author | wenzelm |
Mon, 20 Nov 2006 23:47:10 +0100 | |
changeset 21426 | 87ac12bed1ab |
parent 17892 | 62c397c17d18 |
child 21428 | f84cf8e9cad8 |
permissions | -rw-r--r-- |
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(* Title: Sequents/simpdata.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson |
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Copyright 1999 University of Cambridge |
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|
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Instantiation of the generic simplifier for LK |
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Borrows from the DC simplifier of Soren Heilmann. |
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*) |
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|
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(*** Rewrite rules ***) |
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local |
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val subst = thm "subst" |
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in |
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||
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fun prove_fun s = |
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(writeln s; |
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prove_goal (the_context ()) s |
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(fn prems => [ (cut_facts_tac prems 1), |
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(fast_tac (pack() add_safes [subst]) 1) ])); |
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end; |
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val conj_simps = map prove_fun |
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["|- P & True <-> P", "|- True & P <-> P", |
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"|- P & False <-> False", "|- False & P <-> False", |
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"|- P & P <-> P", " |- P & P & Q <-> P & Q", |
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"|- P & ~P <-> False", "|- ~P & P <-> False", |
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"|- (P & Q) & R <-> P & (Q & R)"]; |
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val disj_simps = map prove_fun |
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["|- P | True <-> True", "|- True | P <-> True", |
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"|- P | False <-> P", "|- False | P <-> P", |
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"|- P | P <-> P", "|- P | P | Q <-> P | Q", |
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"|- (P | Q) | R <-> P | (Q | R)"]; |
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val not_simps = map prove_fun |
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["|- ~ False <-> True", "|- ~ True <-> False"]; |
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val imp_simps = map prove_fun |
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["|- (P --> False) <-> ~P", "|- (P --> True) <-> True", |
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"|- (False --> P) <-> True", "|- (True --> P) <-> P", |
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"|- (P --> P) <-> True", "|- (P --> ~P) <-> ~P"]; |
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val iff_simps = map prove_fun |
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["|- (True <-> P) <-> P", "|- (P <-> True) <-> P", |
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"|- (P <-> P) <-> True", |
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"|- (False <-> P) <-> ~P", "|- (P <-> False) <-> ~P"]; |
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val quant_simps = map prove_fun |
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["|- (ALL x. P) <-> P", |
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"|- (ALL x. x=t --> P(x)) <-> P(t)", |
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"|- (ALL x. t=x --> P(x)) <-> P(t)", |
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"|- (EX x. P) <-> P", |
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"|- (EX x. x=t & P(x)) <-> P(t)", |
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"|- (EX x. t=x & P(x)) <-> P(t)"]; |
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(*** Miniscoping: pushing quantifiers in |
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We do NOT distribute of ALL over &, or dually that of EX over | |
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Baaz and Leitsch, On Skolemization and Proof Complexity (1994) |
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show that this step can increase proof length! |
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***) |
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(*existential miniscoping*) |
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val ex_simps = map prove_fun |
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["|- (EX x. P(x) & Q) <-> (EX x. P(x)) & Q", |
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"|- (EX x. P & Q(x)) <-> P & (EX x. Q(x))", |
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"|- (EX x. P(x) | Q) <-> (EX x. P(x)) | Q", |
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"|- (EX x. P | Q(x)) <-> P | (EX x. Q(x))", |
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"|- (EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q", |
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"|- (EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"]; |
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(*universal miniscoping*) |
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val all_simps = map prove_fun |
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["|- (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q", |
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"|- (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))", |
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"|- (ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q", |
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"|- (ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))", |
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"|- (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q", |
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"|- (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"]; |
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(*These are NOT supplied by default!*) |
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val distrib_simps = map prove_fun |
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["|- P & (Q | R) <-> P&Q | P&R", |
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"|- (Q | R) & P <-> Q&P | R&P", |
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"|- (P | Q --> R) <-> (P --> R) & (Q --> R)"]; |
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(** Conversion into rewrite rules **) |
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local |
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val mp_R = thm "mp_R"; |
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val conjunct1 = thm "conjunct1"; |
