wenzelm@9869
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(* Title: HOL/Tools/meson.ML
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paulson@9840
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ID: $Id$
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paulson@9840
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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paulson@9840
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Copyright 1992 University of Cambridge
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paulson@9840
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wenzelm@9869
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The MESON resolution proof procedure for HOL.
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paulson@9840
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paulson@9840
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When making clauses, avoids using the rewriter -- instead uses RS recursively
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paulson@9840
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paulson@9840
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NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR
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paulson@9840
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FUNCTION nodups -- if done to goal clauses too!
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*)
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paulson@15579
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signature BASIC_MESON =
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paulson@15579
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sig
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val size_of_subgoals : thm -> int
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paulson@15998
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val make_cnf : thm list -> thm -> thm list
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paulson@20417
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val finish_cnf : thm list -> thm list
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paulson@15579
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val make_nnf : thm -> thm
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paulson@17849
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val make_nnf1 : thm -> thm
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paulson@15579
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val skolemize : thm -> thm
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paulson@15579
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val make_clauses : thm list -> thm list
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paulson@15579
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val make_horns : thm list -> thm list
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paulson@15579
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val best_prolog_tac : (thm -> int) -> thm list -> tactic
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paulson@15579
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val depth_prolog_tac : thm list -> tactic
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paulson@15579
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val gocls : thm list -> thm list
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val skolemize_prems_tac : thm list -> int -> tactic
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val MESON : (thm list -> tactic) -> int -> tactic
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val best_meson_tac : (thm -> int) -> int -> tactic
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val safe_best_meson_tac : int -> tactic
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val depth_meson_tac : int -> tactic
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paulson@15579
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val prolog_step_tac' : thm list -> int -> tactic
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val iter_deepen_prolog_tac : thm list -> tactic
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paulson@16563
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val iter_deepen_meson_tac : thm list -> int -> tactic
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paulson@15579
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val meson_tac : int -> tactic
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paulson@15579
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val negate_head : thm -> thm
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val select_literal : int -> thm -> thm
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val skolemize_tac : int -> tactic
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val make_clauses_tac : int -> tactic
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end
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paulson@15579
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structure Meson =
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struct
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paulson@9840
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paulson@15579
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val not_conjD = thm "meson_not_conjD";
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paulson@15579
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val not_disjD = thm "meson_not_disjD";
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paulson@15579
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val not_notD = thm "meson_not_notD";
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paulson@15579
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val not_allD = thm "meson_not_allD";
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paulson@15579
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val not_exD = thm "meson_not_exD";
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paulson@15579
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val imp_to_disjD = thm "meson_imp_to_disjD";
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paulson@15579
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val not_impD = thm "meson_not_impD";
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paulson@15579
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val iff_to_disjD = thm "meson_iff_to_disjD";
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val not_iffD = thm "meson_not_iffD";
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paulson@15579
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val conj_exD1 = thm "meson_conj_exD1";
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paulson@15579
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val conj_exD2 = thm "meson_conj_exD2";
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paulson@15579
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val disj_exD = thm "meson_disj_exD";
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paulson@15579
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val disj_exD1 = thm "meson_disj_exD1";
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paulson@15579
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val disj_exD2 = thm "meson_disj_exD2";
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paulson@15579
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val disj_assoc = thm "meson_disj_assoc";
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paulson@15579
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val disj_comm = thm "meson_disj_comm";
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paulson@15579
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val disj_FalseD1 = thm "meson_disj_FalseD1";
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paulson@15579
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val disj_FalseD2 = thm "meson_disj_FalseD2";
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paulson@9840
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paulson@16563
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val depth_limit = ref 2000;
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paulson@9840
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paulson@15579
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(**** Operators for forward proof ****)
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paulson@9840
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paulson@20417
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paulson@20417
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(** First-order Resolution **)
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paulson@20417
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paulson@20417
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fun typ_pair_of (ix, (sort,ty)) = (TVar (ix,sort), ty);
