393 lines
No EOL
16 KiB
Haskell
393 lines
No EOL
16 KiB
Haskell
module Main where
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import Data.List
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import Data.Maybe
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import Debug.Trace
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{-
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First we define first order predicate logic
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-}
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data Term = Var String | Fun String [Term] deriving (Eq)
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data Formula
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= Pred String [Term]
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| Neg Formula
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| Conj Formula Formula
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| Disj Formula Formula
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| Impl Formula Formula
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| All String Formula
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| Exists String Formula
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| T
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| F
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deriving (Eq)
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instance Show Term where
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show (Var x) = x
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show (Fun f []) = f
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show (Fun f (x : xs)) = f ++ "(" ++ show x ++ foldr ((++) . (", "++) . show) ")" xs
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instance Show Formula where
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show (Pred p []) = p
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show (Pred p (x : xs)) = p ++ "(" ++ show x ++ foldr ((++) . (", "++) . show) ")" xs
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show (Neg f) = "!(" ++ show f ++ ")"
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show (Conj f1 f2) = show f1 ++ " /\\ " ++ show f2
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show (Disj f1 f2) = show f1 ++ " \\/ " ++ show f2
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show (Impl f1 f2) = "(" ++ show f1 ++ " -> " ++ show f2 ++ ")"
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show (All x f) = "forall " ++ x ++ ". " ++ show f
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show (Exists x f) = "exists " ++ x ++ ". " ++ show f
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show T = "true"
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show F = "false"
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-- free variables of a term
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termFreeVars :: Term -> [String]
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termFreeVars (Var x) = [x]
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termFreeVars (Fun _ ts) = concatMap termFreeVars ts
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-- substitution on terms
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termSubst :: Term -> String -> Term -> Term
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termSubst t@(Var x) y s = if x == y then s else t
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termSubst (Fun f ts) y s = Fun f $ map (\t' -> termSubst t' y s) ts
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-- unification algorithm of martelli montanari
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unify :: [(Term, Term)] -> Maybe [(Term, Term)]
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unify [] = Just []
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-- (delete)
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unify ((Var x, Var y) : rest) | x == y = unify rest
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-- (decomp)
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unify ((Fun f es, Fun g ds) : rest) | f == g && length es == length ds = unify $ zip es ds ++ rest
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-- (conflict)
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unify ((Fun _ _, Fun _ _) : _) = Nothing
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-- (orient)
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unify ((Fun f ts, Var x) : rest) = unify $ (Var x, Fun f ts) : rest
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-- (occurs)
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unify ((Var x, t) : _) | x `elem` termFreeVars t && Var x /= t = Nothing
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-- (elim)
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unify ((Var x, t) : rest) | notElem x (termFreeVars t) && x `elem` concatMap (\(t1, t2) -> termFreeVars t1 ++ termFreeVars t2) rest = unify $ (Var x, t) : map (\(t1, t2) -> (termSubst t1 x t, termSubst t2 x t)) rest
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-- decent
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unify ((t, s) : rest) = do
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rest' <- unify rest
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return $ (t, s) : rest'
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{-
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Now we define some functions to convert given terms to normalforms
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i.e. negation normalform, prenex normal, skolemform, conjunctive normalform
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-}
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-- negation normalform
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makeNNF :: Formula -> Formula
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makeNNF form = if form == f' then form else makeNNF f'
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where
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f' = nnfStep form
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nnfStep (Conj f1 f2) = Conj (nnfStep f1) (nnfStep f2)
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nnfStep (Disj f1 f2) = Disj (nnfStep f1) (nnfStep f2)
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nnfStep (Impl f1 f2) = Disj (Neg f1) f2
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nnfStep (All x f) = All x (nnfStep f)
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nnfStep (Exists x f) = Exists x (nnfStep f)
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nnfStep (Neg (Conj f1 f2)) = Disj (Neg f1) (Neg f2)
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nnfStep (Neg (Disj f1 f2)) = Conj (Neg f1) (Neg f2)
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nnfStep (Neg (Impl f1 f2)) = Conj f1 (Neg f2)
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nnfStep (Neg (All x f)) = Exists x (Neg f)
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nnfStep (Neg (Exists x f)) = All x (Neg f)
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nnfStep (Neg (Neg f)) = f
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nnfStep (Neg T) = F
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nnfStep (Neg F) = T
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nnfStep f = f
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-- infinite list of variable names
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vars :: [String]
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vars = ['v' : show n | n <- [(0 :: Int)..]]
