{-# LANGUAGE DeriveDataTypeable, FlexibleContexts, MonoLocalBinds, NoMonomorphismRestriction #-}
{-# LANGUAGE OverloadedStrings, RankNTypes, ScopedTypeVariables, TypeFamilies #-}
{-# OPTIONS -fno-warn-name-shadowing #-}
module Data
( tests
, allFormulas
, proofs
{-
, formulas
, animalKB
, animalConjectures
, chang43KB
, chang43Conjecture
, chang43ConjectureRenamed
-}
) where
import Common (TestFormula(..), TestProof(..), Expected(..), ProofExpected(..), doTest, doProof, TTestProof)
import Data.Boolean (Literal(..))
import Data.Logic.ATP.Apply (HasApply(TermOf, PredOf), pApp, Predicate)
import Data.Logic.ATP.Equate ((.=.), HasEquate)
import Data.Logic.ATP.Formulas (false, IsFormula(AtomOf), true)
import Data.Logic.ATP.Lit ((.~.), IsLiteral)
import Data.Logic.ATP.Prop (IsPropositional(..))
import Data.Logic.ATP.Quantified (IsQuantified(..), for_all, exists)
import Data.Logic.ATP.Skolem (HasSkolem(toSkolem), Formula, SkAtom, SkTerm, Function)
import Data.Logic.ATP.Term (IsTerm(..), V)
import qualified Data.Logic.Instances.Chiou as C
import Data.Logic.KnowledgeBase (WithId(WithId, wiItem, wiIdent), Proof(..), ProofResult(..))
import Data.Logic.Normal.Implicative (ImplicativeForm(INF), makeINF')
import Data.Map as Map (fromList)
import Data.Set as Set (Set, fromList, toList)
import Data.String (IsString)
import Test.HUnit
import Text.PrettyPrint.HughesPJClass (prettyShow)
-- |for_all with a list of variables, for backwards compatibility.
for_all' :: IsQuantified formula => [VarOf formula] -> formula -> formula
for_all' vs f = foldr for_all f vs
-- |exists with a list of variables, for backwards compatibility.
exists' :: IsQuantified formula => [VarOf formula] -> formula -> formula
exists' vs f = foldr for_all f vs
pApp2 :: (atom ~ AtomOf formula, term ~ TermOf atom, p ~ PredOf atom,
IsFormula formula, HasApply atom) => p -> term -> term -> formula
pApp2 p a b = pApp p [a, b]
{-
:m +Data.Logic.Test
:m +Data.Logic.Types.FirstOrder
:m +Data.Set
runNormal (clauseNormalForm (true :: Formula V Predicate Function)) :: Set (Set (Formula V Predicate Function))
runNormal (skolemNormalForm (true :: Formula V Predicate Function)) :: Formula V Predicate Function
:m +Data.Logic.Normal.Prenex
prenexNormalForm true :: Formula V Predicate Function
:m +Data.Logic.Normal.Skolem
:m +Data.Logic.Normal.Negation
-}
tests :: [Test] -> [TTestProof] -> Test
tests fs ps =
TestLabel "Tests.Data" $ TestList (fs ++ map doProof ps)
allFormulas :: [Test]
allFormulas = (formulas ++
map doTest (concatMap snd [animalKB, chang43KB]) ++
animalConjectures ++
[chang43Conjecture, chang43ConjectureRenamed])
formulas :: [Test]
formulas =
let n = (.~.)
p = pApp "p"
q = pApp "q"
r = pApp "r"
s = pApp "s"
t = pApp "t"
p0 = p []
q0 = q []
r0 = r []
s0 = s []
t0 = t []
(x, y, z, u, v, w) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "z" :: SkTerm, vt "u" :: SkTerm, vt "v" :: SkTerm, vt "w" :: SkTerm)
z2 = vt "z'" :: SkTerm in
[ doTest $
TestFormula
{ formula = p0 .|. q0 .&. r0 .|. n s0 .&. n t0
, name = "operator precedence"
, expected = [ FirstOrderFormula (p0 .|. (q0 .&. r0) .|. ((n s0) .&. (n t0))) ] }
, doTest $
TestFormula
{ formula = true
, name = "True"
, expected = [ClauseNormalForm (toSS [[]])] }
, doTest $
TestFormula
{ formula = false
, name = "False"
, expected = [ClauseNormalForm (toSS [])] }
, doTest $
TestFormula
{ formula = true
, name = "True"
, expected = [DisjNormalForm (toSS [[]])] } -- Make sure these are right
, doTest $
TestFormula
{ formula = false
, name = "False"
, expected = [DisjNormalForm (toSS [])] }
, doTest $
TestFormula
{ formula = pApp "p" []
, name = "p"
, expected = [ClauseNormalForm (toSS [[pApp "p" []]])] }
, let p = pApp "p" [] in
doTest $
TestFormula
{ formula = p .&. ((.~.) (p))
, name = "p&~p"
, expected = [ PrenexNormalForm ((pApp ("p") []) .&. (((.~.) (pApp ("p") []))))
, ClauseNormalForm (toSS [[(p)], [((.~.) (p))]])
] }
, doTest $
TestFormula
{ formula = pApp "p" [vt "x"]
, name = "p[x]"
, expected = [ClauseNormalForm (toSS [[pApp "p" [x]]])] }
, let f = pApp "f"
q = pApp "q" in
doTest $
TestFormula
{ name = "iff"
, formula = for_all "x" (for_all "y" (q [x, y] .<=>. for_all "z" (f [z, x] .<=>. f [z, y])))
, expected = [ PrenexNormalForm
(for_all "x"
(for_all "y"
(for_all "z"
(exists "z'"
(((((q [x,y])) .&.
((((((f [z,x])) .&.
((f [z,y])))) .|.
(((((.~.) (f [z,x]))) .&.
(((.~.) (f [z,y]))))))))) .|.
(((((.~.) (q [x,y]))) .&.
((((((f [z2,x])) .&.
(((.~.) (f [z2,y]))))) .|.
(((((.~.) (f [z2,x]))) .&.
((f [z2,y])))))))))))))
, ClauseNormalForm
(toSS [[(pApp2 ("f") (vt ("z")) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("x"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("x"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y")))),
((.~.) (pApp2 ("q") (vt ("x")) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("x"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("x")))),
((.~.) (pApp2 ("q") (vt ("x")) (vt ("y"))))],
[(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),
(pApp2 ("q") (vt ("x")) (vt ("y")))],
[(pApp2 ("q") (vt ("x")) (vt ("y"))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))]])
]
}
, doTest $
TestFormula
{ name = "move quantifiers out"
, formula = (for_all "x" (pApp "p" [x]) .&. (pApp "q" [x]))
, expected = [PrenexNormalForm (for_all "x'" ((pApp "p" [vt ("x'")]) .&. ((pApp "q" [vt ("x")]))))]
}
, doTest $
TestFormula
{ name = "skolemize2"
, formula = exists "x" (for_all "y" (pApp "loves" [x, y]))
, expected = [SkolemNormalForm (pApp ("loves") [fApp (toSkolem "x" 1) [],y])]
}
, doTest $
TestFormula
{ name = "skolemize3"
, formula = for_all "y" (exists "x" (pApp "loves" [x, y]))
, expected = [SkolemNormalForm (pApp ("loves") [fApp (toSkolem "x" 1) [y],y])]
}
, doTest $
TestFormula
{ formula = exists "x" (for_all' ["y", "z"]
(exists "u"
(for_all "v"
(exists "w"
(pApp "P" [x, y, z, u, v, w])))))
, name = "chang example 4.1"
, expected = [ SkolemNormalForm (pApp "P" [fApp (toSkolem "x" 1) [],
vt ("y"),
vt ("z"),
fApp (toSkolem "u" 1) [vt ("y"),vt ("z")],
vt ("v"),
fApp (toSkolem "w" 1) [vt ("v"), vt ("y"),vt ("z")]]) ]
}
, doTest $
TestFormula
{ name = "chang example 4.2"
-- ∀x ∃y∃z ~P(x,y) & Q(x,z) | R(x,y,z)
, formula = for_all "x" (exists' ["y", "z"] (((((.~.) (pApp "P" [x, y])) .&. pApp "Q" [x, z]) .|. pApp "R" [x, y, z])))
-- ∀x ~P(x,Sk1[x]) | R(x,Sk1[x],Sk2[x]) & Q(x,Sk2[x]) | R(x,Sk1[x],Sk2[x])
, expected = [ SkolemNormalForm
((((.~.) (pApp ("P") [vt ("x"),vt ("y")])) .&.