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paulson@20417
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fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);
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paulson@20417
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paulson@20417
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val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
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paulson@20417
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paulson@20417
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(*FIXME: currently does not "rename variables apart"*)
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paulson@20417
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fun first_order_resolve thA thB =
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paulson@20417
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let val thy = theory_of_thm thA
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paulson@20417
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val tmA = concl_of thA
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paulson@20417
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fun match pat = Pattern.first_order_match thy (pat,tmA) (tyenv0,tenv0)
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paulson@20417
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val Const("==>",_) $ tmB $ _ = prop_of thB
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paulson@20417
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val (tyenv,tenv) = match tmB
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paulson@20417
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val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
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paulson@20417
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in thA RS (cterm_instantiate ct_pairs thB) end
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paulson@20417
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handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
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paulson@18175
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paulson@15579
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(*raises exception if no rules apply -- unlike RL*)
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paulson@18141
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fun tryres (th, rls) =
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paulson@18141
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let fun tryall [] = raise THM("tryres", 0, th::rls)
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paulson@20417
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| tryall (rl::rls) = (th RS rl handle THM _ => tryall rls)
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paulson@18141
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in tryall rls end;
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paulson@18141
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paulson@21050
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(*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
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paulson@21050
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e.g. from conj_forward, should have the form
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"[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
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and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
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paulson@15579
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fun forward_res nf st =
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paulson@21050
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let fun forward_tacf [prem] = rtac (nf prem) 1
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| forward_tacf prems =
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error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
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string_of_thm st ^
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"\nPremises:\n" ^
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paulson@21050
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cat_lines (map string_of_thm prems))
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paulson@21050
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in
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case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
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of SOME(th,_) => th
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paulson@21050
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| NONE => raise THM("forward_res", 0, [st])
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paulson@21050
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end;
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paulson@9840
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paulson@20134
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(*Are any of the logical connectives in "bs" present in the term?*)
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paulson@20134
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fun has_conns bs =
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paulson@20134
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let fun has (Const(a,_)) = false
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paulson@20134
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| has (Const("Trueprop",_) $ p) = has p
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paulson@20134
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| has (Const("Not",_) $ p) = has p
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paulson@20134
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| has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
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paulson@20134
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| has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
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paulson@20134
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| has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
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paulson@20134
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| has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
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paulson@15579
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| has _ = false
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in has end;
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paulson@17716
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paulson@9840
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paulson@15579
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(**** Clause handling ****)
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paulson@9840
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paulson@15579
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fun literals (Const("Trueprop",_) $ P) = literals P
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paulson@15579
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| literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
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paulson@15579
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| literals (Const("Not",_) $ P) = [(false,P)]
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paulson@15579
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| literals P = [(true,P)];
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paulson@9840
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paulson@15579
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(*number of literals in a term*)
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paulson@15579
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val nliterals = length o literals;
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paulson@9840
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paulson@18389
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paulson@18389
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(*** Tautology Checking ***)
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paulson@18389
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paulson@18389
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fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
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paulson@18389
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signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
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paulson@18389
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| signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
