer\_hates\_victim},\mbox{axiom})$

$\mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \mbox{hates}(\mbox{butler},x)\qquad\mbox{cnf}(\mbox{same\_hates},\mbox{axiom})$

$\mbox{lives}(x) \ \Rightarrow \ ( \mbox{richer}(x,\mbox{agatha})\mbox{ or }\mbox{hates}(\mbox{butler},x) )\qquad\mbox{cnf}(\mbox{butler\_hates\_poor},\mbox{axiom})$

$\mbox{killed}(\mbox{agatha},\mbox{agatha})\mbox{ or }\mbox{killed}(\mbox{butler},\mbox{agatha})\mbox{ or }\mbox{killed}(\mbox{charles},\mbox{agatha})\qquad\mbox{cnf}(\mbox{somebody\_did\_it},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ069+1.p} Tuesday's Sudoku

1 76
 85
 2    3
 24
 9      6
 53
 7   1
 3    2




$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_1) = n_1\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_3) = n_7\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_4) = n_6\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_6) = n_8\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_7) = n_5\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_3,n_2) = n_2\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_3,n_7) = n_3\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_4) = n_2\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_4\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_9\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_8) = n_6\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_6,n_5) = n_5\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_6,n_6) = n_3\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_7,n_8) = n_1\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_2) = n_3\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_8,n_7) = n_2\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ127$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_5{:}\;\mbox{\$i}\qquad\mbox{thf}(c_5,\mbox{type})$

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

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(c_4,\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(c_3,\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(c_2,\mbox{type})$

$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(c_1,\mbox{type})$

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

$c_8{:}\;\mbox{\$i}\qquad\mbox{thf}(c_8,\mbox{type})$

$\neg\,( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_8 ) ) ) ) ) ) ) ) = c_8\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( s@ ( s@ ( s@ ( s@ \mbox{xx} ) ) ) )\ne \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx}@ \mbox{xy} ) = c_5\iff ( ( \mbox{xx} = c_8\mbox{ and }\mbox{xy} = c_8 )\mbox{ or }( \mbox{xx} = c_1\mbox{ and }\mbox{xy} = c_1 ) ) )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx}@ \mbox{xy} ) = c_1\iff ( g@ ( s@ \mbox{xx} )@ \mbox{xy} ) = c_3 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx}@ \mbox{xy} ) = c_2\iff ( g@ \mbox{xx}@ ( s@ \mbox{xy} ) ) = c_4 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ c_1@ \mbox{xy} )\ne c_3\mbox{ and }( g@ c_8@ \mbox{xy} )\ne c_1\mbox{ and }( g@ \mbox{xx}@ c_1 )\ne c_4\mbox{ and }( g@ \mbox{xx}@ c_8 )\ne c_2 )\mbox{ and }c_1 = ( s@ c_8 )\mbox{ and }c_2 = ( s@ c_1 )\mbox{ and }c_3 = ( s@ c_2 )\mbox{ and }c_4 = ( s@ c_3 )\mbox{ and }c_5 = ( s@ c_4 )\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx}@ \mbox{xy} ) = c_1\mbox{ or }( g@ \mbox{xx}@ \mbox{xy} ) = c_2\mbox{ or }( g@ \mbox{xx}@ \mbox{xy} ) = c_3\mbox{ or }( g@ \mbox{xx}@ \mbox{xy} ) = c_4\mbox{ or }( g@ \mbox{xx}@ \mbox{xy} ) = c_5 )\qquad\mbox{thf}(\mbox{cTOUGHNUT}_2,\mbox{conjecture})$

\medskip\textbf{PUZ036-1.005.p} TopSpin

TopSpin consists of a circular track with 20 pieces numbered
 1..20 placed in the track, with a turnstile in the track that
 always holds four consecutive pieces. There are three legal
 moves in TopSpin: slide all the pieces round the track in
 either direction, or flip the turnstile. Given any initial
 board with scrambled pieces on the track, the problem is to
 find a sequence of moves that unscrambles the pieces.




$\neg\,\mbox{state}(p_1,p_2,p_3,p_4,p_5,p_6,p_7,p_8,p_9,p_{10},p_{11},p_{12},p_{13},p_{14},p_{15},p_{16},p_{17},p_{18},p_{19},p_{20})\qquad\mbox{cnf}(\mbox{make\_like\_this},\mbox{negated\_conjecture})$

$\mbox{state}(p_5,p_4,p_3,p_2,p_1,p_6,p_7,p_8,p_9,p_{10},p_{11},p_{12},p_{13},p_{14},p_{15},p_{16},p_{17},p_{18},p_{19},p_{20})\qquad\mbox{cnf}(\mbox{initial\_configuration},\mbox{hypothesis})$

$\mbox{state}(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20}) \ \Rightarrow \ \mbox{state}(x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20},x_1)\qquad\mbox{cnf}(\mbox{move\_left},\mbox{axiom})$

$\mbox{state}(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20}) \ \Rightarrow \ \mbox{state}(x_{20},x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19})\qquad\mbox{cnf}(\mbox{move\_right},\mbox{axiom})$

$\mbox{state}(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20}) \ \Rightarrow \ \mbox{state}(x_4,x_3,x_2,x_1,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20})\qquad\mbox{cnf}(\mbox{flip},\mbox{axiom})$

\medskip\textbf{PUZ031+1.p} Schubert's Steamroller

Wolves, foxes, birds, caterpillars, and snails are animals, and
 there are some of each of them. Also there are some grains, and
 grains are plants. Every animal either likes to eat all plants
 or all animals much smaller than itself that like to eat some
 plants. Caterpillars and snails are much smaller than birds,
 which are much smaller than foxes, which in turn are much
 smaller than wolves. Wolves do not like to eat foxes or grains,
 while birds like to eat caterpillars but not snails.
 Caterpillars and snails like to eat some plants. Therefore
 there is an animal that likes to eat a grain eating animal.




$\forall x{:}\;( \mbox{wolf}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_1}_1,\mbox{axiom})$

$\exists x_1{:}\;\mbox{wolf}(x_1)\qquad\mbox{fof}(\mbox{pel47\_1}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{fox}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_2}_1,\mbox{axiom})$

$\exists x_1{:}\;\mbox{fox}(x_1)\qquad\mbox{fof}(\mbox{pel47\_2}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{bird}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_3}_1,\mbox{axiom})$

$\exists x_1{:}\;\mbox{bird}(x_1)\qquad\mbox{fof}(\mbox{pel47\_3}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{caterpillar}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_4}_1,\mbox{axiom})$

$\exists x_1{:}\;\mbox{caterpillar}(x_1)\qquad\mbox{fof}(\mbox{pel47\_4}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{snail}(x) \ \Rightarrow \ \mbox{animal}(x) )\qquad\mbox{fof}(\mbox{pel47\_5}_1,\mbox{axiom})$

$\exists x_1{:}\;\mbox{snail}(x_1)\qquad\mbox{fof}(\mbox{pel47\_5}_2,\mbox{axiom})$

$\exists x{:}\;\mbox{grain}(x)\qquad\mbox{fof}(\mbox{pel47\_6}_1,\mbox{axiom})$

$\forall x_1{:}\;( \mbox{grain}(x_1) \ \Rightarrow \ \mbox{plant}(x_1) )\qquad\mbox{fof}(\mbox{pel47\_6}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{animal}(x) \ \Rightarrow \ ( \forall y{:}\;( \mbox{plant}(y) \ \Rightarrow \ \mbox{eats}(x,y) )\mbox{ or }\forall y_1{:}\;( ( \mbox{animal}(y_1)\mbox{ and }\mbox{much\_smaller}(y_1,x)\mbox{ and }\exists z{:}\;( \mbox{plant}(z)\mbox{ and }\mbox{eats}(y_1,z) ) ) \ \Rightarrow \ \mbox{eats}(x,y_1) ) ) )\qquad\mbox{fof}(\mbox{pel47}_7,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(y)\mbox{ and }( \mbox{snail}(x)\mbox{ or }\mbox{caterpillar}(x) ) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_8,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{fox}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_9,\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{fox}(x)\mbox{ and }\mbox{wolf}(y) ) \ \Rightarrow \ \mbox{much\_smaller}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{10},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{wolf}(x)\mbox{ and }( \mbox{fox}(y)\mbox{ or }\mbox{grain}(y) ) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{11},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{caterpillar}(y) ) \ \Rightarrow \ \mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{12},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{bird}(x)\mbox{ and }\mbox{snail}(y) ) \ \Rightarrow \ \neg\,\mbox{eats}(x,y) )\qquad\mbox{fof}(\mbox{pel47}_{13},\mbox{axiom})$

$\forall x{:}\;( ( \mbox{caterpillar}(x)\mbox{ or }\mbox{snail}(x) ) \ \Rightarrow \ \exists y{:}\;( \mbox{plant}(y)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel47}_{14},\mbox{axiom})$

$\exists x,y{:}\;( \mbox{animal}(x)\mbox{ and }\mbox{animal}(y)\mbox{ and }\exists z{:}\;( \mbox{grain}(z)\mbox{ and }\mbox{eats}(y,z)\mbox{ and }\mbox{eats}(x,y) ) )\qquad\mbox{fof}(\mbox{pel}_{47},\mbox{conjecture})$

\medskip\textbf{PUZ024-1.p} Knights and Knaves \#31

There is an island with exactly two types of people -
 truthtellers who always tell the truth and liars who always lie.
 There are a group of three people, A, B, and C on the island. A
 and B make the following statements. A: All of us are liars;
 B: Exactly one of us is a truthteller. What are A, B, and C?
 Answer: A is a liar, B is a truthteller, and C is a liar.




$\mbox{include}(\mbox{'Axioms/PUZ002-0.ax'})$

$( \mbox{a\_truth}(\mbox{says}(x,\mbox{one\_truthteller}))\mbox{ and }\mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(z))\qquad\mbox{cnf}(\mbox{one\_says\_one\_truthteller},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ and }\mbox{people}(x,y,z) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{one\_is\_the\_truth\_teller},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ and }\mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(z))\qquad\mbox{cnf}(\mbox{one\_is\_not\_the\_truthteller},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ and }\mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{liar}(z))\qquad\mbox{cnf}(\mbox{one\_truthteller\_two\_liars},\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{two\_truthtellers},\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(y)) ) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{one\_truthteller})\mbox{ or }\mbox{a\_truth}(\mbox{liar}(z)) )\qquad\mbox{cnf}(\mbox{three\_liars},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{says}(x,\mbox{all\_are\_liars}))\mbox{ and }\mbox{people}(x,y,z) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(x))\qquad\mbox{cnf}(\mbox{speaker\_is\_lying},\mbox{axiom})$

$\mbox{people}(x,y,z) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{all\_are\_liars})\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(z)) )\qquad\mbox{cnf}(\mbox{all\_truthtellers},\mbox{axiom})$

$\mbox{people}(b,c,a)\qquad\mbox{cnf}(\mbox{b\_c\_a\_people},\mbox{hypothesis})$

$\mbox{people}(a,c,b)\qquad\mbox{cnf}(\mbox{a\_c\_b\_people},\mbox{hypothesis})$

$\mbox{people}(c,b,a)\qquad\mbox{cnf}(\mbox{c\_b\_a\_people},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{all\_are\_liars}))\qquad\mbox{cnf}(\mbox{a\_says\_all\_are\_liars},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(b,\mbox{one\_truthteller}))\qquad\mbox{cnf}(\mbox{b\_says\_one\_truthteller},\mbox{hypothesis})$

$\neg\,\mbox{a\_truth}(\mbox{truthteller}(\mbox{truthteller}))\qquad\mbox{cnf}(\mbox{prove\_there\_is\_a\_truthteller},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ043-1.p} Mars and Venus axioms






$\mbox{include}(\mbox{'Axioms/PUZ001-0.ax'})$

\medskip\textbf{PUZ113$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cCKB6\_BLACK}@ ( s@ ( s@ \mbox{xj} ) )@ ( s@ ( s@ \mbox{xk} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{11000},\mbox{conjecture})$

\medskip\textbf{PUZ012-1.p} The Mislabeled Boxes

There are three boxes a, b, and c on a table. Each box contains
 apples or bananas or oranges. No two boxes contain the same
 thing. Each box has a label that says it contains apples or says
 it contains bananas or says it contains oranges. No box contains
 what it says on its label. The label on box a says "apples".
 The label on box b says "oranges". The label on box c says
 "bananas". You pick up box b and it contains apples. What do
 the other two boxes contain?




$\mbox{equal\_fruits}(x,x)\qquad\mbox{cnf}(\mbox{reflexivity\_for\_fruits},\mbox{axiom})$

$\mbox{equal\_boxes}(x,x)\qquad\mbox{cnf}(\mbox{reflexivity\_for\_boxes},\mbox{axiom})$

$\mbox{label}(x,y) \ \Rightarrow \ \neg\,\mbox{contains}(x,y)\qquad\mbox{cnf}(\mbox{label\_is\_wrong},\mbox{axiom})$

$\mbox{contains}(\mbox{boxa},x)\mbox{ or }\mbox{contains}(\mbox{boxb},x)\mbox{ or }\mbox{contains}(\mbox{boxc},x)\qquad\mbox{cnf}(\mbox{each\_thing\_is\_in\_a\_box},\mbox{axiom})$

$\mbox{contains}(x,\mbox{apples})\mbox{ or }\mbox{contains}(x,\mbox{bananas})\mbox{ or }\mbox{contains}(x,\mbox{oranges})\qquad\mbox{cnf}(\mbox{each\_box\_contains\_something},\mbox{axiom})$

$( \mbox{contains}(x,y)\mbox{ and }\mbox{contains}(x,z) ) \ \Rightarrow \ \mbox{equal\_fruits}(y,z)\qquad\mbox{cnf}(\mbox{contains\_is\_well\_defined}_1,\mbox{axiom})$

$( \mbox{contains}(x,y)\mbox{ and }\mbox{contains}(z,y) ) \ \Rightarrow \ \mbox{equal\_boxes}(x,z)\qquad\mbox{cnf}(\mbox{contains\_is\_well\_defined}_2,\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxa},\mbox{boxb})\qquad\mbox{cnf}(\mbox{boxa\_not\_boxb},\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxb},\mbox{boxc})\qquad\mbox{cnf}(\mbox{boxb\_not\_boxc},\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxa},\mbox{boxc})\qquad\mbox{cnf}(\mbox{boxa\_not\_boxc},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{apples},\mbox{bananas})\qquad\mbox{cnf}(\mbox{apples\_not\_bananas},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{bananas},\mbox{oranges})\qquad\mbox{cnf}(\mbox{bananas\_not\_oranges},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{apples},\mbox{oranges})\qquad\mbox{cnf}(\mbox{apples\_not\_oranges},\mbox{axiom})$

$\mbox{label}(\mbox{boxa},\mbox{apples})\qquad\mbox{cnf}(\mbox{boxa\_labelled\_apples},\mbox{hypothesis})$

$\mbox{label}(\mbox{boxb},\mbox{oranges})\qquad\mbox{cnf}(\mbox{boxb\_labelled\_oranges},\mbox{hypothesis})$

$\mbox{label}(\mbox{boxc},\mbox{bananas})\qquad\mbox{cnf}(\mbox{boxc\_labelled\_bananas},\mbox{hypothesis})$

$\mbox{contains}(\mbox{boxb},\mbox{apples})\qquad\mbox{cnf}(\mbox{boxb\_contains\_apples},\mbox{hypothesis})$

$\mbox{contains}(\mbox{boxa},\mbox{bananas}) \ \Rightarrow \ \neg\,\mbox{contains}(\mbox{boxc},\mbox{oranges})\qquad\mbox{cnf}(\mbox{prove\_boxa\_contains\_bananas\_and\_boxc\_oranges},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ009-1.p} Looking for Oona

In another curious incident, when the husband arrived on an
 island looking for Oona, he met 5 natives A,B,C,D,E who all
 guessed his purpose and grinned at meeting him. They said:
 A: Oona is on this island. B: Oona is not on this island.
 C: Oona was here yesterday. D: Oona is not here today, and she
 was not here yesterday. E: Either D is a knave or C is a knight.
 The logician thought for a while, but could get nowhere. `Won't
 one of you please make another statement?' the logician pleaded.
 At this point A said: Either E is a knave or C is a knight. Is
 Oona on the island?"




$\mbox{a\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{b\_is\_a\_knight}\qquad\mbox{cnf}(c_1,\mbox{axiom})$

$\mbox{a\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{d\_is\_a\_knight}\qquad\mbox{cnf}(c_2,\mbox{axiom})$

$\mbox{b\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{a\_is\_a\_knight}\qquad\mbox{cnf}(c_3,\mbox{axiom})$

$\mbox{b\_is\_a\_knight}\mbox{ or }\mbox{a\_is\_a\_knight}\qquad\mbox{cnf}(c_4,\mbox{axiom})$

$\mbox{b\_is\_a\_knight} \ \Rightarrow \ ( \mbox{c\_is\_a\_knight}\mbox{ or }\mbox{d\_is\_a\_knight} )\qquad\mbox{cnf}(c_5,\mbox{axiom})$

$\mbox{c\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{d\_is\_a\_knight}\qquad\mbox{cnf}(c_6,\mbox{axiom})$

$\mbox{d\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{a\_is\_a\_knight}\qquad\mbox{cnf}(c_7,\mbox{axiom})$

$\mbox{d\_is\_a\_knight} \ \Rightarrow \ \mbox{b\_is\_a\_knight}\qquad\mbox{cnf}(c_8,\mbox{axiom})$

$\mbox{d\_is\_a\_knight} \ \Rightarrow \ \neg\,\mbox{c\_is\_a\_knight}\qquad\mbox{cnf}(c_9,\mbox{axiom})$

$( \mbox{e\_is\_a\_knight}\mbox{ and }\mbox{d\_is\_a\_knight} ) \ \Rightarrow \ \mbox{c\_is\_a\_knight}\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{e\_is\_a\_knight}\mbox{ or }\mbox{d\_is\_a\_knight}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{c\_is\_a\_knight} \ \Rightarrow \ \mbox{e\_is\_a\_knight}\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{d\_is\_a\_knight}\mbox{ or }\mbox{a\_is\_a\_knight}\mbox{ or }\mbox{c\_is\_a\_knight}\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{b\_is\_a\_knight} \ \Rightarrow \ ( \mbox{d\_is\_a\_knight}\mbox{ or }\mbox{c\_is\_a\_knight} )\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$( \mbox{a\_is\_a\_knight}\mbox{ and }\mbox{e\_is\_a\_knight} ) \ \Rightarrow \ \mbox{c\_is\_a\_knight}\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{a\_is\_a\_knight}\mbox{ or }\mbox{e\_is\_a\_knight}\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{c\_is\_a\_knight} \ \Rightarrow \ \mbox{a\_is\_a\_knight}\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

$\mbox{b\_is\_a\_knight}\qquad\mbox{cnf}(c_{18},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ093$\wedge$5.p} TPS problem from BASIC-HO-EQ-THMS






$\exists \mbox{xa}{:}\;\mbox{\$i},\mbox{xaa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i},\mbox{xbb}{:}\;\mbox{\$i},\mbox{xc}{:}\;\mbox{\$i},\mbox{xcc}{:}\;\mbox{\$i},\mbox{xd}{:}\;\mbox{\$i},\mbox{xdd}{:}\;\mbox{\$i},\mbox{xe}{:}\;\mbox{\$i},\mbox{xee}{:}\;\mbox{\$i},\mbox{xh}{:}\;\mbox{\$i},\mbox{xhh}{:}\;\mbox{\$i}{:}\;\forall p{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( ( ( ( p@ \mbox{xa} ) = ( p@ \mbox{xaa} )\mbox{ and }( p@ \mbox{xb} ) = ( p@ \mbox{xbb} )\mbox{ and }( p@ \mbox{xe} ) = ( p@ \mbox{xhh} ) ) \ \Rightarrow \ ( p@ \mbox{xc} ) = ( p@ \mbox{xdd} ) )\mbox{ and }( ( ( p@ \mbox{xa} ) = ( p@ \mbox{xaa} )\mbox{ and }( p@ \mbox{xh} ) = ( p@ \mbox{xhh} )\mbox{ and }( p@ \mbox{xb} ) = ( p@ \mbox{xcc} ) ) \ \Rightarrow \ ( p@ \mbox{xd} )\ne ( p@ \mbox{xee} ) )\mbox{ and }( ( ( p@ \mbox{xc} ) = ( p@ \mbox{xcc} )\mbox{ and }( p@ \mbox{xcc} ) = ( p@ \mbox{xd} )\mbox{ and }( p@ \mbox{xd} ) = ( p@ \mbox{xdd} )\mbox{ and }( p@ \mbox{xa} )\ne ( p@ \mbox{xbb} ) ) \ \Rightarrow \ ( p@ \mbox{xe} )\ne ( p@ \mbox{xhh} ) )\mbox{ and }( ( ( p@ \mbox{xa} ) = ( p@ \mbox{xaa} )\mbox{ and }( p@ \mbox{xd} ) = ( p@ \mbox{xdd} )\mbox{ and }( p@ \mbox{xb} )\ne ( p@ \mbox{xcc} ) ) \ \Rightarrow \ ( p@ \mbox{xe} ) = ( p@ \mbox{xhh} ) )\mbox{ and }( ( ( p@ \mbox{xe} ) = ( p@ \mbox{xee} )\mbox{ and }( p@ \mbox{xh} ) = ( p@ \mbox{xhh} )\mbox{ and }( p@ \mbox{xc} ) = ( p@ \mbox{xdd} ) ) \ \Rightarrow \ ( p@ \mbox{xa} )\ne ( p@ \mbox{xbb} ) )\mbox{ and }( ( ( p@ \mbox{xb} ) = ( p@ \mbox{xbb} )\mbox{ and }( p@ \mbox{xbb} ) = ( p@ \mbox{xc} )\mbox{ and }( p@ \mbox{xc} ) = ( p@ \mbox{xcc} )\mbox{ and }( p@ \mbox{xe} )\ne ( p@ \mbox{xhh} ) ) \ \Rightarrow \ ( p@ \mbox{xd} ) = ( p@ \mbox{xee} ) ) ) \ \Rightarrow \ ( ( p@ \mbox{xa} )\ne ( p@ \mbox{xaa} )\mbox{ or }( p@ \mbox{xb} )\ne ( p@ \mbox{xbb} )\mbox{ or }( p@ \mbox{xc} )\ne ( p@ \mbox{xcc} )\mbox{ or }( p@ \mbox{xd} )\ne ( p@ \mbox{xdd} )\mbox{ or }( p@ \mbox{xe} )\ne ( p@ \mbox{xee} )\mbox{ or }( p@ \mbox{xh} )\ne ( p@ \mbox{xhh} ) ) )\qquad\mbox{thf}(\mbox{cSIXFRIENDS\_PTH},\mbox{conjecture})$

\medskip\textbf{PUZ088$\wedge$5.p} TPS problem THM68

If everyone likes Bruce and Lyle likes everyone who likes someone
 then someone likes everyone.




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

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

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

$( \forall x{:}\;\mbox{\$i}{:}\;( \mbox{cLIKES}@ x@ \mbox{cBRUCE} )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( \exists z{:}\;\mbox{\$i}{:}\;( \mbox{cLIKES}@ y@ z ) \ \Rightarrow \ ( \mbox{cLIKES}@ \mbox{cLYLE}@ y ) ) ) \ \Rightarrow \ \exists u{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( \mbox{cLIKES}@ u@ v )\qquad\mbox{thf}(\mbox{cTHM68A},\mbox{conjecture})$

\medskip\textbf{PUZ062-2.p} Problem about mutilated chessboard problem






$\mbox{c\_in}(\mbox{v\_A},\mbox{c\_Finite\_\_Set\_OFinites},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_Finite\_\_Set\_Ocard}(\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) = \mbox{c\_Suc}(\mbox{c\_Finite\_\_Set\_Ocard}(\mbox{v\_A},\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Finite\_\_Set\_Ocard\_\_insert\_\_disjoint}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_G},\mbox{c\_Finite\_\_Set\_OFinites},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_inter}(\mbox{v\_F},\mbox{v\_G},\mbox{t\_a}),\mbox{c\_Finite\_\_Set\_OFinites},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Finite\_\_Set\_Ofinite\_\_Int}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{c\_Mutil\_Odomino},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_t},\mbox{c\_Finite\_\_Set\_OFinites},\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_\_domino\_\_finite}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OInt\_\_iff}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OInt\_\_iff}_2,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_Oinsert\_\_iff}_1,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_c},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_Oempty\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{c\_Mutil\_Odomino},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_Finite\_\_Set\_Ocard}(\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(c_0),\mbox{v\_t},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) = \mbox{c\_Finite\_\_Set\_Ocard}(\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(\mbox{c\_Suc}(c_0)),\mbox{v\_t},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(c_0),\mbox{v\_a},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) = \mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(\mbox{c\_Suc}(c_0)),\mbox{v\_a},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) = \mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_m},\mbox{v\_n},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_Finite\_\_Set\_Ocard}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(c_0),\mbox{v\_t},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\ne \mbox{c\_Finite\_\_Set\_Ocard}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_m},\mbox{v\_n},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_inter}(\mbox{c\_Mutil\_Ocoloured}(\mbox{c\_Suc}(c_0)),\mbox{v\_t},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{PUZ090$\wedge$5.p} TPS problem THM210

Lewis Carroll's problem of the winds and windows; from the Ninth 
 Paper on Logic, November 1892.