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val conjunct2 = thm "conjunct2"; |
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val spec = thm "spec"; |
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in |
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(*Make atomic rewrite rules*) |
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fun atomize r = |
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case concl_of r of |
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Const("Trueprop",_) $ Abs(_,_,a) $ Abs(_,_,c) => |
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(case (forms_of_seq a, forms_of_seq c) of |
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([], [p]) => |
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(case p of |
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Const("op -->",_)$_$_ => atomize(r RS mp_R) |
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| Const("op &",_)$_$_ => atomize(r RS conjunct1) @ |
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atomize(r RS conjunct2) |
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| Const("All",_)$_ => atomize(r RS spec) |
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| Const("True",_) => [] (*True is DELETED*) |
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| Const("False",_) => [] (*should False do something?*) |
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| _ => [r]) |
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| _ => []) (*ignore theorem unless it has precisely one conclusion*) |
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| _ => [r]; |
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end; |
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Goal "|- ~P ==> |- (P <-> False)"; |
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by (etac (thm "thinR" RS (thm "cut")) 1); |
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by (Fast_tac 1); |
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qed "P_iff_F"; |
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bind_thm ("iff_reflection_F", thm "P_iff_F" RS thm "iff_reflection"); |
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Goal "|- P ==> |- (P <-> True)"; |
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by (etac (thm "thinR" RS (thm "cut")) 1); |
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by (Fast_tac 1); |
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qed "P_iff_T"; |
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bind_thm ("iff_reflection_T", thm "P_iff_T" RS thm "iff_reflection"); |
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local |
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val eq_reflection = thm "eq_reflection" |
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val iff_reflection = thm "iff_reflection" |
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in |
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(*Make meta-equalities.*) |
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fun mk_meta_eq th = case concl_of th of |
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Const("==",_)$_$_ => th |
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| Const("Trueprop",_) $ Abs(_,_,a) $ Abs(_,_,c) => |
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(case (forms_of_seq a, forms_of_seq c) of |
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([], [p]) => |
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(case p of |
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(Const("op =",_)$_$_) => th RS eq_reflection |
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| (Const("op <->",_)$_$_) => th RS iff_reflection |
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| (Const("Not",_)$_) => th RS iff_reflection_F |
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| _ => th RS iff_reflection_T) |
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| _ => error ("addsimps: unable to use theorem\n" ^ |
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string_of_thm th)); |
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end; |
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(*Replace premises x=y, X<->Y by X==Y*) |
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val mk_meta_prems = |
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rule_by_tactic |
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(REPEAT_FIRST (resolve_tac [thm "meta_eq_to_obj_eq", thm "def_imp_iff"])); |
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(*Congruence rules for = or <-> (instead of ==)*) |
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fun mk_meta_cong rl = |
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standard(mk_meta_eq (mk_meta_prems rl)) |
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handle THM _ => |
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error("Premises and conclusion of congruence rules must use =-equality or <->"); |
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(*** Named rewrite rules ***) |
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fun prove nm thm = qed_goal nm (the_context ()) thm |
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(fn prems => [ (cut_facts_tac prems 1), |
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(fast_tac LK_pack 1) ]); |
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|
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prove "conj_commute" "|- P&Q <-> Q&P"; |
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prove "conj_left_commute" "|- P&(Q&R) <-> Q&(P&R)"; |
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val conj_comms = [conj_commute, conj_left_commute]; |
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169 |
|
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prove "disj_commute" "|- P|Q <-> Q|P"; |
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prove "disj_left_commute" "|- P|(Q|R) <-> Q|(P|R)"; |
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val disj_comms = [disj_commute, disj_left_commute]; |
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|
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prove "conj_disj_distribL" "|- P&(Q|R) <-> (P&Q | P&R)"; |
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prove "conj_disj_distribR" "|- (P|Q)&R <-> (P&R | Q&R)"; |
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|
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prove "disj_conj_distribL" "|- P|(Q&R) <-> (P|Q) & (P|R)"; |
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prove "disj_conj_distribR" "|- (P&Q)|R <-> (P|R) & (Q|R)"; |
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|
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prove "imp_conj_distrib" "|- (P --> (Q&R)) <-> (P-->Q) & (P-->R)"; |
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prove "imp_conj" "|- ((P&Q)-->R) <-> (P --> (Q --> R))"; |
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prove "imp_disj" "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"; |
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183 |