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paulson@18389
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| signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
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paulson@18389
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paulson@18389
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fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
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paulson@18389
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paulson@18389
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(*Literals like X=X are tautologous*)
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paulson@18389
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fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
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paulson@18389
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| taut_poslit (Const("True",_)) = true
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paulson@18389
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| taut_poslit _ = false;
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paulson@18389
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paulson@18389
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fun is_taut th =
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paulson@18389
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let val (poslits,neglits) = signed_lits th
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paulson@18389
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in exists taut_poslit poslits
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paulson@18389
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orelse
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wenzelm@20073
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exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
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paulson@19894
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end
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paulson@19894
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handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
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paulson@18389
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paulson@18389
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paulson@18389
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(*** To remove trivial negated equality literals from clauses ***)
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paulson@18389
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paulson@18389
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(*They are typically functional reflexivity axioms and are the converses of
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paulson@18389
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injectivity equivalences*)
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paulson@18389
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paulson@18389
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val not_refl_disj_D = thm"meson_not_refl_disj_D";
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paulson@18389
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paulson@20119
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(*Is either term a Var that does not properly occur in the other term?*)
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paulson@20119
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fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
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paulson@20119
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| eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
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paulson@20119
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| eliminable _ = false;
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paulson@20119
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paulson@18389
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fun refl_clause_aux 0 th = th
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paulson@18389
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| refl_clause_aux n th =
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paulson@18389
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case HOLogic.dest_Trueprop (concl_of th) of
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paulson@18389
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(Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
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paulson@18389
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refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
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paulson@18389
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| (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
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paulson@20119
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if eliminable(t,u)
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paulson@20119
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then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
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paulson@18389
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else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
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paulson@18389
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| (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
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paulson@18752
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| _ => (*not a disjunction*) th;
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paulson@18389
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paulson@18389
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fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
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paulson@18389
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notequal_lits_count P + notequal_lits_count Q
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paulson@18389
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| notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
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paulson@18389
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| notequal_lits_count _ = 0;
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paulson@18389
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paulson@18389
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(*Simplify a clause by applying reflexivity to its negated equality literals*)
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paulson@18389
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fun refl_clause th =
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paulson@18389
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let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
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paulson@19894
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in zero_var_indexes (refl_clause_aux neqs th) end
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paulson@19894
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handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
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paulson@18389
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paulson@18389
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paulson@18389
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(*** The basic CNF transformation ***)
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paulson@18389
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paulson@21069
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val max_clauses = ref 20;
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paulson@21069
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paulson@21069
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fun sum x y = if x < !max_clauses andalso y < !max_clauses then x+y else !max_clauses;
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paulson@21069
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fun prod x y = if x < !max_clauses andalso y < !max_clauses then x*y else !max_clauses;
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paulson@21069
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paulson@19894
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(*Estimate the number of clauses in order to detect infeasible theorems*)
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paulson@21069
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fun signed_nclauses b (Const("Trueprop",_) $ t) = signed_nclauses b t