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

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

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

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

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

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

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

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

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

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

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

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

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

$( ( \mbox{cEAST} \ \Rightarrow \ \mbox{cSUNSHINE} )\mbox{ and }( ( \mbox{cCOLD}\mbox{ and }\mbox{cFOGGY} ) \ \Rightarrow \ \mbox{cFLUTE} )\mbox{ and }( ( \mbox{cFIRE}\mbox{ and }\mbox{cSMOKE} ) \ \Rightarrow \ \mbox{cOPEN\_DOOR} )\mbox{ and }( ( \mbox{cCOLD}\mbox{ and }\mbox{cRHEUMATIC} ) \ \Rightarrow \ \mbox{cFIRE} )\mbox{ and }( ( \mbox{cEAST}\mbox{ and }\mbox{cGUSTY} ) \ \Rightarrow \ \mbox{cSMOKES} )\mbox{ and }( \mbox{cOPEN\_DOOR} \ \Rightarrow \ \neg\,\mbox{cHEADACHE} )\mbox{ and }( \mbox{cFOGGY} \ \Rightarrow \ \neg\,\mbox{cOPEN\_WINDOW} )\mbox{ and }( ( \neg\,\mbox{cGUSTY}\mbox{ and }\mbox{cFIRE}\mbox{ and }\neg\,\mbox{cOPEN\_DOOR} ) \ \Rightarrow \ \neg\,\mbox{cRHEUMATIC} )\mbox{ and }( \mbox{cSUNSHINE} \ \Rightarrow \ \mbox{cFOGGY} )\mbox{ and }( \mbox{cFLUTE} \ \Rightarrow \ \neg\,\mbox{cOPEN\_DOOR} )\mbox{ and }( ( \mbox{cFOGGY}\mbox{ and }\mbox{cEAST} ) \ \Rightarrow \ \mbox{cRHEUMATIC} ) ) \ \Rightarrow \ ( \mbox{cEAST} \ \Rightarrow \ \neg\,\mbox{cOPEN\_WINDOW} )\qquad\mbox{thf}(\mbox{cTHM}_{210},\mbox{conjecture})$

\medskip\textbf{PUZ140$\wedge$2.p} A mixture of coffee and syrup that is hot






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

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

$\mbox{coffee}{:}\;\mbox{beverage}\qquad\mbox{thf}(\mbox{coffee\_type},\mbox{type})$

$\mbox{mix}{:}\;\mbox{beverage}\to \mbox{syrup}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{mix\_type},\mbox{type})$

$\mbox{coffee\_mixture}{:}\;\mbox{syrup}\to \mbox{beverage}\qquad\mbox{thf}(\mbox{coffee\_mixture\_type},\mbox{type})$

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

$\mbox{coffee\_mixture} = ( \mbox{mix}@ \mbox{coffee} )\qquad\mbox{thf}(\mbox{coffee\_mixture\_definition},\mbox{definition})$

$\forall s{:}\;\mbox{syrup}{:}\;( ( \mbox{coffee\_mixture}@ s ) = \mbox{coffee}\mbox{ and }\mbox{hot}@ ( \mbox{coffee\_mixture}@ s ) )\qquad\mbox{thf}(\mbox{coffee\_and\_syrup\_is\_hot\_coffee},\mbox{axiom})$

$\exists \mbox{syrupMixer}{:}\;\mbox{syrup}\to \mbox{beverage}{:}\;\forall s{:}\;\mbox{syrup}{:}\;\exists b{:}\;\mbox{beverage}{:}\;( b = ( \mbox{syrupMixer}@ s )\mbox{ and }b = \mbox{coffee}\mbox{ and }\mbox{hot}@ b )\qquad\mbox{thf}(\mbox{there\_is\_hot\_coffee},\mbox{conjecture})$

\medskip\textbf{PUZ038-1.p} Quo vadis 1

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is a simple goal for testing.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(b,v_1,v_2,v_3(s(s(s(o))),o),v_4,h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ091$\wedge$5.p} TPS problem from BASIC-FO-THMS






$\mbox{cIN\_BED\_BEFORE\_FOUR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cIN\_BED\_BEFORE\_FOUR},\mbox{type})$

$\mbox{cRISES\_AT\_FIVE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cRISES\_AT\_FIVE},\mbox{type})$

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

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

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

$\mbox{cLIKELY\_LOSE\_MONEY}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cLIKELY\_LOSE\_MONEY},\mbox{type})$

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

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

$\mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING},\mbox{type})$

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

$\mbox{cHAS\_LOST\_MONEY}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cHAS\_LOST\_MONEY},\mbox{type})$

$( ( ( \mbox{cLOGICIAN}\mbox{ and }\mbox{cEATS\_PORKCHOPS} ) \ \Rightarrow \ \mbox{cLIKELY\_LOSE\_MONEY} )\mbox{ and }( ( \mbox{cGAMBLER}\mbox{ and }\neg\,\mbox{cRAVENOUS} ) \ \Rightarrow \ \mbox{cLIKELY\_LOSE\_MONEY} )\mbox{ and }( ( \neg\,\mbox{cLIVELY}\mbox{ and }\mbox{cHAS\_LOST\_MONEY}\mbox{ and }\mbox{cLIKELY\_LOSE\_MONEY} ) \ \Rightarrow \ \mbox{cRISES\_AT\_FIVE} )\mbox{ and }( ( \neg\,\mbox{cGAMBLER}\mbox{ and }\neg\,\mbox{cEATS\_PORKCHOPS} ) \ \Rightarrow \ \mbox{cRAVENOUS} )\mbox{ and }( ( \mbox{cLIVELY}\mbox{ and }\mbox{cIN\_BED\_BEFORE\_FOUR} ) \ \Rightarrow \ \mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING} )\mbox{ and }( ( \mbox{cRAVENOUS}\mbox{ and }\neg\,\mbox{cHAS\_LOST\_MONEY}\mbox{ and }\neg\,\mbox{cRISES\_AT\_FIVE} ) \ \Rightarrow \ \mbox{cEATS\_PORKCHOPS} )\mbox{ and }( ( \mbox{cLOGICIAN}\mbox{ and }\mbox{cLIKELY\_LOSE\_MONEY} ) \ \Rightarrow \ \mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING} )\mbox{ and }( ( \mbox{cEARNEST}\mbox{ and }\neg\,\mbox{cLIVELY}\mbox{ and }\mbox{cGAMBLER}\mbox{ and }\neg\,\mbox{cHAS\_LOST\_MONEY} ) \ \Rightarrow \ \neg\,\mbox{cLIKELY\_LOSE\_MONEY} )\mbox{ and }( ( \neg\,\mbox{cGAMBLER}\mbox{ and }\neg\,\mbox{cRAVENOUS} ) \ \Rightarrow \ \mbox{cLIVELY} )\mbox{ and }( ( \mbox{cLIVELY}\mbox{ and }\mbox{cLOGICIAN}\mbox{ and }\mbox{cEARNEST} ) \ \Rightarrow \ \neg\,\mbox{cLIKELY\_LOSE\_MONEY} )\mbox{ and }( ( \mbox{cRAVENOUS}\mbox{ and }\mbox{cEARNEST} ) \ \Rightarrow \ \neg\,\mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING} )\mbox{ and }( ( \mbox{cGAMBLER}\mbox{ and }\neg\,\mbox{cLIVELY}\mbox{ and }\neg\,\mbox{cLIKELY\_LOSE\_MONEY} ) \ \Rightarrow \ \neg\,\mbox{cIN\_BED\_BEFORE\_FOUR} )\mbox{ and }( ( \mbox{cHAS\_LOST\_MONEY}\mbox{ and }\neg\,\mbox{cEATS\_PORKCHOPS}\mbox{ and }\neg\,\mbox{cRISES\_AT\_FIVE} ) \ \Rightarrow \ \mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING} )\mbox{ and }( ( \mbox{cGAMBLER}\mbox{ and }\mbox{cIN\_BED\_BEFORE\_FOUR}\mbox{ and }\neg\,\mbox{cRAVENOUS} ) \ \Rightarrow \ \neg\,\mbox{cBETTER\_TAKE\_TO\_CAB\_DRIVING} )\mbox{ and }( ( \mbox{cRAVENOUS}\mbox{ and }\neg\,\mbox{cLIVELY}\mbox{ and }\neg\,\mbox{cLIKELY\_LOSE\_MONEY} ) \ \Rightarrow \ \mbox{cGAMBLER} ) ) \ \Rightarrow \ ( ( \mbox{cEARNEST}\mbox{ and }\mbox{cLOGICIAN} ) \ \Rightarrow \ ( \mbox{cRISES\_AT\_FIVE}\mbox{ and }\neg\,\mbox{cIN\_BED\_BEFORE\_FOUR} ) )\qquad\mbox{thf}(\mbox{cPORKCHOP}_1,\mbox{conjecture})$

\medskip\textbf{PUZ050-1.p} Quo vadis 6 - initial to intermediate

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is the true goal from the puzzle.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(o,o),v_1(s(s(o)),o),v_2(o,s(s(s(o)))),v_3(o,s(s(o))),v_4(s(s(s(o))),s(s(o))),h(s(s(s(s(o)))),o),s_1(s(s(o)),s(o)),s_2(s(s(o)),s(s(o))),s_3(s(s(o)),s(s(s(o)))),s_4(s(s(s(o))),s(o)),e_1(s(s(s(o))),s(s(s(o)))),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{intermediate\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ022-1.p} An ocean that borders on two adjacent Australian states

There is a database of assertions about Australian states and
 oceans and their relationships. Find which ocean borders
 on two adjacent Australian states.




$\mbox{borders}(y,x) \ \Rightarrow \ \mbox{borders}(x,y)\qquad\mbox{cnf}(\mbox{symmetry\_of\_borders},\mbox{axiom})$

$\mbox{ocean}(\mbox{atlantic})\qquad\mbox{cnf}(\mbox{atlantic},\mbox{negated\_conjecture})$

$\mbox{ocean}(\mbox{indian})\qquad\mbox{cnf}(\mbox{indian},\mbox{negated\_conjecture})$

$\mbox{ocean}(\mbox{pacific})\qquad\mbox{cnf}(\mbox{pacific},\mbox{negated\_conjecture})$

$\mbox{ocean}(\mbox{southern})\qquad\mbox{cnf}(\mbox{southern},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{western\_australia})\qquad\mbox{cnf}(\mbox{western\_australia},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{northern\_territory})\qquad\mbox{cnf}(\mbox{northern\_territory},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{queensland})\qquad\mbox{cnf}(\mbox{queensland},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{south\_australia})\qquad\mbox{cnf}(\mbox{south\_australia},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{new\_south\_wales})\qquad\mbox{cnf}(\mbox{new\_south\_wales},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{victoria})\qquad\mbox{cnf}(\mbox{victoria},\mbox{negated\_conjecture})$

$\mbox{state}(\mbox{tasmania})\qquad\mbox{cnf}(\mbox{tasmania},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{western\_australia},\mbox{northern\_territory})\qquad\mbox{cnf}(\mbox{wa\_nt},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{western\_australia},\mbox{south\_australia})\qquad\mbox{cnf}(\mbox{wa\_sa},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{south\_australia},\mbox{northern\_territory})\qquad\mbox{cnf}(\mbox{sa\_nt},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{south\_australia},\mbox{queensland})\qquad\mbox{cnf}(\mbox{sa\_qld},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{south\_australia},\mbox{new\_south\_wales})\qquad\mbox{cnf}(\mbox{sa\_nsw},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{south\_australia},\mbox{victoria})\qquad\mbox{cnf}(\mbox{sa\_vic},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{northern\_territory},\mbox{queensland})\qquad\mbox{cnf}(\mbox{nt\_qld},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{queensland},\mbox{new\_south\_wales})\qquad\mbox{cnf}(\mbox{qld\_nsw},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{new\_south\_wales},\mbox{victoria})\qquad\mbox{cnf}(\mbox{nsw\_vic},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{indian},\mbox{western\_australia})\qquad\mbox{cnf}(\mbox{indian\_wa},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{indian},\mbox{northern\_territory})\qquad\mbox{cnf}(\mbox{indian\_nt},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{indian},\mbox{queensland})\qquad\mbox{cnf}(\mbox{indian\_qld},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{southern},\mbox{western\_australia})\qquad\mbox{cnf}(\mbox{southern\_wa},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{southern},\mbox{south\_australia})\qquad\mbox{cnf}(\mbox{southern\_sa},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{southern},\mbox{victoria})\qquad\mbox{cnf}(\mbox{southern\_vic},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{southern},\mbox{tasmania})\qquad\mbox{cnf}(\mbox{southern\_tas},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{pacific},\mbox{queensland})\qquad\mbox{cnf}(\mbox{pacific\_qld},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{pacific},\mbox{new\_south\_wales})\qquad\mbox{cnf}(\mbox{pacific\_nsw},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{pacific},\mbox{victoria})\qquad\mbox{cnf}(\mbox{pacific\_vic},\mbox{negated\_conjecture})$

$\mbox{borders}(\mbox{pacific},\mbox{tasmania})\qquad\mbox{cnf}(\mbox{pacific\_tas},\mbox{negated\_conjecture})$

$( \mbox{state}(\mbox{state}_1)\mbox{ and }\mbox{state}(\mbox{state}_2)\mbox{ and }\mbox{borders}(\mbox{state}_1,\mbox{state}_2)\mbox{ and }\mbox{borders}(\mbox{state}_1,\mbox{ocean})\mbox{ and }\mbox{borders}(\mbox{state}_2,\mbox{ocean}) ) \ \Rightarrow \ \neg\,\mbox{ocean}(\mbox{ocean})\qquad\mbox{cnf}(\mbox{prove\_borders},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ039-1.p} Quo vadis 2

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is an intermediate goal for testing.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(b,v_1,v_2,v_3,v_4(s(s(o)),s(s(o))),h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ098$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

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

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

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

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

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

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

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

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\mbox{cCKB\_FIN} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{cCKB\_INF}@ \mbox{xk} )\qquad\mbox{thf}(\mbox{cCKB\_FIN\_def},\mbox{definition})$

$\mbox{cCKB\_ODD} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = c_1\mbox{ or }\mbox{xx} = ( s@ ( s@ c_1 ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_ODD\_def},\mbox{definition})$

$\mbox{cCKB\_EVEN} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = ( s@ c_1 )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ c_1 ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_EVEN\_def},\mbox{definition})$

$\mbox{cCKB\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_ODD}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_ODD}@ \mbox{xv} )\mbox{ or }( \mbox{cCKB\_EVEN}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_EVEN}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB\_FIN}@ \mbox{cCKB\_BLACK}\qquad\mbox{thf}(\mbox{cL}_{7000},\mbox{conjecture})$

\medskip\textbf{PUZ122$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






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

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

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

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

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\mbox{cCKB\_FIN} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{cCKB\_INF}@ \mbox{xk} )\qquad\mbox{thf}(\mbox{cCKB\_FIN\_def},\mbox{definition})$

$\forall \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xz}{:}\;\mbox{\$i},\mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_FIN}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cCKB\_FIN}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ or }( \mbox{xu} = \mbox{xz}\mbox{ and }\mbox{xv} = \mbox{xw} ) ) ) )\qquad\mbox{thf}(\mbox{cL2000\_pme},\mbox{conjecture})$

\medskip\textbf{PUZ068+1.p} Monday's Sudoku

47 1
 5 2
 3      8
 6   1 7
 4  3
 5
 7   3
 1
 4




$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_4) = n_4\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_5) = n_7\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_7) = n_1\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_2) = n_5\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_4) = n_2\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_3,n_2) = n_3\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_3,n_9) = n_8\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_1) = n_6\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_1\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_4,n_7) = n_7\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_3) = n_4\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_5,n_6) = n_3\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_6,n_8) = n_5\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_7,n_8) = n_3\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_1) = n_1\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_9) = n_4\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ132+1.p} Crime in beautiful Washington

A capital is a city. USA is a country. Every city has crime.
 Washington is the capital of the USA. Every country has a
 beautiful capital. Therefore, Washington is beautiful but has
 crime.




$\exists a{:}\;\mbox{capital}(a)\qquad\mbox{fof}(\mbox{capital\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{city}(a)\qquad\mbox{fof}(\mbox{city\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{capital}(a) \ \Rightarrow \ \mbox{city}(a) )\qquad\mbox{fof}(\mbox{capital\_city\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{country}(a)\qquad\mbox{fof}(\mbox{country\_type},\mbox{axiom})$

$\mbox{capital}(\mbox{washington})\qquad\mbox{fof}(\mbox{washington\_type},\mbox{axiom})$

$\mbox{country}(\mbox{usa})\qquad\mbox{fof}(\mbox{usa\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{country}(a) \ \Rightarrow \ \mbox{capital}(\mbox{capital\_city}(a)) )\qquad\mbox{fof}(\mbox{country\_capital\_type},\mbox{axiom})$

$\forall x{:}\;( \mbox{city}(x) \ \Rightarrow \ \mbox{has\_crime}(x) )\qquad\mbox{fof}(\mbox{crime\_axiom},\mbox{axiom})$

$\mbox{capital\_city}(\mbox{usa}) = \mbox{washington}\qquad\mbox{fof}(\mbox{usa\_capital\_axiom},\mbox{axiom})$

$\forall x{:}\;( \mbox{country}(x) \ \Rightarrow \ \mbox{beautiful}(\mbox{capital\_city}(x)) )\qquad\mbox{fof}(\mbox{beautiful\_capital\_axiom},\mbox{axiom})$

$\mbox{beautiful}(\mbox{washington})\mbox{ and }\mbox{has\_crime}(\mbox{washington})\qquad\mbox{fof}(\mbox{washington\_conjecture},\mbox{conjecture})$

\medskip\textbf{PUZ047$\wedge$5.p} TPS problem THM100A

A naive formalization of the problem of moving man wolf goat
 cabbage from south to north side of river.




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

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

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

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

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

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

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

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

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

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

$( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cS}@ \mbox{cS}@ \mbox{cO}\mbox{ and }\forall t{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ t ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ ( \mbox{cL}@ t ) ) )\mbox{ and }\forall t_1{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ t_1 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ ( \mbox{cL}@ t_1 ) ) )\mbox{ and }\forall t_2{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ t_2 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ ( \mbox{cL}@ t_2 ) ) )\mbox{ and }\forall t_3{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ t_3 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ ( \mbox{cL}@ t_3 ) ) )\mbox{ and }\forall t_4{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ t_4 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ ( \mbox{cW}@ t_4 ) ) )\mbox{ and }\forall t_5{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ t_5 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ ( \mbox{cW}@ t_5 ) ) )\mbox{ and }\forall t_6{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ t_6 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ ( \mbox{cW}@ t_6 ) ) )\mbox{ and }\forall t_7{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ t_7 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ ( \mbox{cW}@ t_7 ) ) )\mbox{ and }\forall x{:}\;a,y{:}\;a,u{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ x@ \mbox{cS}@ y@ u ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ x@ \mbox{cN}@ y@ ( \mbox{cG}@ u ) ) )\mbox{ and }\forall x_1{:}\;a,y_1{:}\;a,v{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ x_1@ \mbox{cN}@ y_1@ v ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ x_1@ \mbox{cS}@ y_1@ ( \mbox{cG}@ v ) ) )\mbox{ and }\forall t_8{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ \mbox{cS}@ t_8 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ ( \mbox{cD}@ t_8 ) ) )\mbox{ and }\forall t_9{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ t_9 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ \mbox{cS}@ ( \mbox{cD}@ t_9 ) ) )\mbox{ and }\forall u_1{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ u_1 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ \mbox{cN}@ ( \mbox{cD}@ u_1 ) ) )\mbox{ and }\forall v_1{:}\;b{:}\;( ( \mbox{cP}@ \mbox{cN}@ \mbox{cS}@ \mbox{cN}@ \mbox{cN}@ v_1 ) \ \Rightarrow \ ( \mbox{cP}@ \mbox{cS}@ \mbox{cS}@ \mbox{cN}@ \mbox{cS}@ ( \mbox{cD}@ v_1 ) ) ) ) \ \Rightarrow \ \exists z{:}\;b{:}\;( \mbox{cP}@ \mbox{cN}@ \mbox{cN}@ \mbox{cN}@ \mbox{cN}@ z )\qquad\mbox{thf}(\mbox{cTHM100A},\mbox{conjecture})$

\medskip\textbf{PUZ081$\wedge$3.p} 1 of http://philosophy.hku.hk/think/logic/knight.php

A very special island is inhabited only by knights and knaves.
 Knights always tell the truth, and knaves always lie. You meet two
 inhabitants: Zoey and Mel. Zoey tells you that Mel is a knave. Mel
 says, `Neither Zoey nor I are knaves'. Who is a knight and who is
 a knave?




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

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

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

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

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

$\forall p{:}\;\mbox{\$i}{:}\;( \mbox{knight}@ p ) <~> ( \mbox{knave}@ p )\qquad\mbox{thf}(\mbox{knights\_xor\_knaves},\mbox{axiom})$

$\forall p{:}\;\mbox{\$i},s{:}\;\mbox{\$o}{:}\;( ( \mbox{knight}@ p\mbox{ and }\mbox{says}@ p@ s ) \ \Rightarrow \ s )\qquad\mbox{thf}(\mbox{knights\_tell\_truth},\mbox{axiom})$

$\forall p{:}\;\mbox{\$i},s{:}\;\mbox{\$o}{:}\;( ( \mbox{knave}@ p\mbox{ and }\mbox{says}@ p@ s ) \ \Rightarrow \ \neg\,s )\qquad\mbox{thf}(\mbox{knaves\_lie},\mbox{axiom})$

$\mbox{says}@ \mbox{zoey}@ ( \mbox{knave}@ \mbox{mel} )\qquad\mbox{thf}(\mbox{zoey\_speaks},\mbox{axiom})$

$\mbox{says}@ \mbox{mel}@ ( \neg\,\mbox{knave}@ \mbox{zoey}\mbox{ and }\neg\,\mbox{knave}@ \mbox{mel} )\qquad\mbox{thf}(\mbox{mel\_speaks},\mbox{axiom})$

$\exists \mbox{knight}{:}\;\mbox{\$i},\mbox{knave}{:}\;\mbox{\$i}{:}\;( \mbox{knight}@ \mbox{knight}\mbox{ and }\mbox{knave}@ \mbox{knave} )\qquad\mbox{thf}(\mbox{who\_is\_knight\_and\_knave},\mbox{conjecture})$

\medskip\textbf{PUZ044-1.p} Truthtellers and Liars axioms for two types of people

Axioms for two types of people; truthtellers and liars.




$\mbox{include}(\mbox{'Axioms/PUZ002-0.ax'})$

\medskip\textbf{PUZ055-1.p} Show that Sam Loyd's fifteen-puzzle is not solvable

The classic fifteen-puzzle with the 14 and 15 interchanged.
 Show that it is not solvable by showing satisfiability of
 the following problem.




$\mbox{state}(l(n_1,l(n_2,l(n_3,l(n_4,l(\mbox{end},l(n_5,l(n_6,l(n_7,l(n_8,l(\mbox{end},l(n_9,l(n_{10},l(n_{11},l(n_{12},l(\mbox{end},l(n_{13},l(n_{15},l(n_{14},l(x,l(\mbox{end},\mbox{nil})))))))))))))))))))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{axiom})$

$\neg\,\mbox{state}(l(n_1,l(n_2,l(n_3,l(n_4,l(\mbox{end},l(n_5,l(n_6,l(n_7,l(n_8,l(\mbox{end},l(n_9,l(n_{10},l(n_{11},l(n_{12},l(\mbox{end},l(n_{13},l(n_{14},l(n_{15},l(x,l(\mbox{end},\mbox{nil})))))))))))))))))))))\qquad\mbox{cnf}(\mbox{final\_state},\mbox{negated\_conjecture})$

$l(x,l(a,r)) = l(a,l(x,r))\mbox{ or }a = \mbox{end}\qquad\mbox{cnf}(\mbox{shift\_sideways},\mbox{axiom})$

$l(x,l(a,l(b,l(c,l(d,l(e,r)))))) = l(e,l(a,l(b,l(c,l(d,l(x,r))))))\qquad\mbox{cnf}(\mbox{shift\_up\_down},\mbox{axiom})$

\medskip\textbf{PUZ070+1.p} Wednesday's Sudoku

7  59
 6 1
 4   6
 83  2
 95
 
 82   4
 1
 5




$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_2) = n_7\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_5) = n_5\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_1,n_6) = n_9\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_7) = n_6\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_2,n_9) = n_1\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_3,n_1) = n_4\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_3,n_5) = n_6\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_4) = n_8\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_3\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_4,n_8) = n_2\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_9\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_5,n_2) = n_5\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_7,n_3) = n_8\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_2\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_7,n_8) = n_4\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_8,n_4) = n_1\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_7) = n_5\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ141$\wedge$1.p} Labyrinth1

Move 2 to the right.




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

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

$\mbox{left}{:}\;\mbox{direction}\qquad\mbox{thf}(\mbox{left\_type},\mbox{type})$

$\mbox{right}{:}\;\mbox{direction}\qquad\mbox{thf}(\mbox{right\_type},\mbox{type})$

$\top {:}\;\mbox{direction}\qquad\mbox{thf}(\mbox{top\_type},\mbox{type})$

$\mbox{bottom}{:}\;\mbox{direction}\qquad\mbox{thf}(\mbox{bottom\_type},\mbox{type})$

$\mbox{next}{:}\;\mbox{position}\to \mbox{direction}\to \mbox{position}\qquad\mbox{thf}(\mbox{next\_type},\mbox{type})$

$\forall d_1{:}\;\mbox{direction},d_2{:}\;\mbox{direction},p{:}\;\mbox{position}{:}\;( \mbox{next}@ ( \mbox{next}@ p@ d_1 )@ d_2 ) = ( \mbox{next}@ ( \mbox{next}@ p@ d_2 )@ d_1 )\qquad\mbox{thf}(\mbox{next\_comm},\mbox{axiom})$

$\forall p{:}\;\mbox{position}{:}\;( \mbox{next}@ ( \mbox{next}@ p@ \mbox{left} )@ \mbox{right} ) = p\qquad\mbox{thf}(\mbox{left\_right},\mbox{axiom})$

$\forall p{:}\;\mbox{position}{:}\;( \mbox{next}@ ( \mbox{next}@ p@ \top  )@ \mbox{bottom} ) = p\qquad\mbox{thf}(\mbox{top\_bottom},\mbox{axiom})$

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

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

$\mbox{nomove}{:}\;\mbox{movelist}\qquad\mbox{thf}(\mbox{nomove\_type},\mbox{type})$

$\mbox{movedir}{:}\;\mbox{movelist}\to \mbox{direction}\to \mbox{movelist}\qquad\mbox{thf}(\mbox{movedir\_type},\mbox{type})$

$\mbox{playerpos}{:}\;\mbox{movelist}\to \mbox{position}\qquad\mbox{thf}(\mbox{playerpos\_type},\mbox{type})$

$\forall \mbox{pO}{:}\;\mbox{position},m{:}\;\mbox{movelist},d{:}\;\mbox{direction}{:}\;( ( \mbox{playerpos}@ m ) = \mbox{pO} \ \Rightarrow \ ( \neg\,\mbox{wall}@ ( \mbox{next}@ \mbox{pO}@ d ) \ \Rightarrow \ ( \mbox{playerpos}@ ( \mbox{movedir}@ m@ d ) ) = ( \mbox{next}@ \mbox{pO}@ d ) ) )\qquad\mbox{thf}(\mbox{player\_move\_legal},\mbox{axiom})$

$\forall \mbox{pO}{:}\;\mbox{position},m{:}\;\mbox{movelist},d{:}\;\mbox{direction}{:}\;( ( \mbox{playerpos}@ m ) = \mbox{pO} \ \Rightarrow \ ( ( \mbox{wall}@ ( \mbox{next}@ \mbox{pO}@ d ) ) \ \Rightarrow \ ( \mbox{playerpos}@ ( \mbox{movedir}@ m@ d ) ) = \mbox{pO} ) )\qquad\mbox{thf}(\mbox{player\_move\_illegal},\mbox{axiom})$

$c_{00}{:}\;\mbox{position}\qquad\mbox{thf}(\mbox{c00\_type},\mbox{type})$

$c_{10}{:}\;\mbox{position}\qquad\mbox{thf}(\mbox{c10\_type},\mbox{type})$

$c_{20}{:}\;\mbox{position}\qquad\mbox{thf}(\mbox{c20\_type},\mbox{type})$

$c_{10} = ( \mbox{next}@ c_{00}@ \mbox{right} )\qquad\mbox{thf}(\mbox{c10\_defin},\mbox{definition})$

$c_{20} = ( \mbox{next}@ c_{10}@ \mbox{right} )\qquad\mbox{thf}(\mbox{c20\_defin},\mbox{definition})$

$( \mbox{wall}@ c_{00} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{c00\_axiom},\mbox{axiom})$

$( \mbox{wall}@ c_{10} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{c10\_axiom},\mbox{axiom})$

$( \mbox{wall}@ c_{20} ) = \mbox{\$false}\qquad\mbox{thf}(\mbox{c20\_axiom},\mbox{axiom})$

$( \mbox{playerpos}@ \mbox{nomove} ) = c_{00}\qquad\mbox{thf}(\mbox{start\_axiom},\mbox{axiom})$

$\exists m{:}\;\mbox{movelist}{:}\;( \mbox{playerpos}@ m ) = c_{20}\qquad\mbox{thf}(\mbox{exercise},\mbox{conjecture})$

\medskip\textbf{PUZ056-2.010.p} Towers of Hanoi k=10

Each instance encodes Tower of Hanoi with n discs as a
 reachability problem.