((pApp ("Q") [vt ("x"),vt ("z")]))) .|.
((pApp ("R") [vt ("x"),vt ("y"),vt ("z")])))
, ClauseNormalForm
(toSS
[[((.~.) (pApp ("P") [vt ("x"),vt ("y")])),
(pApp ("R") [vt ("x"),vt ("y"),vt ("z")])],
[(pApp ("Q") [vt ("x"),vt ("z")]),
(pApp ("R") [vt ("x"),vt ("y"),vt ("z")])]]) ]
}
, doTest $
TestFormula
{ formula = n p0 .|. q0 .&. p0 .|. r0 .&. n q0 .&. n r0
, name = "chang 7.2.1a - unsat"
, expected = [ SatSolverSat False ] }
, doTest $
TestFormula
{ formula = p0 .|. q0 .|. r0 .&. n p0 .&. n q0 .&. n r0 .|. s0 .&. n s0
, name = "chang 7.2.1b - unsat"
, expected = [ SatSolverSat False ] }
, doTest $
TestFormula
{ formula = p0 .|. q0 .&. q0 .|. r0 .&. r0 .|. s0 .&. n r0 .|. n p0 .&. n s0 .|. n q0 .&. n q0 .|. n r0
, name = "chang 7.2.1c - unsat"
, expected = [ SatSolverSat False ] }
, let q = pApp "q"
f = pApp "f"
sk1 = f [fApp (toSkolem "x" 1) [x,x,y,z],y]
sk2 = f [fApp (toSkolem "x" 1) [x,x,y,z],x] in
doTest $
TestFormula
{ name = "distribute bug test"
, formula = ((((.~.) (q [x,y])) .|.
((((.~.) (sk2)) .|. (sk1)) .&.
(((.~.) (sk1)) .|. (sk2)))) .&.
((((sk2) .&.
((.~.) (sk1))) .|. ((sk1) .&.
((.~.) (sk2)))) .|. (q [x,y])))
, expected = [ClauseNormalForm
(toSS
[[sk2,sk1,pApp ("q") [x,y]],
[sk2,((.~.) (sk1)),((.~.) (q [x,y]))],
[sk1,((.~.) (sk2)),((.~.) (q [x,y]))],
[q [x,y], ((.~.) sk2),((.~.) sk1)]])]
}
, let x = vt "x" :: SkTerm
y = vt "y" :: SkTerm
x' = vt "x" :: C.CTerm V Function
y' = vt "y" :: C.CTerm V Function in
doTest $
TestFormula
{ name = "convert to Chiou 1"
, formula = exists "x" (x .=. y)
, expected = [ConvertToChiou (exists "x" (x' .=. y') :: C.Sentence V Predicate Function)]
}
, let s = pApp "s"
s' = pApp "s"
x' = vt "x"
y' = vt "y" in
doTest $
TestFormula
{ name = "convert to Chiou 2"
, formula = s [fApp ("a") [x, y]]
, expected = [ConvertToChiou (s' [fApp "a" [x', y']])]
}
, let s = pApp "s"
h = pApp "h"
m = pApp "m"
s' = pApp "s"
h' = pApp "h"
m' = pApp "m"
x' = vt "x" in
doTest $
TestFormula
{ name = "convert to Chiou 3"
, formula = for_all "x" (((s [x] .=>. h [x]) .&. (h [x] .=>. m [x])) .=>. (s [x] .=>. m [x]))
, expected = [ConvertToChiou (for_all "x" (((s' [x'] .=>. h' [x']) .&. (h' [x'] .=>. m' [x'])) .=>. (s' [x'] .=>. m' [x'])))]
}
, let taller a b = pApp "taller" [a, b]
wise a = pApp "wise" [a] in
doTest $
TestFormula
{ name = "cnf test 1"
, formula = for_all "y" (for_all "x" (taller y x .|. wise x) .=>. wise y)
, expected = [ClauseNormalForm
(toSS
[[(pApp ("wise") [vt ("y")]),
((.~.) (pApp ("taller") [vt ("y"),fApp (toSkolem "x" 1) [vt ("y")]]))],
[(pApp ("wise") [vt ("y")]),
((.~.) (pApp ("wise") [fApp (toSkolem "x" 1) [vt ("y")]]))]])]
}
, doTest $
TestFormula
{ name = "cnf test 2"
, formula = ((.~.) (exists "x" (pApp "s" [x] .&. pApp "q" [x])))
, expected = [ ClauseNormalForm (toSS
[[((.~.) (pApp ("q") [vt ("x")])),
((.~.) (pApp ("s") [vt ("x")]))]])
, PrenexNormalForm (for_all "x"
(((.~.) (pApp ("s") [vt ("x")])) .|.
(((.~.) (pApp ("q") [vt ("x")])))))
{- [[((.~.) (pApp "s" [vt "x"])),
((.~.) (pApp "q" [vt "x"]))]] -}
]
}
, doTest $
TestFormula
{ name = "cnf test 3"
, formula = (for_all "x" (p [x] .=>. (q [x] .|. r [x])))
, expected = [ClauseNormalForm (toSS [[((.~.) (pApp "p" [vt "x"])),(pApp "q" [vt "x"]),(pApp "r" [vt "x"])]])]
}
, doTest $
TestFormula
{ name = "cnf test 4"
, formula = ((.~.) (exists "x" (p [x] .=>. exists "y" (q [y]))))
, expected = [ClauseNormalForm (toSS [[(pApp "p" [vt "x"])],[((.~.) (pApp "q" [vt "y"]))]])]
}
, doTest $
TestFormula
{ name = "cnf test 5"
, formula = (for_all "x" (q [x] .|. r [x] .=>. s [x]))
, expected = [ClauseNormalForm (toSS [[((.~.) (pApp "q" [vt "x"])),(pApp "s" [vt "x"])],[((.~.) (pApp "r" [vt "x"])),(pApp "s" [vt "x"])]])]
}
, doTest $
TestFormula
{ name = "cnf test 6"
, formula = (exists "x" (p0 .=>. pApp "f" [x]))
, expected = [ClauseNormalForm (toSS [[((.~.) (pApp "p" [])),(pApp "f" [fApp (toSkolem "x" 1) []])]])]
}
, let p = pApp "p" []
f' = pApp "f" [x]
f = pApp "f" [fApp (toSkolem "x" 1) []] in
doTest $
TestFormula
{ name = "cnf test 7"
, formula = exists "x" (p .<=>. f')
, expected = [ PrenexNormalForm (exists "x" ((p .&. f') .|. ((((.~.) p) .&. (((.~.) f'))))))
, SkolemNormalForm ((p .&. f) .|. (((.~.) p) .&. (((.~.) f))))
, TrivialClauses [(False,Set.fromList [((.~.) (pApp ("p") [])),(pApp ("f") [fApp (toSkolem "x" 1) []])]),
(False,Set.fromList [((.~.) (pApp ("f") [fApp (toSkolem "x" 1) []])),(pApp ("p") [])])]
, ClauseNormalForm (toSS [[(f), ((.~.) p)], [p, ((.~.) f)]])]
}
, doTest $
TestFormula
{ name = "cnf test 8"
, formula = (for_all "z" (exists "y" (for_all "x" (pApp "f" [x, y] .<=>. (pApp "f" [x, z] .&. ((.~.) (pApp "f" [x, x])))))))
, expected = [ClauseNormalForm
(toSS [[((.~.) (pApp "f" [vt "x",fApp (toSkolem "y" 1) [vt "z"]])),(pApp "f" [vt "x",vt "z"])],
[((.~.) (pApp "f" [vt "x",fApp (toSkolem "y" 1) [vt "z"]])),((.~.) (pApp "f" [vt "x",vt "x"]))],
[((.~.) (pApp "f" [vt "x",vt "z"])),(pApp "f" [vt "x",vt "x"]),(pApp "f" [vt "x",fApp (toSkolem "y" 1) [vt "z"]])]])]
}
, let f = pApp "f"
q = pApp "q"
(x, y, z) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "z" :: SkTerm) in
doTest $
TestFormula
{ name = "cnf test 9"
, formula = (for_all "x" (for_all "x" (for_all "y" (q [x, y] .<=>. for_all "z" (f [z, x] .<=>. f [z, y])))))
, expected = [ClauseNormalForm
(toSS
[[(pApp2 ("f") (vt ("z")) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("x"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("x"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("y")))),