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paulson@21069
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| signed_nclauses b (Const("Not",_) $ t) = signed_nclauses (not b) t
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paulson@21069
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| signed_nclauses b (Const("op &",_) $ t $ u) =
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paulson@21069
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if b then sum (signed_nclauses b t) (signed_nclauses b u)
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paulson@21069
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else prod (signed_nclauses b t) (signed_nclauses b u)
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paulson@21069
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| signed_nclauses b (Const("op |",_) $ t $ u) =
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paulson@21069
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if b then prod (signed_nclauses b t) (signed_nclauses b u)
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paulson@21069
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else sum (signed_nclauses b t) (signed_nclauses b u)
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paulson@21069
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| signed_nclauses b (Const("op -->",_) $ t $ u) =
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paulson@21069
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if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)
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paulson@21069
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else sum (signed_nclauses (not b) t) (signed_nclauses b u)
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paulson@21069
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| signed_nclauses b (Const("op =",_) $ t $ u) =
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paulson@21069
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if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))
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paulson@21069
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(prod (signed_nclauses (not b) u) (signed_nclauses b t))
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paulson@21069
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else sum (prod (signed_nclauses b t) (signed_nclauses b u))
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paulson@21069
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(prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))
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paulson@21069
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| signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
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paulson@21069
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| signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
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paulson@21069
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| signed_nclauses _ _ = 1; (* literal *)
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paulson@21069
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221 |
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paulson@21069
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222 |
val nclauses = signed_nclauses true;
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paulson@21069
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223 |
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paulson@21069
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fun too_many_clauses t = nclauses t >= !max_clauses;
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paulson@19894
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225 |
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paulson@15579
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226 |
(*Replaces universally quantified variables by FREE variables -- because
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paulson@15579
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227 |
assumptions may not contain scheme variables. Later, call "generalize". *)
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paulson@15579
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fun freeze_spec th =
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paulson@20361
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229 |
let val newname = gensym "mes_"
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paulson@19154
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230 |
val spec' = read_instantiate [("x", newname)] spec
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paulson@19154
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in th RS spec' end;
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paulson@9840
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paulson@15998
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233 |
(*Used with METAHYPS below. There is one assumption, which gets bound to prem
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paulson@15998
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234 |
and then normalized via function nf. The normal form is given to resolve_tac,
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paulson@15998
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235 |
presumably to instantiate a Boolean variable.*)
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paulson@15579
|
236 |
fun resop nf [prem] = resolve_tac (nf prem) 1;
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paulson@9840
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237 |
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paulson@20822
|
238 |
(*Any need to extend this list with
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haftmann@21046
|
239 |
"HOL.type_class","Code_Generator.eq_class","ProtoPure.term"?*)
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paulson@18389
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240 |
val has_meta_conn =
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paulson@18389
|
241 |
exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
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paulson@20417
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242 |
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paulson@20417
|
243 |
fun apply_skolem_ths (th, rls) =
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paulson@20417
|
244 |
let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
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paulson@20417
|
245 |
| tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
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paulson@20417
|
246 |
in tryall rls end;
|
paulson@18389
|
247 |
|
paulson@15998
|
248 |
(*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
|
paulson@15998
|
249 |
Strips universal quantifiers and breaks up conjunctions.
|
paulson@15998
|
250 |
Eliminates existential quantifiers using skoths: Skolemization theorems.*)
|
paulson@15998
|
251 |
fun cnf skoths (th,ths) =
|
paulson@18389
|
252 |
let fun cnf_aux (th,ths) =
|
paulson@20822
|
253 |
if not (HOLogic.is_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
|
paulson@20134
|
254 |
else if not (has_conns ["All","Ex","op &"] (prop_of th))
|
paulson@15998
|
255 |
then th::ths (*no work to do, terminate*)
|
paulson@16588
|
256 |
else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
|
paulson@16588
|
257 |
Const ("op &", _) => (*conjunction*)
|
paulson@20417
|
258 |
cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
|
paulson@16588
|
259 |
| Const ("All", _) => (*universal quantifier*)
|
paulson@18389
|
260 |
cnf_aux (freeze_spec th, ths)
|
paulson@16588
|
261 |
| Const ("Ex", _) =>
|
paulson@16588
|
262 |
(*existential quantifier: Insert Skolem functions*)
|
paulson@20417
|
263 |
cnf_aux (apply_skolem_ths (th,skoths), ths)
|
paulson@16588
|
264 |
| Const ("op |", _) => (*disjunction*)
|
paulson@16588
|
265 |
let val tac =
|
paulson@18389
|
266 |
(METAHYPS (resop cnf_nil) 1) THEN
|
paulson@19154
|
267 |
(fn st' => st' |> METAHYPS (resop cnf_nil) 1)
|
paulson@16588
|
268 |
in Seq.list_of (tac (th RS disj_forward)) @ ths end
|
paulson@16588
|
269 |
| _ => (*no work to do*) th::ths
|
paulson@19154
|
270 |
and cnf_nil th = cnf_aux (th,[])
|
paulson@15998
|
271 |
in
|
paulson@21069
|
272 |
if too_many_clauses (concl_of th)
|
paulson@19894
|
273 |
then (Output.debug ("cnf is ignoring: " ^ string_of_thm th); ths)
|
paulson@19894
|
274 |
else cnf_aux (th,ths)
|
paulson@15998
|
275 |
end;
|
paulson@9840
|
276 |
|
paulson@16012
|
277 |
(*Convert all suitable free variables to schematic variables,
|
paulson@16012
|
278 |
but don't discharge assumptions.*)
|
paulson@16173
|
279 |
fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
|
paulson@16012
|
280 |
|
paulson@20417