$p(i,t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9) \ \Rightarrow \ p(j,t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_1,\mbox{axiom})$

$( p(t_0,i,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j) ) \ \Rightarrow \ p(t_0,j,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_2,\mbox{axiom})$

$( p(t_0,t_1,i,t_3,t_4,t_5,t_6,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j) ) \ \Rightarrow \ p(t_0,t_1,j,t_3,t_4,t_5,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_3,\mbox{axiom})$

$( p(t_0,t_1,t_2,i,t_4,t_5,t_6,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,j,t_4,t_5,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_4,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,i,t_5,t_6,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,j,t_5,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_5,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,t_4,i,t_6,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j)\mbox{ and }\mbox{neq}(t_4,i)\mbox{ and }\mbox{neq}(t_4,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,t_4,j,t_6,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_6,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,t_4,t_5,i,t_7,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j)\mbox{ and }\mbox{neq}(t_4,i)\mbox{ and }\mbox{neq}(t_4,j)\mbox{ and }\mbox{neq}(t_5,i)\mbox{ and }\mbox{neq}(t_5,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,t_4,t_5,j,t_7,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_7,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,i,t_8,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j)\mbox{ and }\mbox{neq}(t_4,i)\mbox{ and }\mbox{neq}(t_4,j)\mbox{ and }\mbox{neq}(t_5,i)\mbox{ and }\mbox{neq}(t_5,j)\mbox{ and }\mbox{neq}(t_6,i)\mbox{ and }\mbox{neq}(t_6,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,j,t_8,t_9)\qquad\mbox{cnf}(\mbox{rule}_8,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7,i,t_9)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j)\mbox{ and }\mbox{neq}(t_4,i)\mbox{ and }\mbox{neq}(t_4,j)\mbox{ and }\mbox{neq}(t_5,i)\mbox{ and }\mbox{neq}(t_5,j)\mbox{ and }\mbox{neq}(t_6,i)\mbox{ and }\mbox{neq}(t_6,j)\mbox{ and }\mbox{neq}(t_7,i)\mbox{ and }\mbox{neq}(t_7,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7,j,t_9)\qquad\mbox{cnf}(\mbox{rule}_9,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,i)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j)\mbox{ and }\mbox{neq}(t_4,i)\mbox{ and }\mbox{neq}(t_4,j)\mbox{ and }\mbox{neq}(t_5,i)\mbox{ and }\mbox{neq}(t_5,j)\mbox{ and }\mbox{neq}(t_6,i)\mbox{ and }\mbox{neq}(t_6,j)\mbox{ and }\mbox{neq}(t_7,i)\mbox{ and }\mbox{neq}(t_7,j)\mbox{ and }\mbox{neq}(t_8,i)\mbox{ and }\mbox{neq}(t_8,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,j)\qquad\mbox{cnf}(\mbox{rule}_{10},\mbox{axiom})$

$\neg\,\mbox{neq}(s_0,s_0)\qquad\mbox{cnf}(\mbox{neq}_1,\mbox{axiom})$

$\mbox{neq}(s_0,s_1)\qquad\mbox{cnf}(\mbox{neq}_2,\mbox{axiom})$

$\mbox{neq}(s_0,s_2)\qquad\mbox{cnf}(\mbox{neq}_3,\mbox{axiom})$

$\mbox{neq}(s_1,s_0)\qquad\mbox{cnf}(\mbox{neq}_4,\mbox{axiom})$

$\neg\,\mbox{neq}(s_1,s_1)\qquad\mbox{cnf}(\mbox{neq}_5,\mbox{axiom})$

$\mbox{neq}(s_1,s_2)\qquad\mbox{cnf}(\mbox{neq}_6,\mbox{axiom})$

$\mbox{neq}(s_2,s_0)\qquad\mbox{cnf}(\mbox{neq}_7,\mbox{axiom})$

$\mbox{neq}(s_2,s_1)\qquad\mbox{cnf}(\mbox{neq}_8,\mbox{axiom})$

$\neg\,\mbox{neq}(s_2,s_2)\qquad\mbox{cnf}(\mbox{neq}_9,\mbox{axiom})$

$p(s_0,s_0,s_0,s_0,s_0,s_0,s_0,s_0,s_0,s_0)\qquad\mbox{cnf}(\mbox{init},\mbox{axiom})$

$\neg\,p(s_2,s_2,s_2,s_2,s_2,s_2,s_2,s_2,s_2,s_2)\qquad\mbox{cnf}(\mbox{goal},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ081$\wedge$2.p} 1 of http://philosophy.hku.hk/think/logic/knight.php

A very special island is inhabited only by knights and knaves.
 Knights always tell the truth, and knaves always lie. You meet two
 inhabitants: Zoey and Mel. Zoey tells you that Mel is a knave. Mel
 says, `Neither Zoey nor I are knaves'. Who is a knight and who is
 a knave?




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

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

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

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

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

$\forall p{:}\;\mbox{\$i}{:}\;( \mbox{knight}@ p ) <~> ( \mbox{knave}@ p )\qquad\mbox{thf}(\mbox{knights\_xor\_knaves},\mbox{axiom})$

$\forall p{:}\;\mbox{\$i},s{:}\;\mbox{\$o}{:}\;( ( \mbox{knight}@ p\mbox{ and }\mbox{says}@ p@ s ) \ \Rightarrow \ s )\qquad\mbox{thf}(\mbox{knights\_tell\_truth},\mbox{axiom})$

$\forall p{:}\;\mbox{\$i},s{:}\;\mbox{\$o}{:}\;( ( \mbox{knave}@ p\mbox{ and }\mbox{says}@ p@ s ) \ \Rightarrow \ \neg\,s )\qquad\mbox{thf}(\mbox{knaves\_lie},\mbox{axiom})$

$\mbox{says}@ \mbox{zoey}@ ( \mbox{knave}@ \mbox{mel} )\qquad\mbox{thf}(\mbox{zoey\_speaks},\mbox{axiom})$

$\mbox{says}@ \mbox{mel}@ ( \neg\,\mbox{knave}@ \mbox{zoey}\mbox{ and }\neg\,\mbox{knave}@ \mbox{mel} )\qquad\mbox{thf}(\mbox{mel\_speaks},\mbox{axiom})$

$\exists \mbox{tZ}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{tM}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{tZ}@ \mbox{zoey}\mbox{ and }\mbox{tM}@ \mbox{mel} )\qquad\mbox{thf}(\mbox{what\_are\_zoey\_and\_mel},\mbox{conjecture})$

\medskip\textbf{PUZ121$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

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

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

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

$\mbox{cCKB\_EVEN} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = ( s@ c_1 )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ c_1 ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_EVEN\_def},\mbox{definition})$

$\mbox{cCKB\_ODD} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = c_1\mbox{ or }\mbox{xx} = ( s@ ( s@ c_1 ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_ODD\_def},\mbox{definition})$

$\mbox{cCKB\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_ODD}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_ODD}@ \mbox{xv} )\mbox{ or }( \mbox{cCKB\_EVEN}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_EVEN}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_BLACK}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{cCKB\_BLACK}@ ( s@ \mbox{xu} )@ ( s@ \mbox{xv} )\mbox{ and }\mbox{cCKB\_BLACK}@ \mbox{xu}@ ( s@ ( s@ \mbox{xv} ) )\mbox{ and }\mbox{cCKB\_BLACK}@ ( s@ ( s@ \mbox{xu} ) )@ \mbox{xv} ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{11000},\mbox{conjecture})$

\medskip\textbf{PUZ035-5.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(p,\mbox{knight})) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ and }\mbox{says}(p,s) ) \ \Rightarrow \ \mbox{truth}(s)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_1,\mbox{axiom})$

$( \mbox{truth}(s)\mbox{ and }\mbox{says}(p,s) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_2,\mbox{axiom})$

$\mbox{truth}(\mbox{or}(a,b)) \ \Rightarrow \ ( \mbox{truth}(a)\mbox{ or }\mbox{truth}(b) )\qquad\mbox{cnf}(\mbox{or}_1,\mbox{axiom})$

$\mbox{truth}(a) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_2,\mbox{axiom})$

$\mbox{truth}(b) \ \Rightarrow \ \mbox{truth}(\mbox{or}(a,b))\qquad\mbox{cnf}(\mbox{or}_3,\mbox{axiom})$

$\mbox{says}(\mbox{asked},\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight})))\qquad\mbox{cnf}(\mbox{says\_yes},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(\mbox{asked},x)) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(\mbox{other},y))\qquad\mbox{cnf}(\mbox{query},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ133+1.p} N queens problem has the variable symmetry property






$\mbox{queens\_p} \ \Rightarrow \ \forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n) ) \ \Rightarrow \ ( p(i)\ne p(j)\mbox{ and }p(i) + i\ne p(j) + j\mbox{ and }-p(i)\ne -p(j) ) )\qquad\mbox{fof}(\mbox{queens\_p},\mbox{axiom})$

$\forall i{:}\;\mbox{perm}(i) = -s(n)\qquad\mbox{fof}(\mbox{permutation},\mbox{axiom})$

$\forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n)\mbox{ and }( \mbox{le}(s(i),j)\iff \mbox{le}(s(\mbox{perm}(j)),\mbox{perm}(i)) ) ) \ \Rightarrow \ ( q(i)\ne q(j)\mbox{ and }q(i) + i\ne q(j) + j\mbox{ and }-q(i)\ne -q(j) ) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_q},\mbox{axiom})$

$( \mbox{queens\_p}\mbox{ and }\forall i{:}\;q(i) = p(\mbox{perm}(i)) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_sym},\mbox{conjecture})$

$\forall i{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n) ) \ \Rightarrow \ ( \mbox{le}(s(n_0),\mbox{perm}(i))\mbox{ and }\mbox{le}(\mbox{perm}(i),n) ) )\qquad\mbox{fof}(\mbox{permutation\_range},\mbox{axiom})$

$\forall j,i{:}\;-i = -\mbox{perm}(j)\qquad\mbox{fof}(\mbox{permutation\_another\_one},\mbox{axiom})$

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

$\forall x{:}\;\mbox{le}(x,s(x))\qquad\mbox{fof}(\mbox{succ\_le},\mbox{axiom})$

$\forall i,j,k,l{:}\;( i + j = k + l\iff -i = -l )\qquad\mbox{fof}(\mbox{plus}_1,\mbox{axiom})$

$\forall i,j,k,l{:}\;( -i = -k\iff -i = -j )\qquad\mbox{fof}(\mbox{minus}_1,\mbox{axiom})$

\medskip\textbf{PUZ028-2.p} People at a party

We can always choose 3 persons who are either familiar with
 each other or not familiar with each other, from 6 persons
 who meet at a party.




$\mbox{person}(n_1)\qquad\mbox{cnf}(\mbox{person}_1,\mbox{axiom})$

$\mbox{person}(n_2)\qquad\mbox{cnf}(\mbox{person}_2,\mbox{axiom})$

$\mbox{person}(n_3)\qquad\mbox{cnf}(\mbox{person}_3,\mbox{axiom})$

$\mbox{person}(n_4)\qquad\mbox{cnf}(\mbox{person}_4,\mbox{axiom})$

$\mbox{person}(n_5)\qquad\mbox{cnf}(\mbox{person}_5,\mbox{axiom})$

$\mbox{person}(n_6)\qquad\mbox{cnf}(\mbox{person}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_5,\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{person}(y)\mbox{ and }\mbox{not\_equal}(x,y) ) \ \Rightarrow \ ( \mbox{familiar}(x,y)\mbox{ or }\mbox{not\_familiar}(x,y) )\qquad\mbox{cnf}(\mbox{familiar\_or\_not},\mbox{axiom})$

$( \mbox{familiar}(x_1,x_2)\mbox{ and }\mbox{familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_familiar},\mbox{negated\_conjecture})$

$( \mbox{not\_familiar}(x_1,x_2)\mbox{ and }\mbox{not\_familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{not\_familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_not\_familiar},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ129+2.p} The grocer is not a cyclist

If every honest and industrious person is healthy, and no grocer 
 is healthy, and every industrious grocer is honest, and every 
 cyclist is industrious, and every unhealthy cyclist is dishonest,
 and no healthy person is unhealthy, and no honest person is
 dishonest, and every grocer is a person, and every cyclist is a
 person then no grocer is a cyclist.




$( \forall a{:}\;( ( \mbox{person}(a)\mbox{ and }\mbox{property}_1(a,\mbox{honest},\mbox{pos})\mbox{ and }\mbox{property}_1(a,\mbox{industrious},\mbox{pos}) ) \ \Rightarrow \ \exists b{:}\;( \mbox{property}_1(b,\mbox{healthy},\mbox{pos})\mbox{ and }a = b ) )\mbox{ and }\forall c{:}\;( \mbox{grocer}(c) \ \Rightarrow \ \neg\,\exists d{:}\;( \mbox{property}_1(d,\mbox{healthy},\mbox{pos})\mbox{ and }c = d ) )\mbox{ and }\forall e{:}\;( ( \mbox{grocer}(e)\mbox{ and }\mbox{property}_1(e,\mbox{industrious},\mbox{pos}) ) \ \Rightarrow \ \exists f{:}\;( \mbox{property}_1(f,\mbox{honest},\mbox{pos})\mbox{ and }e = f ) )\mbox{ and }\forall g{:}\;( \mbox{cyclist}(g) \ \Rightarrow \ \exists h{:}\;( \mbox{property}_1(h,\mbox{industrious},\mbox{pos})\mbox{ and }g = h ) )\mbox{ and }\forall i{:}\;( ( \mbox{cyclist}(i)\mbox{ and }\mbox{property}_1(i,\mbox{unhealthy},\mbox{pos}) ) \ \Rightarrow \ \exists j{:}\;( \mbox{property}_1(j,\mbox{dishonest},\mbox{pos})\mbox{ and }i = j ) )\mbox{ and }\forall k{:}\;( ( \mbox{person}(k)\mbox{ and }\mbox{property}_1(k,\mbox{healthy},\mbox{pos}) ) \ \Rightarrow \ \neg\,\exists l{:}\;( \mbox{property}_1(l,\mbox{unhealthy},\mbox{pos})\mbox{ and }k = l ) )\mbox{ and }\forall m{:}\;( ( \mbox{person}(m)\mbox{ and }\mbox{property}_1(m,\mbox{honest},\mbox{pos}) ) \ \Rightarrow \ \neg\,\exists n{:}\;( \mbox{property}_1(n,\mbox{dishonest},\mbox{pos})\mbox{ and }m = n ) )\mbox{ and }\forall o{:}\;( \mbox{grocer}(o) \ \Rightarrow \ \exists p{:}\;( \mbox{person}(p)\mbox{ and }o = p ) )\mbox{ and }\forall q{:}\;( \mbox{cyclist}(q) \ \Rightarrow \ \exists r{:}\;( \mbox{person}(r)\mbox{ and }q = r ) ) ) \ \Rightarrow \ \forall s{:}\;( \mbox{grocer}(s) \ \Rightarrow \ \neg\,\exists t{:}\;( \mbox{cyclist}(t)\mbox{ and }s = t ) )\qquad\mbox{fof}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{PUZ080+2.p} Another Sudoku

3 1
 7  9  6
 9       4
 4  8 6  2
 4 7
 1
 67 21
 3 1   9 7
 8   3




$\mbox{include}(\mbox{'Axioms/PUZ006+0.ax'})$

$p(n_1,n_4,n_3)\qquad\mbox{fof}(\mbox{ax}_{143},\mbox{axiom})$

$p(n_1,n_6,n_1)\qquad\mbox{fof}(\mbox{ax}_{161},\mbox{axiom})$

$p(n_2,n_2,n_7)\qquad\mbox{fof}(\mbox{ax}_{227},\mbox{axiom})$

$p(n_2,n_5,n_9)\qquad\mbox{fof}(\mbox{ax}_{259},\mbox{axiom})$

$p(n_2,n_8,n_6)\qquad\mbox{fof}(\mbox{ax}_{286},\mbox{axiom})$

$p(n_3,n_1,n_9)\qquad\mbox{fof}(\mbox{ax}_{319},\mbox{axiom})$

$p(n_3,n_9,n_4)\qquad\mbox{fof}(\mbox{ax}_{394},\mbox{axiom})$

$p(n_4,n_1,n_4)\qquad\mbox{fof}(\mbox{ax}_{414},\mbox{axiom})$

$p(n_4,n_4,n_8)\qquad\mbox{fof}(\mbox{ax}_{448},\mbox{axiom})$

$p(n_4,n_6,n_6)\qquad\mbox{fof}(\mbox{ax}_{466},\mbox{axiom})$

$p(n_4,n_9,n_2)\qquad\mbox{fof}(\mbox{ax}_{492},\mbox{axiom})$

$p(n_5,n_4,n_4)\qquad\mbox{fof}(\mbox{ax}_{544},\mbox{axiom})$

$p(n_5,n_6,n_7)\qquad\mbox{fof}(\mbox{ax}_{567},\mbox{axiom})$

$p(n_6,n_5,n_1)\qquad\mbox{fof}(\mbox{ax}_{651},\mbox{axiom})$

$p(n_7,n_3,n_6)\qquad\mbox{fof}(\mbox{ax}_{736},\mbox{axiom})$

$p(n_7,n_4,n_7)\qquad\mbox{fof}(\mbox{ax}_{747},\mbox{axiom})$

$p(n_7,n_6,n_2)\qquad\mbox{fof}(\mbox{ax}_{762},\mbox{axiom})$

$p(n_7,n_7,n_1)\qquad\mbox{fof}(\mbox{ax}_{771},\mbox{axiom})$

$p(n_8,n_1,n_3)\qquad\mbox{fof}(\mbox{ax}_{813},\mbox{axiom})$

$p(n_8,n_3,n_1)\qquad\mbox{fof}(\mbox{ax}_{831},\mbox{axiom})$

$p(n_8,n_7,n_9)\qquad\mbox{fof}(\mbox{ax}_{879},\mbox{axiom})$

$p(n_8,n_9,n_7)\qquad\mbox{fof}(\mbox{ax}_{897},\mbox{axiom})$

$p(n_9,n_3,n_8)\qquad\mbox{fof}(\mbox{ax}_{938},\mbox{axiom})$

$p(n_9,n_7,n_3)\qquad\mbox{fof}(\mbox{ax}_{973},\mbox{axiom})$

\medskip\textbf{PUZ007-1.p} Mixed couples on Mars and Venus

Here's the situation: human observers in this exclusive club
 on Ganymede can't distinguish Martians from Venusians, males
 from females, except for the fact that Venusian women and
 Martian men always tell the truth and Venusian men and
 Martian women always lie.
 A says "I'm from Mars" B exclaims "That's not true!" A and
 B are married; are they a mixed couple?




$\mbox{include}(\mbox{'Axioms/PUZ001-0.ax'})$

$\mbox{says}(a,\mbox{a\_from\_mars})\qquad\mbox{cnf}(\mbox{a\_says\_hes\_from\_mars},\mbox{hypothesis})$

$\mbox{says}(b,\mbox{a\_has\_lied})\qquad\mbox{cnf}(\mbox{b\_says\_a\_lies},\mbox{hypothesis})$

$\mbox{from\_mars}(a) \ \Rightarrow \ \mbox{a\_truth}(\mbox{a\_from\_mars})\qquad\mbox{cnf}(\mbox{a\_from\_mars}_1,\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{a\_from\_mars}) \ \Rightarrow \ \mbox{from\_mars}(a)\qquad\mbox{cnf}(\mbox{a\_from\_mars}_2,\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{a\_from\_mars}) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{statement\_by}(b))\qquad\mbox{cnf}(\mbox{a\_from\_mars}_3,\mbox{hypothesis})$

$\mbox{statement\_by}(a) = \mbox{a\_from\_mars}\qquad\mbox{cnf}(\mbox{a\_states},\mbox{hypothesis})$

$\mbox{statement\_by}(b) = \mbox{a\_has\_lied}\qquad\mbox{cnf}(\mbox{b\_states},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{a\_from\_mars})\mbox{ or }\mbox{a\_truth}(\mbox{statement\_by}(b))\qquad\mbox{cnf}(\mbox{truth\_of\_bs\_statement},\mbox{hypothesis})$

$\mbox{female}(a) \ \Rightarrow \ \mbox{male}(b)\qquad\mbox{cnf}(\mbox{different\_sex}_1,\mbox{hypothesis})$

$\mbox{male}(a) \ \Rightarrow \ \mbox{female}(b)\qquad\mbox{cnf}(\mbox{different\_sex}_2,\mbox{hypothesis})$

$\mbox{from\_mars}(b)\mbox{ or }\mbox{from\_mars}(a)\qquad\mbox{cnf}(\mbox{one\_from\_mars},\mbox{negated\_conjecture})$

$\mbox{from\_venus}(a)\mbox{ or }\mbox{from\_venus}(b)\qquad\mbox{cnf}(\mbox{one\_from\_venus},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ138+2.p} Platinum Blonde

+---+---+---+
 |   |   | 12|
 |   |   |  3|
 |  2|3  |4  |
 +---+---+---+
 |  1|8  |  5|
 | 6 | 7 |8  |
 |   |  9|   |
 +---+---+---+
 |  8|5  |   |
 |9  | 4 |5  |
 |47 |  6|   |
 +---+---+---+




$\mbox{include}(\mbox{'Axioms/PUZ006+0.ax'})$

$p(n_1,n_8,n_1)\qquad\mbox{fof}(\mbox{ax}_{181},\mbox{axiom})$

$p(n_1,n_9,n_2)\qquad\mbox{fof}(\mbox{ax}_{192},\mbox{axiom})$

$p(n_2,n_9,n_3)\qquad\mbox{fof}(\mbox{ax}_{293},\mbox{axiom})$

$p(n_3,n_3,n_2)\qquad\mbox{fof}(\mbox{ax}_{332},\mbox{axiom})$

$p(n_3,n_4,n_3)\qquad\mbox{fof}(\mbox{ax}_{343},\mbox{axiom})$

$p(n_3,n_7,n_4)\qquad\mbox{fof}(\mbox{ax}_{374},\mbox{axiom})$

$p(n_4,n_3,n_1)\qquad\mbox{fof}(\mbox{ax}_{431},\mbox{axiom})$

$p(n_4,n_4,n_8)\qquad\mbox{fof}(\mbox{ax}_{448},\mbox{axiom})$

$p(n_4,n_9,n_5)\qquad\mbox{fof}(\mbox{ax}_{495},\mbox{axiom})$

$p(n_5,n_2,n_6)\qquad\mbox{fof}(\mbox{ax}_{526},\mbox{axiom})$

$p(n_5,n_5,n_7)\qquad\mbox{fof}(\mbox{ax}_{557},\mbox{axiom})$

$p(n_5,n_7,n_8)\qquad\mbox{fof}(\mbox{ax}_{578},\mbox{axiom})$

$p(n_6,n_6,n_9)\qquad\mbox{fof}(\mbox{ax}_{669},\mbox{axiom})$

$p(n_7,n_3,n_8)\qquad\mbox{fof}(\mbox{ax}_{738},\mbox{axiom})$

$p(n_7,n_4,n_5)\qquad\mbox{fof}(\mbox{ax}_{745},\mbox{axiom})$

$p(n_8,n_1,n_9)\qquad\mbox{fof}(\mbox{ax}_{819},\mbox{axiom})$

$p(n_8,n_5,n_4)\qquad\mbox{fof}(\mbox{ax}_{854},\mbox{axiom})$

$p(n_8,n_7,n_5)\qquad\mbox{fof}(\mbox{ax}_{875},\mbox{axiom})$

$p(n_9,n_1,n_4)\qquad\mbox{fof}(\mbox{ax}_{914},\mbox{axiom})$

$p(n_9,n_2,n_7)\qquad\mbox{fof}(\mbox{ax}_{927},\mbox{axiom})$

$p(n_9,n_6,n_6)\qquad\mbox{fof}(\mbox{ax}_{966},\mbox{axiom})$

\medskip\textbf{PUZ133=2.p} N queens problem has the variable symmetry property






$\mbox{queens\_q}{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{queens\_q\_type},\mbox{type})$

$\mbox{queens\_p}{:}\;\mbox{\$o}\qquad\mbox{tff}(\mbox{queens\_p\_type},\mbox{type})$

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

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

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

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

$\mbox{queens\_p} \ \Rightarrow \ \forall i{:}\;\mbox{\$int},j{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(1,i)\mbox{ and }\mbox{\$lesseq}(i,n)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(1,i),j)\mbox{ and }\mbox{\$lesseq}(j,n) ) \ \Rightarrow \ ( p(i)\ne p(j)\mbox{ and }\mbox{\$sum}(p(i),i)\ne \mbox{\$sum}(p(j),j)\mbox{ and }\mbox{\$difference}(p(i),i)\ne \mbox{\$difference}(p(j),j) ) )\qquad\mbox{tff}(\mbox{queens\_p},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int}{:}\;\mbox{perm}(i) = \mbox{\$difference}(\mbox{\$sum}(1,n),i)\qquad\mbox{tff}(\mbox{permutation},\mbox{axiom})$

$\forall i{:}\;\mbox{\$int},j{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(1,i)\mbox{ and }\mbox{\$lesseq}(i,n)\mbox{ and }\mbox{\$lesseq}(\mbox{\$sum}(1,i),j)\mbox{ and }\mbox{\$lesseq}(j,n)\mbox{ and }( \mbox{\$lesseq}(\mbox{\$sum}(1,i),j)\iff \mbox{\$lesseq}(\mbox{\$sum}(1,\mbox{perm}(j)),\mbox{perm}(i)) ) ) \ \Rightarrow \ ( q(i)\ne q(j)\mbox{ and }\mbox{\$sum}(q(i),i)\ne \mbox{\$sum}(q(j),j)\mbox{ and }\mbox{\$difference}(q(i),i)\ne \mbox{\$difference}(q(j),j) ) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{tff}(\mbox{queens\_q},\mbox{axiom})$

$( \mbox{queens\_p}\mbox{ and }\forall i{:}\;\mbox{\$int}{:}\;q(i) = p(\mbox{perm}(i)) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{tff}(\mbox{queens\_sym},\mbox{conjecture})$

$\forall i{:}\;\mbox{\$int}{:}\;( ( \mbox{\$lesseq}(1,i)\mbox{ and }\mbox{\$lesseq}(i,n) ) \ \Rightarrow \ ( \mbox{\$lesseq}(1,\mbox{perm}(i))\mbox{ and }\mbox{\$lesseq}(\mbox{perm}(i),n) ) )\qquad\mbox{tff}(\mbox{permutation\_range},\mbox{axiom})$

$\forall j{:}\;\mbox{\$int},i{:}\;\mbox{\$int}{:}\;\mbox{\$difference}(i,j) = \mbox{\$difference}(\mbox{perm}(j),\mbox{perm}(i))\qquad\mbox{tff}(\mbox{permutation\_another\_one},\mbox{axiom})$

\medskip\textbf{PUZ096$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_INJ}@ \mbox{cCKB6\_H}\qquad\mbox{thf}(\mbox{cCKB6\_L}_{25000},\mbox{conjecture})$

\medskip\textbf{PUZ069-1.p} Tuesday's Sudoku

1 76
 85
 2    3
 24
 9      6
 53
 7   1
 3    2




$\mbox{include}(\mbox{'Axioms/PUZ005-0.ax'})$

$\mbox{el}(s(n_0),s(n_0),s(n_0))\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(n_0))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(n_0)))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(n_0)))))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(n_0)),s(s(n_0)))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(s(s(s(s(s(n_0))))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(n_0)))),s(s(n_0)))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(n_0))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(n_0),s(s(s(s(s(s(s(s(s(n_0))))))))))\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(n_0)))))),s(s(s(s(s(n_0))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(n_0)))))),s(s(s(s(s(s(n_0)))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(s(n_0)))))))),s(n_0))\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(n_0)),s(s(s(n_0))))\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(s(s(n_0))))))),s(s(n_0)))\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

\medskip\textbf{PUZ001+2.p} Dreadbury Mansion

Someone who lives in DreadburyMansion kills AuntAgatha. If 
 somebody X lives in DreadburyMansion then X is AuntAgatha or X
 is the Butler or X is Charles. Everyone hates everyone that he
 kills. Noone is richer than someone that he kills. Charles hates
 noone who is hated by AuntAgatha. AuntAgatha does not hate the
 Butler. Everyone that is not the Butler is hated by AuntAgatha.
 The Butler hates everyone who is not richer than AuntAgatha. The
 Butler hates everyone who is hated by AuntAgatha. Noone hates
 everyone. AuntAgatha is not the Butler. Therefore, AuntAgatha
 kills AuntAgatha.