((.~.) (pApp2 ("q") (vt ("x")) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),((.~.) (pApp2 ("f") (vt ("z")) (vt ("x"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))],
[(pApp2 ("f") (vt ("z")) (vt ("y"))),
((.~.) (pApp2 ("f") (vt ("z")) (vt ("x")))),
((.~.) (pApp2 ("q") (vt ("x")) (vt ("y"))))],
[(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x"))),
(pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))),
(pApp2 ("q") (vt ("x")) (vt ("y")))],
[(pApp2 ("q") (vt ("x")) (vt ("y"))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("x")))),
((.~.) (pApp2 ("f") (fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]) (vt ("y"))))]])
]
}
, doTest $
TestFormula
{ name = "cnf test 10"
, formula = (for_all "x" (exists "y" ((for_all "x" (exists "z" (q [y, x, z]) .=>. r [y]) .=>. p [x, y]))))
, expected = [ClauseNormalForm
(toSS
[[(pApp ("p") [vt ("x"),fApp (toSkolem "y" 1) [vt ("x")]]),
(pApp ("q") [fApp (toSkolem "y" 1) [vt "x"],fApp (toSkolem "x'" 1) [vt "x"],fApp (toSkolem "z" 1) [vt "x"]])],
[(pApp ("p") [vt ("x"),fApp (toSkolem "y" 1) [vt ("x")]]),
((.~.) (pApp ("r") [fApp (toSkolem "y" 1) [vt "x"]]))]])
]
}
, doTest $
TestFormula
{ name = "cnf test 11"
, formula = (for_all "x" (for_all "z" (p [x, z] .=>. exists "y" ((.~.) (q [x, y] .|. ((.~.) (r [y, z])))))))
, expected = [ClauseNormalForm
(toSS
[[((.~.) (pApp "p" [vt "x",vt "z"])),((.~.) (pApp "q" [vt "x",fApp (toSkolem "y" 1) [vt "x",vt "z"]]))],
[((.~.) (pApp "p" [vt "x",vt "z"])),(pApp "r" [fApp (toSkolem "y" 1) [vt "x",vt "z"],vt "z"])]])]
}
, doTest $
TestFormula
{ name = "cnf test 12"
, formula = ((p0 .=>. q0) .=>. (((.~.) r0) .=>. (s0 .&. t0)))
, expected = [ClauseNormalForm
(toSS
[[(pApp "p" []),(pApp "r" []),(pApp "s" [])],
[((.~.) (pApp "q" [])),(pApp "r" []),(pApp "s" [])],
[(pApp "p" []),(pApp "r" []),(pApp "t" [])],
[((.~.) (pApp "q" [])),(pApp "r" []),(pApp "t" [])]])]
}
, let (f :: Formula) = for_all "x" ( x .=. x) .=>. for_all "x" (exists "y" ((x .=. y))) in
doTest $
TestFormula
{ name = "cnf test 13 " ++ prettyShow f
, formula = f
-- [[x = sKy[x], ¬sKx[] = sKx[]]]
, expected = [ClauseNormalForm (toSS [[x .=. fApp (toSkolem "y" 1) [x], (.~.) (fApp (toSkolem "x" 1) [] .=. fApp (toSkolem "x" 1) [])]])]
}
, let p = pApp "p" [] in
doTest $
TestFormula
{ name = "psimplify 50"
, formula = true .=>. (p .<=>. (p .<=>. false))
, expected = [ SimplifiedForm (p .<=>. (.~.) p) ] }
, doTest $
TestFormula
{ name = "psimplify 51"
, formula = (((pApp "x" [] .=>. pApp "y" []) .=>. true) .|. false)
, expected = [ SimplifiedForm true ] }
, let q = pApp "q" [] in
doTest $
TestFormula
{ name = "simplify 140.3"
, formula = (for_all "x"
(for_all "y"
(pApp "p" [vt "x"] .|. (pApp "p" [vt "y"] .&. false))) .=>.
(exists "z" q))
, expected = [ SimplifiedForm ((for_all "x" (pApp "p" [vt "x"])) .=>.
(pApp "q" [])) ] }
, doTest $
TestFormula
{ name = "nnf 141.1"
, formula = ((for_all "x" (pApp "p" [vt "x"])) .=>. ((exists "y" (pApp "q" [vt "y"])) .<=>. (exists "z" (pApp "p" [vt "z"] .&. pApp "q" [vt "z"]))))
, expected = [ NegationNormalForm
((exists "x" ((.~.) (pApp "p" [vt "x"]))) .|.
((((exists "y" (pApp "q" [vt "y"])) .&. ((exists "z" ((pApp "p" [vt "z"]) .&. ((pApp "q" [vt "z"])))))) .|.
(((for_all "y" ((.~.) (pApp "q" [vt "y"]))) .&.
((for_all "z" (((.~.) (pApp "p" [vt "z"])) .|. (((.~.) (pApp "q" [vt "z"]))))))))))) ] }
, doTest $
TestFormula
{ name = "pnf 144.1"
, formula = (for_all "x" (pApp "p" [vt "x"] .|. pApp "r" [vt "y"]) .=>.
(exists "y" (exists "z" (pApp "q" [vt "y"] .|. ((.~.) (exists "z" (pApp "p" [vt "z"] .&. pApp "q" [vt "z"])))))))
, expected = [ PrenexNormalForm
(exists "x"
(for_all "z"
((((.~.) (pApp "p" [vt "x"])) .&. (((.~.) (pApp "r" [vt "y"])))) .|.
(((pApp "q" [vt "x"]) .|. ((((.~.) (pApp "p" [vt "z"])) .|. (((.~.) (pApp "q" [vt "z"])))))))))) ] }
, let (x, y, u, v) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "u" :: SkTerm, vt "v" :: SkTerm)
fv = fApp (toSkolem "v" 1) [u,x]
fy = fApp (toSkolem "y" 1) [x] in
doTest $
TestFormula
{ name = "snf 150.1"
, formula = (exists "y" (pApp "<" [x, y] .=>. for_all "u" (exists "v" (pApp "<" [fApp "*" [x, u], fApp "*" [y, v]]))))
, expected = [ SkolemNormalForm (((.~.) (pApp "<" [x, fy])) .|. pApp "<" [fApp "*" [x, u], fApp "*" [fy, fv]]) ] }
, let p x = pApp "p" [x]
q x = pApp "q" [x]
(x, y, z) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "z" :: SkTerm) in
doTest $
TestFormula
{ name = "snf 150.2"
, formula = (for_all "x" (p x .=>. (exists "y" (exists "z" (q y .|. (.~.) (exists "z" (p z .&. (q z))))))))
, expected = [ SkolemNormalForm (((.~.) (p x)) .|. (q (fApp (toSkolem "y" 1) []) .|. (((.~.) (p z)) .|. ((.~.) (q z))))) ] }
]
animalKB :: (String, [TestFormula Formula SkAtom V])
animalKB =
let x = vt "x"
y = vt "y"
dog = pApp "Dog"
cat = pApp "Cat"
owns = pApp "Owns"
kills = pApp "Kills"
animal = pApp "Animal"
animalLover = pApp "AnimalLover"
jack = fApp "Jack" []
tuna = fApp "Tuna" []
curiosity = fApp "Curiosity" [] in
("animal"
, [ TestFormula
{ formula = exists "x" (dog [x] .&. owns [jack, x]) -- [[Pos 1],[Pos 2]]
, name = "jack owns a dog"
, expected = [ClauseNormalForm (toSS [[(pApp "Dog" [fApp (toSkolem "x" 1) []])],[(pApp "Owns" [fApp "Jack" [],fApp (toSkolem "x" 1) []])]])]
-- owns(jack,sK0)
-- dog (SK0)
}
, TestFormula
{ formula = for_all "x" ((exists "y" (dog [y] .&. (owns [x, y]))) .=>. (animalLover [x])) -- [[Neg 1,Neg 2,Pos 3]]
, name = "dog owners are animal lovers"
, expected = [ PrenexNormalForm (for_all "x" (for_all "y" ((((.~.) (pApp "Dog" [vt "y"])) .|.
(((.~.) (pApp "Owns" [vt "x",vt "y"])))) .|.
