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(* Title: HOL/ex/meson
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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The MESON resolution proof procedure for HOL
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When making clauses, avoids using the rewriter -- instead uses RS recursively
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*)
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writeln"File HOL/ex/meson.";
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(*Prove theorems using fast_tac*)
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fun prove_fun s =
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prove_goal HOL.thy s
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(fn prems => [ cut_facts_tac prems 1, fast_tac HOL_cs 1 ]);
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(**** Negation Normal Form ****)
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(*** de Morgan laws ***)
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val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
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val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
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val not_notD = prove_fun "~~P ==> P";
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val not_allD = prove_fun "~(! x.P(x)) ==> ? x. ~P(x)";
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val not_exD = prove_fun "~(? x.P(x)) ==> ! x. ~P(x)";
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(*** Removal of --> and <-> (positive and negative occurrences) ***)
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val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
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val not_impD = prove_fun "~(P-->Q) ==> P & ~Q";
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val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
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(*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
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val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
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(**** Pulling out the existential quantifiers ****)
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(*** Conjunction ***)
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val conj_exD1 = prove_fun "(? x.P(x)) & Q ==> ? x. P(x) & Q";
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val conj_exD2 = prove_fun "P & (? x.Q(x)) ==> ? x. P & Q(x)";
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(*** Disjunction ***)
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(*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
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With ex-Skolemization, makes fewer Skolem constants*)
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val disj_exD = prove_fun "(? x.P(x)) | (? x.Q(x)) ==> ? x. P(x) | Q(x)";
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val disj_exD1 = prove_fun "(? x.P(x)) | Q ==> ? x. P(x) | Q";
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val disj_exD2 = prove_fun "P | (? x.Q(x)) ==> ? x. P | Q(x)";
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(**** Skolemization -- pulling "?" over "!" ****)
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(*"Axiom" of Choice, proved using the description operator*)
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val [major] = goal HOL.thy
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"! x. ? y. Q x y ==> ? f. ! x. Q x (f x)";
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by (cut_facts_tac [major] 1);
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by (fast_tac (HOL_cs addEs [selectI]) 1);
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qed "choice";
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(***** Generating clauses for the Meson Proof Procedure *****)
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(*** Disjunctions ***)
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val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
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val disj_comm = prove_fun "P|Q ==> Q|P";
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val disj_FalseD1 = prove_fun "False|P ==> P";
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val disj_FalseD2 = prove_fun "P|False ==> P";
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(*** Generation of contrapositives ***)
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(*Inserts negated disjunct after removing the negation; P is a literal*)
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val [major,minor] = goal HOL.thy "~P|Q ==> ((~P==>P) ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (etac minor 2);
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by (ALLGOALS assume_tac);
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qed "make_neg_rule";
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(*For Plaisted's "Postive refinement" of the MESON procedure*)
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val [major,minor] = goal HOL.thy "~P|Q ==> (P ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (rtac minor 2);
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by (ALLGOALS assume_tac);
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qed "make_refined_neg_rule";
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(*P should be a literal*)
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val [major,minor] = goal HOL.thy "P|Q ==> ((P==>~P) ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (etac minor 1);
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by (ALLGOALS assume_tac);
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qed "make_pos_rule";
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(*** Generation of a goal clause -- put away the final literal ***)
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val [major,minor] = goal HOL.thy "~P ==> ((~P==>P) ==> False)";
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by (rtac notE 1);
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by (rtac minor 2);
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by (ALLGOALS (rtac major));
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qed "make_neg_goal";
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val [major,minor] = goal HOL.thy "P ==> ((P==>~P) ==> False)";
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by (rtac notE 1);
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by (rtac minor 1);
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by (ALLGOALS (rtac major));
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qed "make_pos_goal";
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(**** Lemmas for forward proof (like congruence rules) ****)
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(*NOTE: could handle conjunctions (faster?) by
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nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
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val major::prems = goal HOL.thy
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"[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q";
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by (rtac (major RS conjE) 1);
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by (rtac conjI 1);
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by (ALLGOALS (eresolve_tac prems));
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qed "conj_forward";
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val major::prems = goal HOL.thy
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"[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q";
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by (rtac (major RS disjE) 1);
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by (ALLGOALS (dresolve_tac prems));
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by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
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qed "disj_forward";
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val major::prems = goal HOL.thy
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"[| ! x. P'(x); !!x. P'(x) ==> P(x) |] ==> ! x. P(x)";
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by (rtac allI 1);
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by (resolve_tac prems 1);
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by (rtac (major RS spec) 1);
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qed "all_forward";
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val major::prems = goal HOL.thy
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"[| ? x. P'(x); !!x. P'(x) ==> P(x) |] ==> ? x. P(x)";
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by (rtac (major RS exE) 1);
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by (rtac exI 1);
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by (eresolve_tac prems 1);
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qed "ex_forward";
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(**** Operators for forward proof ****)