$\exists a,b,c{:}\;( \mbox{\$true}\mbox{ and }\mbox{predicate}_1(b,\mbox{live},a)\mbox{ and }\mbox{modifier\_pp}(b,\mbox{in},\mbox{'DreadburyMansion'})\mbox{ and }\mbox{predicate}_2(c,\mbox{kill},a,\mbox{'AuntAgatha'})\mbox{ and }\forall d,e{:}\;( ( \mbox{\$true}\mbox{ and }\mbox{predicate}_1(e,\mbox{live},d)\mbox{ and }\mbox{modifier\_pp}(e,\mbox{in},\mbox{'DreadburyMansion'}) ) \ \Rightarrow \ ( d = \mbox{'AuntAgatha'}\mbox{ or }d = \mbox{'Butler'}\mbox{ or }d = \mbox{'Charles'} ) )\mbox{ and }\forall f{:}\;( \mbox{\$true} \ \Rightarrow \ \forall g,h{:}\;( ( \mbox{\$true}\mbox{ and }\mbox{predicate}_2(h,\mbox{kill},f,g) ) \ \Rightarrow \ \exists i{:}\;\mbox{predicate}_2(i,\mbox{hate},f,g) ) )\mbox{ and }\forall j{:}\;( \mbox{\$true} \ \Rightarrow \ \neg\,\exists k,l,m{:}\;( \mbox{\$true}\mbox{ and }\mbox{predicate}_2(l,\mbox{kill},j,k)\mbox{ and }\mbox{property}_2(m,\mbox{rich},\mbox{comp\_than},k)\mbox{ and }j = m ) )\mbox{ and }\forall n,o{:}\;( ( \mbox{\$true}\mbox{ and }\mbox{predicate}_2(o,\mbox{hate},\mbox{'AuntAgatha'},n) ) \ \Rightarrow \ \neg\,\exists p{:}\;\mbox{predicate}_2(p,\mbox{hate},\mbox{'Charles'},n) )\mbox{ and }\neg\,\exists q{:}\;\mbox{predicate}_2(q,\mbox{hate},\mbox{'AuntAgatha'},\mbox{'Butler'})\mbox{ and }\forall r{:}\;( ( \mbox{\$true}\mbox{ and }r\ne \mbox{'Butler'} ) \ \Rightarrow \ \exists s{:}\;\mbox{predicate}_2(s,\mbox{hate},\mbox{'AuntAgatha'},r) )\mbox{ and }\forall t{:}\;( ( \mbox{\$true}\mbox{ and }\neg\,\exists u{:}\;( \mbox{property}_2(u,\mbox{rich},\mbox{comp\_than},\mbox{'AuntAgatha'})\mbox{ and }t = u ) ) \ \Rightarrow \ \exists v{:}\;\mbox{predicate}_2(v,\mbox{hate},\mbox{'Butler'},t) )\mbox{ and }\forall w,x{:}\;( ( \mbox{\$true}\mbox{ and }\mbox{predicate}_2(x,\mbox{hate},\mbox{'AuntAgatha'},w) ) \ \Rightarrow \ \exists y{:}\;\mbox{predicate}_2(y,\mbox{hate},\mbox{'Butler'},w) )\mbox{ and }\forall z{:}\;( \mbox{\$true} \ \Rightarrow \ \neg\,\forall a_1{:}\;( \mbox{\$true} \ \Rightarrow \ \exists b_1{:}\;\mbox{predicate}_2(b_1,\mbox{hate},z,a_1) ) )\mbox{ and }\mbox{'AuntAgatha'}\ne \mbox{'Butler'} )\qquad\mbox{fof}(\mbox{background},\mbox{axiom})$

$\exists a{:}\;\mbox{predicate}_2(a,\mbox{kill},\mbox{'AuntAgatha'},\mbox{'AuntAgatha'})\qquad\mbox{fof}(\mbox{prove},\mbox{conjecture})$

\medskip\textbf{PUZ028-6.p} People at a party

We can always choose 3 persons who are either familiar with
 each other or not familiar with each other, from 6 persons
 who meet at a party.




$\mbox{person}(n_1)\qquad\mbox{cnf}(\mbox{person}_1,\mbox{axiom})$

$\mbox{person}(n_2)\qquad\mbox{cnf}(\mbox{person}_2,\mbox{axiom})$

$\mbox{person}(n_3)\qquad\mbox{cnf}(\mbox{person}_3,\mbox{axiom})$

$\mbox{person}(n_4)\qquad\mbox{cnf}(\mbox{person}_4,\mbox{axiom})$

$\mbox{person}(n_5)\qquad\mbox{cnf}(\mbox{person}_5,\mbox{axiom})$

$\mbox{person}(n_6)\qquad\mbox{cnf}(\mbox{person}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_1,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_1}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_2,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_2}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_3,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_3}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_5,\mbox{axiom})$

$\mbox{not\_equal}(n_4,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_4}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_5,n_6)\qquad\mbox{cnf}(\mbox{not\_equal\_5}_6,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_1)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_1,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_2)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_2,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_3)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_3,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_4)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_4,\mbox{axiom})$

$\mbox{not\_equal}(n_6,n_5)\qquad\mbox{cnf}(\mbox{not\_equal\_6}_5,\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{person}(y)\mbox{ and }\mbox{not\_equal}(x,y) ) \ \Rightarrow \ ( \mbox{familiar}(x,y)\mbox{ or }\mbox{not\_familiar}(x,y) )\qquad\mbox{cnf}(\mbox{familiar\_or\_not},\mbox{axiom})$

$\mbox{familiar}(x_1,x_2) \ \Rightarrow \ \mbox{familiar}(x_2,x_1)\qquad\mbox{cnf}(\mbox{symmetry\_of\_familiar},\mbox{axiom})$

$\mbox{not\_familiar}(x_1,x_2) \ \Rightarrow \ \mbox{not\_familiar}(x_2,x_1)\qquad\mbox{cnf}(\mbox{symmetry\_of\_not\_familiar},\mbox{axiom})$

$( \mbox{familiar}(x_1,x_2)\mbox{ and }\mbox{familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_familiar},\mbox{negated\_conjecture})$

$( \mbox{not\_familiar}(x_1,x_2)\mbox{ and }\mbox{not\_familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{not\_familiar}(x_3,x_1)\qquad\mbox{cnf}(\mbox{three\_not\_familiar},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ058-1.p} Show the Hanoi problem is not solvable anymore

Move blocks from pin to pin, until a goal configuration
 is reached. Some rules are removed: one cannot move the medium
 block onto the second pin directly.




$\mbox{state}(\mbox{nextto}(\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_1)))),\mbox{nextto}(\mbox{pin}(n_2),\mbox{pin}(n_3))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{axiom})$

$\neg\,\mbox{state}(\mbox{nextto}(\mbox{pin}(n_1),\mbox{nextto}(\mbox{pin}(n_2),\mbox{on}(\mbox{small},\mbox{on}(\mbox{medium},\mbox{on}(\mbox{large},\mbox{pin}(n_3)))))))\qquad\mbox{cnf}(\mbox{final\_state},\mbox{negated\_conjecture})$

$\mbox{nextto}(x,y) = \mbox{nextto}(y,x)\qquad\mbox{cnf}(\mbox{nextto\_commutative},\mbox{axiom})$

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

$( \mbox{fits}(x,b)\mbox{ and }\mbox{fits}(x,a) ) \ \Rightarrow \ \mbox{nextto}(\mbox{on}(x,a),b) = \mbox{nextto}(a,\mbox{on}(x,b))\qquad\mbox{cnf}(\mbox{a\_move},\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_1,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{pin}(n))\mbox{ or }n = n_2\qquad\mbox{cnf}(\mbox{fits}_2,\mbox{axiom})$

$\mbox{fits}(\mbox{large},\mbox{pin}(n))\qquad\mbox{cnf}(\mbox{fits}_3,\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{on}(\mbox{medium},a))\qquad\mbox{cnf}(\mbox{fits}_4,\mbox{axiom})$

$\mbox{fits}(\mbox{medium},\mbox{on}(\mbox{large},a))\qquad\mbox{cnf}(\mbox{fits}_5,\mbox{axiom})$

$\mbox{fits}(\mbox{small},\mbox{on}(\mbox{large},a))\qquad\mbox{cnf}(\mbox{fits}_6,\mbox{axiom})$

\medskip\textbf{PUZ114$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cCKB6\_BLACK}@ ( s@ ( s@ ( s@ \mbox{xj} ) ) )@ ( s@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{10000},\mbox{conjecture})$

\medskip\textbf{PUZ001-2.p} Dreadbury Mansion

Someone who lives in Dreadbury Mansion killed Aunt Agatha.
 Agatha, the butler, and Charles live in Dreadbury Mansion,
 and are the only people who live therein. A killer always
 hates his victim, and is never richer than his victim.
 Charles hates no one that Aunt Agatha hates. Agatha hates
 everyone except the butler. The butler hates everyone not
 richer than Aunt Agatha. The butler hates everyone Aunt
 Agatha hates. No one hates everyone. Agatha is not the
 butler. Therefore : Agatha killed herself.




$\mbox{lives\_at\_dreadsbury}(\mbox{someone})\qquad\mbox{cnf}(\mbox{someone\_in\_mansion},\mbox{axiom})$

$\mbox{killed}(\mbox{someone},\mbox{aunt\_agatha})\qquad\mbox{cnf}(\mbox{someone\_killed\_agatha},\mbox{axiom})$

$\mbox{lives\_at\_dreadsbury}(\mbox{aunt\_agatha})\qquad\mbox{cnf}(\mbox{agatha\_lives\_at\_mansion},\mbox{axiom})$

$\mbox{lives\_at\_dreadsbury}(\mbox{butler})\qquad\mbox{cnf}(\mbox{butler\_lives\_at\_mansion},\mbox{axiom})$

$\mbox{lives\_at\_dreadsbury}(\mbox{charles})\qquad\mbox{cnf}(\mbox{charles\_lives\_at\_mansion},\mbox{axiom})$

$\mbox{lives\_at\_dreadsbury}(\mbox{person}) \ \Rightarrow \ ( \mbox{person} = \mbox{aunt\_agatha}\mbox{ or }\mbox{person} = \mbox{butler}\mbox{ or }\mbox{person} = \mbox{charles} )\qquad\mbox{cnf}(\mbox{noone\_else\_lives\_at\_mansion},\mbox{axiom})$

$\mbox{killed}(\mbox{killer},\mbox{victim}) \ \Rightarrow \ \mbox{hates}(\mbox{killer},\mbox{victim})\qquad\mbox{cnf}(\mbox{killer\_hates\_victim},\mbox{axiom})$

$\mbox{killed}(\mbox{killer},\mbox{victim}) \ \Rightarrow \ \neg\,\mbox{richer}(\mbox{killer},\mbox{victim})\qquad\mbox{cnf}(\mbox{killer\_poorer\_than\_victim},\mbox{axiom})$

$\mbox{hates}(\mbox{aunt\_agatha},\mbox{person}) \ \Rightarrow \ \neg\,\mbox{hates}(\mbox{charles},\mbox{person})\qquad\mbox{cnf}(\mbox{charles\_and\_agatha\_hate\_different\_people},\mbox{axiom})$

$\mbox{person} = \mbox{butler}\mbox{ or }\mbox{hates}(\mbox{aunt\_agatha},\mbox{person})\qquad\mbox{cnf}(\mbox{agatha\_likes\_only\_butler},\mbox{axiom})$

$\mbox{richer}(\mbox{person},\mbox{aunt\_agatha})\mbox{ or }\mbox{hates}(\mbox{butler},\mbox{person})\qquad\mbox{cnf}(\mbox{butler\_hates\_poor\_people},\mbox{axiom})$

$\mbox{hates}(\mbox{aunt\_agatha},\mbox{person}) \ \Rightarrow \ \mbox{hates}(\mbox{butler},\mbox{person})\qquad\mbox{cnf}(\mbox{butler\_and\_agatha\_hate\_the\_same\_people},\mbox{axiom})$

$\neg\,\mbox{hates}(\mbox{person},\mbox{every\_one\_but}(\mbox{person}))\qquad\mbox{cnf}(\mbox{noone\_hates\_everyone},\mbox{axiom})$

$\mbox{aunt\_agatha}\ne \mbox{butler}\qquad\mbox{cnf}(\mbox{agatha\_is\_not\_the\_butler},\mbox{axiom})$

$\neg\,\mbox{killed}(\mbox{aunt\_agatha},\mbox{aunt\_agatha})\qquad\mbox{cnf}(\mbox{prove\_agatha\_killed\_herself},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ026-1.p} Knights and Knaves \#39

There is an island with exactly three types of people -
 truthtellers who always tell the truth, and liars who always
 lie, and normals who sometimes tell the truth and sometimes
 lie. We are given three people, A, B, C, one of whom is a
 truthteller, one a liar, and one a normal (but not neccesarily
 in that order). They make the following statements. A: I am
 normal; B: That is true. C: I am not normal. What are A,B,
 and C? Answer: A is a liar, B is a normal, and C is a truthteller.




$\mbox{include}(\mbox{'Axioms/PUZ003-0.ax'})$

$\mbox{a\_truth}(\mbox{not\_normal}(x)) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{normal}(x))\qquad\mbox{cnf}(\mbox{not\_normal\_and\_not\_normal},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{not\_normal}(x))\mbox{ or }\mbox{a\_truth}(\mbox{normal}(x))\qquad\mbox{cnf}(\mbox{normal\_or\_not\_normal},\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(y))\qquad\mbox{cnf}(\mbox{not\_two\_truthtellers}_1,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(z))\qquad\mbox{cnf}(\mbox{not\_two\_truthtellers}_2,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{liar}(y))\qquad\mbox{cnf}(\mbox{not\_two\_liars}_1,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{liar}(z))\qquad\mbox{cnf}(\mbox{not\_two\_liars}_2,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{normal}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{normal}(y))\qquad\mbox{cnf}(\mbox{not\_two\_normal}_1,\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{normal}(x)) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{normal}(z))\qquad\mbox{cnf}(\mbox{not\_two\_normal}_2,\mbox{axiom})$

$\mbox{people}(a,b,c)\qquad\mbox{cnf}(\mbox{a\_b\_c\_are\_people},\mbox{hypothesis})$

$\mbox{people}(b,c,a)\qquad\mbox{cnf}(\mbox{b\_c\_a\_are\_people},\mbox{hypothesis})$

$\mbox{people}(c,b,a)\qquad\mbox{cnf}(\mbox{c\_b\_a\_are\_people},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{normal}(a)))\qquad\mbox{cnf}(\mbox{a\_says\_a\_normal},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(b,\mbox{normal}(a)))\qquad\mbox{cnf}(\mbox{b\_says\_a\_normal},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(c,\mbox{not\_normal}(c)))\qquad\mbox{cnf}(\mbox{c\_says\_c\_not\_normal},\mbox{hypothesis})$

$( \mbox{a\_truth}(\mbox{liar}(\mbox{liar}))\mbox{ and }\mbox{a\_truth}(\mbox{normal}(\mbox{normal})) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{truthteller}(\mbox{truthteller}))\qquad\mbox{cnf}(\mbox{prove\_one\_of\_each},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ063-1.p} Problem about mutilated chessboard problem






$\mbox{include}(\mbox{'Axioms/MSC001-1.ax'})$

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

$\mbox{c\_in}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{c\_Suc}(\mbox{v\_j}),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{c\_Mutil\_Odomino},\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Odomino\_Ohoriz}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{v\_i},\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_insert}(\mbox{c\_Pair}(\mbox{c\_Suc}(\mbox{v\_i}),\mbox{v\_j},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{c\_emptyset},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})),\mbox{c\_Mutil\_Odomino},\mbox{tc\_set}(\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Odomino\_Overtl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}) = \mbox{c\_emptyset} ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_OUn}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_emptyset},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_Oempty}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{t\_a}) = \mbox{c\_union}(\mbox{v\_A},\mbox{c\_union}(\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_assoc}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{c\_emptyset},\mbox{v\_u},\mbox{t\_a}) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_emptyset},\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{PUZ028-5.p} People at a party

We can always choose 3 persons who are either familiar with
 each other or not familiar with each other, from 6 persons
 who meet at a party.




$\mbox{person}(1)\qquad\mbox{cnf}(\mbox{person}_1,\mbox{axiom})$

$\mbox{person}(\mbox{two})\qquad\mbox{cnf}(\mbox{person}_2,\mbox{axiom})$

$\mbox{person}(\mbox{three})\qquad\mbox{cnf}(\mbox{person}_3,\mbox{axiom})$

$\mbox{person}(\mbox{four})\qquad\mbox{cnf}(\mbox{person}_4,\mbox{axiom})$

$\mbox{person}(\mbox{five})\qquad\mbox{cnf}(\mbox{person}_5,\mbox{axiom})$

$\mbox{person}(\mbox{six})\qquad\mbox{cnf}(\mbox{person}_6,\mbox{axiom})$

$\mbox{after}(1,\mbox{two})\qquad\mbox{cnf}(\mbox{order}_1,\mbox{axiom})$

$\mbox{after}(\mbox{two},\mbox{three})\qquad\mbox{cnf}(\mbox{order}_2,\mbox{axiom})$

$\mbox{after}(\mbox{three},\mbox{four})\qquad\mbox{cnf}(\mbox{order}_3,\mbox{axiom})$

$\mbox{after}(\mbox{four},\mbox{five})\qquad\mbox{cnf}(\mbox{order}_4,\mbox{axiom})$

$\mbox{after}(\mbox{five},\mbox{six})\qquad\mbox{cnf}(\mbox{order}_5,\mbox{axiom})$

$( \mbox{after}(\mbox{large},\mbox{medium})\mbox{ and }\mbox{after}(\mbox{medium},\mbox{small}) ) \ \Rightarrow \ \mbox{after}(\mbox{large},\mbox{small})\qquad\mbox{cnf}(\mbox{transitivity\_of\_order},\mbox{axiom})$

$( \mbox{person}(x)\mbox{ and }\mbox{person}(y)\mbox{ and }\mbox{after}(x,y) ) \ \Rightarrow \ ( \mbox{familiar}(x,y)\mbox{ or }\mbox{not\_familiar}(x,y) )\qquad\mbox{cnf}(\mbox{familiar\_or\_not},\mbox{axiom})$

$( \mbox{familiar}(x_1,x_2)\mbox{ and }\mbox{familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{familiar}(x_1,x_3)\qquad\mbox{cnf}(\mbox{three\_familiar},\mbox{negated\_conjecture})$

$( \mbox{not\_familiar}(x_1,x_2)\mbox{ and }\mbox{not\_familiar}(x_2,x_3) ) \ \Rightarrow \ \neg\,\mbox{not\_familiar}(x_1,x_3)\qquad\mbox{cnf}(\mbox{three\_not\_familiar},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ056-2.005.p} Towers of Hanoi k=05

Each instance encodes Tower of Hanoi with n discs as a
 reachability problem.




$p(i,t_1,t_2,t_3,t_4) \ \Rightarrow \ p(j,t_1,t_2,t_3,t_4)\qquad\mbox{cnf}(\mbox{rule}_1,\mbox{axiom})$

$( p(t_0,i,t_2,t_3,t_4)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j) ) \ \Rightarrow \ p(t_0,j,t_2,t_3,t_4)\qquad\mbox{cnf}(\mbox{rule}_2,\mbox{axiom})$

$( p(t_0,t_1,i,t_3,t_4)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j) ) \ \Rightarrow \ p(t_0,t_1,j,t_3,t_4)\qquad\mbox{cnf}(\mbox{rule}_3,\mbox{axiom})$

$( p(t_0,t_1,t_2,i,t_4)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,j,t_4)\qquad\mbox{cnf}(\mbox{rule}_4,\mbox{axiom})$

$( p(t_0,t_1,t_2,t_3,i)\mbox{ and }\mbox{neq}(t_0,i)\mbox{ and }\mbox{neq}(t_0,j)\mbox{ and }\mbox{neq}(t_1,i)\mbox{ and }\mbox{neq}(t_1,j)\mbox{ and }\mbox{neq}(t_2,i)\mbox{ and }\mbox{neq}(t_2,j)\mbox{ and }\mbox{neq}(t_3,i)\mbox{ and }\mbox{neq}(t_3,j) ) \ \Rightarrow \ p(t_0,t_1,t_2,t_3,j)\qquad\mbox{cnf}(\mbox{rule}_5,\mbox{axiom})$

$\neg\,\mbox{neq}(s_0,s_0)\qquad\mbox{cnf}(\mbox{neq}_1,\mbox{axiom})$

$\mbox{neq}(s_0,s_1)\qquad\mbox{cnf}(\mbox{neq}_2,\mbox{axiom})$

$\mbox{neq}(s_0,s_2)\qquad\mbox{cnf}(\mbox{neq}_3,\mbox{axiom})$

$\mbox{neq}(s_1,s_0)\qquad\mbox{cnf}(\mbox{neq}_4,\mbox{axiom})$

$\neg\,\mbox{neq}(s_1,s_1)\qquad\mbox{cnf}(\mbox{neq}_5,\mbox{axiom})$

$\mbox{neq}(s_1,s_2)\qquad\mbox{cnf}(\mbox{neq}_6,\mbox{axiom})$

$\mbox{neq}(s_2,s_0)\qquad\mbox{cnf}(\mbox{neq}_7,\mbox{axiom})$

$\mbox{neq}(s_2,s_1)\qquad\mbox{cnf}(\mbox{neq}_8,\mbox{axiom})$

$\neg\,\mbox{neq}(s_2,s_2)\qquad\mbox{cnf}(\mbox{neq}_9,\mbox{axiom})$

$p(s_0,s_0,s_0,s_0,s_0)\qquad\mbox{cnf}(\mbox{init},\mbox{axiom})$

$\neg\,p(s_2,s_2,s_2,s_2,s_2)\qquad\mbox{cnf}(\mbox{goal},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ035-4.p} Knights and Knaves \#36

On an island, there live exactly two types of people: knights
 and knaves. Knights always tell the truth and knaves always
 lie. I landed on the island, met two inhabitants, asked one of
 them: "Is one of you a knight?" and he answered me. What can
 be said about the types of the asked and the other person
 depending on the answer I get?




$\mbox{person}(p) \ \Rightarrow \ ( \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\mbox{ or }\mbox{truth}(\mbox{isa}(p,\mbox{knave})) )\qquad\mbox{cnf}(\mbox{everyone\_a\_knight\_or\_knave},\mbox{axiom})$

$\mbox{truth}(\mbox{isa}(p,\mbox{knight})) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{isa}(p,\mbox{knave}))\qquad\mbox{cnf}(\mbox{not\_both\_a\_knight\_and\_knave},\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(\mbox{isa}(p,\mbox{knight})) ) \ \Rightarrow \ \mbox{truth}(s)\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_1,\mbox{axiom})$

$( \mbox{says}(p,s)\mbox{ and }\mbox{truth}(s) ) \ \Rightarrow \ \mbox{truth}(\mbox{isa}(p,\mbox{knight}))\qquad\mbox{cnf}(\mbox{knights\_make\_true\_statements}_2,\mbox{axiom})$

$\mbox{says}(\mbox{asked},\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight})))\mbox{ or }\mbox{says}(\mbox{asked},\mbox{not}(\mbox{or}(\mbox{isa}(\mbox{asked},\mbox{knight}),\mbox{isa}(\mbox{other},\mbox{knight}))))\qquad\mbox{cnf}(\mbox{one\_is\_knight\_answer},\mbox{axiom})$

$\mbox{says}(\mbox{anyone},s) \ \Rightarrow \ ( \mbox{truth}(s)\mbox{ or }\mbox{truth}(\mbox{not}(s)) )\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false},\mbox{axiom})$

$\mbox{says}(\mbox{anyone},\mbox{not}(s)) \ \Rightarrow \ ( \mbox{truth}(s)\mbox{ or }\mbox{truth}(\mbox{not}(s)) )\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false}_2,\mbox{axiom})$

$\mbox{truth}(s) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(s))\qquad\mbox{cnf}(\mbox{statements\_are\_true\_or\_false}_3,\mbox{axiom})$

$\mbox{truth}(\mbox{or}(s_1,s_2)) \ \Rightarrow \ ( \mbox{truth}(s_1)\mbox{ or }\mbox{truth}(s_2) )\qquad\mbox{cnf}(\mbox{or\_def},\mbox{axiom})$

$\mbox{truth}(s_1) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(\mbox{or}(s_1,\mbox{anything})))\qquad\mbox{cnf}(\mbox{not}_1,\mbox{axiom})$

$\mbox{truth}(s_2) \ \Rightarrow \ \neg\,\mbox{truth}(\mbox{not}(\mbox{or}(\mbox{anything},s_2)))\qquad\mbox{cnf}(\mbox{not}_2,\mbox{axiom})$

$\mbox{person}(\mbox{asked})\qquad\mbox{cnf}(\mbox{asked\_person},\mbox{axiom})$

$\mbox{person}(\mbox{other})\qquad\mbox{cnf}(\mbox{other\_person},\mbox{axiom})$

$( \mbox{truth}(\mbox{isa}(\mbox{asked},p_1))\mbox{ and }\mbox{truth}(\mbox{isa}(\mbox{other},p_2)) ) \ \Rightarrow \ \neg\,\mbox{says}(\mbox{asked},s)\qquad\mbox{cnf}(\mbox{prove\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ123$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_8{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c8\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,( ( s@ ( s@ \mbox{xx} ) ) = c_1\mbox{ and }( s@ \mbox{xy} ) = c_1 )\mbox{ or }( ( s@ ( s@ \mbox{xx} ) ) = c_8\mbox{ and }( s@ \mbox{xy} ) = c_8 ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{39000},\mbox{conjecture})$

\medskip\textbf{PUZ116$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_5{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c5\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) )\ne c_5 )\qquad\mbox{thf}(\mbox{cCKB\_L}_{39100},\mbox{conjecture})$

\medskip\textbf{PUZ082$\wedge$1.p} Peter the liar

Peter says that everything he says is false. Show that not
 everything Peter says is false.




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

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

$\mbox{says}@ \mbox{peter}@ \forall x{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ \mbox{peter}@ x ) \ \Rightarrow \ \neg\,x )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\neg\,\forall x{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ \mbox{peter}@ x ) \ \Rightarrow \ \neg\,x )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{PUZ079+2.p} Another Sudoku

1  6 4  9
 5 7
 4 9 5
 47     53
 3   1
 91     68
 7 5 2
 9 3
 2  1 6  4




$\mbox{include}(\mbox{'Axioms/PUZ006+0.ax'})$

$p(n_1,n_1,n_1)\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$p(n_1,n_4,n_6)\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$p(n_1,n_6,n_4)\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$p(n_1,n_9,n_9)\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$p(n_2,n_4,n_5)\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$p(n_2,n_6,n_7)\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$p(n_3,n_3,n_4)\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$p(n_3,n_5,n_9)\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$p(n_3,n_7,n_5)\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

$p(n_4,n_1,n_4)\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$p(n_4,n_2,n_7)\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$p(n_4,n_8,n_5)\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$p(n_4,n_9,n_3)\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$p(n_5,n_3,n_3)\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$p(n_5,n_7,n_1)\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$p(n_6,n_1,n_9)\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$p(n_6,n_2,n_1)\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$p(n_6,n_8,n_6)\qquad\mbox{fof}(\mbox{ax369}_1,\mbox{axiom})$

$p(n_6,n_9,n_8)\qquad\mbox{fof}(\mbox{ax369}_2,\mbox{axiom})$

$p(n_7,n_3,n_7)\qquad\mbox{fof}(\mbox{ax369}_3,\mbox{axiom})$

$p(n_7,n_5,n_5)\qquad\mbox{fof}(\mbox{ax369}_4,\mbox{axiom})$

$p(n_7,n_7,n_2)\qquad\mbox{fof}(\mbox{ax369}_5,\mbox{axiom})$

$p(n_8,n_4,n_9)\qquad\mbox{fof}(\mbox{ax369}_6,\mbox{axiom})$

$p(n_8,n_6,n_3)\qquad\mbox{fof}(\mbox{ax369}_7,\mbox{axiom})$

$p(n_9,n_1,n_2)\qquad\mbox{fof}(\mbox{ax369}_8,\mbox{axiom})$

$p(n_9,n_4,n_1)\qquad\mbox{fof}(\mbox{ax369}_9,\mbox{axiom})$

$p(n_9,n_6,n_6)\qquad\mbox{fof}(\mbox{ax369}_{10},\mbox{axiom})$

$p(n_9,n_9,n_4)\qquad\mbox{fof}(\mbox{ax369}_{11},\mbox{axiom})$

\medskip\textbf{PUZ001+1.p} Dreadbury Mansion

Someone who lives in Dreadbury Mansion killed Aunt Agatha.
 Agatha, the butler, and Charles live in Dreadbury Mansion,
 and are the only people who live therein. A killer always
 hates his victim, and is never richer than his victim.
 Charles hates no one that Aunt Agatha hates. Agatha hates
 everyone except the butler. The butler hates everyone not
 richer than Aunt Agatha. The butler hates everyone Aunt
 Agatha hates. No one hates everyone. Agatha is not the
 butler. Therefore : Agatha killed herself.