((pApp "AnimalLover" [vt "x"])))))
, ClauseNormalForm (toSS [[((.~.) (pApp "Dog" [vt "y"])),((.~.) (pApp "Owns" [vt "x",vt "y"])),(pApp "AnimalLover" [vt "x"])]]) ]
-- animalLover(X0) | ~owns(X0,sK1(X0)) | ~dog(sK1(X0))
}
, TestFormula
{ formula = for_all "x" (animalLover [x] .=>. (for_all "y" ((animal [y]) .=>. ((.~.) (kills [x, y]))))) -- [[Neg 3,Neg 4,Neg 5]]
, name = "animal lovers don't kill animals"
, expected = [ClauseNormalForm (toSS [[((.~.) (pApp "AnimalLover" [vt "x"])),((.~.) (pApp "Animal" [vt "y"])),((.~.) (pApp "Kills" [vt "x",vt "y"]))]])]
-- ~kills(X0,X2) | ~animal(X2) | ~animalLover(sK2(X0))
}
, TestFormula
{ formula = (kills [jack, tuna]) .|. (kills [curiosity, tuna]) -- [[Pos 5,Pos 5]]
, name = "Either jack or curiosity kills tuna"
, expected = [ClauseNormalForm (toSS [[(pApp "Kills" [fApp "Jack" [],fApp "Tuna" []]),(pApp "Kills" [fApp "Curiosity" [],fApp "Tuna" []])]])]
-- kills(curiosity,tuna) | kills(jack,tuna)
}
, TestFormula
{ formula = cat [tuna] -- [[Pos 6]]
, name = "tuna is a cat"
, expected = [ClauseNormalForm (toSS [[(pApp "Cat" [fApp "Tuna" []])]])]
-- cat(tuna)
}
, TestFormula
{ formula = for_all "x" ((cat [x]) .=>. (animal [x])) -- [[Neg 6,Pos 4]]
, name = "a cat is an animal"
, expected = [ClauseNormalForm (toSS [[((.~.) (pApp "Cat" [vt "x"])),(pApp "Animal" [vt "x"])]])]
-- animal(X0) | ~cat(X0)
}
])
animalConjectures :: [Test]
animalConjectures =
let kills = pApp "Kills"
jack = fApp "Jack" []
tuna = fApp "Tuna" []
curiosity = fApp "Curiosity" [] in
map (doTest . withKB animalKB) $
[ TestFormula
{ formula = kills [jack, tuna] -- False
, name = "jack kills tuna"
, expected =
[ FirstOrderFormula ((.~.) (((exists "x" ((pApp "Dog" [vt ("x")]) .&. ((pApp "Owns" [fApp ("Jack") [],vt ("x")])))) .&.
(((for_all "x" ((exists "y" ((pApp "Dog" [vt ("y")]) .&. ((pApp "Owns" [vt ("x"),vt ("y")])))) .=>.
((pApp "AnimalLover" [vt ("x")])))) .&.
(((for_all "x" ((pApp "AnimalLover" [vt ("x")]) .=>.
((for_all "y" ((pApp "Animal" [vt ("y")]) .=>.
(((.~.) (pApp "Kills" [vt ("x"),vt ("y")])))))))) .&.
((((pApp "Kills" [fApp ("Jack") [],fApp ("Tuna") []]) .|. ((pApp "Kills" [fApp ("Curiosity") [],fApp ("Tuna") []]))) .&.
(((pApp "Cat" [fApp ("Tuna") []]) .&.
((for_all "x" ((pApp "Cat" [vt ("x")]) .=>.
((pApp "Animal" [vt ("x")])))))))))))))) .=>.
((pApp "Kills" [fApp ("Jack") [],fApp ("Tuna") []]))))
, PrenexNormalForm
(for_all "x"
(for_all "y"
(exists "x'"
((((pApp ("Dog") [vt ("x'")]) .&.
((pApp ("Owns") [fApp ("Jack") [],vt ("x'")]))) .&.
((((((.~.) (pApp ("Dog") [vt ("y")])) .|.
(((.~.) (pApp ("Owns") [vt ("x"),vt ("y")])))) .|.
((pApp ("AnimalLover") [vt ("x")]))) .&.
(((((.~.) (pApp ("AnimalLover") [vt ("x")])) .|.
((((.~.) (pApp ("Animal") [vt ("y")])) .|.
(((.~.) (pApp ("Kills") [vt ("x"),vt ("y")])))))) .&.
((((pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []]) .|.
((pApp ("Kills") [fApp ("Curiosity") [],fApp ("Tuna") []]))) .&.
(((pApp ("Cat") [fApp ("Tuna") []]) .&.
((((.~.) (pApp ("Cat") [vt ("x")])) .|.
((pApp ("Animal") [vt ("x")]))))))))))))) .&.
(((.~.) (pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []])))))))
, ClauseNormalForm
(toSS
[[(pApp ("Animal") [vt ("x")]),
((.~.) (pApp ("Cat") [vt ("x")]))],
[(pApp ("AnimalLover") [vt ("x")]),
((.~.) (pApp ("Dog") [vt ("y")])),
((.~.) (pApp ("Owns") [vt ("x"),vt ("y")]))],
[(pApp ("Cat") [fApp ("Tuna") []])],
[(pApp ("Dog") [fApp (toSkolem "x" 1) []])],
[(pApp ("Kills") [fApp ("Curiosity") [],fApp ("Tuna") []]),
(pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []])],
[(pApp ("Owns") [fApp ("Jack") [],fApp (toSkolem "x" 1) []])],
[((.~.) (pApp ("Animal") [vt ("y")])),
((.~.) (pApp ("AnimalLover") [vt ("x")])),
((.~.) (pApp ("Kills") [vt ("x"),vt ("y")]))],
[((.~.) (pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []]))]])
, ChiouKB1
(Proof
Invalid
(Set.fromList
[makeINF' ([]) ([(pApp ("Cat") [fApp ("Tuna") []])]),
makeINF' ([]) ([(pApp ("Dog") [fApp (toSkolem "x" 1) []])]),
makeINF' ([]) ([(pApp ("Kills") [fApp ("Curiosity") [],fApp ("Tuna") []]),(pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []])]),
makeINF' ([]) ([(pApp ("Owns") [fApp ("Jack") [],fApp (toSkolem "x" 1) []])]),
makeINF' ([(pApp ("Animal") [vt ("y")]),(pApp ("AnimalLover") [vt ("x")]),(pApp ("Kills") [vt ("x"),vt ("y")])]) ([]),
makeINF' ([(pApp ("Cat") [vt ("x")])]) ([(pApp ("Animal") [vt ("x")])]),
makeINF' ([(pApp ("Dog") [vt ("y")]),(pApp ("Owns") [vt ("x"),vt ("y")])]) ([(pApp ("AnimalLover") [vt ("x")])]),
makeINF' ([(pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []])]) ([])]))
]
}
, TestFormula
{ formula = kills [curiosity, tuna] -- True
, name = "curiosity kills tuna"
, expected =
[ ClauseNormalForm
(toSS
[[(pApp "Dog" [fApp (toSkolem "x" 1) []])],
[(pApp "Owns" [fApp ("Jack") [],fApp (toSkolem "x" 1) []])],
[((.~.) (pApp "Dog" [vt ("y")])),
((.~.) (pApp "Owns" [vt ("x"),vt ("y")])),
(pApp "AnimalLover" [vt ("x")])],
[((.~.) (pApp "AnimalLover" [vt ("x")])),
((.~.) (pApp "Animal" [vt ("y")])),
((.~.) (pApp "Kills" [vt ("x"),vt ("y")]))],
[(pApp "Kills" [fApp ("Jack") [],fApp ("Tuna") []]),
(pApp "Kills" [fApp ("Curiosity") [],fApp ("Tuna") []])],
[(pApp "Cat" [fApp ("Tuna") []])],
[((.~.) (pApp "Cat" [vt ("x")])),
(pApp "Animal" [vt ("x")])],
[((.~.) (pApp "Kills" [fApp "Curiosity" [],fApp "Tuna" []]))]])
, PropLogicSat True
{-
, SatSolverCNF [ [Neg 1,Neg 2,Neg 3] -- animallover(x)|animal(y)|kills(x,y)
, [Neg 4,Pos 5] -- ~cat(x)|animal(x)
, [Neg 6,Neg 7,Pos 2] -- ~dog(y)|~owns(x,y)|animallover(x)
, [Neg 8] -- ~kills(curisity,tuna)
, [Pos 8,Pos 11] -- kills(curiosity,tuna)|kills(jack,tuna)
, [Pos 9] -- cat(tuna)
, [Pos 10] -- owns(jack,sk1)
, [Pos 12] -- dog(sk1)
]
-}
-- I haven't tried to figure out if this is correct, it
-- probably is because things are working.