|
281 |
fun make_cnf skoths th = cnf skoths (th, []);
|
paulson@20417
|
282 |
|
paulson@20417
|
283 |
(*Generalization, removal of redundant equalities, removal of tautologies.*)
|
paulson@20417
|
284 |
fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
|
paulson@15998
|
285 |
|
paulson@9840
|
286 |
|
paulson@15579
|
287 |
(**** Removal of duplicate literals ****)
|
paulson@9840
|
288 |
|
paulson@15579
|
289 |
(*Forward proof, passing extra assumptions as theorems to the tactic*)
|
paulson@15579
|
290 |
fun forward_res2 nf hyps st =
|
paulson@15579
|
291 |
case Seq.pull
|
paulson@15579
|
292 |
(REPEAT
|
paulson@15579
|
293 |
(METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
|
paulson@15579
|
294 |
st)
|
paulson@15579
|
295 |
of SOME(th,_) => th
|
paulson@15579
|
296 |
| NONE => raise THM("forward_res2", 0, [st]);
|
paulson@9840
|
297 |
|
paulson@15579
|
298 |
(*Remove duplicates in P|Q by assuming ~P in Q
|
paulson@15579
|
299 |
rls (initially []) accumulates assumptions of the form P==>False*)
|
paulson@15579
|
300 |
fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
|
paulson@15579
|
301 |
handle THM _ => tryres(th,rls)
|
paulson@15579
|
302 |
handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
|
paulson@15579
|
303 |
[disj_FalseD1, disj_FalseD2, asm_rl])
|
paulson@15579
|
304 |
handle THM _ => th;
|
paulson@9840
|
305 |
|
paulson@15579
|
306 |
(*Remove duplicate literals, if there are any*)
|
paulson@15579
|
307 |
fun nodups th =
|
haftmann@20972
|
308 |
if has_duplicates (op =) (literals (prop_of th))
|
haftmann@20972
|
309 |
then nodups_aux [] th
|
haftmann@20972
|
310 |
else th;
|
paulson@9840
|
311 |
|
paulson@9840
|
312 |
|
paulson@15579
|
313 |
(**** Generation of contrapositives ****)
|
paulson@9840
|
314 |
|
paulson@15579
|
315 |
(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
|
paulson@15579
|
316 |
fun assoc_right th = assoc_right (th RS disj_assoc)
|
paulson@15579
|
317 |
handle THM _ => th;
|
paulson@9840
|
318 |
|
paulson@15579
|
319 |
(*Must check for negative literal first!*)
|
paulson@15579
|
320 |
val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
|
paulson@9840
|
321 |
|
paulson@15579
|
322 |
(*For ordinary resolution. *)
|
paulson@15579
|
323 |
val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
|
paulson@9840
|
324 |
|
paulson@15579
|
325 |
(*Create a goal or support clause, conclusing False*)
|
paulson@15579
|
326 |
fun make_goal th = (*Must check for negative literal first!*)
|
paulson@15579
|
327 |
make_goal (tryres(th, clause_rules))
|
paulson@15579
|
328 |
handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
|
paulson@9840
|
329 |
|
paulson@15579
|
330 |
(*Sort clauses by number of literals*)
|
paulson@15579
|
331 |
fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
|
paulson@9840
|
332 |
|
paulson@18389
|
333 |
fun sort_clauses ths = sort (make_ord fewerlits) ths;
|
paulson@9840
|
334 |
|
paulson@15581
|
335 |
(*True if the given type contains bool anywhere*)
|
paulson@15581
|
336 |
fun has_bool (Type("bool",_)) = true
|
paulson@15581
|
337 |
| has_bool (Type(_, Ts)) = exists has_bool Ts
|
paulson@15581
|
338 |
| has_bool _ = false;
|
paulson@15581
|
339 |
|
paulson@20524
|
340 |
(*Is the string the name of a connective? Really only | and Not can remain,
|
paulson@20524
|
341 |
since this code expects to be called on a clause form.*)
|
wenzelm@19875
|
342 |
val is_conn = member (op =)
|
paulson@20524
|
343 |
["Trueprop", "op &", "op |", "op -->", "Not",
|
paulson@15613
|
344 |
"All", "Ex", "Ball", "Bex"];
|
paulson@15613
|
345 |
|
paulson@20524
|
346 |
(*True if the term contains a function--not a logical connective--where the type
|
paulson@20524
|
347 |
of any argument contains bool.*)
|
paulson@15613
|
348 |
val has_bool_arg_const =
|
paulson@15613
|
349 |
exists_Const
|
paulson@15613
|
350 |
(fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
|
paulson@15908
|
351 |
|
paulson@16588
|
352 |
(*Raises an exception if any Vars in the theorem mention type bool; they
|
paulson@16588
|
353 |
could cause make_horn to loop! Also rejects functions whose arguments are
|
paulson@16588
|
354 |
Booleans or other functions.*)
|
paulson@19204
|
355 |
fun is_fol_term t =
|
paulson@19204
|
356 |
not (exists (has_bool o fastype_of) (term_vars t) orelse
|
paulson@19204
|
357 |
not (Term.is_first_order ["all","All","Ex"] t) orelse
|
paulson@19204
|
358 |
has_bool_arg_const t orelse
|
paulson@19204
|
359 |
has_meta_conn t);
|
paulson@19204
|
360 |
|
paulson@15579
|
361 |
(*Create a meta-level Horn clause*)
|
paulson@15579
|
362 |
fun make_horn crules th = make_horn crules (tryres(th,crules))
|
paulson@15579
|
363 |
handle THM _ => th;
|
paulson@9840
|
364 |
|
paulson@16563
|
365 |
(*Generate Horn clauses for all contrapositives of a clause. The input, th,
|
paulson@16563
|
366 |
is a HOL disjunction.*)
|
paulson@15579
|
367 |
fun add_contras crules (th,hcs) =
|
paulson@15579
|
368 |
let fun rots (0,th) = hcs
|
paulson@15579
|
369 |
| rots (k,th) = zero_var_indexes (make_horn crules th) ::
|
paulson@15579
|
370 |
rots(k-1, assoc_right (th RS disj_comm))
|
paulson@15862
|
371 |
in case nliterals(prop_of th) of
|
paulson@15579
|
372 |
1 => th::hcs
|
paulson@15579
|
373 |
| n => rots(n, assoc_right th)
|
paulson@15579
|
374 |
end;
|
paulson@9840
|
375 |
|
paulson@15579
|
376 |
(*Use "theorem naming" to label the clauses*)
|
paulson@15579
|
377 |
fun name_thms label =
|
paulson@15579
|
378 |
let fun name1 (th, (k,ths)) =
|
paulson@15579
|
379 |
(k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
|
paulson@9840
|
380 |
|
paulson@15579
|
381 |
in fn ths => #2 (foldr name1 (length ths, []) ths) end;
|
paulson@9840
|
382 |
|
paulson@16563
|
383 |
(*Is the given disjunction an all-negative support clause?*)
|
paulson@15579
|
384 |
fun is_negative th = forall (not o #1) (literals (prop_of th));
|
paulson@9840
|
385 |
|
paulson@15579
|
386 |
val neg_clauses = List.filter is_negative;
|
paulson@9840
|
387 |
|
paulson@9840
|
388 |
|
paulson@15579
|
389 |
(***** MESON PROOF PROCEDURE *****)
|
paulson@9840
|
390 |
|
paulson@15579
|
391 |
fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
|
paulson@15579
|
392 |
As) = rhyps(phi, A::As)
|
paulson@15579
|
393 |
| rhyps (_, As) = As;
|
paulson@9840
|
394 |
|
paulson@15579
|
395 |
(** Detecting repeated assumptions in a subgoal **)
|
paulson@9840
|
396 |
|
paulson@15579
|
397 |
(*The stringtree detects repeated assumptions.*)
|
wenzelm@16801
|
398 |
fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
|
paulson@9840
|
399 |
|
paulson@15579
|
400 |
(*detects repetitions in a list of terms*)
|
paulson@15579
|
401 |
fun has_reps [] = false
|
paulson@15579
|
402 |
| has_reps [_] = false
|
paulson@15579
|
403 |
| has_reps [t,u] = (t aconv u)
|
paulson@15579
|
404 |
| has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
|
wenzelm@19875
|
405 |
handle Net.INSERT => true;
|
paulson@9840
|
406 |
|
paulson@15579
|
407 |
(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
|
paulson@18508
|
408 |
fun TRYING_eq_assume_tac 0 st = Seq.single st
|
paulson@18508
|
409 |
| TRYING_eq_assume_tac i st =
|
paulson@18508
|
410 |
TRYING_eq_assume_tac (i-1) (eq_assumption i st)
|
paulson@18508
|
411 |
handle THM _ => TRYING_eq_assume_tac (i-1) st;
|
paulson@18508
|
412 |
|
paulson@18508
|
413 |
fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
|
paulson@9840
|
414 |
|
paulson@15579
|
415 |
(*Loop checking: FAIL if trying to prove the same thing twice
|
paulson@15579
|
416 |
-- if *ANY* subgoal has repeated literals*)
|
paulson@15579
|
417 |
fun check_tac st =
|
paulson@15579
|
418 |
if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
|
paulson@15579
|
419 |
then Seq.empty else Seq.single st;
|
paulson@9840
|
420 |
|
paulson@9840
|
421 |
|
paulson@15579
|
422 |
(* net_resolve_tac actually made it slower... *)
|
paulson@15579
|
423 |
fun prolog_step_tac horns i =
|
paulson@15579
|
424 |
(assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
|
paulson@18508
|
425 |
TRYALL_eq_assume_tac;
|
paulson@15579
|
426 |
|
paulson@9840
|
427 |
(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
|
paulson@15579
|
428 |
fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
|
paulson@15579
|
429 |
|
paulson@15579
|
430 |
fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
|
paulson@15579
|
431 |
|
paulson@9840
|
432 |
|
paulson@9840
|
433 |
(*Negation Normal Form*)
|
paulson@9840
|
434 |
val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
|
wenzelm@9869
|
435 |
not_impD, not_iffD, not_allD, not_exD, not_notD];
|
paulson@15581
|
436 |
|
paulson@15581
|
437 |
fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
|
wenzelm@9869
|
438 |
handle THM _ =>
|
paulson@15581
|
439 |
forward_res make_nnf1
|
wenzelm@9869
|
440 |
(tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
|
paulson@9840
|
441 |
handle THM _ => th;
|
paulson@9840
|
442 |
|
paulson@20018
|
443 |
(*The simplification removes defined quantifiers and occurrences of True and False.