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-- return *all* vars in a formula, i.e. bound ones and free ones
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formulaVars :: Formula -> [String]
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formulaVars (Pred _ ts) = concatMap termFreeVars ts
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formulaVars (Neg f) = formulaVars f
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formulaVars (Conj f1 f2) = formulaVars f1 ++ formulaVars f2
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formulaVars (Disj f1 f2) = formulaVars f1 ++ formulaVars f2
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formulaVars (Impl f1 f2) = formulaVars f1 ++ formulaVars f2
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formulaVars (All x f) = x : formulaVars f
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formulaVars (Exists x f) = x : formulaVars f
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formulaVars _ = []
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-- renames all occurences of variable v with v' in a term
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renameTerm :: Term -> String -> String -> Term
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renameTerm t@(Var x) v v' = if x == v then Var v' else t
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renameTerm (Fun g ts) v v' = Fun g (map (\t -> renameTerm t v v') ts)
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-- renames all occurences of free variable v with v' in a formula
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renameFormula :: Formula -> String -> String -> Formula
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renameFormula (Pred p ts) v v' = Pred p (map (\t -> renameTerm t v v') ts)
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renameFormula (Neg f') v v' = Neg $ renameFormula f' v v'
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renameFormula (Conj f1 f2) v v' = Conj (renameFormula f1 v v') (renameFormula f2 v v')
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renameFormula (Disj f1 f2) v v' = Disj (renameFormula f1 v v') (renameFormula f2 v v')
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renameFormula (Impl f1 f2) v v' = Impl (renameFormula f1 v v') (renameFormula f2 v v')
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renameFormula (All y f') v v' | y /= v = All y $ renameFormula f' v v'
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renameFormula (Exists y f') v v' | y /= v = Exists y $ renameFormula f' v v'
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renameFormula f' _ _ = f'
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-- finds a fresh variable i.e. a variable not occuring in vs
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findFresh :: [String] -> String
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findFresh vs = fromJust $ find (`notElem` vs) vars
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-- rename every binder in formula to be disjunct
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renameBinders :: Formula -> Formula
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renameBinders f = fst $ go f []
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where
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go :: Formula -> [String] -> (Formula, [String])
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go (All x f') vs = (All x' f'', vs')
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where
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(f'', vs') = go (renameFormula f' x x') (x' : vs)
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x' = findFresh (formulaVars f ++ vs)
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go (Exists x f') vs = (Exists x' f'', vs')
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where
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(f'', vs') = go (renameFormula f' x x') (x' : vs)
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x' = findFresh (formulaVars f ++ vs)
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go (Neg f') vs = (Neg f'', vs')
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where
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(f'', vs') = go f' vs
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go (Conj f1 f2) vs = (Conj f1' f2', vs'')
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where
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(f1', vs') = go f1 vs
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(f2', vs'') = go f2 vs'
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go (Disj f1 f2) vs = (Disj f1' f2', vs'')
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where
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(f1', vs') = go f1 vs
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(f2', vs'') = go f2 vs'
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go (Impl f1 f2) vs = (Impl f1' f2', vs'')
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where
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(f1', vs') = go f1 vs
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(f2', vs'') = go f2 vs'
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go f' vs = (f', vs)
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-- prenex normalform
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-- TODO make it so that forall has higher priority to be moved left (makes skolem functions smaller)
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makePNF :: Formula -> Formula
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makePNF form = go $ renameBinders . makeNNF $ form
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where
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go f = let f' = pnfStep f in
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if f == f' then f else go f'
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-- swapping rules
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pnfStep (Conj phi (All x psi)) = All x (Conj phi psi)
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pnfStep (Disj phi (All x psi)) = All x (Disj phi psi)
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pnfStep (Conj (All x psi) phi) = All x (Conj psi phi)
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pnfStep (Disj (All x psi) phi) = All x (Disj psi phi)
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pnfStep (Conj phi (Exists x psi)) = Exists x (Conj phi psi)
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pnfStep (Disj phi (Exists x psi)) = Exists x (Disj phi psi)
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pnfStep (Conj (Exists x psi) phi) = Exists x (Conj psi phi)
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pnfStep (Disj (Exists x psi) phi) = Exists x (Disj psi phi)
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{-
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pnfStep (Conj phi (Exists x psi)) = Exists x (Conj phi psi)
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pnfStep (Conj phi (All x psi)) = All x (Conj phi psi)
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pnfStep (Disj phi (Exists x psi)) = Exists x (Disj phi psi)
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pnfStep (Disj phi (All x psi)) = All x (Disj phi psi)
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pnfStep (Conj (Exists x psi) phi) = Exists x (Conj psi phi)
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pnfStep (Conj (All x psi) phi) = All x (Conj psi phi)
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pnfStep (Disj (Exists x psi) phi) = Exists x (Disj psi phi)
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pnfStep (Disj (All x psi) phi) = All x (Disj psi phi)
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-}
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-- descent rules
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pnfStep (All x f) = All x (pnfStep f)
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pnfStep (Exists x f) = Exists x (pnfStep f)
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pnfStep (Conj f1 f2) = Conj (pnfStep f1) (pnfStep f2)
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pnfStep (Disj f1 f2) = Disj (pnfStep f1) (pnfStep f2)
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pnfStep (Impl f1 f2) = Impl (pnfStep f1) (pnfStep f2) -- can't happen, gets removed in makeNNF
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pnfStep (Neg f) = Neg (pnfStep f)
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pnfStep f = f
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-- infinite function names for skolemizing
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skolemFuns :: [String]
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skolemFuns = ["sk" ++ show n | n <- [(0 :: Int)..]]