, SatSolverCNF [[Neg 2,Pos 1],[Neg 3,Neg 11,Neg 12],[Neg 4,Neg 5,Pos 3],[Neg 8],[Pos 6],[Pos 7],[Pos 8,Pos 9],[Pos 10]]
-- It seems like this should be True.
, SatSolverSat False
]
}
]
socratesKB :: forall t formula atom predicate v term.
(atom ~ AtomOf formula, v ~ VarOf formula, term ~ TermOf atom, predicate ~ PredOf atom,
Ord formula, IsString t,
IsQuantified formula,
HasApply atom,
IsTerm term) =>
(t, [TestFormula formula atom v])
socratesKB =
let x = vt "x"
socrates x = pApp "Socrates" [x]
human x = pApp "Human" [x]
mortal x = pApp "Mortal" [x] in
("socrates"
, [ TestFormula
{ name = "all humans are mortal"
, formula = for_all "x" (human x .=>. mortal x)
, expected = [ClauseNormalForm (toSS [[((.~.) (human x)), mortal x]])] }
, TestFormula
{ name = "socrates is human"
, formula = for_all "x" (socrates x .=>. human x)
, expected = [ClauseNormalForm (toSS [[(.~.) (socrates x), human x]])] }
])
{-
socratesConjectures =
map (withKB socratesKB)
[ TestFormula
{ formula = for_all' [V "x"] (socrates x .=>. mortal x)
, name = "socrates is mortal"
, expected = [ FirstOrderFormula ((.~.) (((for_all' [V "x"] ((pApp "Human" [vt "x"]) .=>. ((pApp "Mortal" [vt "x"])))) .&.
((for_all' [V "x"] ((pApp "Socrates" [vt "x"]) .=>. ((pApp "Human" [vt "x"])))))) .=>.
((for_all' [V "x"] ((pApp "Socrates" [vt "x"]) .=>. ((pApp "Mortal" [vt "x"])))))))
, ClauseNormalForm [[((.~.) (pApp "Human" [vt "x'"])),(pApp "Mortal" [vt "x'"])],
[((.~.) (pApp "Socrates" [vt "x'"])),(pApp "Human" [vt "x'"])],
[(pApp "Socrates" [fApp (toSkolem "x" 1) [vt "x'",vt "x'"]])],
[((.~.) (pApp "Mortal" [fApp (toSkolem "x" 1) [vt "x'",vt "x'"]]))]]
, SatPropLogic True ]
}
, TestFormula
{ formula = (.~.) (for_all' [V "x"] (socrates x .=>. mortal x))
, name = "not (socrates is mortal)"
, expected = [ SatPropLogic False
, FirstOrderFormula ((.~.) (((for_all' [V "x"] ((pApp "Human" [vt "x"]) .=>. ((pApp "Mortal" [vt "x"])))) .&.
((for_all' [V "x"] ((pApp "Socrates" [vt "x"]) .=>. ((pApp "Human" [vt "x"])))))) .=>.
(((.~.) (for_all' [V "x"] ((pApp "Socrates" [vt "x"]) .=>. ((pApp "Mortal" [vt "x"]))))))))
-- [~human(x) | mortal(x)], [~socrates(Sk1(x,y)) | human(Sk1(x,y))], socrates(Sk1(x,y)), ~mortal(Sk1(x,y))
-- ~1 | 2, ~3 | 4, 3, ~5?
, ClauseNormalForm [[((.~.) (pApp "Human" [x])), (pApp "Mortal" [x])],
[((.~.) (pApp "Socrates" [fApp (toSkolem "x" 1) [x,y]])), (pApp "Human" [fApp (toSkolem "x" 1) [x,y]])],
[(pApp "Socrates" [fApp (toSkolem "x" 1) [x,y]])], [((.~.) (pApp "Mortal" [fApp (toSkolem "x" 1) [x,y]]))]]
, ClauseNormalForm [[((.~.) (pApp "Human" [vt "x'"])), (pApp "Mortal" [vt "x'"])],
[((.~.) (pApp "Socrates" [vt "x'"])), (pApp "Human" [vt "x'"])],
[((.~.) (pApp "Socrates" [vt "x"])), (pApp "Mortal" [vt "x"])]] ]
}
]
-}
chang43KB :: (String, [TestFormula Formula SkAtom V])
chang43KB =
let e = fApp "e" []
(x, y, z, u, v, w) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "z" :: SkTerm, vt "u" :: SkTerm, vt "v" :: SkTerm, vt "w" :: SkTerm) in
("chang example 4.3"
, [ TestFormula { name = "closure property"
, formula = for_all' ["x", "y"] (exists "z" (pApp "P" [x,y,z]))
, expected = [] }
, TestFormula { name = "associativity property"
, formula = for_all' ["x", "y", "z", "u", "v", "w"] (pApp "P" [x, y, u] .&. pApp "P" [y, z, v] .&. pApp "P" [u, z, w] .=>. pApp "P" [x, v, w]) .&.
for_all' ["x", "y", "z", "u", "v", "w"] (pApp "P" [x, y, u] .&. pApp "P" [y, z, v] .&. pApp "P" [x, v, w] .=>. pApp "P" [u, z, w])
, expected = [] }
, TestFormula { name = "identity property"
, formula = (for_all "x" (pApp "P" [x,e,x])) .&. (for_all "x" (pApp "P" [e,x,x]))
, expected = [] }
, TestFormula { name = "inverse property"
, formula = (for_all "x" (pApp "P" [x,fApp "i" [x], e])) .&. (for_all "x" (pApp "P" [fApp "i" [x], x, e]))
, expected = [] }
])
chang43Conjecture :: Test
chang43Conjecture =
let e = (fApp "e" [])
(x, u, v, w) = (vt "x" :: SkTerm, vt "u" :: SkTerm, vt "v" :: SkTerm, vt "w" :: SkTerm) in
doTest . withKB chang43KB $
TestFormula { name = "G is commutative"
, formula = for_all "x" (pApp "P" [x, x, e] .=>. (for_all' ["u", "v", "w"] (pApp "P" [u, v, w] .=>. pApp "P" [v, u, w])))
, expected =
[ FirstOrderFormula
((.~.) (((for_all' ["x","y"] (exists "z" (pApp "P" [vt ("x"),vt ("y"),vt ("z")]))) .&. ((((for_all' ["x","y","z","u","v","w"] ((((pApp "P" [vt ("x"),vt ("y"),vt ("u")]) .&. ((pApp "P" [vt ("y"),vt ("z"),vt ("v")]))) .&. ((pApp "P" [vt ("u"),vt ("z"),vt ("w")]))) .=>. ((pApp "P" [vt ("x"),vt ("v"),vt ("w")])))) .&. ((for_all' ["x","y","z","u","v","w"] ((((pApp "P" [vt ("x"),vt ("y"),vt ("u")]) .&. ((pApp "P" [vt ("y"),vt ("z"),vt ("v")]))) .&. ((pApp "P" [vt ("x"),vt ("v"),vt ("w")]))) .=>. ((pApp "P" [vt ("u"),vt ("z"),vt ("w")])))))) .&. ((((for_all "x" (pApp "P" [vt ("x"),fApp ("e") [],vt ("x")])) .&. ((for_all "x" (pApp "P" [fApp ("e") [],vt ("x"),vt ("x")])))) .&. (((for_all "x" (pApp "P" [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []])) .&. ((for_all "x" (pApp "P" [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []])))))))))) .=>. ((for_all "x" ((pApp "P" [vt ("x"),vt ("x"),fApp ("e") []]) .=>. ((for_all' ["u","v","w"] ((pApp "P" [vt ("u"),vt ("v"),vt ("w")]) .=>. ((pApp "P" [vt ("v"),vt ("u"),vt ("w")]))))))))))
-- (∀x ∀y ∃z P(x,y,z)) &
-- (∀x∀y∀z∀u∀v∀w ~P(x,y,u) | ~P(y,z,v) | ~P(u,z,w) | P(x,v,w)) &
-- (∀x∀y∀z∀u∀v∀w ~P(x,y,u) | ~P(y,z,v) | ~P(x,v,w) | P(u,z,w)) &
-- (∀x P(x,e,x)) &
-- (∀x P(e,x,x)) &
-- (∀x P(x,i[x],e)) &
-- (∀x P(i[x],x,e)) &
-- (∃x P(x,x,e) & (∃u∃v∃w P(u,v,w) & ~P(v,u,w)))
, NegationNormalForm
(((for_all "x"
(for_all "y"
(exists "z"
(pApp ("P") [vt ("x"),vt ("y"),vt ("z")])))) .&.
