author | wenzelm |
Mon, 03 Nov 1997 12:13:18 +0100 | |
changeset 4089 | 96fba19bcbe2 |
parent 3842 | b55686a7b22c |
child 4153 | e534c4c32d54 |
permissions | -rw-r--r-- |
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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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NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR |
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FUNCTION nodups -- if done to goal clauses too! |
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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 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 (claset() 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 st = |
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case Sequence.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st) |
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of Some(th,_) => th |
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| None => raise THM("forward_res", 0, [st]); |
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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 detect, and remove, 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 term_False = term_of (read_cterm (sign_of HOL.thy) |
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("False", Type("bool",[]))); |
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(*Include False as a literal: an occurrence of ~False is a tautology*) |
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fun is_taut th = taut_lits ((true,term_False) :: literals (prop_of th)); |
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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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(fn st' => st' |> |
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METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1) |
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in Sequence.list_of_s (tac (th RS disj_forward)) @ ths 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 st = |
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case Sequence.pull |
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(REPEAT |
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(METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1) |
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st) |
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of Some(th,_) => th |
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| None => raise THM("forward_res2", 0, [st]); |
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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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(*For Plaisted's postive refinement. [currently unused] *) |
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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, previously Skolemized and put into nnf. |
321 |
The resulting clauses are HOL disjunctions.*) |
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fun make_clauses ths = |
323 |
sort_clauses (map (generalize o nodups) (foldr cnf (ths,[]))); |
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(*Create a meta-level Horn clause*) |
969 | 326 |
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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331 |
let fun rots (0,th) = hcs |
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| rots (k,th) = zero_var_indexes (make_horn crules th) :: |
333 |
rots(k-1, assoc_right (th RS disj_comm)) |
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969 | 334 |
in case nliterals(prop_of th) of |
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1 => th::hcs |
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| n => rots(n, assoc_right th) |
337 |
end; |
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(*Use "theorem naming" to label the clauses*) |
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fun name_thms label = |
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let fun name1 (th, (k,ths)) = |
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(k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths) |
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in fn ths => #2 (foldr name1 (ths, (length ths, []))) end; |
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969 | 346 |
(*Convert a list of clauses to (contrapositive) Horn clauses*) |
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fun make_horns ths = |
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name_thms "Horn#" |
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349 |
(gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[]))); |
969 | 350 |
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351 |
(*Find an all-negative support clause*) |
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352 |
fun is_negative th = forall (not o #1) (literals (prop_of th)); |
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354 |
val neg_clauses = filter is_negative; |
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356 |
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357 |
(***** MESON PROOF PROCEDURE *****) |
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358 |
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359 |
fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi, |
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1465 | 360 |