$\exists x{:}\;( \mbox{lives}(x)\mbox{ and }\mbox{killed}(x,\mbox{agatha}) )\qquad\mbox{fof}(\mbox{pel55}_1,\mbox{axiom})$

$\mbox{lives}(\mbox{agatha})\qquad\mbox{fof}(\mbox{pel55\_2}_1,\mbox{axiom})$

$\mbox{lives}(\mbox{butler})\qquad\mbox{fof}(\mbox{pel55\_2}_2,\mbox{axiom})$

$\mbox{lives}(\mbox{charles})\qquad\mbox{fof}(\mbox{pel55\_2}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{lives}(x) \ \Rightarrow \ ( x = \mbox{agatha}\mbox{ or }x = \mbox{butler}\mbox{ or }x = \mbox{charles} ) )\qquad\mbox{fof}(\mbox{pel55}_3,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{killed}(x,y) \ \Rightarrow \ \mbox{hates}(x,y) )\qquad\mbox{fof}(\mbox{pel55}_4,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{killed}(x,y) \ \Rightarrow \ \neg\,\mbox{richer}(x,y) )\qquad\mbox{fof}(\mbox{pel55}_5,\mbox{axiom})$

$\forall x{:}\;( \mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \neg\,\mbox{hates}(\mbox{charles},x) )\qquad\mbox{fof}(\mbox{pel55}_6,\mbox{axiom})$

$\forall x{:}\;( x\ne \mbox{butler} \ \Rightarrow \ \mbox{hates}(\mbox{agatha},x) )\qquad\mbox{fof}(\mbox{pel55}_7,\mbox{axiom})$

$\forall x{:}\;( \neg\,\mbox{richer}(x,\mbox{agatha}) \ \Rightarrow \ \mbox{hates}(\mbox{butler},x) )\qquad\mbox{fof}(\mbox{pel55}_8,\mbox{axiom})$

$\forall x{:}\;( \mbox{hates}(\mbox{agatha},x) \ \Rightarrow \ \mbox{hates}(\mbox{butler},x) )\qquad\mbox{fof}(\mbox{pel55}_9,\mbox{axiom})$

$\forall x{:}\;\exists y{:}\;\neg\,\mbox{hates}(x,y)\qquad\mbox{fof}(\mbox{pel55}_{10},\mbox{axiom})$

$\mbox{agatha}\ne \mbox{butler}\qquad\mbox{fof}(\mbox{pel55}_{11},\mbox{axiom})$

$\mbox{killed}(\mbox{agatha},\mbox{agatha})\qquad\mbox{fof}(\mbox{pel}_{55},\mbox{conjecture})$

\medskip\textbf{PUZ034-1.004.p} N queens problem

The problem is to place 4 queens on an 4x4 chess board,
 so that no queen can attack another.




$( \mbox{less}(\mbox{low},\mbox{high})\mbox{ and }\mbox{low} + s(n_0){=}\mbox{newLow}\mbox{ and }\mbox{range}(\mbox{newLow},\mbox{high},\mbox{restOfNumbers}) ) \ \Rightarrow \ \mbox{range}(\mbox{low},\mbox{high},\mbox{cons}(\mbox{low},\mbox{restOfNumbers}))\qquad\mbox{cnf}(\mbox{make\_list\_of\_numbers}_1,\mbox{axiom})$

$\mbox{range}(\mbox{same},\mbox{same},\mbox{cons}(\mbox{same},\mbox{empty\_list}))\qquad\mbox{cnf}(\mbox{make\_list\_of\_numbers}_2,\mbox{axiom})$

$\mbox{less}(n_0,s(x))\qquad\mbox{cnf}(\mbox{less}_1,\mbox{axiom})$

$\mbox{less}(x,y) \ \Rightarrow \ \mbox{less}(s(x),s(y))\qquad\mbox{cnf}(\mbox{less}_2,\mbox{axiom})$

$x + n_0{=}x\qquad\mbox{cnf}(\mbox{add}_0,\mbox{axiom})$

$x + y{=}z \ \Rightarrow \ x + s(y){=}s(z)\qquad\mbox{cnf}(\mbox{add},\mbox{axiom})$

$\mbox{select}(\mbox{head},\mbox{cons}(\mbox{head},\mbox{tail}),\mbox{tail})\qquad\mbox{cnf}(\mbox{select}_1,\mbox{axiom})$

$\mbox{select}(\mbox{element},\mbox{tail},\mbox{newTail}) \ \Rightarrow \ \mbox{select}(\mbox{element},\mbox{cons}(\mbox{head},\mbox{tail}),\mbox{cons}(\mbox{head},\mbox{newTail}))\qquad\mbox{cnf}(\mbox{select}_2,\mbox{axiom})$

$\neg\,\mbox{same}(s(x),n_0)\qquad\mbox{cnf}(\mbox{same\_definition}_1,\mbox{axiom})$

$\neg\,\mbox{same}(n_0,s(x))\qquad\mbox{cnf}(\mbox{same\_definition}_2,\mbox{axiom})$

$\mbox{same}(s(x),s(y)) \ \Rightarrow \ \mbox{same}(x,y)\qquad\mbox{cnf}(\mbox{same\_definition}_3,\mbox{axiom})$

$\mbox{attack}(\mbox{queen},\mbox{placedQueens}) \ \Rightarrow \ \mbox{diagonal\_attack}(\mbox{queen},s(n_0),\mbox{placedQueens})\qquad\mbox{cnf}(\mbox{attack},\mbox{axiom})$

$( \mbox{diagonal\_attack}(\mbox{queen},\mbox{queenNumber},\mbox{cons}(\mbox{placedQueen},\mbox{otherPlacedQueens}))\mbox{ and }\mbox{diagonal}_1 + \mbox{queenNumber}{=}\mbox{placedQueen}\mbox{ and }\mbox{placedQueen} + \mbox{queenNumber}{=}\mbox{diagonal}_2\mbox{ and }\mbox{queenNumber} + s(n_0){=}\mbox{nextQueenNumber} ) \ \Rightarrow \ ( \mbox{same}(\mbox{diagonal}_1,\mbox{queen})\mbox{ or }\mbox{same}(\mbox{diagonal}_2,\mbox{queen})\mbox{ or }\mbox{diagonal\_attack}(\mbox{queen},\mbox{nextQueenNumber},\mbox{otherPlacedQueens}) )\qquad\mbox{cnf}(\mbox{check\_diagonals}_1,\mbox{axiom})$

$\neg\,\mbox{diagonal\_attack}(\mbox{queen},\mbox{lastQueen},\mbox{empty\_list})\qquad\mbox{cnf}(\mbox{check\_diagonals}_2,\mbox{axiom})$

$( \mbox{select}(\mbox{aQueen},\mbox{unplacedQueens},\mbox{restOfUnplacedQueens})\mbox{ and }\mbox{do\_queens}(\mbox{restOfUnplacedQueens},\mbox{cons}(\mbox{aQueen},\mbox{safeQueens}),\mbox{placement}) ) \ \Rightarrow \ ( \mbox{do\_queens}(\mbox{unplacedQueens},\mbox{safeQueens},\mbox{placement})\mbox{ or }\mbox{attack}(\mbox{aQueen},\mbox{safeQueens}) )\qquad\mbox{cnf}(\mbox{place\_a\_queen}_1,\mbox{axiom})$

$\mbox{do\_queens}(\mbox{empty\_list},\mbox{placement},\mbox{placement})\qquad\mbox{cnf}(\mbox{place\_a\_queen}_2,\mbox{axiom})$

$( \mbox{numberOfQueens} + s(n_0){=}\mbox{low}\mbox{ and }\mbox{numberOfQueens} + \mbox{numberOfQueens}{=}\mbox{high}\mbox{ and }\mbox{range}(\mbox{low},\mbox{high},\mbox{positions})\mbox{ and }\mbox{do\_queens}(\mbox{positions},\mbox{empty\_list},\mbox{placement}) ) \ \Rightarrow \ \mbox{queens}(\mbox{numberOfQueens},\mbox{placement})\qquad\mbox{cnf}(\mbox{set\_up\_queens},\mbox{axiom})$

$\neg\,\mbox{queens}(s(s(s(s(n_0)))),\mbox{placement})\qquad\mbox{cnf}(\mbox{place\_queens},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ124$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_H}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }( g@ \mbox{xu}@ \mbox{xv} ) = c_3 ) \ \Rightarrow \ ( \mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{43000},\mbox{conjecture})$

\medskip\textbf{PUZ119$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx}\mbox{ and }\mbox{cCKB6\_NUM}@ \mbox{xy}\mbox{ and }( s@ ( s@ \mbox{xx} ) ) = ( s@ ( s@ \mbox{xy} ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cCKB6\_L25000A},\mbox{conjecture})$

\medskip\textbf{PUZ025-1.p} Knights and Knaves \#35

There is an island with exactly two types of people -
 truthtellers who always tell the truth and liars who always lie.
 There are a group of three people, A, B, and C on the island.
 A and B make the following statements. A: "B and C are the same
 type". Someone asks "Are A and B of the same type ? " What does
 C answer? Answer: "yes"




$\mbox{include}(\mbox{'Axioms/PUZ002-0.ax'})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(y)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{equal\_type}(x,y))\qquad\mbox{cnf}(\mbox{two\_liars\_are\_equal},\mbox{axiom})$

$( \mbox{people}(x,y,z)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(y)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{equal\_type}(x,y))\qquad\mbox{cnf}(\mbox{two\_truthtellers\_are\_equal},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{equal\_type}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{truthteller}(y))\qquad\mbox{cnf}(\mbox{truthteller\_equals\_truthtller},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{equal\_type}(x,y))\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{liar}(y))\qquad\mbox{cnf}(\mbox{liar\_equals\_liar},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{truthteller}(x)) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{equal\_type}(x,y))\mbox{ or }\mbox{a\_truth}(\mbox{liar}(y)) )\qquad\mbox{cnf}(\mbox{truthteller\_not\_equal\_liar},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{liar}(x)) \ \Rightarrow \ ( \mbox{a\_truth}(\mbox{equal\_type}(x,y))\mbox{ or }\mbox{a\_truth}(\mbox{truthteller}(y)) )\qquad\mbox{cnf}(\mbox{liar\_not\_equal\_truthteller},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{equal\_type}(x,y)) \ \Rightarrow \ \mbox{a\_truth}(\mbox{equal\_type}(y,x))\qquad\mbox{cnf}(\mbox{symmetry\_of\_equal\_type},\mbox{axiom})$

$( \mbox{ask\_1\_if}_2(x,y)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x))\mbox{ and }\mbox{a\_truth}(y) ) \ \Rightarrow \ \mbox{answer}(\mbox{yes})\qquad\mbox{cnf}(\mbox{truthteller\_identifies\_truths},\mbox{axiom})$

$( \mbox{ask\_1\_if}_2(x,y)\mbox{ and }\mbox{a\_truth}(\mbox{truthteller}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(y)\mbox{ or }\mbox{answer}(\mbox{no}) )\qquad\mbox{cnf}(\mbox{truthteller\_denies\_lies},\mbox{axiom})$

$( \mbox{ask\_1\_if}_2(x,y)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x))\mbox{ and }\mbox{a\_truth}(y) ) \ \Rightarrow \ \mbox{answer}(\mbox{no})\qquad\mbox{cnf}(\mbox{liar\_denies\_truths},\mbox{axiom})$

$( \mbox{ask\_1\_if}_2(x,y)\mbox{ and }\mbox{a\_truth}(\mbox{liar}(x)) ) \ \Rightarrow \ ( \mbox{a\_truth}(y)\mbox{ or }\mbox{answer}(\mbox{yes}) )\qquad\mbox{cnf}(\mbox{liar\_supports\_lies},\mbox{axiom})$

$\mbox{people}(b,c,a)\qquad\mbox{cnf}(\mbox{b\_c\_a\_are\_people},\mbox{hypothesis})$

$\mbox{people}(a,b,a)\qquad\mbox{cnf}(\mbox{a\_b\_c\_are\_people},\mbox{hypothesis})$

$\mbox{people}(a,c,b)\qquad\mbox{cnf}(\mbox{a\_c\_b\_are\_people},\mbox{hypothesis})$

$\mbox{people}(c,b,a)\qquad\mbox{cnf}(\mbox{c\_b\_a\_are\_people},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{equal\_type}(b,c)))\qquad\mbox{cnf}(\mbox{a\_says\_b\_and\_c\_equal},\mbox{hypothesis})$

$\mbox{ask\_1\_if}_2(c,\mbox{equal\_type}(a,b))\qquad\mbox{cnf}(\mbox{ask\_c\_if\_a\_and\_b\_equal},\mbox{hypothesis})$

$\neg\,\mbox{answer}(\mbox{answer})\qquad\mbox{cnf}(\mbox{prove\_there\_is\_an\_answer},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ032-1.p} Knights and Knaves \#26

On a certain island the inhabitants are partitioned into those
 who always tell the truth and those who always lie. I landed on
 the island and met three inhabitants A, B, and C. I asked A,
 'Are you a truthteller or a liar?' He mumbled something which I
 couldn't make out. I asked B what A had said. B replied, 'A
 said he was a liar'. C then volunteered, 'Don't believe B, he's
 lying!' Prove C is a truthteller.




$\mbox{include}(\mbox{'Axioms/PUZ002-0.ax'})$

$\mbox{a\_truth}(\mbox{says}(a,\mbox{mumble}))\qquad\mbox{cnf}(\mbox{a\_mumbles},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(b,\mbox{says}(a,\mbox{liar}(a))))\qquad\mbox{cnf}(\mbox{b\_says\_a\_says\_hes\_a\_liar},\mbox{hypothesis})$

$\mbox{a\_truth}(\mbox{says}(c,\mbox{liar}(b)))\qquad\mbox{cnf}(\mbox{c\_says\_b\_is\_a\_liar},\mbox{hypothesis})$

$\neg\,\mbox{a\_truth}(\mbox{truthteller}(c))\qquad\mbox{cnf}(\mbox{prove\_c\_is\_a\_truthteller},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ131+1.p} Victor teaches Michael

Every student is enrolled in at least one course. Every professor
 teaches at least one course. Every course has at least one student
 enrolled. Every course has at least one professor teaching. The
 coordinator of a course teaches the course. If a student is
 enroled in a course then the student is taught by every professor
 who teaches the course. Michael is enrolled in CSC410. Victor is
 the coordinator of CSC410. Therefore, Michael is taught by Victor.




$\exists a{:}\;\mbox{student}(a)\qquad\mbox{fof}(\mbox{student\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{professor}(a)\qquad\mbox{fof}(\mbox{professor\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{course}(a)\qquad\mbox{fof}(\mbox{course\_type},\mbox{axiom})$

$\mbox{student}(\mbox{michael})\qquad\mbox{fof}(\mbox{michael\_type},\mbox{axiom})$

$\mbox{professor}(\mbox{victor})\qquad\mbox{fof}(\mbox{victor\_type},\mbox{axiom})$

$\mbox{course}(\mbox{csc}_{410})\qquad\mbox{fof}(\mbox{csc410\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{course}(a) \ \Rightarrow \ \mbox{professor}(\mbox{coordinatorof}(a)) )\qquad\mbox{fof}(\mbox{coordinator\_of\_type},\mbox{axiom})$

$\forall x{:}\;( \mbox{student}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{course}(y)\mbox{ and }\mbox{enrolled}(x,y) ) )\qquad\mbox{fof}(\mbox{student\_enrolled\_axiom},\mbox{axiom})$

$\forall x{:}\;( \mbox{professor}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{course}(y)\mbox{ and }\mbox{teaches}(x,y) ) )\qquad\mbox{fof}(\mbox{professor\_teaches},\mbox{axiom})$

$\forall x{:}\;( \mbox{course}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{student}(y)\mbox{ and }\mbox{enrolled}(y,x) ) )\qquad\mbox{fof}(\mbox{course\_enrolled},\mbox{axiom})$

$\forall x{:}\;( \mbox{course}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{professor}(y)\mbox{ and }\mbox{teaches}(y,x) ) )\qquad\mbox{fof}(\mbox{course\_teaches},\mbox{axiom})$

$\forall x{:}\;( \mbox{course}(x) \ \Rightarrow \ \mbox{teaches}(\mbox{coordinatorof}(x),x) )\qquad\mbox{fof}(\mbox{coordinator\_teaches},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{student}(x)\mbox{ and }\mbox{course}(y) ) \ \Rightarrow \ ( \mbox{enrolled}(x,y) \ \Rightarrow \ \forall z{:}\;( \mbox{professor}(z) \ \Rightarrow \ ( \mbox{teaches}(z,y) \ \Rightarrow \ \mbox{taughtby}(x,z) ) ) ) )\qquad\mbox{fof}(\mbox{student\_enrolled\_taught},\mbox{axiom})$

$\mbox{enrolled}(\mbox{michael},\mbox{csc}_{410})\qquad\mbox{fof}(\mbox{michael\_enrolled\_csc410\_axiom},\mbox{axiom})$

$\mbox{coordinatorof}(\mbox{csc}_{410}) = \mbox{victor}\qquad\mbox{fof}(\mbox{victor\_coordinator\_csc410\_axiom},\mbox{axiom})$

$\mbox{taughtby}(\mbox{michael},\mbox{victor})\qquad\mbox{fof}(\mbox{teaching\_conjecture},\mbox{conjecture})$

\medskip\textbf{PUZ040-1.p} Quo vadis 3

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is an intermediate goal for testing.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(o,o),v_1,v_2,v_3,v_4,h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ016-1.p} Checkerboard and Dominoes : Row 1, columns 2 and 3 removed

There is a checker board whose second and third squares from
 the first row have been removed. There is a box of dominoes
 that are one square by two squares in size. Can you exactly
 cover the checker board with dominoes?




$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(\mbox{not\_covered},\mbox{not\_covered},y_3,y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(\mbox{covered},\mbox{covered},y_3,y_4,y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_1\_and}_2,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,\mbox{not\_covered},\mbox{not\_covered},y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,\mbox{covered},\mbox{covered},y_4,y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_2\_and}_3,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,\mbox{not\_covered},\mbox{not\_covered},y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,\mbox{covered},\mbox{covered},y_5,y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_3\_and}_4,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,\mbox{not\_covered},\mbox{not\_covered},y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,\mbox{covered},\mbox{covered},y_6,y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_4\_and}_5,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,\mbox{not\_covered},\mbox{not\_covered},y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,\mbox{covered},\mbox{covered},y_7,y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_5\_and}_6,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,\mbox{not\_covered},\mbox{not\_covered},y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,\mbox{covered},\mbox{covered},y_8))\qquad\mbox{cnf}(\mbox{cover\_columns\_6\_and}_7,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,\mbox{not\_covered},\mbox{not\_covered})) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,\mbox{covered},\mbox{covered}))\qquad\mbox{cnf}(\mbox{cover\_columns\_7\_and}_8,\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(x),\mbox{squares}(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8)) \ \Rightarrow \ \mbox{achievable}(\mbox{row}(\mbox{successor}(x)),\mbox{squares}(y_1',y_2',y_3',y_4',y_5',y_6',y_7',y_8'))\qquad\mbox{cnf}(\mbox{place\_vertical\_pieces},\mbox{axiom})$

$\mbox{successor}(n_1) = n_2\qquad\mbox{cnf}(\mbox{successor\_of\_1\_is}_2,\mbox{axiom})$

$\mbox{successor}(n_2) = n_3\qquad\mbox{cnf}(\mbox{successor\_of\_2\_is}_3,\mbox{axiom})$

$\mbox{successor}(n_3) = n_4\qquad\mbox{cnf}(\mbox{successor\_of\_3\_is}_4,\mbox{axiom})$

$\mbox{successor}(n_4) = n_5\qquad\mbox{cnf}(\mbox{successor\_of\_4\_is}_5,\mbox{axiom})$

$\mbox{successor}(n_5) = n_6\qquad\mbox{cnf}(\mbox{successor\_of\_5\_is}_6,\mbox{axiom})$

$\mbox{successor}(n_6) = n_7\qquad\mbox{cnf}(\mbox{successor\_of\_6\_is}_7,\mbox{axiom})$

$\mbox{successor}(n_7) = n_8\qquad\mbox{cnf}(\mbox{successor\_of\_7\_is}_8,\mbox{axiom})$

$\mbox{successor}(n_8) = n_9\qquad\mbox{cnf}(\mbox{successor\_of\_8\_is}_9,\mbox{axiom})$

$\mbox{covered}' = \mbox{not\_covered}\qquad\mbox{cnf}(\mbox{complement\_of\_covered\_is\_not\_covered},\mbox{axiom})$

$\mbox{not\_covered}' = \mbox{covered}\qquad\mbox{cnf}(\mbox{complement\_of\_not\_covered\_is\_covered},\mbox{axiom})$

$\mbox{removed}' = \mbox{not\_covered}\qquad\mbox{cnf}(\mbox{complement\_of\_removed\_is\_not\_covered},\mbox{axiom})$

$\mbox{achievable}(\mbox{row}(n_1),\mbox{squares}(\mbox{not\_covered},\mbox{removed},\mbox{removed},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered},\mbox{not\_covered}))\qquad\mbox{cnf}(\mbox{first\_row\_initially},\mbox{hypothesis})$

$\neg\,\mbox{achievable}(\mbox{row}(n_8),\mbox{squares}(\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered},\mbox{covered}))\qquad\mbox{cnf}(\mbox{prove\_row\_8\_can\_be\_covered},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ125$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_H}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( ( g@ \mbox{xu}@ \mbox{xv} ) = c_1\mbox{ or }( g@ \mbox{xu}@ \mbox{xv} ) = c_2\mbox{ or }( g@ \mbox{xu}@ \mbox{xv} ) = c_3\mbox{ or }( g@ \mbox{xu}@ \mbox{xv} ) = c_4 ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{30000},\mbox{conjecture})$

\medskip\textbf{PUZ086$\wedge$1.p} The friends puzzle - they both know

(i) Peter is a friend of John, so if Peter knows that John knows
 something then John knows that Peter knows the same thing. 
 (ii) Peter is married, so if Peter's wife knows something, then
 Peter knows the same thing. John and Peter have an appointment,
 let us consider the following situation: (a) Peter knows the time
 of their appointment. (b) Peter also knows that John knows the
 place of their appointment. Moreover, (c) Peter's wife knows that
 if Peter knows the time of their appointment, then John knows
 that too (since John and Peter are friends). Finally, (d) Peter
 knows that if John knows the place and the time of their
 appointment, then John knows that he has an appointment. From
 this situation we want to prove (e) that each of the two friends
 knows that the other one knows that he has an appointment.




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

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

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

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

$\mbox{mreflexive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{refl\_peter},\mbox{axiom})$

$\mbox{mreflexive}@ \mbox{john}\qquad\mbox{thf}(\mbox{refl\_john},\mbox{axiom})$

$\mbox{mreflexive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{refl\_wife\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{trans\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{john}\qquad\mbox{thf}(\mbox{trans\_john},\mbox{axiom})$

$\mbox{mtransitive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{trans\_wife\_peter},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ \mbox{peter}@ ( \mbox{mbox}@ \mbox{john}@ a ) )@ ( \mbox{mbox}@ \mbox{john}@ ( \mbox{mbox}@ \mbox{peter}@ a ) ) ) )\qquad\mbox{thf}(\mbox{ax\_i},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_prop}@ \lambda a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{mimplies}@ ( \mbox{mbox}@ ( \mbox{wife}@ \mbox{peter} )@ a )@ ( \mbox{mbox}@ \mbox{peter}@ a ) ) )\qquad\mbox{thf}(\mbox{ax\_ii},\mbox{axiom})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mbox}@ \mbox{peter}@ \mbox{time} )\qquad\mbox{thf}(\mbox{ax\_a},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox}@ \mbox{peter}@ ( \mbox{mbox}@ \mbox{john}@ \mbox{place} ) )\qquad\mbox{thf}(\mbox{ax\_b},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox}@ ( \mbox{wife}@ \mbox{peter} )@ ( \mbox{mimplies}@ ( \mbox{mbox}@ \mbox{peter}@ \mbox{time} )@ ( \mbox{mbox}@ \mbox{john}@ \mbox{time} ) ) )\qquad\mbox{thf}(\mbox{ax\_c},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox}@ \mbox{peter}@ ( \mbox{mbox}@ \mbox{john}@ ( \mbox{mimplies}@ ( \mbox{mand}@ \mbox{place}@ \mbox{time} )@ \mbox{appointment} ) ) )\qquad\mbox{thf}(\mbox{ax\_d},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mand}@ ( \mbox{mbox}@ \mbox{peter}@ ( \mbox{mbox}@ \mbox{john}@ \mbox{appointment} ) )@ ( \mbox{mbox}@ \mbox{john}@ ( \mbox{mbox}@ \mbox{peter}@ \mbox{appointment} ) ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{PUZ101$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_XPL}@ \mbox{cCKB6\_H}@ \mbox{cCKB6\_BLACK}@ c_1@ c_1\qquad\mbox{thf}(\mbox{cCKB6\_L}_{40000},\mbox{conjecture})$

\medskip\textbf{PUZ047+1.p} Taking the wolf, goat, and cabbage across river






$( p(\mbox{south},\mbox{south},\mbox{south},\mbox{south},\mbox{start})\mbox{ and }\forall t{:}\;( p(\mbox{south},\mbox{north},\mbox{south},\mbox{north},t) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{go\_alone}(t)) )\mbox{ and }\forall t_1{:}\;( p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},t_1) \ \Rightarrow \ p(\mbox{south},\mbox{north},\mbox{south},\mbox{north},\mbox{go\_alone}(t_1)) )\mbox{ and }\forall t_2{:}\;( p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},t_2) \ \Rightarrow \ p(\mbox{north},\mbox{south},\mbox{north},\mbox{south},\mbox{go\_alone}(t_2)) )\mbox{ and }\forall t_3{:}\;( p(\mbox{north},\mbox{south},\mbox{north},\mbox{south},t_3) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{go\_alone}(t_3)) )\mbox{ and }\forall t_4{:}\;( p(\mbox{south},\mbox{south},\mbox{south},\mbox{north},t_4) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{take\_wolf}(t_4)) )\mbox{ and }\forall t_5{:}\;( p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},t_5) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{south},\mbox{north},\mbox{take\_wolf}(t_5)) )\mbox{ and }\forall t_6{:}\;( p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},t_6) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{north},\mbox{south},\mbox{take\_wolf}(t_6)) )\mbox{ and }\forall t_7{:}\;( p(\mbox{north},\mbox{north},\mbox{north},\mbox{south},t_7) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{take\_wolf}(t_7)) )\mbox{ and }\forall x,y,u{:}\;( p(\mbox{south},x,\mbox{south},y,u) \ \Rightarrow \ p(\mbox{north},x,\mbox{north},y,\mbox{take\_goat}(u)) )\mbox{ and }\forall x_1,y_1,v{:}\;( p(\mbox{north},x_1,\mbox{north},y_1,v) \ \Rightarrow \ p(\mbox{south},x_1,\mbox{south},y_1,\mbox{take\_goat}(v)) )\mbox{ and }\forall t_8{:}\;( p(\mbox{south},\mbox{north},\mbox{south},\mbox{south},t_8) \ \Rightarrow \ p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},\mbox{take\_cabbage}(t_8)) )\mbox{ and }\forall t_9{:}\;( p(\mbox{north},\mbox{north},\mbox{south},\mbox{north},t_9) \ \Rightarrow \ p(\mbox{south},\mbox{north},\mbox{south},\mbox{south},\mbox{take\_cabbage}(t_9)) )\mbox{ and }\forall u_1{:}\;( p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},u_1) \ \Rightarrow \ p(\mbox{north},\mbox{south},\mbox{north},\mbox{north},\mbox{take\_cabbage}(u_1)) )\mbox{ and }\forall v_1{:}\;( p(\mbox{north},\mbox{south},\mbox{north},\mbox{north},v_1) \ \Rightarrow \ p(\mbox{south},\mbox{south},\mbox{north},\mbox{south},\mbox{take\_cabbage}(v_1)) ) ) \ \Rightarrow \ \exists z{:}\;p(\mbox{north},\mbox{north},\mbox{north},\mbox{north},z)\qquad\mbox{fof}(\mbox{thm}_{100},\mbox{conjecture})$

\medskip\textbf{PUZ031\_1.p} Schubert's Steamroller

Wolves, foxes, birds, caterpillars, and snails are animals, and
 there are some of each of them. Also there are some grains, and
 grains are plants. Every animal either likes to eat all plants
 or all animals much smaller than itself that like to eat some
 plants. Caterpillars and snails are much smaller than birds,
 which are much smaller than foxes, which in turn are much
 smaller than wolves. Wolves do not like to eat foxes or grains,
 while birds like to eat caterpillars but not snails.
 Caterpillars and snails like to eat some plants. Therefore
 there is an animal that likes to eat a grain eating animal.