|
paulson@20018
|
444 |
nnf_ss also includes the one-point simprocs,
|
paulson@18405
|
445 |
which are needed to avoid the various one-point theorems from generating junk clauses.*)
|
paulson@19894
|
446 |
val nnf_simps =
|
paulson@20018
|
447 |
[simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
|
paulson@19894
|
448 |
if_False, if_cancel, if_eq_cancel, cases_simp];
|
paulson@19894
|
449 |
val nnf_extra_simps =
|
paulson@19894
|
450 |
thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
|
paulson@18405
|
451 |
|
paulson@18405
|
452 |
val nnf_ss =
|
paulson@19894
|
453 |
HOL_basic_ss addsimps nnf_extra_simps
|
paulson@19894
|
454 |
addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
|
paulson@15872
|
455 |
|
paulson@21050
|
456 |
fun make_nnf th = case prems_of th of
|
paulson@21050
|
457 |
[] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
|
paulson@21050
|
458 |
|> simplify nnf_ss (*But this doesn't simplify premises...*)
|
paulson@21050
|
459 |
|> make_nnf1
|
paulson@21050
|
460 |
| _ => raise THM ("make_nnf: premises in argument", 0, [th]);
|
paulson@15581
|
461 |
|
paulson@15965
|
462 |
(*Pull existential quantifiers to front. This accomplishes Skolemization for
|
paulson@15965
|
463 |
clauses that arise from a subgoal.*)
|
wenzelm@9869
|
464 |
fun skolemize th =
|
paulson@20134
|
465 |
if not (has_conns ["Ex"] (prop_of th)) then th
|
quigley@15773
|
466 |
else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
|
quigley@15679
|
467 |
disj_exD, disj_exD1, disj_exD2])))
|
wenzelm@9869
|
468 |
handle THM _ =>
|
wenzelm@9869
|
469 |
skolemize (forward_res skolemize
|
wenzelm@9869
|
470 |
(tryres (th, [conj_forward, disj_forward, all_forward])))
|
paulson@9840
|
471 |
handle THM _ => forward_res skolemize (th RS ex_forward);
|
paulson@9840
|
472 |
|
paulson@9840
|
473 |
|
paulson@9840
|
474 |
(*Make clauses from a list of theorems, previously Skolemized and put into nnf.
|
paulson@9840
|
475 |
The resulting clauses are HOL disjunctions.*)
|
wenzelm@9869
|
476 |
fun make_clauses ths =
|
paulson@15998
|
477 |
(sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
|
quigley@15773
|
478 |
|
paulson@16563
|
479 |
(*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
|
wenzelm@9869
|
480 |
fun make_horns ths =
|
paulson@9840
|
481 |
name_thms "Horn#"
|
wenzelm@19046
|
482 |
(distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
|
paulson@9840
|
483 |
|
paulson@9840
|
484 |
(*Could simply use nprems_of, which would count remaining subgoals -- no
|
paulson@9840
|
485 |
discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
|
paulson@9840
|
486 |
|
wenzelm@9869
|
487 |
fun best_prolog_tac sizef horns =
|
paulson@9840
|
488 |
BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
|
paulson@9840
|
489 |
|
wenzelm@9869
|
490 |
fun depth_prolog_tac horns =
|
paulson@9840
|
491 |
DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
|
paulson@9840
|
492 |
|
paulson@9840
|
493 |
(*Return all negative clauses, as possible goal clauses*)
|
paulson@9840
|
494 |
fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
|
paulson@9840
|
495 |
|
paulson@15008
|
496 |
fun skolemize_prems_tac prems =
|
paulson@9840
|
497 |
cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
|
paulson@9840
|
498 |
REPEAT o (etac exE);
|
paulson@9840
|
499 |
|
paulson@18141
|
500 |
(*Expand all definitions (presumably of Skolem functions) in a proof state.*)
|
paulson@18141
|
501 |
fun expand_defs_tac st =
|
paulson@18141
|
502 |
let val defs = filter (can dest_equals) (#hyps (crep_thm st))
|
wenzelm@20288
|
503 |
in PRIMITIVE (LocalDefs.def_export false defs) st end;
|
paulson@18141
|
504 |
|
paulson@16588
|
505 |
(*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
|
paulson@16588
|
506 |
fun MESON cltac i st =
|
paulson@16588
|
507 |
SELECT_GOAL
|
paulson@18141
|
508 |
(EVERY [rtac ccontr 1,
|
paulson@16588
|
509 |
METAHYPS (fn negs =>
|
paulson@16588
|
510 |
EVERY1 [skolemize_prems_tac negs,
|
paulson@18141
|
511 |