((((for_all "x"
(for_all "y"
(for_all "z"
(for_all "u"
(for_all "v"
(for_all "w"
(((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("u"),vt ("z"),vt ("w")])))) .|.
((pApp ("P") [vt ("x"),vt ("v"),vt ("w")]))))))))) .&.
((for_all "x"
(for_all "y"
(for_all "z"
(for_all "u"
(for_all "v"
(for_all "w"
(((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("x"),vt ("v"),vt ("w")])))) .|.
((pApp ("P") [vt ("u"),vt ("z"),vt ("w")]))))))))))) .&.
((((for_all "x" (pApp ("P") [vt ("x"),fApp ("e") [],vt ("x")])) .&.
((for_all "x" (pApp ("P") [fApp ("e") [],vt ("x"),vt ("x")])))) .&.
(((for_all "x" (pApp ("P") [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []])) .&.
((for_all "x" (pApp ("P") [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []])))))))))) .&.
((exists "x"
((pApp ("P") [vt ("x"),vt ("x"),fApp ("e") []]) .&.
((exists "u"
(exists "v"
(exists "w"
((pApp ("P") [vt ("u"),vt ("v"),vt ("w")]) .&.
(((.~.) (pApp ("P") [vt ("v"),vt ("u"),vt ("w")]))))))))))))
, PrenexNormalForm
(for_all "x"
(for_all "y"
(for_all "z"
(for_all "u"
(for_all "v"
(for_all "w"
(exists "z'"
(exists "x'"
(exists "u'"
(exists "v'"
(exists "w'"
(((pApp ("P") [vt ("x"),vt ("y"),vt ("z'")]) .&.
((((((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("u"),vt ("z"),vt ("w")])))) .|.
((pApp ("P") [vt ("x"),vt ("v"),vt ("w")]))) .&.
((((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("x"),vt ("v"),vt ("w")])))) .|.
((pApp ("P") [vt ("u"),vt ("z"),vt ("w")]))))) .&.
((((pApp ("P") [vt ("x"),fApp ("e") [],vt ("x")]) .&.
((pApp ("P") [fApp ("e") [],vt ("x"),vt ("x")]))) .&.
(((pApp ("P") [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []]) .&.
((pApp ("P") [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []]))))))))) .&.
(((pApp ("P") [vt ("x'"),vt ("x'"),fApp ("e") []]) .&.
(((pApp ("P") [vt ("u'"),vt ("v'"),vt ("w'")]) .&.
(((.~.) (pApp ("P") [vt ("v'"),vt ("u'"),vt ("w'")])))))))))))))))))))
, SkolemNormalForm
(((pApp ("P") [vt ("x"),vt ("y"),fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]]) .&.
((((((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("u"),vt ("z"),vt ("w")])))) .|.
((pApp ("P") [vt ("x"),vt ("v"),vt ("w")]))) .&.
((((((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])) .|.
(((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])))) .|.
(((.~.) (pApp ("P") [vt ("x"),vt ("v"),vt ("w")])))) .|.
((pApp ("P") [vt ("u"),vt ("z"),vt ("w")]))))) .&.
((((pApp ("P") [vt ("x"),fApp ("e") [],vt ("x")]) .&.
((pApp ("P") [fApp ("e") [],vt ("x"),vt ("x")]))) .&.
(((pApp ("P") [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []]) .&.
((pApp ("P") [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []]))))))))) .&.
(((pApp ("P") [fApp (toSkolem "x" 1) [],fApp (toSkolem "x" 1) [],fApp ("e") []]) .&.
(((pApp ("P") [fApp (toSkolem "u" 1) [],fApp (toSkolem "v" 1) [],fApp (toSkolem "w" 1) []]) .&.
(((.~.) (pApp ("P") [fApp (toSkolem "v" 1) [],fApp (toSkolem "u" 1) [],fApp (toSkolem "w" 1) []]))))))))
, SkolemNumbers (Set.fromList [toSkolem "u" 1,toSkolem "v" 1,toSkolem "w" 1,toSkolem "x" 1,toSkolem "z" 1])
-- From our algorithm
, ClauseNormalForm
(toSS
[[(pApp ("P") [vt ("x"),vt ("y"),fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]])],
[((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])),
((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])),
((.~.) (pApp ("P") [vt ("u"),vt ("z"),vt ("w")])),
(pApp ("P") [vt ("x"),vt ("v"),vt ("w")])],
[((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])),
((.~.) (pApp ("P") [vt ("y"),vt ("z"),vt ("v")])),
((.~.) (pApp ("P") [vt ("x"),vt ("v"),vt ("w")])),
(pApp ("P") [vt ("u"),vt ("z"),vt ("w")])],
[(pApp ("P") [vt ("x"),fApp ("e") [],vt ("x")])],
[(pApp ("P") [fApp ("e") [],vt ("x"),vt ("x")])],
[(pApp ("P") [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []])],
[(pApp ("P") [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []])],
[(pApp ("P") [fApp (toSkolem "x" 1) [],fApp (toSkolem "x" 1) [],fApp ("e") []])],
[(pApp ("P") [fApp (toSkolem "u" 1) [],fApp (toSkolem "v" 1) [],fApp (toSkolem "w" 1) []])],
[((.~.) (pApp ("P") [fApp (toSkolem "v" 1) [],fApp (toSkolem "u" 1) [],fApp (toSkolem "w" 1) []]))]])
-- From the book
{-
, let (a, b, c) =
(fApp (toSkolem "x" 1) [vt ("x"),vt ("y"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u"),vt ("v"),vt ("w"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u'"),vt ("v'"),vt ("w'"),vt ("x''"),vt ("x''"),vt ("x''"),vt ("x''")],
fApp (toSkolem "x" 1) [vt ("x"),vt ("y"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u"),vt ("v"),vt ("w"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u'"),vt ("v'"),vt ("w'"),vt ("x''"),vt ("x''"),vt ("x''"),vt ("x''")],
fApp (toSkolem "x" 1) [vt ("x"),vt ("y"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u"),vt ("v"),vt ("w"),vt ("x'"),vt ("y'"),vt ("z'"),vt ("u'"),vt ("v'"),vt ("w'"),vt ("x''"),vt ("x''"),vt ("x''"),vt ("x''")]) in
ClauseNormalForm
[[(pApp "P" [vt "x",vt "y",fApp (toSkolem "x" 1) [vt "x",vt "y"]])],
[((.~.) (pApp "P" [vt "x",vt "y",vt "u"])),
((.~.) (pApp "P" [vt "y",vt "z",vt "v"])),
((.~.) (pApp "P" [vt "u",vt "z",vt "w"])),
(pApp "P" [vt "x",vt "v",vt "w"])],
[((.~.) (pApp "P" [vt "x",vt "y",vt "u"])),
((.~.) (pApp "P" [vt "y",vt "z",vt "v"])),
((.~.) (pApp "P" [vt "x",vt "v",vt "w"])),
(pApp "P" [vt "u",vt "z",vt "w"])],
[(pApp "P" [vt "x",fApp "e" [],vt "x"])],
[(pApp "P" [fApp "e" [],vt "x",vt "x"])],
[(pApp "P" [vt "x",fApp "i" [vt "x"],fApp "e" []])],
[(pApp "P" [fApp "i" [vt "x"],vt "x",fApp "e" []])],
[(pApp "P" [vt "x",
vt "x",
fApp "e" []])],
[(pApp "P" [a, b, c])],
[((.~.) (pApp "P" [b, a, c]))]]
-}
]
}
{-
% ghci
> :load Test/Data.hs
> :m +Logic.FirstOrder
> :m +Logic.Normal
> let f = (.~.) (conj (map formula (snd chang43KB)) .=>. formula chang43Conjecture)
> putStrLn (runNormal (cnfTrace f))
-}
chang43ConjectureRenamed :: Test
chang43ConjectureRenamed =
let e = fApp "e" []
(x, y, z, u, v, w) = (vt "x" :: SkTerm, vt "y" :: SkTerm, vt "z" :: SkTerm, vt "u" :: SkTerm, vt "v" :: SkTerm, vt "w" :: SkTerm)
(u2, v2, w2, x2, y2, z2, u3, v3, w3, x3, y3, z3, x4, x5, x6, x7, x8) =
(vt "u'" :: SkTerm, vt "v'" :: SkTerm, vt "w'" :: SkTerm, vt "x'" :: SkTerm, vt "y'" :: SkTerm, vt "z'" :: SkTerm, vt "u3" :: SkTerm, vt "v3" :: SkTerm, vt "w3" :: SkTerm, vt "x3" :: SkTerm, vt "y3" :: SkTerm, vt "z3" :: SkTerm, vt "x4" :: SkTerm, vt "x5" :: SkTerm, vt "x6" :: SkTerm, vt "x7" :: SkTerm, vt "x8" :: SkTerm) in
doTest $
TestFormula { name = "chang 43 renamed"
, formula = (.~.) ((for_all' ["x", "y"] (exists "z" (pApp "P" [x,y,z])) .&.