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-- returns all function names in a formula
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usedFunctions :: Formula -> [String]
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usedFunctions (Pred _ ts) = concatMap termFunctions ts
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where
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termFunctions (Var _) = []
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termFunctions (Fun f ts') = f : concatMap termFunctions ts'
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usedFunctions (Neg f) = usedFunctions f
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usedFunctions (Conj f1 f2) = usedFunctions f1 ++ usedFunctions f2
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usedFunctions (Disj f1 f2) = usedFunctions f1 ++ usedFunctions f2
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usedFunctions (Impl f1 f2) = usedFunctions f1 ++ usedFunctions f2
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usedFunctions (All _ f) = usedFunctions f
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usedFunctions (Exists _ f) = usedFunctions f
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usedFunctions _ = []
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-- skolem form of a formula
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makeSkolem :: Formula -> Formula
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makeSkolem form = go (makePNF form) [] []
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where
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substTermInFormula :: Formula -> String -> Term -> Formula
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substTermInFormula (Pred p ts) x s = Pred p $ map (\t -> termSubst t x s) ts
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substTermInFormula (Neg f) x s = Neg $ substTermInFormula f x s
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substTermInFormula (Conj f1 f2) x s = Conj (substTermInFormula f1 x s) (substTermInFormula f2 x s)
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substTermInFormula (Disj f1 f2) x s = Disj (substTermInFormula f1 x s) (substTermInFormula f2 x s)
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substTermInFormula (Impl f1 f2) x s = Impl (substTermInFormula f1 x s) (substTermInFormula f2 x s)
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substTermInFormula f@(All y f') x s = if x == y then f else All y (substTermInFormula f' x s)
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substTermInFormula f@(Exists y f') x s = if x == y then f else Exists y (substTermInFormula f' x s)
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substTermInFormula f _ _ = f
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go (All x f) vs skolems = All x (go f (vs ++ [Var x]) skolems)
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go (Exists x f) vs skolems = go (substTermInFormula f x (Fun newSkolem vs)) vs (newSkolem : skolems)
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where newSkolem = fromJust $ find (`notElem` skolems) skolemFuns
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go f _ _ = f
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-- conjunctive normalform, removes all quantors, so its ready for resolution
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makeCNF :: Formula -> Formula
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makeCNF form = go $ makeSkolem form
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where
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go f = let f' = cnfStep f in
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if f == f' then f else go f'
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cnfStep (Conj phi psi) = Conj (makeCNF phi) (makeCNF psi)
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cnfStep (Disj (Conj phi psi) xi) = Conj (makeCNF $ Disj phi xi) (makeCNF $ Disj psi xi)
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cnfStep (Disj xi (Conj phi psi)) = Conj (makeCNF $ Disj xi phi) (makeCNF $ Disj xi psi)
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cnfStep (All _ f) = makeCNF f
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cnfStep (Exists _ f) = makeCNF f
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cnfStep f = f
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-- create the list of clauses from a formula
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-- TODO not working correctly for `makeCNFList formula5`
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makeCNFList :: Formula -> [[Formula]]
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makeCNFList form = go $ makeCNF form
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where
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go (Conj f1 f2) = go f1 ++ go f2
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go (Disj f1 f2) = [collectDisjs f1 ++ collectDisjs f2]
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where
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collectDisjs (Disj f1' f2') = collectDisjs f1' ++ collectDisjs f2'
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collectDisjs f' = [f']
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go f = [[f]]
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-- unifies predicates, e.g. P(x,y) == P(f(a), z)
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unifyPredicates :: (Formula, [Formula]) -> (Formula, [Formula]) -> Maybe (([Formula], [Formula]), [(Term, Term)])
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unifyPredicates (e1@(Pred p1 ts1), c1) (e2@(Neg (Pred p2 ts2)), c2) | p1 == p2 && length p1 == length p2 = do
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mgu <- unify $ zip ts1 ts2
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return ((c1', c2'), mgu)
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where
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c1' = filter (/= e1) c1
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c2' = filter (/= e2) c2
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unifyPredicates _ _ = Nothing
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-- applies an mgu to a given formula, asserts that the formula contains no quantifiers
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applyMgu :: Formula -> [(Term, Term)] -> Formula