METAHYPS (cltac o make_clauses)]) 1,
|
paulson@18141
|
512 |
expand_defs_tac]) i st
|
paulson@20417
|
513 |
handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
|
paulson@9840
|
514 |
|
paulson@9840
|
515 |
(** Best-first search versions **)
|
paulson@9840
|
516 |
|
paulson@16563
|
517 |
(*ths is a list of additional clauses (HOL disjunctions) to use.*)
|
wenzelm@9869
|
518 |
fun best_meson_tac sizef =
|
wenzelm@9869
|
519 |
MESON (fn cls =>
|
paulson@9840
|
520 |
THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
|
paulson@9840
|
521 |
(has_fewer_prems 1, sizef)
|
paulson@9840
|
522 |
(prolog_step_tac (make_horns cls) 1));
|
paulson@9840
|
523 |
|
paulson@9840
|
524 |
(*First, breaks the goal into independent units*)
|
paulson@9840
|
525 |
val safe_best_meson_tac =
|
wenzelm@9869
|
526 |
SELECT_GOAL (TRY Safe_tac THEN
|
paulson@9840
|
527 |
TRYALL (best_meson_tac size_of_subgoals));
|
paulson@9840
|
528 |
|
paulson@9840
|
529 |
(** Depth-first search version **)
|
paulson@9840
|
530 |
|
paulson@9840
|
531 |
val depth_meson_tac =
|
wenzelm@9869
|
532 |
MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
|
paulson@9840
|
533 |
depth_prolog_tac (make_horns cls)]);
|
paulson@9840
|
534 |
|
paulson@9840
|
535 |
|
paulson@9840
|
536 |
(** Iterative deepening version **)
|
paulson@9840
|
537 |
|
paulson@9840
|
538 |
(*This version does only one inference per call;
|
paulson@9840
|
539 |
having only one eq_assume_tac speeds it up!*)
|
wenzelm@9869
|
540 |
fun prolog_step_tac' horns =
|
paulson@9840
|
541 |
let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
|
paulson@9840
|
542 |
take_prefix Thm.no_prems horns
|
paulson@9840
|
543 |
val nrtac = net_resolve_tac horns
|
paulson@9840
|
544 |
in fn i => eq_assume_tac i ORELSE
|
paulson@9840
|
545 |
match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
|
paulson@9840
|
546 |
((assume_tac i APPEND nrtac i) THEN check_tac)
|
paulson@9840
|
547 |
end;
|
paulson@9840
|
548 |
|
wenzelm@9869
|
549 |
fun iter_deepen_prolog_tac horns =
|
paulson@9840
|
550 |
ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
|
paulson@9840
|
551 |
|
paulson@16563
|
552 |
fun iter_deepen_meson_tac ths =
|
wenzelm@9869
|
553 |
MESON (fn cls =>
|
paulson@16563
|
554 |
case (gocls (cls@ths)) of
|
paulson@16563
|
555 |
[] => no_tac (*no goal clauses*)
|
paulson@16563
|
556 |
| goes =>
|
paulson@16563
|
557 |
(THEN_ITER_DEEPEN (resolve_tac goes 1)
|
paulson@16563
|
558 |
(has_fewer_prems 1)
|
paulson@16563
|
559 |
(prolog_step_tac' (make_horns (cls@ths)))));
|
paulson@9840
|
560 |
|
paulson@16563
|
561 |
fun meson_claset_tac ths cs =
|
paulson@16563
|
562 |
SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
|
wenzelm@9869
|
563 |
|
paulson@16563
|
564 |
val meson_tac = CLASET' (meson_claset_tac []);
|
wenzelm@9869
|
565 |
|
wenzelm@9869
|
566 |
|
paulson@14813
|
567 |
(**** Code to support ordinary resolution, rather than Model Elimination ****)
|
paulson@14744
|
568 |
|
paulson@15008
|
569 |
(*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
|
paulson@15008
|
570 |
with no contrapositives, for ordinary resolution.*)
|
paulson@14744
|
571 |
|
paulson@14744
|
572 |
(*Rules to convert the head literal into a negated assumption. If the head
|
paulson@14744
|
573 |
literal is already negated, then using notEfalse instead of notEfalse'
|
paulson@14744
|
574 |
prevents a double negation.*)
|
paulson@14744
|
575 |
val notEfalse = read_instantiate [("R","False")] notE;
|
paulson@14744
|
576 |
val notEfalse' = rotate_prems 1 notEfalse;
|
paulson@14744
|
577 |
|
paulson@15448
|
578 |
fun negated_asm_of_head th =
|
paulson@14744
|
579 |
th RS notEfalse handle THM _ => th RS notEfalse';
|
paulson@14744
|
580 |
|
paulson@14744
|
581 |
(*Converting one clause*)
|
paulson@15581
|
582 |
fun make_meta_clause th =
|
paulson@20417
|
583 |
if is_fol_term (prop_of th)
|
paulson@20417
|
584 |
then negated_asm_of_head (make_horn resolution_clause_rules th)
|
paulson@20417
|
585 |
else raise THM("make_meta_clause", 0, [th]);