for_all' ["x'", "y'", "z'", "u", "v", "w"] (pApp "P" [x2, y2, u] .&. pApp "P" [y2, z2, v] .&. pApp "P" [u, z2, w] .=>. pApp "P" [x2, v, w]) .&.
for_all' ["x3", "y3", "z3", "u'", "v'", "w'"] (pApp "P" [x3, y3, u2] .&. pApp "P" [y3, z3, v2] .&. pApp "P" [x3, v2, w2] .=>. pApp "P" [u2, z3, w2]) .&.
for_all "x4" (pApp "P" [x4,e,x4]) .&.
for_all "x5" (pApp "P" [e,x5,x5]) .&.
for_all "x6" (pApp "P" [x6,fApp "i" [x6], e]) .&.
for_all "x7" (pApp "P" [fApp "i" [x7], x7, e])) .=>.
(for_all "x8" (pApp "P" [x8, x8, e] .=>. (for_all' ["u3", "v3", "w3"] (pApp "P" [u3, v3, w3] .=>. pApp "P" [v3, u3, w3])))))
, expected =
[ FirstOrderFormula
((.~.) ((((((((for_all' ["x","y"] (exists "z" (pApp "P" [vt "x",vt "y",vt "z"]))) .&.
((for_all' ["x'","y'","z'","u","v","w"] ((((pApp "P" [vt "x'",vt "y'",vt "u"]) .&.
((pApp "P" [vt "y'",vt "z'",vt "v"]))) .&.
((pApp "P" [vt "u",vt "z'",vt "w"]))) .=>.
((pApp "P" [vt "x'",vt "v",vt "w"])))))) .&.
((for_all' ["x3","y3","z3","u'","v'","w'"] ((((pApp "P" [vt "x3",vt "y3",vt "u'"]) .&.
((pApp "P" [vt "y3",vt "z3",vt "v'"]))) .&.
((pApp "P" [vt "x3",vt "v'",vt "w'"]))) .=>.
((pApp "P" [vt "u'",vt "z3",vt "w'"])))))) .&.
((for_all "x4" (pApp "P" [vt "x4",fApp "e" [],vt "x4"])))) .&.
((for_all "x5" (pApp "P" [fApp "e" [],vt "x5",vt "x5"])))) .&.
((for_all "x6" (pApp "P" [vt "x6",fApp "i" [vt "x6"],fApp "e" []])))) .&.
((for_all "x7" (pApp "P" [fApp "i" [vt "x7"],vt "x7",fApp "e" []])))) .=>.
((for_all "x8" ((pApp "P" [vt "x8",vt "x8",fApp "e" []]) .=>.
((for_all' ["u3","v3","w3"] ((pApp "P" [vt "u3",vt "v3",vt "w3"]) .=>.
((pApp "P" [vt "v3",vt "u3",vt "w3"]))))))))))
, let a = fApp (toSkolem "u3" 1) []
b = fApp (toSkolem "v3" 1) []
c = fApp (toSkolem "w3" 1) [] in
ClauseNormalForm
(toSS
[[(pApp ("P") [vt ("x"),vt ("y"),fApp (toSkolem "z" 1) [vt ("x"),vt ("y")]])],
[((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])),
((.~.) (pApp ("P") [vt ("y"),vt ("z'"),vt ("v")])),
((.~.) (pApp ("P") [vt ("u"),vt ("z'"),vt ("w")])),
(pApp ("P") [vt ("x"),vt ("v"),vt ("w")])],
[((.~.) (pApp ("P") [vt ("x"),vt ("y"),vt ("u")])),
((.~.) (pApp ("P") [vt ("y"),vt ("z'"),vt ("v")])),
((.~.) (pApp ("P") [vt ("x"),vt ("v"),vt ("w")])),
(pApp ("P") [vt ("u"),vt ("z'"),vt ("w")])],
[(pApp ("P") [vt ("x"),fApp ("e") [],vt ("x")])],
[(pApp ("P") [fApp ("e") [],vt ("x"),vt ("x")])],
[(pApp ("P") [vt ("x"),fApp ("i") [vt ("x")],fApp ("e") []])],
[(pApp ("P") [fApp ("i") [vt ("x")],vt ("x"),fApp ("e") []])],
[(pApp ("P") [fApp (toSkolem "x8" 1) [],fApp (toSkolem "x8" 1) [],fApp ("e") []])],
[(pApp ("P") [a,b,c])],
[((.~.) (pApp ("P") [b,a,c]))]])
]
}
withKB :: forall formula atom term v.
(formula ~ Formula, atom ~ SkAtom, v ~ V,
term ~ TermOf atom,
IsQuantified formula, HasEquate atom, IsTerm term) =>
(String, [TestFormula formula atom v]) -> TestFormula formula atom v -> TestFormula formula atom v
withKB (kbName, knowledge) conjecture =
conjecture { name = name conjecture ++ " with " ++ kbName ++ " knowledge base"
-- Here we say that the conjunction of the knowledge
-- base formula implies the conjecture. We prove the
-- theorem by showing that the negation is
-- unsatisfiable.
, formula = (.~.) (conj (map formula knowledge) .=>. formula conjecture)}
where
conj [] = error "conj []"
conj [x] = x
conj (x:xs) = x .&. conj xs
kbKnowledge :: forall formula atom term v.
(formula ~ Formula, atom ~ SkAtom, v ~ V, term ~ TermOf atom,
IsQuantified formula, HasEquate atom, IsTerm term) =>
(String, [TestFormula formula atom v]) -> (String, [formula])
kbKnowledge kb = (fst (kb :: (String, [TestFormula formula atom v])), map formula (snd kb))
proofs :: [TestProof Formula SkAtom SkTerm V]
proofs =
let -- dog = pApp "Dog" :: [term] -> formula
-- cat = pApp "Cat" :: [term] -> formula
-- owns = pApp "Owns" :: [term] -> formula
kills = pApp "Kills"
-- animal = pApp "Animal" :: [term] -> formula
-- animalLover = pApp "AnimalLover" :: [term] -> formula
socrates = pApp "Socrates"
-- human = pApp "Human" :: [term] -> formula
mortal = pApp "Mortal"
jack = fApp "Jack" []
tuna = fApp "Tuna" []
curiosity = fApp "Curiosity" [] in
[ TestProof
{ proofName = "prove jack kills tuna"
, proofKnowledge = kbKnowledge animalKB
, conjecture = kills [jack, tuna]
, proofExpected =
[ ChiouKB (Set.fromList
[WithId {wiItem = INF (Set.fromList []) (Set.fromList [(pApp "Dog" [fApp (toSkolem "x" 1) []])]), wiIdent = 1},
WithId {wiItem = INF (Set.fromList []) (Set.fromList [(pApp "Owns" [fApp "Jack" [],fApp (toSkolem "x" 1) []])]), wiIdent = 1},
WithId {wiItem = INF (Set.fromList [(pApp "Dog" [vt "y"]),(pApp "Owns" [vt "x",vt "y"])]) (Set.fromList [(pApp "AnimalLover" [vt "x"])]), wiIdent = 2},
WithId {wiItem = INF (Set.fromList [(pApp "Animal" [vt "y"]),(pApp "AnimalLover" [vt "x"]),(pApp "Kills" [vt "x",vt "y"])]) (Set.fromList []), wiIdent = 3},
WithId {wiItem = INF (Set.fromList []) (Set.fromList [(pApp "Kills" [fApp "Curiosity" [],fApp "Tuna" []]),(pApp "Kills" [fApp "Jack" [],fApp "Tuna" []])]), wiIdent = 4},
WithId {wiItem = INF (Set.fromList []) (Set.fromList [(pApp "Cat" [fApp "Tuna" []])]), wiIdent = 5},
WithId {wiItem = INF (Set.fromList [(pApp "Cat" [vt "x"])]) (Set.fromList [(pApp "Animal" [vt "x"])]), wiIdent = 6}])
, ChiouResult (False,
(Set.fromList
[(inf' [(pApp "Kills" [fApp "Jack" [],fApp "Tuna" []])] [],Map.fromList []),
(inf' [] [(pApp "Kills" [fApp "Curiosity" [],fApp "Tuna" []])],Map.fromList []),
(inf' [(pApp "Animal" [fApp "Tuna" []]),(pApp "AnimalLover" [fApp "Curiosity" []])] [],Map.fromList []),
(inf' [(pApp "Animal" [fApp "Tuna" []]),(pApp "Dog" [vt "y"]),(pApp "Owns" [fApp "Curiosity" [],vt "y"])] [],Map.fromList []),
(inf' [(pApp "AnimalLover" [fApp "Curiosity" []]),(pApp "Cat" [fApp "Tuna" []])] [],Map.fromList []),
(inf' [(pApp "Animal" [fApp "Tuna" []]),(pApp "Owns" [fApp "Curiosity" [],fApp (toSkolem "x" 1) []])] [],Map.fromList []),