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(*raises exception if no rules apply -- unlike RL*)
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fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
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| tryres (th, []) = raise THM("tryres", 0, [th]);
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val prop_of = #prop o rep_thm;
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(*Permits forward proof from rules that discharge assumptions*)
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fun forward_res nf state =
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case Sequence.pull
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(tapply(ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)),
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state))
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of Some(th,_) => th
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| None => raise THM("forward_res", 0, [state]);
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(*Negation Normal Form*)
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val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
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not_impD, not_iffD, not_allD, not_exD, not_notD];
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fun make_nnf th = make_nnf (tryres(th, nnf_rls))
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handle THM _ =>
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forward_res make_nnf
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(tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
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handle THM _ => th;
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(*Are any of the constants in "bs" present in the term?*)
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fun has_consts bs =
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let fun has (Const(a,_)) = a mem bs
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| has (f$u) = has f orelse has u
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| has (Abs(_,_,t)) = has t
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| has _ = false
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in has end;
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(*Pull existential quantifiers (Skolemization)*)
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fun skolemize th =
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if not (has_consts ["Ex"] (prop_of th)) then th
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else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
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disj_exD, disj_exD1, disj_exD2]))
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handle THM _ =>
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skolemize (forward_res skolemize
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(tryres (th, [conj_forward, disj_forward, all_forward])))
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handle THM _ => forward_res skolemize (th RS ex_forward);
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(**** Clause handling ****)
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fun literals (Const("Trueprop",_) $ P) = literals P
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| literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
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| literals (Const("not",_) $ P) = [(false,P)]
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| literals P = [(true,P)];
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(*number of literals in a term*)
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val nliterals = length o literals;
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(*to delete tautologous clauses*)
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fun taut_lits [] = false
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| taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
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val is_taut = taut_lits o literals o prop_of;
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(*Generation of unique names -- maxidx cannot be relied upon to increase!
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Cannot rely on "variant", since variables might coincide when literals
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are joined to make a clause...
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19 chooses "U" as the first variable name*)
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val name_ref = ref 19;
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(*Replaces universally quantified variables by FREE variables -- because
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assumptions may not contain scheme variables. Later, call "generalize". *)
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fun freeze_spec th =
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let val sth = th RS spec
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val newname = (name_ref := !name_ref + 1;
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radixstring(26, "A", !name_ref))
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in read_instantiate [("x", newname)] sth end;
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fun resop nf [prem] = resolve_tac (nf prem) 1;
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(*Conjunctive normal form, detecting tautologies early.
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Strips universal quantifiers and breaks up conjunctions. *)
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fun cnf_aux seen (th,ths) =
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if taut_lits (literals(prop_of th) @ seen) then ths
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else if not (has_consts ["All","op &"] (prop_of th)) then th::ths
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else (*conjunction?*)
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cnf_aux seen (th RS conjunct1,
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cnf_aux seen (th RS conjunct2, ths))
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handle THM _ => (*universal quant?*)
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cnf_aux seen (freeze_spec th, ths)
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handle THM _ => (*disjunction?*)
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let val tac =
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(METAHYPS (resop (cnf_nil seen)) 1) THEN
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(STATE (fn st' =>
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METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1))
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in Sequence.list_of_s (tapply(tac, th RS disj_forward)) @ ths
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end
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and cnf_nil seen th = cnf_aux seen (th,[]);
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(*Top-level call to cnf -- it's safe to reset name_ref*)
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fun cnf (th,ths) =
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(name_ref := 19; cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
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handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
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(**** Removal of duplicate literals ****)
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(*Version for removal of duplicate literals*)
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val major::prems = goal HOL.thy
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"[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q";
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by (rtac (major RS disjE) 1);
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by (rtac disjI1 1);
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by (rtac (disjCI RS disj_comm) 2);
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by (ALLGOALS (eresolve_tac prems));
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by (etac notE 1);
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by (assume_tac 1);
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qed "disj_forward2";
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(*Forward proof, passing extra assumptions as theorems to the tactic*)
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fun forward_res2 nf hyps state =
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case Sequence.pull
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(tapply(REPEAT
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(METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1),
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state))
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of Some(th,_) => th
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| None => raise THM("forward_res2", 0, [state]);