|
paulson@14744
|
586 |
|
paulson@14744
|
587 |
fun make_meta_clauses ths =
|
paulson@14744
|
588 |
name_thms "MClause#"
|
wenzelm@19046
|
589 |
(distinct Drule.eq_thm_prop (map make_meta_clause ths));
|
paulson@14744
|
590 |
|
paulson@14744
|
591 |
(*Permute a rule's premises to move the i-th premise to the last position.*)
|
paulson@14744
|
592 |
fun make_last i th =
|
paulson@14744
|
593 |
let val n = nprems_of th
|
paulson@14744
|
594 |
in if 1 <= i andalso i <= n
|
paulson@14744
|
595 |
then Thm.permute_prems (i-1) 1 th
|
paulson@15118
|
596 |
else raise THM("select_literal", i, [th])
|
paulson@14744
|
597 |
end;
|
paulson@14744
|
598 |
|
paulson@14744
|
599 |
(*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
|
paulson@14744
|
600 |
double-negations.*)
|
paulson@14744
|
601 |
val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
|
paulson@14744
|
602 |
|
paulson@14744
|
603 |
(*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
|
paulson@14744
|
604 |
fun select_literal i cl = negate_head (make_last i cl);
|
paulson@14744
|
605 |
|
paulson@18508
|
606 |
|
paulson@14813
|
607 |
(*Top-level Skolemization. Allows part of the conversion to clauses to be
|
paulson@14813
|
608 |
expressed as a tactic (or Isar method). Each assumption of the selected
|
paulson@14813
|
609 |
goal is converted to NNF and then its existential quantifiers are pulled
|
paulson@14813
|
610 |
to the front. Finally, all existential quantifiers are eliminated,
|
paulson@14813
|
611 |
leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
|
paulson@14813
|
612 |
might generate many subgoals.*)
|
mengj@18194
|
613 |
|
paulson@19204
|
614 |
fun skolemize_tac i st =
|
paulson@19204
|
615 |
let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
|
paulson@19204
|
616 |
in
|
paulson@19204
|
617 |
EVERY' [METAHYPS
|
quigley@15773
|
618 |
(fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
|
paulson@14813
|
619 |
THEN REPEAT (etac exE 1))),
|
paulson@19204
|
620 |
REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
|
paulson@19204
|
621 |
end
|
paulson@19204
|
622 |
handle Subscript => Seq.empty;
|
mengj@18194
|
623 |
|
paulson@15118
|
624 |
(*Top-level conversion to meta-level clauses. Each clause has
|
paulson@15118
|
625 |
leading !!-bound universal variables, to express generality. To get
|
paulson@15118
|
626 |
disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
|
paulson@15008
|
627 |
val make_clauses_tac =
|
paulson@15008
|
628 |
SUBGOAL
|
paulson@15008
|
629 |
(fn (prop,_) =>
|
paulson@15008
|
630 |
let val ts = Logic.strip_assums_hyp prop
|
paulson@15008
|
631 |
in EVERY1
|
paulson@15008
|
632 |
[METAHYPS
|
paulson@15008
|
633 |
(fn hyps =>
|
paulson@15151
|
634 |
(Method.insert_tac
|
paulson@15118
|
635 |
(map forall_intr_vars
|
paulson@15118
|
636 |
(make_meta_clauses (make_clauses hyps))) 1)),
|
paulson@15008
|
637 |
REPEAT_DETERM_N (length ts) o (etac thin_rl)]
|
paulson@15008
|
638 |
end);
|
paulson@16563
|
639 |
|
paulson@16563
|
640 |
|
paulson@16563
|
641 |
(*** setup the special skoklemization methods ***)
|
paulson@15008
|
642 |
|
paulson@16563
|
643 |
(*No CHANGED_PROP here, since these always appear in the preamble*)
|
paulson@14744
|
644 |
|
paulson@16563
|
645 |
val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
|
wenzelm@9869
|
646 |
|
paulson@16563
|
647 |
val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
|
wenzelm@9869
|
648 |
|
paulson@16563
|
649 |
val skolemize_setup =
|
wenzelm@18708
|
650 |
Method.add_methods
|
wenzelm@18708
|
651 |
[("skolemize", Method.no_args skolemize_meth,
|
wenzelm@18708
|
652 |
"Skolemization into existential quantifiers"),
|
wenzelm@18708
|
653 |
("make_clauses", Method.no_args make_clauses_meth,
|
wenzelm@18708
|
654 |
"Conversion to !!-quantified meta-level clauses")];
|
paulson@9840
|
655 |
|
paulson@9840
|
656 |
end;
|
wenzelm@9869
|
657 |
|
paulson@15579
|
658 |
structure BasicMeson: BASIC_MESON = Meson;
|
paulson@15579
|
659 |
open BasicMeson;
|