(inf' [(pApp "Cat" [fApp "Tuna" []]),(pApp "Dog" [vt "y"]),(pApp "Owns" [fApp "Curiosity" [],vt "y"])] [],Map.fromList []),
(inf' [(pApp "AnimalLover" [fApp "Curiosity" []])] [],Map.fromList []),
(inf' [(pApp "Cat" [fApp "Tuna" []]),(pApp "Owns" [fApp "Curiosity" [],fApp (toSkolem "x" 1) []])] [],Map.fromList []),
(inf' [(pApp "Dog" [vt "y"]),(pApp "Owns" [fApp "Curiosity" [],vt "y"])] [],Map.fromList []),
(inf' [(pApp "Owns" [fApp "Curiosity" [],fApp (toSkolem "x" 1) []])] [],Map.fromList [])]))
]
}
, TestProof
{ proofName = "prove curiosity kills tuna"
, proofKnowledge = kbKnowledge animalKB
, conjecture = kills [curiosity, tuna]
, proofExpected =
[ ChiouKB (Set.fromList
[WithId {wiItem = inf' [] [(pApp "Dog" [fApp (toSkolem "x" 1) []])], wiIdent = 1},
WithId {wiItem = inf' [] [(pApp "Owns" [fApp "Jack" [],fApp (toSkolem "x" 1) []])], wiIdent = 1},
WithId {wiItem = inf' [(pApp "Dog" [vt "y"]),
(pApp "Owns" [vt "x",vt "y"])] [(pApp "AnimalLover" [vt "x"])], wiIdent = 2},
WithId {wiItem = inf' [(pApp "Animal" [vt "y"]),
(pApp "AnimalLover" [vt "x"]),
(pApp "Kills" [vt "x",vt "y"])] [], wiIdent = 3},
WithId {wiItem = inf' [] [(pApp "Kills" [fApp "Curiosity" [],fApp "Tuna" []]),
(pApp "Kills" [fApp "Jack" [],fApp "Tuna" []])], wiIdent = 4},
WithId {wiItem = inf' [] [(pApp "Cat" [fApp "Tuna" []])], wiIdent = 5},
WithId {wiItem = inf' [(pApp "Cat" [vt "x"])] [(pApp "Animal" [vt "x"])], wiIdent = 6}])
, ChiouResult (True,
Set.fromList
[(makeINF' ([]) ([]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Kills") [fApp ("Jack") [],fApp ("Tuna") []])]),Map.fromList []),
(makeINF' ([(pApp ("Animal") [fApp ("Tuna") []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Animal") [fApp ("Tuna") []]),(pApp ("AnimalLover") [fApp ("Jack") []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Animal") [fApp ("Tuna") []]),(pApp ("Dog") [vt ("y")]),(pApp ("Owns") [fApp ("Jack") [],vt ("y")])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Animal") [fApp ("Tuna") []]),(pApp ("Dog") [fApp (toSkolem "x" 1) []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Animal") [fApp ("Tuna") []]),(pApp ("Owns") [fApp ("Jack") [],fApp (toSkolem "x" 1) []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("AnimalLover") [fApp ("Jack") []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("AnimalLover") [fApp ("Jack") []]),(pApp ("Cat") [fApp ("Tuna") []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Cat") [fApp ("Tuna") []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Cat") [fApp ("Tuna") []]),(pApp ("Dog") [vt ("y")]),(pApp ("Owns") [fApp ("Jack") [],vt ("y")])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Cat") [fApp ("Tuna") []]),(pApp ("Dog") [fApp (toSkolem "x" 1) []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Cat") [fApp ("Tuna") []]),(pApp ("Owns") [fApp ("Jack") [],fApp (toSkolem "x" 1) []])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Dog") [vt ("y")]),(pApp ("Owns") [fApp ("Jack") [],vt ("y")])]) ([]),Map.fromList []),
(makeINF' ([(pApp ("Kills") [fApp ("Curiosity") [],fApp ("Tuna") []])]) ([]),Map.fromList [])])
]
}
{-
-- Seems not to terminate
, let (x, u, v, w, e) = (vt "x", vt "u", vt "v", vt "w", vt "e") in
TestProof
{ proofName = "chang example 4.3"
, proofKnowledge = (fst chang43KB, map (convertFOF id id id . formula) (snd chang43KB))
, conjecture = for_all' ["x"] (pApp "P" [x, x, e] .=>. (for_all' ["u", "v", "w"] (pApp "P" [u, v, w] .=>. pApp "P" [v, u, w])))
, proofExpected =
[ChiouResult (True, [])]
}
-}
, let x = vt "x" in
TestProof
{ proofName = "socrates is mortal"
, proofKnowledge = kbKnowledge (socratesKB)
, conjecture = for_all "x" (socrates [x] .=>. mortal [x])
, proofExpected =
[ ChiouKB (Set.fromList
[WithId {wiItem = inf' [(pApp "Human" [vt "x"])] [(pApp "Mortal" [vt "x"])], wiIdent = 1},
WithId {wiItem = inf' [(pApp "Socrates" [vt "x"])] [(pApp "Human" [vt "x"])], wiIdent = 2}])
, ChiouResult (True,
Set.fromList
[(makeINF' ([]) ([]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Human") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Mortal") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Socrates") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([(pApp ("Mortal") [fApp (toSkolem "x" 1) []])]) ([]),Map.fromList [])])]
}
, let x = vt "x" in
TestProof
{ proofName = "socrates is not mortal"
, proofKnowledge = kbKnowledge (socratesKB)
, conjecture = (.~.) (for_all "x" (socrates [x] .=>. mortal [x]))
, proofExpected =
[ ChiouKB (Set.fromList
[WithId {wiItem = inf' [(pApp "Human" [vt "x"])] [(pApp "Mortal" [vt "x"])], wiIdent = 1},
WithId {wiItem = inf' [(pApp "Socrates" [vt "x"])] [(pApp "Human" [vt "x"])], wiIdent = 2}])
, ChiouResult (False
,(Set.fromList [(inf' [(pApp "Socrates" [vt "x"])] [(pApp "Mortal" [vt "x"])],Map.fromList [("x",vt "x")])]))]
}
, let x = vt "x" in
TestProof
{ proofName = "socrates exists and is not mortal"
, proofKnowledge = kbKnowledge (socratesKB)
, conjecture = (.~.) (exists "x" (socrates [x]) .&. for_all "x" (socrates [x] .=>. mortal [x]))
, proofExpected =
[ ChiouKB (Set.fromList
[WithId {wiItem = inf' [(pApp "Human" [vt "x"])] [(pApp "Mortal" [vt "x"])], wiIdent = 1},
WithId {wiItem = inf' [(pApp "Socrates" [vt "x"])] [(pApp "Human" [vt "x"])], wiIdent = 2}])
, ChiouResult (False,
Set.fromList [(makeINF' ([]) ([(pApp ("Human") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Mortal") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([]) ([(pApp ("Socrates") [fApp (toSkolem "x" 1) []])]),Map.fromList []),
(makeINF' ([(pApp ("Socrates") [vt ("x")])]) ([(pApp ("Mortal") [vt ("x")])]),Map.fromList [("x",vt ("x"))])])
]
}
]
inf' :: (IsLiteral lit, Ord lit) => [lit] -> [lit] -> ImplicativeForm lit
inf' = makeINF'
toLL :: Set (Set a) -> [[a]]
toLL = map Set.toList . Set.toList
toSS :: Ord a => [[a]] -> Set (Set a)
toSS = Set.fromList . map Set.fromList