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applyMgu (Pred p ts) mgu = Pred p $ map (`applyMguTerm` mgu) ts
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applyMgu (Neg f) mgu = Neg $ applyMgu f mgu
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applyMgu (Conj f1 f2) mgu = Conj (applyMgu f1 mgu) (applyMgu f2 mgu)
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applyMgu (Disj f1 f2) mgu = Disj (applyMgu f1 mgu) (applyMgu f2 mgu)
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applyMgu (Impl f1 f2) mgu = Impl (applyMgu f1 mgu) (applyMgu f2 mgu)
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applyMgu f _ = f
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-- applies mgu to term
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applyMguTerm :: Term -> [(Term, Term)] -> Term
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applyMguTerm (Var x) [] = Var x
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applyMguTerm (Var x) ((Var y, t) : rest) = if x == y then t else applyMguTerm (Var x) rest
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applyMguTerm (Fun f ts) mgu = Fun f $ map (`applyMguTerm` mgu) ts
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applyMguTerm t _ = t
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-- a single resolution step as described in gloin
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rifStep :: [[Formula]] -> Either () [[Formula]]
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rifStep clauses | trace (show clauses) True = if [] `elem` clauses then Left () else Right newClauses
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where
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resolveClauses :: [Formula] -> [Formula] -> [(([Formula], [Formula]), [(Term, Term)])]
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resolveClauses c1 c2 = let zippedElems = [((e1, c1), (e2, c2)) | e1 <- c1, e2 <- c2] in mapMaybe (uncurry unifyPredicates) zippedElems
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zippedClauses = [(c1, c2) | c1 <- clauses, c2 <- clauses, c1 /= c2]
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clausesWithMgus = concatMap (uncurry resolveClauses) zippedClauses
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newClauses = clauses ++ map (\((f1, f2), mgu) -> map (`applyMgu` mgu) f1 ++ map (`applyMgu` mgu) f2) clausesWithMgus
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rifStep _ = undefined
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-- do resolution until we have proven unfulfillability of formula set
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doResolution :: [[Formula]] -> Either () [[Formula]]
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doResolution f= do
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f' <- rifStep f
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-- TODO after every resolution step make variable names of clauses disjunct
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doResolution f'
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{-
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To prove a formula we:
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1. Construct `Neg phi`
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2. Transform `Neg phi` to a clause list
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3. doResolution on clause list
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-}
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proveFormula :: Formula -> Either () [[Formula]]
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proveFormula form = doResolution $ makeCNFList (Neg form)
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-- unification examples
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terma1 :: Term
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terma1 = Fun "f" [Var "x", Fun "g" [Var "y"]]
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termb1 :: Term
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termb1 = Fun "f" [Fun "g" [Var "z"], Var "z"]
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terma2 :: Term
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terma2 = Fun "f" [Var "x", Fun "g" [Var "x"], Fun "h" [Var "y"]]
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termb2 :: Term
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termb2 = Fun "f" [Fun "k" [Var "y"], Fun "g" [Var "z"], Var "z"]
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terma3 :: Term
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terma3 = Fun "f" [Var "x", Fun "g" [Var "x"]]
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termb3 :: Term
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termb3 = Fun "f" [Var "z", Var "z"]
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-- NNF example from gloin
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formula1 :: Formula
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formula1 = Neg (Conj (Disj (Pred "A" []) (Neg $ Pred "B" [])) (Pred "C" []))
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-- PNF and skolem example from gloin
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formula2 :: Formula
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formula2 = All "x" $ Impl (All "y" $ Pred "L" [Var "y", Var "x"]) (Exists "y" $ Pred "M" [Var "x", Var "y"])
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-- Resolution example from gloin script
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formula3 :: Formula
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formula3 = Impl (Conj (Pred "P" [Fun "a" []]) (All "x" $ Impl (Pred "P" [Var "x"]) (Pred "P" [Fun "f" [Var "x"]]))) (Exists "x" $ Pred "P" [Fun "f" [Fun "f" [Var "x"]]])
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-- Resolution example from gloin exercises
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formula4 :: Formula
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formula4 = Conj (Disj (Pred "P" [Fun "f" [Var "x"], Var "y"]) (Disj (Pred "S" [Var "y", Var "z"]) (Pred "P" [Var "y"]))) (Conj (Neg $ Pred "S" [Fun "f" [Fun "f" [Var "x"]], Var "x"]) (Neg $ Pred "P" [Fun "f" [Var "z"]]))
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-- now a big example, sheet 11, exercise 6: Drogenschmuggel, this doesn't work yet but I'm sure its just the exercise thats wrong...