$\mbox{animal}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{animal\_type},\mbox{type})$

$\mbox{wolf}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{wolf\_type},\mbox{type})$

$\mbox{wolf\_to\_animal}{:}\;\mbox{wolf}\to \mbox{animal}\qquad\mbox{tff}(\mbox{wolf\_is\_animal},\mbox{type})$

$\mbox{fox}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{fox\_type},\mbox{type})$

$\mbox{fox\_to\_animal}{:}\;\mbox{fox}\to \mbox{animal}\qquad\mbox{tff}(\mbox{fox\_is\_animal},\mbox{type})$

$\mbox{bird}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{bird\_type},\mbox{type})$

$\mbox{bird\_to\_animal}{:}\;\mbox{bird}\to \mbox{animal}\qquad\mbox{tff}(\mbox{bird\_is\_animal},\mbox{type})$

$\mbox{caterpillar}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{caterpillar\_type},\mbox{type})$

$\mbox{caterpillar\_to\_animal}{:}\;\mbox{caterpillar}\to \mbox{animal}\qquad\mbox{tff}(\mbox{caterpillar\_is\_animal},\mbox{type})$

$\mbox{snail}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{snail\_type},\mbox{type})$

$\mbox{snail\_to\_animal}{:}\;\mbox{snail}\to \mbox{animal}\qquad\mbox{tff}(\mbox{snail\_is\_animal},\mbox{type})$

$\mbox{plant}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{plant\_type},\mbox{type})$

$\mbox{grain}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{grain\_type},\mbox{type})$

$\mbox{grain\_to\_plant}{:}\;\mbox{grain}\to \mbox{plant}\qquad\mbox{tff}(\mbox{grain\_is\_plant},\mbox{type})$

$\mbox{edible}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{edible\_type},\mbox{type})$

$\mbox{animal\_to\_edible}{:}\;\mbox{animal}\to \mbox{edible}\qquad\mbox{tff}(\mbox{animal\_is\_edible},\mbox{type})$

$\mbox{plant\_to\_edible}{:}\;\mbox{plant}\to \mbox{edible}\qquad\mbox{tff}(\mbox{plant\_is\_edible},\mbox{type})$

$\mbox{eats}{:}\;( \mbox{animal}\times \mbox{edible} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{eats\_type},\mbox{type})$

$\mbox{much\_smaller}{:}\;( \mbox{animal}\times \mbox{animal} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{much\_smaller\_type},\mbox{type})$

$\forall x{:}\;\mbox{animal}{:}\;( \forall y{:}\;\mbox{plant}{:}\;\mbox{eats}(x,\mbox{plant\_to\_edible}(y))\mbox{ or }\forall y_1{:}\;\mbox{animal}{:}\;( ( \mbox{much\_smaller}(y_1,x)\mbox{ and }\exists z{:}\;\mbox{plant}{:}\;\mbox{eats}(y_1,\mbox{plant\_to\_edible}(z)) ) \ \Rightarrow \ \mbox{eats}(x,\mbox{animal\_to\_edible}(y_1)) ) )\qquad\mbox{tff}(\mbox{pel47}_7,\mbox{axiom})$

$\forall x{:}\;\mbox{snail},y{:}\;\mbox{bird}{:}\;\mbox{much\_smaller}(\mbox{snail\_to\_animal}(x),\mbox{bird\_to\_animal}(y))\qquad\mbox{tff}(\mbox{pel47}_8,\mbox{axiom})$

$\forall x{:}\;\mbox{caterpillar},y{:}\;\mbox{bird}{:}\;\mbox{much\_smaller}(\mbox{caterpillar\_to\_animal}(x),\mbox{bird\_to\_animal}(y))\qquad\mbox{tff}(\mbox{pel47\_8a},\mbox{axiom})$

$\forall x{:}\;\mbox{bird},y{:}\;\mbox{fox}{:}\;\mbox{much\_smaller}(\mbox{bird\_to\_animal}(x),\mbox{fox\_to\_animal}(y))\qquad\mbox{tff}(\mbox{pel47}_9,\mbox{axiom})$

$\forall x{:}\;\mbox{fox},y{:}\;\mbox{wolf}{:}\;\mbox{much\_smaller}(\mbox{fox\_to\_animal}(x),\mbox{wolf\_to\_animal}(y))\qquad\mbox{tff}(\mbox{pel47}_{10},\mbox{axiom})$

$\forall x{:}\;\mbox{wolf},y{:}\;\mbox{fox}{:}\;\neg\,\mbox{eats}(\mbox{wolf\_to\_animal}(x),\mbox{animal\_to\_edible}(\mbox{fox\_to\_animal}(y)))\qquad\mbox{tff}(\mbox{pel47}_{11},\mbox{axiom})$

$\forall x{:}\;\mbox{wolf},y{:}\;\mbox{grain}{:}\;\neg\,\mbox{eats}(\mbox{wolf\_to\_animal}(x),\mbox{plant\_to\_edible}(\mbox{grain\_to\_plant}(y)))\qquad\mbox{tff}(\mbox{pel47\_11a},\mbox{axiom})$

$\forall x{:}\;\mbox{bird},y{:}\;\mbox{caterpillar}{:}\;\mbox{eats}(\mbox{bird\_to\_animal}(x),\mbox{animal\_to\_edible}(\mbox{caterpillar\_to\_animal}(y)))\qquad\mbox{tff}(\mbox{pel47}_{12},\mbox{axiom})$

$\forall x{:}\;\mbox{bird},y{:}\;\mbox{snail}{:}\;\neg\,\mbox{eats}(\mbox{bird\_to\_animal}(x),\mbox{animal\_to\_edible}(\mbox{snail\_to\_animal}(y)))\qquad\mbox{tff}(\mbox{pel47}_{13},\mbox{axiom})$

$\forall x{:}\;\mbox{caterpillar}{:}\;\exists y{:}\;\mbox{plant}{:}\;\mbox{eats}(\mbox{caterpillar\_to\_animal}(x),\mbox{plant\_to\_edible}(y))\qquad\mbox{tff}(\mbox{pel47}_{14},\mbox{axiom})$

$\forall x{:}\;\mbox{snail}{:}\;\exists y{:}\;\mbox{plant}{:}\;\mbox{eats}(\mbox{snail\_to\_animal}(x),\mbox{plant\_to\_edible}(y))\qquad\mbox{tff}(\mbox{pel47\_14a},\mbox{axiom})$

$\exists x{:}\;\mbox{animal},y{:}\;\mbox{animal},z{:}\;\mbox{grain}{:}\;( \mbox{eats}(y,\mbox{plant\_to\_edible}(\mbox{grain\_to\_plant}(z)))\mbox{ and }\mbox{eats}(x,\mbox{animal\_to\_edible}(y)) )\qquad\mbox{tff}(\mbox{pel}_{47},\mbox{conjecture})$

\medskip\textbf{PUZ105$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cCKB\_E}_2@ ( s@ ( s@ \mbox{xx} ) )@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{36000},\mbox{conjecture})$

\medskip\textbf{PUZ130+1.p} Garfield and Odie

Garfield is a cat and Odie is a dog. Cats and dogs are pets. John
 is a human. Every pet has a human owner. Jon owns Garfield and
 Odie, and they are the only cat and dog he owns. If a dog chases
 a cat, then the cat's owner hates the dog's owner. Odie has chased
 Garfield. Therefore, Jon hates himself.




$\exists a{:}\;\mbox{cat}(a)\qquad\mbox{fof}(\mbox{cat\_type},\mbox{axiom})$

$\mbox{cat}(\mbox{garfield})\qquad\mbox{fof}(\mbox{garfield\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{dog}(a)\qquad\mbox{fof}(\mbox{dog\_type},\mbox{axiom})$

$\mbox{dog}(\mbox{odie})\qquad\mbox{fof}(\mbox{odie\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{pet}(a)\qquad\mbox{fof}(\mbox{pet\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{dog}(a) \ \Rightarrow \ \mbox{pet}(a) )\qquad\mbox{fof}(\mbox{dog\_pet\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{cat}(a) \ \Rightarrow \ \mbox{pet}(a) )\qquad\mbox{fof}(\mbox{cat\_pet\_type},\mbox{axiom})$

$\exists a{:}\;\mbox{human}(a)\qquad\mbox{fof}(\mbox{human\_type},\mbox{axiom})$

$\mbox{human}(\mbox{jon})\qquad\mbox{fof}(\mbox{jon\_type},\mbox{axiom})$

$\forall a{:}\;( \mbox{pet}(a) \ \Rightarrow \ \mbox{human}(\mbox{owner\_of}(a)) )\qquad\mbox{fof}(\mbox{owner\_of\_type},\mbox{axiom})$

$\forall x{:}\;( \mbox{pet}(x) \ \Rightarrow \ \exists y{:}\;( \mbox{human}(y)\mbox{ and }\mbox{owner}(x,y) ) )\qquad\mbox{fof}(\mbox{pet\_owner\_axiom},\mbox{axiom})$

$\mbox{owner}(\mbox{jon},\mbox{odie})\qquad\mbox{fof}(\mbox{jon\_o\_owner\_axiom},\mbox{axiom})$

$\mbox{owner}(\mbox{jon},\mbox{garfield})\qquad\mbox{fof}(\mbox{jon\_g\_owner\_axiom},\mbox{axiom})$

$\forall x{:}\;( \forall x{:}\;( \mbox{cat}(x) \ \Rightarrow \ \mbox{owner}(\mbox{jon},x) ) \ \Rightarrow \ x = \mbox{garfield} )\qquad\mbox{fof}(\mbox{jon\_only\_g\_owner\_axiom},\mbox{axiom})$

$\forall x{:}\;( \forall x{:}\;( \mbox{dog}(x) \ \Rightarrow \ \mbox{owner}(\mbox{jon},x) ) \ \Rightarrow \ x = \mbox{odie} )\qquad\mbox{fof}(\mbox{jon\_only\_o\_owner\_axiom},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{cat}(x)\mbox{ and }\mbox{dog}(y) ) \ \Rightarrow \ ( \mbox{chased}(y,x) \ \Rightarrow \ \mbox{hates}(\mbox{owner\_of}(x),\mbox{owner\_of}(y)) ) )\qquad\mbox{fof}(\mbox{cat\_chase\_axiom},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{human}(x)\mbox{ and }\mbox{pet}(y) ) \ \Rightarrow \ ( \mbox{owner}(x,y)\iff x = \mbox{owner\_of}(y) ) )\qquad\mbox{fof}(\mbox{owner\_def},\mbox{axiom})$

$\mbox{chased}(\mbox{odie},\mbox{garfield})\qquad\mbox{fof}(\mbox{odie\_chase\_axiom},\mbox{axiom})$

$\mbox{hates}(\mbox{jon},\mbox{jon})\qquad\mbox{fof}(\mbox{jon\_conjecture},\mbox{conjecture})$

\medskip\textbf{PUZ117$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx}\mbox{ and }\mbox{cCKB6\_NUM}@ \mbox{xy}\mbox{ and }( s@ \mbox{xx} ) = ( s@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{2000},\mbox{conjecture})$

\medskip\textbf{PUZ115$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx} ) \ \Rightarrow \ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ \mbox{xx} ) ) ) ) ) ) ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{1000},\mbox{conjecture})$

\medskip\textbf{PUZ094$\wedge$5.p} TPS problem from BASIC-FO-THMS

Reduced version of Schubert's Steamroller.




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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\neg\,\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{wolf}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{fox}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{bird}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{snail}@ x )\mbox{ and }\mbox{wolf}@ \mbox{a\_wolf}\mbox{ and }\mbox{fox}@ \mbox{a\_fox}\mbox{ and }\mbox{bird}@ \mbox{a\_bird}\mbox{ and }\mbox{snail}@ \mbox{a\_snail}\mbox{ and }\mbox{grain}@ \mbox{a\_grain}\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{plant}@ x\mbox{ or }\neg\,\mbox{grain}@ x )\mbox{ and }\forall a{:}\;\mbox{\$i},p{:}\;\mbox{\$i},s{:}\;\mbox{\$i},o{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ a@ p\mbox{ or }\mbox{eats}@ a@ s\mbox{ or }\neg\,\mbox{animal}@ a\mbox{ or }\neg\,\mbox{plant}@ p\mbox{ or }\neg\,\mbox{animal}@ s\mbox{ or }\neg\,\mbox{plant}@ o\mbox{ or }\neg\,\mbox{msmaller}@ s@ a\mbox{ or }\neg\,\mbox{eats}@ s@ o )\mbox{ and }\forall s{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ s@ b\mbox{ or }\neg\,\mbox{snail}@ s\mbox{ or }\neg\,\mbox{bird}@ b )\mbox{ and }\forall b{:}\;\mbox{\$i},f{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ b@ f\mbox{ or }\neg\,\mbox{bird}@ b\mbox{ or }\neg\,\mbox{fox}@ f )\mbox{ and }\forall f{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ f@ w\mbox{ or }\neg\,\mbox{fox}@ f\mbox{ or }\neg\,\mbox{wolf}@ w )\mbox{ and }\forall f{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{wolf}@ w\mbox{ or }\neg\,\mbox{fox}@ f\mbox{ or }\neg\,\mbox{eats}@ w@ f )\mbox{ and }\forall w{:}\;\mbox{\$i},g{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{wolf}@ w\mbox{ or }\neg\,\mbox{grain}@ g\mbox{ or }\neg\,\mbox{eats}@ w@ g )\mbox{ and }\forall b{:}\;\mbox{\$i},s{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{bird}@ b\mbox{ or }\neg\,\mbox{snail}@ s\mbox{ or }\neg\,\mbox{eats}@ b@ s )\mbox{ and }\forall s{:}\;\mbox{\$i}{:}\;( \mbox{plant}@ ( \mbox{sf}@ s )\mbox{ or }\neg\,\mbox{snail}@ s )\mbox{ and }\forall s{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ s@ ( \mbox{sf}@ s )\mbox{ or }\neg\,\mbox{snail}@ s )\mbox{ and }\forall a{:}\;\mbox{\$i},e{:}\;\mbox{\$i},g{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{animal}@ a\mbox{ or }\neg\,\mbox{animal}@ e\mbox{ or }\neg\,\mbox{grain}@ g\mbox{ or }\neg\,\mbox{eats}@ a@ e\mbox{ or }\neg\,\mbox{eats}@ e@ g )\qquad\mbox{thf}(\mbox{cPUZ031}_2,\mbox{conjecture})$

\medskip\textbf{PUZ068-1.p} Monday's Sudoku

47 1
 5 2
 3      8
 6   1 7
 4  3
 5
 7   3
 1
 4




$\mbox{include}(\mbox{'Axioms/PUZ005-0.ax'})$

$\mbox{el}(s(n_0),s(s(s(s(n_0)))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(n_0))))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(s(s(n_0))))))),s(n_0))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(n_0)),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(n_0)))),s(s(n_0)))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(n_0)),s(s(s(n_0))))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(n_0),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(n_0))))),s(n_0))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(n_0))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(s(s(s(n_0)))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(n_0)))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(n_0),s(n_0))\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

\medskip\textbf{PUZ021-1.p} How to Win a Bride

Suppose you are an inhabitant of the island of 'knights' and
 'knaves'. The knights always tell the truth and the knaves
 always lie. You fall in love with a girl there and wish
 to marry her. However, this girl has strange tastes; for some
 odd reason she does not wish to marry a knight; she wants
 to marry only a knave. But she wants a rich knave, not a poor
 one. (We assume for convenience that everyone is classified
 as either rich or poor.) Suppose, in fact, that you are
 a rich knave. You are allowed to make only one statement, can
 you convince her that you are a rich knave?




$\mbox{a\_truth}(\mbox{knight}(x),y) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{knave}(x),y)\qquad\mbox{cnf}(\mbox{not\_knight\_and\_knave},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{knight}(x),y)\mbox{ or }\mbox{a\_truth}(\mbox{knave}(x),y)\qquad\mbox{cnf}(\mbox{knight\_or\_knave},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{rich}(x),y) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{poor}(x),y)\qquad\mbox{cnf}(\mbox{not\_rich\_and\_poor},\mbox{axiom})$

$\mbox{a\_truth}(\mbox{rich}(x),y)\mbox{ or }\mbox{a\_truth}(\mbox{poor}(x),y)\qquad\mbox{cnf}(\mbox{rich\_or\_poor},\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{knight}(x),z)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \mbox{a\_truth}(y,z)\qquad\mbox{cnf}(\mbox{knights\_tell\_truth}_1,\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{knight}(x),z)\mbox{ and }\mbox{a\_truth}(y,z) ) \ \Rightarrow \ \mbox{says}(x,y)\qquad\mbox{cnf}(\mbox{knights\_tell\_truth}_2,\mbox{axiom})$

$( \mbox{a\_truth}(\mbox{knave}(x),z)\mbox{ and }\mbox{says}(x,y) ) \ \Rightarrow \ \neg\,\mbox{a\_truth}(y,z)\qquad\mbox{cnf}(\mbox{knaves\_lie}_1,\mbox{axiom})$

$\mbox{a\_truth}(\mbox{knave}(x),z) \ \Rightarrow \ ( \mbox{says}(x,y)\mbox{ or }\mbox{a\_truth}(y,z) )\qquad\mbox{cnf}(\mbox{knaves\_lie}_2,\mbox{axiom})$

$\mbox{a\_truth}(\mbox{and}(x,y),z) \ \Rightarrow \ \mbox{a\_truth}(x,z)\qquad\mbox{cnf}(\mbox{conjunction}_1,\mbox{axiom})$

$\mbox{a\_truth}(\mbox{and}(x,y),z) \ \Rightarrow \ \mbox{a\_truth}(y,z)\qquad\mbox{cnf}(\mbox{conjunction}_2,\mbox{axiom})$

$( \mbox{a\_truth}(x,z)\mbox{ and }\mbox{a\_truth}(y,z) ) \ \Rightarrow \ \mbox{a\_truth}(\mbox{and}(x,y),z)\qquad\mbox{cnf}(\mbox{conjunction}_3,\mbox{axiom})$

$\mbox{says}(\mbox{me},x) \ \Rightarrow \ \neg\,\mbox{a\_truth}(\mbox{and}(\mbox{knave}(\mbox{me}),\mbox{rich}(\mbox{me})),x)\qquad\mbox{cnf}(\mbox{prove\_statement\_exists}_1,\mbox{negated\_conjecture})$

$\mbox{says}(\mbox{me},x)\mbox{ or }\mbox{a\_truth}(\mbox{and}(\mbox{knave}(\mbox{me}),\mbox{rich}(\mbox{me})),x)\qquad\mbox{cnf}(\mbox{prove\_statement\_exists}_2,\mbox{negated\_conjecture})$

\medskip\textbf{PUZ063-2.p} Problem about mutilated chessboard problem






$\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_emptyset},\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_union}(\mbox{c\_emptyset},\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_empty\_\_left}_0,\mbox{axiom})$

\medskip\textbf{PUZ051-1.p} Quo vadis 6 - intermediate to final

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is the true goal from the puzzle.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,o),v_1(s(s(o)),o),v_2(o,s(s(s(o)))),v_3(o,s(s(o))),v_4(s(s(s(o))),s(s(o))),h(s(s(s(s(o)))),o),s_1(s(s(o)),s(o)),s_2(s(s(o)),s(s(o))),s_3(s(s(o)),s(s(s(o)))),s_4(s(s(s(o))),s(o)),e_1(s(s(s(o))),s(s(s(o)))),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{intermediate\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(s(s(s(o))),s(o)),v_1,v_2,v_3,v_4,h,s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ071-1.p} Thursday's Sudoku

45  8
 71
 2
 16   3
 5   4 2
 
 2     8
 1 3




$\mbox{include}(\mbox{'Axioms/PUZ005-0.ax'})$

$\mbox{el}(s(n_0),s(n_0),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(n_0)),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(n_0))))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(s(s(n_0))))))))),s(n_0))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(n_0)),s(s(n_0)))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(n_0))),s(n_0))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(n_0)))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(n_0),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(s(s(n_0))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(s(s(s(s(s(n_0))))))),s(s(n_0)))\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(n_0),s(s(n_0)))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(n_0)))),s(n_0))\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(s(s(n_0)))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(n_0)))),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

\medskip\textbf{PUZ137$\wedge$1.p} Peter the liar says everything

Peter says that everything he says is false. Show that Peter says
 everything.




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

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

$\mbox{says}@ \mbox{peter}@ \forall x{:}\;\mbox{\$o}{:}\;( ( \mbox{says}@ \mbox{peter}@ x ) \ \Rightarrow \ \neg\,x )\qquad\mbox{thf}(\mbox{ax}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{\$o}{:}\;( \mbox{says}@ \mbox{peter}@ x )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{PUZ031$\wedge$5.p} TPS problem from BASIC-FO-THMS






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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\neg\,\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{wolf}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{fox}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{bird}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{caterpillar}@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{animal}@ x\mbox{ or }\neg\,\mbox{snail}@ x )\mbox{ and }\mbox{wolf}@ \mbox{a\_wolf}\mbox{ and }\mbox{fox}@ \mbox{a\_fox}\mbox{ and }\mbox{bird}@ \mbox{a\_bird}\mbox{ and }\mbox{caterpillar}@ \mbox{a\_caterpillar}\mbox{ and }\mbox{snail}@ \mbox{a\_snail}\mbox{ and }\mbox{grain}@ \mbox{a\_grain}\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( \mbox{plant}@ x\mbox{ or }\neg\,\mbox{grain}@ x )\mbox{ and }\forall a{:}\;\mbox{\$i},p{:}\;\mbox{\$i},s{:}\;\mbox{\$i},o{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ a@ p\mbox{ or }\mbox{eats}@ a@ s\mbox{ or }\neg\,\mbox{animal}@ a\mbox{ or }\neg\,\mbox{plant}@ p\mbox{ or }\neg\,\mbox{animal}@ s\mbox{ or }\neg\,\mbox{plant}@ o\mbox{ or }\neg\,\mbox{msmaller}@ s@ a\mbox{ or }\neg\,\mbox{eats}@ s@ o )\mbox{ and }\forall c{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ c@ b\mbox{ or }\neg\,\mbox{caterpillar}@ c\mbox{ or }\neg\,\mbox{bird}@ b )\mbox{ and }\forall s{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ s@ b\mbox{ or }\neg\,\mbox{snail}@ s\mbox{ or }\neg\,\mbox{bird}@ b )\mbox{ and }\forall b{:}\;\mbox{\$i},f{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ b@ f\mbox{ or }\neg\,\mbox{bird}@ b\mbox{ or }\neg\,\mbox{fox}@ f )\mbox{ and }\forall f{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( \mbox{msmaller}@ f@ w\mbox{ or }\neg\,\mbox{fox}@ f\mbox{ or }\neg\,\mbox{wolf}@ w )\mbox{ and }\forall f{:}\;\mbox{\$i},w{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{wolf}@ w\mbox{ or }\neg\,\mbox{fox}@ f\mbox{ or }\neg\,\mbox{eats}@ w@ f )\mbox{ and }\forall w{:}\;\mbox{\$i},g{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{wolf}@ w\mbox{ or }\neg\,\mbox{grain}@ g\mbox{ or }\neg\,\mbox{eats}@ w@ g )\mbox{ and }\forall b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ b@ c\mbox{ or }\neg\,\mbox{bird}@ b\mbox{ or }\neg\,\mbox{caterpillar}@ c )\mbox{ and }\forall b{:}\;\mbox{\$i},s{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{bird}@ b\mbox{ or }\neg\,\mbox{snail}@ s\mbox{ or }\neg\,\mbox{eats}@ b@ s )\mbox{ and }\forall c{:}\;\mbox{\$i}{:}\;( \mbox{plant}@ ( \mbox{cf}@ c )\mbox{ or }\neg\,\mbox{caterpillar}@ c )\mbox{ and }\forall c{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ c@ ( \mbox{cf}@ c )\mbox{ or }\neg\,\mbox{caterpillar}@ c )\mbox{ and }\forall s{:}\;\mbox{\$i}{:}\;( \mbox{plant}@ ( \mbox{sf}@ s )\mbox{ or }\neg\,\mbox{snail}@ s )\mbox{ and }\forall s{:}\;\mbox{\$i}{:}\;( \mbox{eats}@ s@ ( \mbox{sf}@ s )\mbox{ or }\neg\,\mbox{snail}@ s )\mbox{ and }\forall a{:}\;\mbox{\$i},e{:}\;\mbox{\$i},g{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{animal}@ a\mbox{ or }\neg\,\mbox{animal}@ e\mbox{ or }\neg\,\mbox{grain}@ g\mbox{ or }\neg\,\mbox{eats}@ a@ e\mbox{ or }\neg\,\mbox{eats}@ e@ g )\qquad\mbox{thf}(\mbox{cPUZ031}_1,\mbox{conjecture})$

\medskip\textbf{PUZ131\_1.p} Victor teaches Michael

Every student is enrolled in at least one course. Every professor
 teaches at least one course. Every course has at least one student
 enrolled. Every course has at least one professor teaching. The
 coordinator of a course teaches the course. If a student is
 enroled in a course then the student is taught by every professor
 who teaches the course. Michael is enrolled in CSC410. Victor is
 the coordinator of CSC410. Therefore, Michael is taught by Victor.