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(*Remove duplicates in P|Q by assuming ~P in Q
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rls (initially []) accumulates assumptions of the form P==>False*)
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fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
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handle THM _ => tryres(th,rls)
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handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
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[disj_FalseD1, disj_FalseD2, asm_rl])
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handle THM _ => th;
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(*Remove duplicate literals, if there are any*)
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fun nodups th =
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if null(findrep(literals(prop_of th))) then th
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else nodups_aux [] th;
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(**** Generation of contrapositives ****)
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(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
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fun assoc_right th = assoc_right (th RS disj_assoc)
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handle THM _ => th;
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(*Must check for negative literal first!*)
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val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
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val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
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(*Create a goal or support clause, conclusing False*)
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fun make_goal th = (*Must check for negative literal first!*)
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make_goal (tryres(th, clause_rules))
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handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
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(*Sort clauses by number of literals*)
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fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
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(*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
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fun sort_clauses ths = sort fewerlits (filter (not o is_taut) ths);
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(*Convert all suitable free variables to schematic variables*)
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fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
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(*make clauses from a list of theorems*)
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fun make_clauses ths =
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sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
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(*Create a Horn clause*)
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fun make_horn crules th = make_horn crules (tryres(th,crules))
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handle THM _ => th;
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(*Generate Horn clauses for all contrapositives of a clause*)
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fun add_contras crules (th,hcs) =
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let fun rots (0,th) = hcs
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| rots (k,th) = zero_var_indexes (make_horn crules th) ::
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rots(k-1, assoc_right (th RS disj_comm))
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in case nliterals(prop_of th) of
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1 => th::hcs
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| n => rots(n, assoc_right th)
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end;
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(*Convert a list of clauses to (contrapositive) Horn clauses*)
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fun make_horns ths = foldr (add_contras clause_rules) (ths,[]);
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(*Find an all-negative support clause*)
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fun is_negative th = forall (not o #1) (literals (prop_of th));
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val neg_clauses = filter is_negative;
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(***** MESON PROOF PROCEDURE *****)
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fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
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As) = rhyps(phi, A::As)
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| rhyps (_, As) = As;
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(** Detecting repeated assumptions in a subgoal **)
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(*The stringtree detects repeated assumptions.*)
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fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
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(*detects repetitions in a list of terms*)
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fun has_reps [] = false
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| has_reps [_] = false
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| has_reps [t,u] = (t aconv u)
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| has_reps ts = (foldl ins_term (Net.empty, ts); false)
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handle INSERT => true;
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(*Loop checking: FAIL if trying to prove the same thing twice
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-- repeated literals*)
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val check_tac = SUBGOAL (fn (prem,_) =>
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if has_reps (rhyps(prem,[])) then no_tac else all_tac);
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|
365 |
(* net_resolve_tac actually made it slower... *)
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|
366 |
fun prolog_step_tac horns i =
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|
367 |
(assume_tac i APPEND resolve_tac horns i) THEN
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|
368 |
(ALLGOALS check_tac) THEN
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|
369 |
(TRYALL eq_assume_tac);
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|
370 |
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|
371 |
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|
372 |
(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
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|
373 |
local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz
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374 |
in
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|
375 |
fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
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|
376 |
end;
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|
377 |
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|
378 |
(*Could simply use nprems_of, which would count remaining subgoals -- no
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|
379 |
discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
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|
380 |
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|
381 |
fun best_prolog_tac sizef horns =
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|
382 |
BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
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|
383 |
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|
384 |
fun depth_prolog_tac horns =
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|
385 |
DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
|
|
386 |
|
|
387 |
(*Return all negative clauses, as possible goal clauses*)
|
|
388 |
fun gocls cls = map make_goal (neg_clauses cls);
|
|
389 |
|
|
390 |
|
|
391 |
fun skolemize_tac prems =
|
|
392 |
cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
|
|
393 |
REPEAT o (etac exE);
|
|
394 |
|
|
395 |
fun MESON sko_tac = SELECT_GOAL
|
|
396 |
(EVERY1 [rtac ccontr,
|
|
397 |
METAHYPS (fn negs =>
|
|
398 |
EVERY1 [skolemize_tac negs,
|
|
399 |
METAHYPS (sko_tac o make_clauses)])]);
|
|
400 |
|
|
401 |
fun best_meson_tac sizef =
|
|
402 |
MESON (fn cls =>
|
|
403 |
resolve_tac (gocls cls) 1
|
|
404 |
THEN_BEST_FIRST
|
|
405 |
(has_fewer_prems 1, sizef,
|
|
406 |
prolog_step_tac (make_horns cls) 1));
|
|
407 |
|
|
408 |
(*First, breaks the goal into independent units*)
|
|
409 |
val safe_meson_tac =
|
|
410 |
SELECT_GOAL (TRY (safe_tac HOL_cs) THEN
|
|
411 |
TRYALL (best_meson_tac size_of_subgoals));
|
|
412 |
|
|
413 |
val depth_meson_tac =
|
|
414 |
MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
|
|
415 |
depth_prolog_tac (make_horns cls)]);
|
|
416 |
|
|
417 |
writeln"Reached end of file.";
|