As) = rhyps(phi, A::As) |
969 | 361 |
| rhyps (_, As) = As; |
362 |
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363 |
(** Detecting repeated assumptions in a subgoal **) |
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364 |
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365 |
(*The stringtree detects repeated assumptions.*) |
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366 |
fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv); |
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367 |
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368 |
(*detects repetitions in a list of terms*) |
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369 |
fun has_reps [] = false |
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| has_reps [_] = false |
|
371 |
| has_reps [t,u] = (t aconv u) |
|
372 |
| has_reps ts = (foldl ins_term (Net.empty, ts); false) |
|
1465 | 373 |
handle INSERT => true; |
969 | 374 |
|
1585 | 375 |
(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*) |
376 |
fun TRYALL_eq_assume_tac 0 st = Sequence.single st |
|
377 |
| TRYALL_eq_assume_tac i st = TRYALL_eq_assume_tac (i-1) (eq_assumption i st) |
|
378 |
handle THM _ => TRYALL_eq_assume_tac (i-1) st; |
|
379 |
||
969 | 380 |
(*Loop checking: FAIL if trying to prove the same thing twice |
1585 | 381 |
-- if *ANY* subgoal has repeated literals*) |
382 |
fun check_tac st = |
|
383 |
if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st) |
|
384 |
then Sequence.null else Sequence.single st; |
|
385 |
||
969 | 386 |
|
387 |
(* net_resolve_tac actually made it slower... *) |
|
388 |
fun prolog_step_tac horns i = |
|
1585 | 389 |
(assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN |
390 |
TRYALL eq_assume_tac; |
|
969 | 391 |
|
392 |
||
393 |
(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*) |
|
394 |
local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz |
|
395 |
in |
|
396 |
fun size_of_subgoals st = foldr addconcl (prems_of st, 0) |
|
397 |
end; |
|
398 |
||
399 |
(*Could simply use nprems_of, which would count remaining subgoals -- no |
|
400 |
discrimination as to their size! With BEST_FIRST, fails for problem 41.*) |
|
401 |
||
402 |
fun best_prolog_tac sizef horns = |
|
403 |
BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1); |
|
404 |
||
405 |
fun depth_prolog_tac horns = |
|
406 |
DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1); |
|
407 |
||
408 |
(*Return all negative clauses, as possible goal clauses*) |
|
1599
b11ac7072422
Now labels the Horn and goal clauses to make the proof
paulson
parents:
1585
diff
changeset
|
409 |
fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls)); |
969 | 410 |
|
411 |
||
412 |
fun skolemize_tac prems = |
|
413 |
cut_facts_tac (map (skolemize o make_nnf) prems) THEN' |
|
414 |
REPEAT o (etac exE); |
|
415 |
||
1599
b11ac7072422
Now labels the Horn and goal clauses to make the proof
paulson
parents:
1585
diff
changeset
|
416 |
(*Shell of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*) |
b11ac7072422
Now labels the Horn and goal clauses to make the proof
paulson
parents:
1585
diff
changeset
|
417 |
fun MESON cltac = SELECT_GOAL |
969 | 418 |
(EVERY1 [rtac ccontr, |
1465 | 419 |
METAHYPS (fn negs => |
420 |
EVERY1 [skolemize_tac negs, |
|
1599
b11ac7072422
Now labels the Horn and goal clauses to make the proof
paulson
parents:
1585
diff
changeset
|
421 |
METAHYPS (cltac o make_clauses)])]); |
969 | 422 |
|
1585 | 423 |
(** Best-first search versions **) |
424 |
||
969 | 425 |
fun best_meson_tac sizef = |
426 |
MESON (fn cls => |
|
1585 | 427 |
THEN_BEST_FIRST (resolve_tac (gocls cls) 1) |
428 |
(has_fewer_prems 1, sizef) |
|
2031 | 429 |
(prolog_step_tac (make_horns cls) 1)); |
969 | 430 |
|
431 |
(*First, breaks the goal into independent units*) |
|
1585 | 432 |
val safe_best_meson_tac = |
4089 | 433 |
SELECT_GOAL (TRY (safe_tac (claset())) THEN |
1465 | 434 |
TRYALL (best_meson_tac size_of_subgoals)); |
969 | 435 |
|
1585 | 436 |
(** Depth-first search version **) |
437 |
||
969 | 438 |
val depth_meson_tac = |
439 |
MESON (fn cls => EVERY [resolve_tac (gocls cls) 1, |
|
1465 | 440 |
depth_prolog_tac (make_horns cls)]); |
969 | 441 |
|
1585 | 442 |
|
443 |
||
444 |
(** Iterative deepening version **) |
|
445 |
||
446 |
(*This version does only one inference per call; |
|
447 |
having only one eq_assume_tac speeds it up!*) |
|
448 |
fun prolog_step_tac' horns = |
|
449 |
let val (horn0s, hornps) = (*0 subgoals vs 1 or more*) |
|
450 |
take_prefix (fn rl => nprems_of rl=0) horns |
|
451 |
val nrtac = net_resolve_tac horns |
|
452 |
in fn i => eq_assume_tac i ORELSE |
|
453 |
match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*) |
|
2031 | 454 |
((assume_tac i APPEND nrtac i) THEN check_tac) |
1585 | 455 |
end; |
456 |
||
457 |
fun iter_deepen_prolog_tac horns = |
|
458 |
ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns); |
|
459 |
||
460 |
val iter_deepen_meson_tac = |
|
461 |
MESON (fn cls => |
|
462 |
(THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1) |
|
2031 | 463 |
(has_fewer_prems 1) |
464 |
(prolog_step_tac' (make_horns cls)))); |
|
1585 | 465 |
|
466 |
val safe_meson_tac = |
|
4089 | 467 |
SELECT_GOAL (TRY (safe_tac (claset())) THEN |
1585 | 468 |
TRYALL (iter_deepen_meson_tac)); |
469 |
||
470 |
||
969 | 471 |
writeln"Reached end of file."; |