$\mbox{student}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{student\_type},\mbox{type})$

$\mbox{professor}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{professor\_type},\mbox{type})$

$\mbox{course}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{course\_type},\mbox{type})$

$\mbox{michael}{:}\;\mbox{student}\qquad\mbox{tff}(\mbox{michael\_type},\mbox{type})$

$\mbox{victor}{:}\;\mbox{professor}\qquad\mbox{tff}(\mbox{victor\_type},\mbox{type})$

$\mbox{csc}_{410}{:}\;\mbox{course}\qquad\mbox{tff}(\mbox{csc410\_type},\mbox{type})$

$\mbox{enrolled}{:}\;( \mbox{student}\times \mbox{course} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{enrolled\_type},\mbox{type})$

$\mbox{teaches}{:}\;( \mbox{professor}\times \mbox{course} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{teaches\_type},\mbox{type})$

$\mbox{taughtby}{:}\;( \mbox{student}\times \mbox{professor} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{taught\_by\_type},\mbox{type})$

$\mbox{coordinatorof}{:}\;\mbox{course}\to \mbox{professor}\qquad\mbox{tff}(\mbox{coordinator\_of\_type},\mbox{type})$

$\forall x{:}\;\mbox{student}{:}\;\exists y{:}\;\mbox{course}{:}\;\mbox{enrolled}(x,y)\qquad\mbox{tff}(\mbox{student\_enrolled\_axiom},\mbox{axiom})$

$\forall x{:}\;\mbox{professor}{:}\;\exists y{:}\;\mbox{course}{:}\;\mbox{teaches}(x,y)\qquad\mbox{tff}(\mbox{professor\_teaches},\mbox{axiom})$

$\forall x{:}\;\mbox{course}{:}\;\exists y{:}\;\mbox{student}{:}\;\mbox{enrolled}(y,x)\qquad\mbox{tff}(\mbox{course\_enrolled},\mbox{axiom})$

$\forall x{:}\;\mbox{course}{:}\;\exists y{:}\;\mbox{professor}{:}\;\mbox{teaches}(y,x)\qquad\mbox{tff}(\mbox{course\_teaches},\mbox{axiom})$

$\forall x{:}\;\mbox{course}{:}\;\mbox{teaches}(\mbox{coordinatorof}(x),x)\qquad\mbox{tff}(\mbox{coordinator\_teaches},\mbox{axiom})$

$\forall x{:}\;\mbox{student},y{:}\;\mbox{course}{:}\;( \mbox{enrolled}(x,y) \ \Rightarrow \ \forall z{:}\;\mbox{professor}{:}\;( \mbox{teaches}(z,y) \ \Rightarrow \ \mbox{taughtby}(x,z) ) )\qquad\mbox{tff}(\mbox{student\_enrolled\_taught},\mbox{axiom})$

$\mbox{enrolled}(\mbox{michael},\mbox{csc}_{410})\qquad\mbox{tff}(\mbox{michael\_enrolled\_csc410\_axiom},\mbox{axiom})$

$\mbox{coordinatorof}(\mbox{csc}_{410}) = \mbox{victor}\qquad\mbox{tff}(\mbox{victor\_coordinator\_csc410\_axiom},\mbox{axiom})$

$\mbox{taughtby}(\mbox{michael},\mbox{victor})\qquad\mbox{tff}(\mbox{teaching\_conjecture},\mbox{conjecture})$

\medskip\textbf{PUZ133+3.p} N queens problem has the variable symmetry property






$\mbox{queens\_p} \ \Rightarrow \ \forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n) ) \ \Rightarrow \ ( p(i)\ne p(j)\mbox{ and }p(i) + i\ne p(j) + j\mbox{ and }-p(i)\ne -p(j) ) )\qquad\mbox{fof}(\mbox{queens\_p},\mbox{axiom})$

$\forall i{:}\;\mbox{perm}(i) = -s(n)\qquad\mbox{fof}(\mbox{permutation},\mbox{axiom})$

$\forall i,j{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n)\mbox{ and }\mbox{le}(s(i),j)\mbox{ and }\mbox{le}(j,n) ) \ \Rightarrow \ ( q(i)\ne q(j)\mbox{ and }q(i) + i\ne q(j) + j\mbox{ and }-q(i)\ne -q(j) ) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_q},\mbox{axiom})$

$( \mbox{queens\_p}\mbox{ and }\forall i{:}\;q(i) = p(\mbox{perm}(i)) ) \ \Rightarrow \ \mbox{queens\_q}\qquad\mbox{fof}(\mbox{queens\_sym},\mbox{conjecture})$

$\forall i{:}\;( ( \mbox{le}(s(n_0),i)\mbox{ and }\mbox{le}(i,n) ) \ \Rightarrow \ ( \mbox{le}(s(n_0),\mbox{perm}(i))\mbox{ and }\mbox{le}(\mbox{perm}(i),n) ) )\qquad\mbox{fof}(\mbox{permutation\_range},\mbox{axiom})$

$\forall i,j{:}\;( \mbox{lt}(i,j)\iff \mbox{lt}(\mbox{perm}(j),\mbox{perm}(i)) )\qquad\mbox{fof}(\mbox{permutation\_anti\_monotone},\mbox{axiom})$

$\forall j,i{:}\;-i = -\mbox{perm}(j)\qquad\mbox{fof}(\mbox{permutation\_another\_one},\mbox{axiom})$

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

$\forall x{:}\;\mbox{le}(x,s(x))\qquad\mbox{fof}(\mbox{succ\_le},\mbox{axiom})$

$\forall i,j,k,l{:}\;( i + j = k + l\iff -i = -l )\qquad\mbox{fof}(\mbox{plus}_1,\mbox{axiom})$

$\forall i,j,k,l{:}\;( -i = -k\iff -i = -j )\qquad\mbox{fof}(\mbox{minus}_1,\mbox{axiom})$

\medskip\textbf{PUZ004-1.p} The Letters Puzzle

(1) All the dated letters in this room are written on blue paper.
 (2) None of them are in black ink except those that are written
 in the third person.
 (3) I have not filed any of them that I can read.
 (4) None of them that are written on one sheet are undated.
 (5) All of them that are not crossed are in black ink.  :
 (6) All of them written by Brown begin with "Dear Sir" :
 (7) All of them written on blue paper are filed.        :
 (8) None of them written on more than one sheet are crossed.
 (9) None of them that begin with "Dear Sir" are written
 in third person.
 Prove that letters by Brown cannot be read.




$\mbox{dated} \ \Rightarrow \ \mbox{on\_blue\_paper}\qquad\mbox{cnf}(\mbox{dated\_on\_blue\_paper},\mbox{axiom})$

$\mbox{in\_third\_person} \ \Rightarrow \ \mbox{in\_black\_ink}\qquad\mbox{cnf}(\mbox{third\_person\_in\_black\_ink},\mbox{axiom})$

$\mbox{in\_black\_ink} \ \Rightarrow \ \mbox{in\_third\_person}\qquad\mbox{cnf}(\mbox{black\_ink\_in\_third\_person},\mbox{axiom})$

$\mbox{can\_be\_read} \ \Rightarrow \ \neg\,\mbox{filed}\qquad\mbox{cnf}(\mbox{not\_filed\_if\_read},\mbox{axiom})$

$\mbox{on\_one\_sheet} \ \Rightarrow \ \mbox{dated}\qquad\mbox{cnf}(\mbox{dated\_if\_on\_one\_sheet},\mbox{axiom})$

$\mbox{crossed}\mbox{ or }\mbox{in\_black\_ink}\qquad\mbox{cnf}(\mbox{not\_crossed\_in\_black\_ink},\mbox{axiom})$

$\mbox{by\_brown} \ \Rightarrow \ \mbox{begins\_with\_dear\_sir}\qquad\mbox{cnf}(\mbox{brown\_starts\_with\_sir},\mbox{axiom})$

$\mbox{on\_blue\_paper} \ \Rightarrow \ \mbox{filed}\qquad\mbox{cnf}(\mbox{filed\_if\_on\_blue\_paper},\mbox{axiom})$

$\mbox{crossed} \ \Rightarrow \ \mbox{on\_one\_sheet}\qquad\mbox{cnf}(\mbox{on\_one\_sheet\_if\_crossed},\mbox{axiom})$

$\mbox{begins\_with\_dear\_sir} \ \Rightarrow \ \neg\,\mbox{in\_third\_person}\qquad\mbox{cnf}(\mbox{no\_dear\_sirs\_in\_third\_person},\mbox{axiom})$

$\mbox{by\_brown}\qquad\mbox{cnf}(\mbox{letter\_by\_brown},\mbox{hypothesis})$

$\mbox{can\_be\_read}\qquad\mbox{cnf}(\mbox{prove\_it\_cannot\_be\_read},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ070-1.p} Wednesday's Sudoku

7  59
 6 1
 4   6
 83  2
 95
 
 82   4
 1
 5




$\mbox{include}(\mbox{'Axioms/PUZ005-0.ax'})$

$\mbox{el}(s(n_0),s(s(n_0)),s(s(s(s(s(s(s(n_0))))))))\qquad\mbox{cnf}(c_{01},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(n_0))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{02},\mbox{axiom})$

$\mbox{el}(s(n_0),s(s(s(s(s(s(n_0)))))),s(s(s(s(s(s(s(s(s(n_0))))))))))\qquad\mbox{cnf}(c_{03},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{04},\mbox{axiom})$

$\mbox{el}(s(s(n_0)),s(s(s(s(s(s(s(s(s(n_0))))))))),s(n_0))\qquad\mbox{cnf}(c_{05},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(n_0),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{06},\mbox{axiom})$

$\mbox{el}(s(s(s(n_0))),s(s(s(s(s(n_0))))),s(s(s(s(s(s(n_0)))))))\qquad\mbox{cnf}(c_{07},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(n_0)))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{08},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(n_0))))),s(s(s(n_0))))\qquad\mbox{cnf}(c_{09},\mbox{axiom})$

$\mbox{el}(s(s(s(s(n_0)))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(n_0)))\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(n_0),s(s(s(s(s(s(s(s(s(n_0))))))))))\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(n_0))))),s(s(n_0)),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{12},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(n_0))),s(s(s(s(s(s(s(s(n_0)))))))))\qquad\mbox{cnf}(c_{13},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(n_0)))),s(s(n_0)))\qquad\mbox{cnf}(c_{14},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(n_0)))))\qquad\mbox{cnf}(c_{15},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(n_0)))))))),s(s(s(s(n_0)))),s(n_0))\qquad\mbox{cnf}(c_{16},\mbox{axiom})$

$\mbox{el}(s(s(s(s(s(s(s(s(s(n_0))))))))),s(s(s(s(s(s(s(n_0))))))),s(s(s(s(s(n_0))))))\qquad\mbox{cnf}(c_{17},\mbox{axiom})$

\medskip\textbf{PUZ109$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{cCKB6\_BLACK}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{9000},\mbox{conjecture})$

\medskip\textbf{PUZ112$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

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

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

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

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

$\mbox{cCKB\_EVEN} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = ( s@ c_1 )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ c_1 ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_EVEN\_def},\mbox{definition})$

$\mbox{cCKB\_ODD} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = c_1\mbox{ or }\mbox{xx} = ( s@ ( s@ c_1 ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_ODD\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_ODD}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cCKB\_ODD}@ ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{cCKB\_EVEN}@ ( \mbox{cS}@ \mbox{cX} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{9000},\mbox{conjecture})$

\medskip\textbf{PUZ060+1.p} Food problems






$\forall \mbox{peanuts},\mbox{john},\mbox{bill},\mbox{sue},\mbox{apples},\mbox{chicken}{:}\;( ( \forall x{:}\;( \mbox{food}(x) \ \Rightarrow \ \mbox{likes}(\mbox{john},x) )\mbox{ and }\mbox{food}(\mbox{apples})\mbox{ and }\mbox{food}(\mbox{chicken})\mbox{ and }\forall x{:}\;( \exists y{:}\;( \mbox{eats}(y,x)\mbox{ and }\mbox{not\_killed\_by}(y,x) ) \ \Rightarrow \ \mbox{food}(x) )\mbox{ and }\mbox{eats}(\mbox{bill},\mbox{peanuts})\mbox{ and }\mbox{alive}(\mbox{bill})\mbox{ and }\forall x{:}\;( \mbox{eats}(\mbox{bill},x) \ \Rightarrow \ \mbox{eats}(\mbox{sue},x) )\mbox{ and }\forall y{:}\;( \mbox{alive}(y) \ \Rightarrow \ \forall x{:}\;\mbox{not\_killed\_by}(y,x) ) ) \ \Rightarrow \ \mbox{likes}(\mbox{john},\mbox{peanuts}) )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{PUZ012\_1.p} The Mislabeled Boxes

There are three boxes a, b, and c on a table. Each box contains
 apples or bananas or oranges. No two boxes contain the same
 thing. Each box has a label that says it contains apples or says
 it contains bananas or says it contains oranges. No box contains
 what it says on its label. The label on box a says "apples".
 The label on box b says "oranges". The label on box c says
 "bananas". You pick up box b and it contains apples. What do
 the other two boxes contain?




$\mbox{box}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{box\_type},\mbox{type})$

$\mbox{fruit}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{fruit\_type},\mbox{type})$

$\mbox{boxa}{:}\;\mbox{box}\qquad\mbox{tff}(\mbox{boxa\_type},\mbox{type})$

$\mbox{boxb}{:}\;\mbox{box}\qquad\mbox{tff}(\mbox{boxb\_type},\mbox{type})$

$\mbox{boxc}{:}\;\mbox{box}\qquad\mbox{tff}(\mbox{boxc\_type},\mbox{type})$

$\mbox{apples}{:}\;\mbox{fruit}\qquad\mbox{tff}(\mbox{apples\_type},\mbox{type})$

$\mbox{bananas}{:}\;\mbox{fruit}\qquad\mbox{tff}(\mbox{bananas\_type},\mbox{type})$

$\mbox{oranges}{:}\;\mbox{fruit}\qquad\mbox{tff}(\mbox{oranges\_type},\mbox{type})$

$\mbox{equal\_fruits}{:}\;( \mbox{fruit}\times \mbox{fruit} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{equal\_fruits\_type},\mbox{type})$

$\mbox{equal\_boxes}{:}\;( \mbox{box}\times \mbox{box} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{equal\_boxes\_type},\mbox{type})$

$\mbox{contains}{:}\;( \mbox{box}\times \mbox{fruit} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{contains\_type},\mbox{type})$

$\mbox{label}{:}\;( \mbox{box}\times \mbox{fruit} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{label\_type},\mbox{type})$

$\forall x{:}\;\mbox{fruit}{:}\;\mbox{equal\_fruits}(x,x)\qquad\mbox{tff}(\mbox{reflexivity\_for\_fruits},\mbox{axiom})$

$\forall x{:}\;\mbox{box}{:}\;\mbox{equal\_boxes}(x,x)\qquad\mbox{tff}(\mbox{reflexivity\_for\_boxes},\mbox{axiom})$

$\forall x{:}\;\mbox{box},y{:}\;\mbox{fruit}{:}\;\neg\,\mbox{label}(x,y)\mbox{ and }\mbox{contains}(x,y)\qquad\mbox{tff}(\mbox{label\_is\_wrong},\mbox{axiom})$

$\forall x{:}\;\mbox{fruit}{:}\;( \mbox{contains}(\mbox{boxa},x)\mbox{ or }\mbox{contains}(\mbox{boxb},x)\mbox{ or }\mbox{contains}(\mbox{boxc},x) )\qquad\mbox{tff}(\mbox{each\_thing\_is\_in\_a\_box},\mbox{axiom})$

$\forall x{:}\;\mbox{box}{:}\;( \mbox{contains}(x,\mbox{apples})\mbox{ or }\mbox{contains}(x,\mbox{bananas})\mbox{ or }\mbox{contains}(x,\mbox{oranges}) )\qquad\mbox{tff}(\mbox{each\_box\_contains\_something},\mbox{axiom})$

$\forall x{:}\;\mbox{box},y{:}\;\mbox{fruit},z{:}\;\mbox{fruit}{:}\;( ( \mbox{contains}(x,y)\mbox{ and }\mbox{contains}(x,z) ) \ \Rightarrow \ \mbox{equal\_fruits}(y,z) )\qquad\mbox{tff}(\mbox{contains\_is\_well\_defined}_1,\mbox{axiom})$

$\forall x{:}\;\mbox{box},y{:}\;\mbox{fruit},z{:}\;\mbox{box}{:}\;( ( \mbox{contains}(x,y)\mbox{ and }\mbox{contains}(z,y) ) \ \Rightarrow \ \mbox{equal\_boxes}(x,z) )\qquad\mbox{tff}(\mbox{contains\_is\_well\_defined}_2,\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxa},\mbox{boxb})\qquad\mbox{tff}(\mbox{boxa\_not\_boxb},\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxb},\mbox{boxc})\qquad\mbox{tff}(\mbox{boxb\_not\_boxc},\mbox{axiom})$

$\neg\,\mbox{equal\_boxes}(\mbox{boxa},\mbox{boxc})\qquad\mbox{tff}(\mbox{boxa\_not\_boxc},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{apples},\mbox{bananas})\qquad\mbox{tff}(\mbox{apples\_not\_bananas},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{bananas},\mbox{oranges})\qquad\mbox{tff}(\mbox{bananas\_not\_oranges},\mbox{axiom})$

$\neg\,\mbox{equal\_fruits}(\mbox{apples},\mbox{oranges})\qquad\mbox{tff}(\mbox{apples\_not\_oranges},\mbox{axiom})$

$\mbox{label}(\mbox{boxa},\mbox{apples})\qquad\mbox{tff}(\mbox{boxa\_labelled\_apples},\mbox{hypothesis})$

$\mbox{label}(\mbox{boxb},\mbox{oranges})\qquad\mbox{tff}(\mbox{boxb\_labelled\_oranges},\mbox{hypothesis})$

$\mbox{label}(\mbox{boxc},\mbox{bananas})\qquad\mbox{tff}(\mbox{boxc\_labelled\_bananas},\mbox{hypothesis})$

$\mbox{contains}(\mbox{boxb},\mbox{apples})\qquad\mbox{tff}(\mbox{boxb\_contains\_apples},\mbox{hypothesis})$

$\mbox{contains}(\mbox{boxa},\mbox{bananas})\mbox{ and }\mbox{contains}(\mbox{boxc},\mbox{oranges})\qquad\mbox{tff}(\mbox{prove\_boxa\_contains\_bananas\_and\_boxc\_oranges},\mbox{conjecture})$

\medskip\textbf{PUZ072+1.p} Friday's Sudoku

1      9
 3  8
 6
 124
 7 3
 5
 8  6
 4  2
 7   5




$\mbox{include}(\mbox{'Axioms/PUZ005+0.ax'})$

$\mbox{ssA}(n_1,n_2) = n_1\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$\mbox{ssA}(n_1,n_9) = n_9\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$\mbox{ssA}(n_2,n_4) = n_3\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$\mbox{ssA}(n_2,n_7) = n_8\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$\mbox{ssA}(n_3,n_7) = n_6\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$\mbox{ssA}(n_4,n_5) = n_1\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$\mbox{ssA}(n_4,n_6) = n_2\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$\mbox{ssA}(n_4,n_7) = n_4\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$\mbox{ssA}(n_5,n_1) = n_7\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$\mbox{ssA}(n_5,n_3) = n_3\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$\mbox{ssA}(n_6,n_1) = n_5\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$\mbox{ssA}(n_7,n_1) = n_8\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$\mbox{ssA}(n_7,n_4) = n_6\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$\mbox{ssA}(n_8,n_5) = n_4\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$\mbox{ssA}(n_8,n_8) = n_2\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$\mbox{ssA}(n_9,n_4) = n_7\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$\mbox{ssA}(n_9,n_8) = n_5\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ097$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

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

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

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

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

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

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_EVEN} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = ( s@ c_1 )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ c_1 ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_EVEN\_def},\mbox{definition})$

$\mbox{cCKB\_ODD} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = c_1\mbox{ or }\mbox{xx} = ( s@ ( s@ c_1 ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) )\mbox{ or }\mbox{xx} = ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ c_1 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_ODD\_def},\mbox{definition})$

$\mbox{cCKB\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_ODD}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_ODD}@ \mbox{xv} )\mbox{ or }( \mbox{cCKB\_EVEN}@ \mbox{xu}\mbox{ and }\mbox{cCKB\_EVEN}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_H\_def},\mbox{definition})$

$\mbox{cCKB\_INJ}@ \mbox{cCKB\_H}\qquad\mbox{thf}(\mbox{cL}_{2500},\mbox{conjecture})$

\medskip\textbf{PUZ146$\wedge$1.p} Peter and Mary have different hobbies






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

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

$\mbox{peter}{:}\;\mbox{earthling}\qquad\mbox{thf}(\mbox{peter},\mbox{type})$

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

$\mbox{beer\_drinking}{:}\;\mbox{hobby}\qquad\mbox{thf}(\mbox{beer\_drinking},\mbox{type})$

$\mbox{belly\_dancing}{:}\;\mbox{hobby}\qquad\mbox{thf}(\mbox{belly\_dancing},\mbox{type})$

$\mbox{weight\_lifting}{:}\;\mbox{hobby}\qquad\mbox{thf}(\mbox{weight\_lifting},\mbox{type})$

$\mbox{has\_hobby}{:}\;\mbox{earthling}\to \mbox{hobby}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{has\_hobby},\mbox{type})$

$\mbox{peters\_hobbies}{:}\;\mbox{hobby}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{peters\_hobbies},\mbox{type})$

$\mbox{marys\_hobbies}{:}\;\mbox{hobby}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{marys\_hobbies},\mbox{type})$

$\mbox{peter}\ne \mbox{mary}\qquad\mbox{thf}(\mbox{not\_the\_same}_1,\mbox{axiom})$

$\mbox{beer\_drinking}\ne \mbox{belly\_dancing}\mbox{ and }\mbox{belly\_dancing}\ne \mbox{weight\_lifting}\mbox{ and }\mbox{beer\_drinking}\ne \mbox{weight\_lifting}\qquad\mbox{thf}(\mbox{not\_the\_same}_2,\mbox{axiom})$

$\mbox{peters\_hobbies} = ( \mbox{has\_hobby}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{peters\_hobbies}_{001},\mbox{definition})$

$\mbox{marys\_hobbies} = ( \mbox{has\_hobby}@ \mbox{mary} )\qquad\mbox{thf}(\mbox{marys\_hobbies}_{002},\mbox{definition})$

$\mbox{marys\_hobbies}@ \mbox{belly\_dancing}\qquad\mbox{thf}(\mbox{mary\_does\_belly\_dancing},\mbox{axiom})$

$\neg\,\mbox{marys\_hobbies}@ \mbox{beer\_drinking}\qquad\mbox{thf}(\mbox{mary\_does\_not\_do\_beer\_drinking},\mbox{axiom})$

$\mbox{peters\_hobbies}@ \mbox{beer\_drinking}\qquad\mbox{thf}(\mbox{peter\_does\_beer\_drinking},\mbox{axiom})$

$\mbox{peters\_hobbies}@ \mbox{weight\_lifting}\qquad\mbox{thf}(\mbox{peter\_does\_weight\_lifting},\mbox{axiom})$

$\mbox{peters\_hobbies}\ne \mbox{marys\_hobbies}\qquad\mbox{thf}(\mbox{peter\_and\_mary\_have\_different\_hobbies},\mbox{conjecture})$

\medskip\textbf{PUZ033-1.p} The Winds and the Windows Puzzle

(1) There is always sunshine when the wind is in the East.
 (2) When it is cold and foggy, my neighbor practices the flute.
 (3) When my fire smokes, I set the door open.
 (4) When it is cold and I feel rheumatic, I light my fire.
 (5) When the wind is in the East and comes in gusts, my fire
 smokes.
 (6) When I keep the door open, I am free from headache.
 (7) Even when the sun is shining and it is not cold, I keep my
 window shut if it is foggy.
 (8) When the wind does not come in gusts, and when I have a
 fire and keep the door shut, I do not feel rheumatic.
 (9) Sunshine always brings on fog.
 (10) When my neighbor practices the flute, I shut the door,
 even if I have no headache.
 (11) When there is a fog and the wind is in the East, I feel
 rheumatic.
 Show that when the wind is in the East, I keep my windows shut.




$\mbox{wind\_in\_east} \ \Rightarrow \ \mbox{sunshine}\qquad\mbox{cnf}(c_1,\mbox{axiom})$

$( \mbox{cold}\mbox{ and }\mbox{foggy} ) \ \Rightarrow \ \mbox{neighbor\_practices\_flute}\qquad\mbox{cnf}(c_2,\mbox{axiom})$

$\mbox{fire\_smokes} \ \Rightarrow \ \mbox{door\_is\_open}\qquad\mbox{cnf}(c_3,\mbox{axiom})$

$( \mbox{cold}\mbox{ and }\mbox{i\_feel\_rheumatic} ) \ \Rightarrow \ \mbox{fire\_is\_lit}\qquad\mbox{cnf}(c_4,\mbox{axiom})$

$( \mbox{wind\_in\_east}\mbox{ and }\mbox{wind\_in\_gusts} ) \ \Rightarrow \ \mbox{fire\_smokes}\qquad\mbox{cnf}(c_5,\mbox{axiom})$

$\mbox{door\_is\_open} \ \Rightarrow \ \neg\,\mbox{headache}\qquad\mbox{cnf}(c_6,\mbox{axiom})$

$( \mbox{sunshine}\mbox{ and }\mbox{foggy} ) \ \Rightarrow \ ( \mbox{cold}\mbox{ or }\mbox{window\_is\_shut} )\qquad\mbox{cnf}(c_7,\mbox{axiom})$

$( \mbox{fire\_is\_lit}\mbox{ and }\mbox{i\_feel\_rheumatic} ) \ \Rightarrow \ ( \mbox{wind\_in\_gusts}\mbox{ or }\mbox{door\_is\_open} )\qquad\mbox{cnf}(c_8,\mbox{axiom})$

$\mbox{sunshine} \ \Rightarrow \ \mbox{foggy}\qquad\mbox{cnf}(c_9,\mbox{axiom})$

$\mbox{neighbor\_practices\_flute} \ \Rightarrow \ \neg\,\mbox{door\_is\_open}\qquad\mbox{cnf}(c_{10},\mbox{axiom})$

$( \mbox{foggy}\mbox{ and }\mbox{wind\_in\_east} ) \ \Rightarrow \ \mbox{i\_feel\_rheumatic}\qquad\mbox{cnf}(c_{11},\mbox{axiom})$

$\mbox{wind\_in\_east}\qquad\mbox{cnf}(c_{12},\mbox{hypothesis})$

$\neg\,\mbox{window\_is\_shut}\qquad\mbox{cnf}(\mbox{prove\_window\_is\_shut},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ103$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_NUM}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cCKB6\_NUM}@ ( s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L}_{3000},\mbox{conjecture})$

\medskip\textbf{PUZ130\_1.p} Garfield and Odie

Garfield is a cat and Odie is a dog. Cats and dogs are pets. John
 is a human. Every pet has a human owner. Jon owns Garfield and
 Odie, and they are the only cat and dog he owns. If a dog chases
 a cat, then the cat's owner hates the dog's owner. Odie has chased
 Garfield. Therefore, Jon hates himself.




$\mbox{animal}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{animal\_type},\mbox{type})$

$\mbox{cat}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{cat\_type},\mbox{type})$

$\mbox{dog}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{dog\_type},\mbox{type})$

$\mbox{human}{:}\;\mbox{\$tType}\qquad\mbox{tff}(\mbox{human\_type},\mbox{type})$

$\mbox{cat\_to\_animal}{:}\;\mbox{cat}\to \mbox{animal}\qquad\mbox{tff}(\mbox{cat\_to\_animal\_type},\mbox{type})$

$\mbox{dog\_to\_animal}{:}\;\mbox{dog}\to \mbox{animal}\qquad\mbox{tff}(\mbox{dog\_to\_animal\_type},\mbox{type})$

$\mbox{garfield}{:}\;\mbox{cat}\qquad\mbox{tff}(\mbox{garfield\_type},\mbox{type})$

$\mbox{odie}{:}\;\mbox{dog}\qquad\mbox{tff}(\mbox{odie\_type},\mbox{type})$

$\mbox{jon}{:}\;\mbox{human}\qquad\mbox{tff}(\mbox{jon\_type},\mbox{type})$

$\mbox{owner\_of}{:}\;\mbox{animal}\to \mbox{human}\qquad\mbox{tff}(\mbox{owner\_of\_type},\mbox{type})$

$\mbox{chased}{:}\;( \mbox{dog}\times \mbox{cat} )\to \mbox{\$o}\qquad\mbox{tff}(\mbox{chased\_type},\mbox{type})$

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

$\forall a{:}\;\mbox{animal}{:}\;\exists h{:}\;\mbox{human}{:}\;h = \mbox{owner\_of}(a)\qquad\mbox{tff}(\mbox{human\_owner},\mbox{axiom})$

$\mbox{jon} = \mbox{owner\_of}(\mbox{cat\_to\_animal}(\mbox{garfield}))\qquad\mbox{tff}(\mbox{jon\_owns\_garfield},\mbox{axiom})$

$\mbox{jon} = \mbox{owner\_of}(\mbox{dog\_to\_animal}(\mbox{odie}))\qquad\mbox{tff}(\mbox{jon\_owns\_odie},\mbox{axiom})$

$\forall a{:}\;\mbox{animal}{:}\;( \mbox{jon} = \mbox{owner\_of}(a) \ \Rightarrow \ ( a = \mbox{cat\_to\_animal}(\mbox{garfield})\mbox{ or }a = \mbox{dog\_to\_animal}(\mbox{odie}) ) )\qquad\mbox{tff}(\mbox{jon\_owns\_only},\mbox{axiom})$

$\forall c{:}\;\mbox{cat},d{:}\;\mbox{dog}{:}\;( \mbox{chased}(d,c) \ \Rightarrow \ \mbox{hates}(\mbox{owner\_of}(\mbox{cat\_to\_animal}(c)),\mbox{owner\_of}(\mbox{dog\_to\_animal}(d))) )\qquad\mbox{tff}(\mbox{dog\_chase\_cat},\mbox{axiom})$

$\mbox{chased}(\mbox{odie},\mbox{garfield})\qquad\mbox{tff}(\mbox{odie\_chased\_garfield},\mbox{axiom})$

$\mbox{hates}(\mbox{jon},\mbox{jon})\qquad\mbox{tff}(\mbox{jon\_hates\_jon},\mbox{conjecture})$

\medskip\textbf{PUZ048-1.p} Quo vadis 6 - initial to intermediate

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is the true goal from the puzzle.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(\mbox{bb}(s(s(o)),s(s(o))),v_1(o,o),v_2(o,s(o)),v_3(o,s(s(o))),v_4(o,s(s(s(o)))),h(s(s(s(s(o)))),o),s_1(s(s(o)),o),s_2(s(s(o)),s(o)),s_3(s(s(o)),o),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(o))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{intermediate\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ031-1.p} Schubert's Steamroller

Wolves, foxes, birds, caterpillars, and snails are animals, and
 there are some of each of them. Also there are some grains, and
 grains are plants. Every animal either likes to eat all plants
 or all animals much smaller than itself that like to eat some
 plants. Caterpillars and snails are much smaller than birds,
 which are much smaller than foxes, which in turn are much
 smaller than wolves. Wolves do not like to eat foxes or grains,
 while birds like to eat caterpillars but not snails.
 Caterpillars and snails like to eat some plants. Therefore
 there is an animal that likes to eat a grain eating animal.