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phi1 :: Formula
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phi1 = All "x" $ Impl (Conj (Pred "E" [Var "x"]) (Neg $ Pred "I" [Var "x"])) (Exists "y" $ Conj (Pred "Z" [Var "y"]) (Pred "S" [Var "y", Var "x"]))
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phi2 :: Formula
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phi2 = Exists "x" $ Conj (Conj (Pred "D" [Var "x"]) (Pred "E" [Var "x"])) (All "y" $ Impl (Pred "S" [Var "y", Var "x"]) (Pred "D" [Var "y"]))
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phi3 :: Formula
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phi3 = All "x" $ Impl (Pred "I" [Var "x"]) (Neg $ Pred "D" [Var "x"])
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psi' :: Formula
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psi' = Exists "x" $ Conj (Pred "Z" [Var "x"]) (Pred "D" [Var "x"])
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formula5 :: Formula
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formula5 = Impl (Conj (Conj phi1 phi2) phi3) psi'
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-- exercise 2: Ärzte und Quacksalber
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formula6 :: [[Formula]]
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formula6 = [
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[Neg $ Pred "D" [Var "x1"], Pred "L" [Fun "f" [Var "x1"], Var "x1"]],
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[Pred "P" [Fun "f" [Var "x2"]], Neg $ Pred "L" [Fun "f" [Var "x2"], Var "x2"]],
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[Neg $ Pred "P" [Var "x3"], Neg $ Pred "Q" [Var "y3"], Neg $ Pred "L" [Var "x3", Var "y3"]],
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[Pred "D" [Fun "a" []]],
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[Pred "Q" [Fun "a" []]]]
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-- Drogenschmuggel but already as clauses
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formula7 :: [[Formula]]
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formula7 = [
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[Neg $ Pred "E" [Var "x2"], Pred "I" [Var "x2"], Pred "Z" [Fun "f" [Var "x2"]]],
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[Neg $ Pred "E" [Var "x3"], Pred "I" [Var "x3"], Pred "S" [Var "x3", Fun "f" [Var "x3"]]],
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[Pred "D" [Fun "c" []]],
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[Pred "E" [Fun "c" []]],
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[Neg $ Pred "S" [Fun "c" [], Var "y"], Pred "D" [Var "y"]],
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[Neg $ Pred "I" [Var "x4"], Neg $ Pred "D" [Var "x4"]],
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[Neg $ Pred "Z" [Var "x5"], Neg $ Pred "D" [Var "x5"]]
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]
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main :: IO ()
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main = do
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putStrLn $ "Now making NNF of formula: " ++ show formula1
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print $ makeNNF formula1
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putStrLn $ "Now making PNF of formula: " ++ show formula2
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print $ makePNF formula2
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putStrLn $ "Now making Skolemform of formula: " ++ show formula2
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print $ makeSkolem formula2
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putStrLn $ "Now proving formula by resolution: " ++ show formula3
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case proveFormula formula3 of
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Left _ -> putStrLn "Success!"
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Right _ -> return ()
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putStrLn $ "Now Proving formula by resolution: " ++ show formula4
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case doResolution $ makeCNFList formula4 of
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Left _ -> putStrLn "Success!"
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Right _ -> return ()
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putStrLn $ "Now Proving formula by resolution: " ++ show formula6
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case doResolution formula6 of
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Left _ -> putStrLn "Success!"
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Right _ -> return () |