$\mbox{wolf}(x) \ \Rightarrow \ \mbox{animal}(x)\qquad\mbox{cnf}(\mbox{wolf\_is\_an\_animal},\mbox{axiom})$

$\mbox{fox}(x) \ \Rightarrow \ \mbox{animal}(x)\qquad\mbox{cnf}(\mbox{fox\_is\_an\_animal},\mbox{axiom})$

$\mbox{bird}(x) \ \Rightarrow \ \mbox{animal}(x)\qquad\mbox{cnf}(\mbox{bird\_is\_an\_animal},\mbox{axiom})$

$\mbox{caterpillar}(x) \ \Rightarrow \ \mbox{animal}(x)\qquad\mbox{cnf}(\mbox{caterpillar\_is\_an\_animal},\mbox{axiom})$

$\mbox{snail}(x) \ \Rightarrow \ \mbox{animal}(x)\qquad\mbox{cnf}(\mbox{snail\_is\_an\_animal},\mbox{axiom})$

$\mbox{wolf}(\mbox{a\_wolf})\qquad\mbox{cnf}(\mbox{there\_is\_a\_wolf},\mbox{axiom})$

$\mbox{fox}(\mbox{a\_fox})\qquad\mbox{cnf}(\mbox{there\_is\_a\_fox},\mbox{axiom})$

$\mbox{bird}(\mbox{a\_bird})\qquad\mbox{cnf}(\mbox{there\_is\_a\_bird},\mbox{axiom})$

$\mbox{caterpillar}(\mbox{a\_caterpillar})\qquad\mbox{cnf}(\mbox{there\_is\_a\_caterpillar},\mbox{axiom})$

$\mbox{snail}(\mbox{a\_snail})\qquad\mbox{cnf}(\mbox{there\_is\_a\_snail},\mbox{axiom})$

$\mbox{grain}(\mbox{a\_grain})\qquad\mbox{cnf}(\mbox{there\_is\_a\_grain},\mbox{axiom})$

$\mbox{grain}(x) \ \Rightarrow \ \mbox{plant}(x)\qquad\mbox{cnf}(\mbox{grain\_is\_a\_plant},\mbox{axiom})$

$( \mbox{animal}(\mbox{animal})\mbox{ and }\mbox{plant}(\mbox{plant})\mbox{ and }\mbox{animal}(\mbox{small\_animal})\mbox{ and }\mbox{plant}(\mbox{other\_plant})\mbox{ and }\mbox{much\_smaller}(\mbox{small\_animal},\mbox{animal})\mbox{ and }\mbox{eats}(\mbox{small\_animal},\mbox{other\_plant}) ) \ \Rightarrow \ ( \mbox{eats}(\mbox{animal},\mbox{plant})\mbox{ or }\mbox{eats}(\mbox{animal},\mbox{small\_animal}) )\qquad\mbox{cnf}(\mbox{eating\_habits},\mbox{axiom})$

$( \mbox{caterpillar}(\mbox{catapillar})\mbox{ and }\mbox{bird}(\mbox{bird}) ) \ \Rightarrow \ \mbox{much\_smaller}(\mbox{catapillar},\mbox{bird})\qquad\mbox{cnf}(\mbox{caterpillar\_smaller\_than\_bird},\mbox{axiom})$

$( \mbox{snail}(\mbox{snail})\mbox{ and }\mbox{bird}(\mbox{bird}) ) \ \Rightarrow \ \mbox{much\_smaller}(\mbox{snail},\mbox{bird})\qquad\mbox{cnf}(\mbox{snail\_smaller\_than\_bird},\mbox{axiom})$

$( \mbox{bird}(\mbox{bird})\mbox{ and }\mbox{fox}(\mbox{fox}) ) \ \Rightarrow \ \mbox{much\_smaller}(\mbox{bird},\mbox{fox})\qquad\mbox{cnf}(\mbox{bird\_smaller\_than\_fox},\mbox{axiom})$

$( \mbox{fox}(\mbox{fox})\mbox{ and }\mbox{wolf}(\mbox{wolf}) ) \ \Rightarrow \ \mbox{much\_smaller}(\mbox{fox},\mbox{wolf})\qquad\mbox{cnf}(\mbox{fox\_smaller\_than\_wolf},\mbox{axiom})$

$( \mbox{wolf}(\mbox{wolf})\mbox{ and }\mbox{fox}(\mbox{fox}) ) \ \Rightarrow \ \neg\,\mbox{eats}(\mbox{wolf},\mbox{fox})\qquad\mbox{cnf}(\mbox{wolf\_dont\_eat\_fox},\mbox{axiom})$

$( \mbox{wolf}(\mbox{wolf})\mbox{ and }\mbox{grain}(\mbox{grain}) ) \ \Rightarrow \ \neg\,\mbox{eats}(\mbox{wolf},\mbox{grain})\qquad\mbox{cnf}(\mbox{wolf\_dont\_eat\_grain},\mbox{axiom})$

$( \mbox{bird}(\mbox{bird})\mbox{ and }\mbox{caterpillar}(\mbox{catapillar}) ) \ \Rightarrow \ \mbox{eats}(\mbox{bird},\mbox{catapillar})\qquad\mbox{cnf}(\mbox{bird\_eats\_caterpillar},\mbox{axiom})$

$( \mbox{bird}(\mbox{bird})\mbox{ and }\mbox{snail}(\mbox{snail}) ) \ \Rightarrow \ \neg\,\mbox{eats}(\mbox{bird},\mbox{snail})\qquad\mbox{cnf}(\mbox{bird\_dont\_eat\_snail},\mbox{axiom})$

$\mbox{caterpillar}(\mbox{catapillar}) \ \Rightarrow \ \mbox{plant}(\mbox{caterpillar\_food\_of}(\mbox{catapillar}))\qquad\mbox{cnf}(\mbox{caterpillar\_food\_is\_a\_plant},\mbox{axiom})$

$\mbox{caterpillar}(\mbox{catapillar}) \ \Rightarrow \ \mbox{eats}(\mbox{catapillar},\mbox{caterpillar\_food\_of}(\mbox{catapillar}))\qquad\mbox{cnf}(\mbox{caterpillar\_eats\_caterpillar\_food},\mbox{axiom})$

$\mbox{snail}(\mbox{snail}) \ \Rightarrow \ \mbox{plant}(\mbox{snail\_food\_of}(\mbox{snail}))\qquad\mbox{cnf}(\mbox{snail\_food\_is\_a\_plant},\mbox{axiom})$

$\mbox{snail}(\mbox{snail}) \ \Rightarrow \ \mbox{eats}(\mbox{snail},\mbox{snail\_food\_of}(\mbox{snail}))\qquad\mbox{cnf}(\mbox{snail\_eats\_snail\_food},\mbox{axiom})$

$( \mbox{animal}(\mbox{animal})\mbox{ and }\mbox{animal}(\mbox{grain\_eater})\mbox{ and }\mbox{grain}(\mbox{grain})\mbox{ and }\mbox{eats}(\mbox{animal},\mbox{grain\_eater}) ) \ \Rightarrow \ \neg\,\mbox{eats}(\mbox{grain\_eater},\mbox{grain})\qquad\mbox{cnf}(\mbox{prove\_the\_animal\_exists},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ078+1.p} Leo the Liar

Leo the Liar is a strange liar. On six days of the week, he lies.
 On the seventh day he always tells the truth. He made the
 following statements on three consecutive days:
 Day 1. I lie on Monday and Tuesday.
 Day 2. Today it is Thursday, Saturday, or Sunday.
 Day 3. I lie on Wednesday and Friday.
 Prove that Leo tells the truth on Tuesday.




$\forall x{:}\;( x = \mbox{monday}\mbox{ or }x = \mbox{tuesday}\mbox{ or }x = \mbox{wednesday}\mbox{ or }x = \mbox{thursday}\mbox{ or }x = \mbox{friday}\mbox{ or }x = \mbox{saturday}\mbox{ or }x = \mbox{sunday} )\qquad\mbox{fof}(\mbox{days\_of\_week},\mbox{axiom})$

$\mbox{monday}\ne \mbox{tuesday}\mbox{ and }\mbox{monday}\ne \mbox{wednesday}\mbox{ and }\mbox{monday}\ne \mbox{thursday}\mbox{ and }\mbox{monday}\ne \mbox{friday}\mbox{ and }\mbox{monday}\ne \mbox{saturday}\mbox{ and }\mbox{monday}\ne \mbox{sunday}\mbox{ and }\mbox{tuesday}\ne \mbox{wednesday}\mbox{ and }\mbox{tuesday}\ne \mbox{thursday}\mbox{ and }\mbox{tuesday}\ne \mbox{friday}\mbox{ and }\mbox{tuesday}\ne \mbox{saturday}\mbox{ and }\mbox{tuesday}\ne \mbox{sunday}\mbox{ and }\mbox{wednesday}\ne \mbox{thursday}\mbox{ and }\mbox{wednesday}\ne \mbox{friday}\mbox{ and }\mbox{wednesday}\ne \mbox{saturday}\mbox{ and }\mbox{wednesday}\ne \mbox{sunday}\mbox{ and }\mbox{thursday}\ne \mbox{friday}\mbox{ and }\mbox{thursday}\ne \mbox{saturday}\mbox{ and }\mbox{thursday}\ne \mbox{sunday}\mbox{ and }\mbox{friday}\ne \mbox{saturday}\mbox{ and }\mbox{friday}\ne \mbox{sunday}\mbox{ and }\mbox{saturday}\ne \mbox{sunday}\qquad\mbox{fof}(\mbox{uniquenames},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{truthday}(x)\mbox{ and }\mbox{truthday}(y) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{truth\_unique},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{day}_1(x)\mbox{ and }\mbox{day}_1(y) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{day1\_unique},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{day}_2(x)\mbox{ and }\mbox{day}_2(y) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{day2\_unique},\mbox{axiom})$

$\forall x,y{:}\;( ( \mbox{day}_3(x)\mbox{ and }\mbox{day}_3(y) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{day3\_unique},\mbox{axiom})$

$\exists x{:}\;\mbox{truthday}(x)\qquad\mbox{fof}(\mbox{truth\_once},\mbox{axiom})$

$\exists x{:}\;\mbox{day}_1(x)\qquad\mbox{fof}(\mbox{day1\_once},\mbox{axiom})$

$\exists x{:}\;\mbox{day}_2(x)\qquad\mbox{fof}(\mbox{day2\_once},\mbox{axiom})$

$\exists x{:}\;\mbox{day}_3(x)\qquad\mbox{fof}(\mbox{day3\_once},\mbox{axiom})$

$\mbox{consecutive}(\mbox{sunday}) = \mbox{monday}\qquad\mbox{fof}(\mbox{monday\_follows\_sunday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{monday}) = \mbox{tuesday}\qquad\mbox{fof}(\mbox{tuesday\_follows\_monday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{tuesday}) = \mbox{wednesday}\qquad\mbox{fof}(\mbox{wednesday\_follows\_tuesday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{wednesday}) = \mbox{thursday}\qquad\mbox{fof}(\mbox{thursday\_follows\_wednesday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{thursday}) = \mbox{friday}\qquad\mbox{fof}(\mbox{friday\_follows\_thursday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{friday}) = \mbox{saturday}\qquad\mbox{fof}(\mbox{saturday\_follows\_friday},\mbox{axiom})$

$\mbox{consecutive}(\mbox{saturday}) = \mbox{sunday}\qquad\mbox{fof}(\mbox{sunday\_follows\_saturday},\mbox{axiom})$

$\forall x{:}\;( \mbox{day}_1(x)\iff \mbox{day}_2(\mbox{consecutive}(x)) )\qquad\mbox{fof}(\mbox{day2\_follows\_day}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{day}_2(x)\iff \mbox{day}_3(\mbox{consecutive}(x)) )\qquad\mbox{fof}(\mbox{day3\_follows\_day}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{day}_1(x) \ \Rightarrow \ ( \mbox{truthday}(x)\iff ( \neg\,\mbox{truthday}(\mbox{monday})\mbox{ and }\neg\,\mbox{truthday}(\mbox{tuesday}) ) ) )\qquad\mbox{fof}(\mbox{statement}_1,\mbox{axiom})$

$\forall x{:}\;( \mbox{day}_2(x) \ \Rightarrow \ ( \mbox{truthday}(x)\iff ( \mbox{day}_2(\mbox{thursday})\mbox{ or }\mbox{day}_2(\mbox{saturday})\mbox{ or }\mbox{day}_2(\mbox{sunday}) ) ) )\qquad\mbox{fof}(\mbox{statement}_2,\mbox{axiom})$

$\forall x{:}\;( \mbox{day}_3(x) \ \Rightarrow \ ( \mbox{truthday}(x)\iff ( \neg\,\mbox{truthday}(\mbox{wednesday})\mbox{ and }\neg\,\mbox{truthday}(\mbox{friday}) ) ) )\qquad\mbox{fof}(\mbox{statement}_3,\mbox{axiom})$

$\mbox{truthday}(\mbox{tuesday})\qquad\mbox{fof}(\mbox{tuesday\_leo\_tells\_truths},\mbox{conjecture})$

\medskip\textbf{PUZ042-1.p} Quo vadis 5

bb is big block (square, size=4 tiles).
 s1-s4 : 4 small square blocks, size=1 tile
 v1-v4: 4 vertical blocks, size= 2 tiles
 b1: horizontal block, size= 2 tiles
 e1, e2 are the 2 blank tiles
 It's a 5x4 playing field to move from the start state to
 the goal state. This is a medium goal for testing.




$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

$\mbox{state}(\mbox{bb}(o,s(o)),v_1(o,o),v_2(o,s(s(s(o)))),v_3(s(s(o)),o),v_4(s(s(o)),s(s(s(o)))),h(s(s(o)),s(o)),s_1(s(s(s(o))),s(o)),s_2(s(s(s(o))),s(s(o))),s_3(s(s(s(s(o)))),s(o)),s_4(s(s(s(s(o)))),s(s(o))),e_1(s(s(s(s(o)))),o),e_2(s(s(s(s(o)))),s(s(s(o)))))\qquad\mbox{cnf}(\mbox{initial\_state},\mbox{hypothesis})$

$\neg\,\mbox{state}(b,v_1,v_2,v_3,v_4,h(s(s(s(o))),s(o)),s_1,s_2,s_3,s_4,e_1,e_2)\qquad\mbox{cnf}(\mbox{goal\_state},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ003-1.p} The Barber Puzzle

There is a barbers' club that obeys the following three
 conditions:
 (1) If any member A has shaved any other member B - whether
 himself or another - then all members have shaved A,
 though not necessarily at the same time.
 (2) Four of the members are named Guido, Lorenzo, Petrucio,
 and Cesare.
 (3) Guido has shaved Cesare.
 Prove Petrucio has shaved Lorenzo




$( \mbox{\mbox{member}}(x)\mbox{ and }\mbox{\mbox{member}}(y)\mbox{ and }\mbox{shaved}(x,y) ) \ \Rightarrow \ \mbox{shaved}(\mbox{members},x)\qquad\mbox{cnf}(\mbox{one\_shaved\_then\_all\_shaved},\mbox{axiom})$

$( \mbox{shaved}(\mbox{members},x)\mbox{ and }\mbox{\mbox{member}}(y) ) \ \Rightarrow \ \mbox{shaved}(y,x)\qquad\mbox{cnf}(\mbox{all\_shaved\_then\_one\_shaved},\mbox{axiom})$

$\mbox{\mbox{member}}(\mbox{guido})\qquad\mbox{cnf}(\mbox{guido},\mbox{hypothesis})$

$\mbox{\mbox{member}}(\mbox{lorenzo})\qquad\mbox{cnf}(\mbox{lorenzo},\mbox{hypothesis})$

$\mbox{\mbox{member}}(\mbox{petruchio})\qquad\mbox{cnf}(\mbox{petruchio},\mbox{hypothesis})$

$\mbox{\mbox{member}}(\mbox{cesare})\qquad\mbox{cnf}(\mbox{cesare},\mbox{hypothesis})$

$\mbox{shaved}(\mbox{guido},\mbox{cesare})\qquad\mbox{cnf}(\mbox{guido\_has\_shaved\_cesare},\mbox{hypothesis})$

$\neg\,\mbox{shaved}(\mbox{petruchio},\mbox{lorenzo})\qquad\mbox{cnf}(\mbox{prove\_petruchio\_has\_shaved\_lorenzo},\mbox{negated\_conjecture})$

\medskip\textbf{PUZ085$\wedge$1.p} The friends puzzle - transitivity for Peter's wife

(i) Peter is a friend of John, so if Peter knows that John knows
 something then John knows that Peter knows the same thing. 
 (ii) Peter is married, so if Peter's wife knows something, then
 Peter knows the same thing. John and Peter have an appointment,
 let us consider the following situation: (a) Peter knows the time
 of their appointment. (b) Peter also knows that John knows the
 place of their appointment. Moreover, (c) Peter's wife knows that
 if Peter knows the time of their appointment, then John knows
 that too (since John and Peter are friends). Finally, (d) Peter
 knows that if John knows the place and the time of their
 appointment, then John knows that he has an appointment. From
 this situation we want to prove (e) that each of the two friends
 knows that the other one knows that he has an appointment.




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

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

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

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

$\mbox{mreflexive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{refl\_peter},\mbox{axiom})$

$\mbox{mreflexive}@ \mbox{john}\qquad\mbox{thf}(\mbox{refl\_john},\mbox{axiom})$

$\mbox{mreflexive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{refl\_wife\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{peter}\qquad\mbox{thf}(\mbox{trans\_peter},\mbox{axiom})$

$\mbox{mtransitive}@ \mbox{john}\qquad\mbox{thf}(\mbox{trans\_john},\mbox{axiom})$

$\mbox{mtransitive}@ ( \mbox{wife}@ \mbox{peter} )\qquad\mbox{thf}(\mbox{trans\_wife\_peter},\mbox{axiom})$

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

\medskip\textbf{PUZ126$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$c_2{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c2\_type},\mbox{type})$

$c_3{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c3\_type},\mbox{type})$

$c_4{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c4\_type},\mbox{type})$

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

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_H}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_H\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_H} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy}\mbox{ and }( ( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_1\mbox{ and }\mbox{xu} = ( s@ ( s@ ( s@ \mbox{xx} ) ) )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_2\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = ( s@ ( s@ \mbox{xy} ) ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_3\mbox{ and }\mbox{xu} = ( s@ \mbox{xx} )\mbox{ and }\mbox{xv} = ( s@ \mbox{xy} ) )\mbox{ or }( ( g@ ( s@ ( s@ \mbox{xx} ) )@ ( s@ \mbox{xy} ) ) = c_4\mbox{ and }\mbox{xu} = ( s@ ( s@ \mbox{xx} ) )\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_H\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_H}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }( g@ \mbox{xu}@ \mbox{xv} ) = c_3 ) \ \Rightarrow \ ( ( s@ ( s@ ( s@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) = \mbox{xx}\mbox{ and }( s@ ( s@ ( s@ ( s@ ( s@ ( s@ ( s@ \mbox{xv} ) ) ) ) ) ) ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cCKB\_L}_{43100},\mbox{conjecture})$

\medskip\textbf{PUZ046-1.p} Quo vadis axioms






$\mbox{include}(\mbox{'Axioms/PUZ004-0.ax'})$

\medskip\textbf{PUZ064-2.p} Problem about mutilated chessboard problem






$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_inter}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{v\_u},\mbox{t\_a}) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_inter}(\mbox{v\_t},\mbox{v\_u},\mbox{t\_a}) = \mbox{c\_emptyset}\mbox{ and }\mbox{c\_in}(\mbox{v\_u},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{v\_t},\mbox{v\_u},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_t},\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_inter}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}) = \mbox{c\_emptyset} ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{v\_a},\mbox{v\_t},\mbox{t\_a}),\mbox{c\_Mutil\_Otiling}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Mutil\_Otiling\_OUn}_0,\mbox{axiom})$

$\mbox{c\_Relation\_OImage}(\mbox{c\_Relation\_Odiag}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a},\mbox{t\_a}) = \mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Relation\_OImage\_\_diag}_0,\mbox{axiom})$

$\mbox{c\_uminus}(\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_union}(\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OCompl\_\_Diff\_\_eq}_0,\mbox{axiom})$

$\mbox{c\_uminus}(\mbox{c\_UNIV},\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_emptyset}\qquad\mbox{cnf}(\mbox{cls\_Set\_OCompl\_\_UNIV\_\_eq}_0,\mbox{axiom})$

$\mbox{c\_minus}(\mbox{v\_A},\mbox{c\_uminus}(\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_ODiff\_\_Compl}_0,\mbox{axiom})$

$\mbox{c\_inter}(\mbox{c\_UNIV},\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_OInt\_\_UNIV\_\_left}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OInt\_\_iff}_1,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_minus}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_A},\mbox{t\_a}) = \mbox{c\_union}(\mbox{v\_B},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_Diff\_\_cancel2}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{v\_A},\mbox{c\_minus}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}) = \mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_Diff\_\_cancel}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_UNIV},\mbox{v\_B},\mbox{t\_a}) = \mbox{c\_UNIV}\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_UNIV\_\_left}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{v\_y},\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_absorb}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{t\_a}) = \mbox{c\_union}(\mbox{v\_A},\mbox{c\_union}(\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_assoc}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{c\_emptyset},\mbox{v\_y},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_empty\_\_left}_0,\mbox{axiom})$

$\mbox{c\_union}(\mbox{v\_y},\mbox{c\_emptyset},\mbox{t\_a}) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_empty\_\_right}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_subset\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_subset\_\_iff}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUn\_\_subset\_\_iff}_2,\mbox{axiom})$

$\mbox{c\_uminus}(\mbox{c\_uminus}(\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{v\_y}\qquad\mbox{cnf}(\mbox{cls\_Set\_Odouble\_\_complement}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_c},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_Oempty\_\_iff}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_emptyset},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Oempty\_\_subsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

\medskip\textbf{PUZ069+2.p} Tuesday's Sudoku

1 76
 85
 2    3
 24
 9      6
 53
 7   1
 3    2




$\mbox{include}(\mbox{'Axioms/PUZ006+0.ax'})$

$p(n_1,n_1,n_1)\qquad\mbox{fof}(\mbox{ax}_{353},\mbox{axiom})$

$p(n_1,n_3,n_7)\qquad\mbox{fof}(\mbox{ax}_{354},\mbox{axiom})$

$p(n_1,n_4,n_6)\qquad\mbox{fof}(\mbox{ax}_{355},\mbox{axiom})$

$p(n_2,n_6,n_8)\qquad\mbox{fof}(\mbox{ax}_{356},\mbox{axiom})$

$p(n_2,n_7,n_5)\qquad\mbox{fof}(\mbox{ax}_{357},\mbox{axiom})$

$p(n_3,n_2,n_2)\qquad\mbox{fof}(\mbox{ax}_{358},\mbox{axiom})$

$p(n_3,n_7,n_3)\qquad\mbox{fof}(\mbox{ax}_{359},\mbox{axiom})$

$p(n_4,n_4,n_2)\qquad\mbox{fof}(\mbox{ax}_{360},\mbox{axiom})$

$p(n_4,n_5,n_4)\qquad\mbox{fof}(\mbox{ax}_{361},\mbox{axiom})$

$p(n_5,n_1,n_9)\qquad\mbox{fof}(\mbox{ax}_{362},\mbox{axiom})$

$p(n_5,n_8,n_6)\qquad\mbox{fof}(\mbox{ax}_{363},\mbox{axiom})$

$p(n_6,n_5,n_5)\qquad\mbox{fof}(\mbox{ax}_{364},\mbox{axiom})$

$p(n_6,n_6,n_3)\qquad\mbox{fof}(\mbox{ax}_{365},\mbox{axiom})$

$p(n_7,n_4,n_7)\qquad\mbox{fof}(\mbox{ax}_{366},\mbox{axiom})$

$p(n_7,n_8,n_1)\qquad\mbox{fof}(\mbox{ax}_{367},\mbox{axiom})$

$p(n_8,n_2,n_3)\qquad\mbox{fof}(\mbox{ax}_{368},\mbox{axiom})$

$p(n_8,n_7,n_2)\qquad\mbox{fof}(\mbox{ax}_{369},\mbox{axiom})$

\medskip\textbf{PUZ095$\wedge$5.p} TPS problem from CHECKERBOARD-THMS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

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

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

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

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

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

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\mbox{cCKB\_INF}@ \mbox{cCKB6\_BLACK}\qquad\mbox{thf}(\mbox{cCKB6\_L}_{50000},\mbox{conjecture})$

\medskip\textbf{PUZ128+1.p} Iokaste patricide triangle

Iokaste is a parent of Oedipus. Iokaste is a parent of Polyneikes.
 Oedipus is a parent of Polyneikes. Polyneikes is a parent of 
 Thersandros. Oedipus is a patricide. Thersandros is not a 
 patricide. Therefore, Iokaste is a parent of a patricide who is 
 a parent of somebody who is not a patricide.




$\mbox{parent\_of}(\mbox{iokaste},\mbox{oedipus})\qquad\mbox{fof}(\mbox{iokaste\_oedipus},\mbox{axiom})$

$\mbox{parent\_of}(\mbox{iokaste},\mbox{polyneikes})\qquad\mbox{fof}(\mbox{iokaste\_polyneikes},\mbox{axiom})$

$\mbox{parent\_of}(\mbox{oedipus},\mbox{polyneikes})\qquad\mbox{fof}(\mbox{oedipus\_polyneikes},\mbox{axiom})$

$\mbox{parent\_of}(\mbox{polyneikes},\mbox{thersandros})\qquad\mbox{fof}(\mbox{polyneikes\_thersandros},\mbox{axiom})$

$\mbox{patricide}(\mbox{oedipus})\qquad\mbox{fof}(\mbox{oedipus\_patricidal},\mbox{axiom})$

$\neg\,\mbox{patricide}(\mbox{thersandros})\qquad\mbox{fof}(\mbox{thersandros\_not\_patricidal},\mbox{axiom})$

$\exists p,\mbox{nP}{:}\;( \mbox{parent\_of}(\mbox{iokaste},p)\mbox{ and }\mbox{patricide}(p)\mbox{ and }\mbox{parent\_of}(p,\mbox{nP})\mbox{ and }\neg\,\mbox{patricide}(\mbox{nP}) )\qquad\mbox{fof}(\mbox{iokaste\_parent\_particide\_parent\_not\_patricide},\mbox{conjecture})$

\medskip
\end{document}