|
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prove "imp_disj1" "|- (P-->Q) | R <-> (P-->Q | R)"; |
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prove "imp_disj2" "|- Q | (P-->R) <-> (P-->Q | R)"; |
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|
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prove "de_Morgan_disj" "|- (~(P | Q)) <-> (~P & ~Q)"; |
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prove "de_Morgan_conj" "|- (~(P & Q)) <-> (~P | ~Q)"; |
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|
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prove "not_iff" "|- ~(P <-> Q) <-> (P <-> ~Q)"; |
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val [p1,p2] = Goal |
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"[| |- P <-> P'; |- P' ==> |- Q <-> Q' |] ==> |- (P-->Q) <-> (P'-->Q')"; |
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by (lemma_tac p1 1); |
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by (Safe_tac 1); |
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by (REPEAT (rtac (thm "cut") 1 |
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THEN |
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DEPTH_SOLVE_1 (resolve_tac [thm "thinL", thm "thinR", p2 COMP thm "monotonic"] 1) |
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THEN |
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Safe_tac 1)); |
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qed "imp_cong"; |
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203 |
|
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val [p1,p2] = Goal |
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"[| |- P <-> P'; |- P' ==> |- Q <-> Q' |] ==> |- (P&Q) <-> (P'&Q')"; |
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by (lemma_tac p1 1); |
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by (Safe_tac 1); |
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by (REPEAT (rtac (thm "cut") 1 |
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THEN |
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DEPTH_SOLVE_1 (resolve_tac [thm "thinL", thm "thinR", p2 COMP thm "monotonic"] 1) |
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THEN |
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Safe_tac 1)); |
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qed "conj_cong"; |
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214 |
|
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Goal "|- (x=y) <-> (y=x)"; |
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by (fast_tac (pack() add_safes [thm "subst"]) 1); |
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qed "eq_sym_conv"; |
218 |
||
219 |
||
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|
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(*** Standard simpsets ***) |
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222 |
|
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val triv_rls = [thm "FalseL", thm "TrueR", thm "basic", thm "refl", |
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thm "iff_refl", reflexive_thm]; |
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225 |
|
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fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems), |
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assume_tac]; |
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(*No premature instantiation of variables during simplification*) |
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fun safe_solver prems = FIRST'[fn i => DETERM (match_tac (triv_rls@prems) i), |
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eq_assume_tac]; |
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231 |
|
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(*No simprules, but basic infrastructure for simplification*) |
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val LK_basic_ss = |
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Simplifier.theory_context (the_context ()) empty_ss |
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setsubgoaler asm_simp_tac |
|
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setSSolver (mk_solver "safe" safe_solver) |
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setSolver (mk_solver "unsafe" unsafe_solver) |
|
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setmksimps (map mk_meta_eq o atomize o gen_all) |
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setmkcong mk_meta_cong; |
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240 |
|
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val LK_simps = |
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[triv_forall_equality, (* prunes params *) |
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thm "refl" RS P_iff_T] @ |
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conj_simps @ disj_simps @ not_simps @ |
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imp_simps @ iff_simps @quant_simps @ all_simps @ ex_simps @ |
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[de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2] @ |
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map prove_fun |
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["|- P | ~P", "|- ~P | P", |
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"|- ~ ~ P <-> P", "|- (~P --> P) <-> P", |
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"|- (~P <-> ~Q) <-> (P<->Q)"]; |
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251 |
|
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val LK_ss = |
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LK_basic_ss addsimps LK_simps |
|
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addeqcongs [left_cong] |
|
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addcongs [imp_cong]; |
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256 |
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change_simpset (fn _ => LK_ss); |
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258 |
|
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(* To create substition rules *) |
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qed_goal "eq_imp_subst" (the_context ()) "|- a=b ==> $H, A(a), $G |- $E, A(b), $F" |
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(fn prems => |
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[cut_facts_tac prems 1, |
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asm_simp_tac LK_basic_ss 1]); |
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266 |
|
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Goal "|- P(if Q then x else y) <-> ((Q --> P(x)) & (~Q --> P(y)))"; |
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by (res_inst_tac [ ("P","Q") ] (thm "cut") 1); |
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by (simp_tac (simpset() addsimps [thm "if_P"]) 2); |
|
270 |
by (res_inst_tac [ ("P","~Q") ] (thm "cut") 1); |
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by (simp_tac (simpset() addsimps [thm "if_not_P"]) 2); |
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by (Fast_tac 1); |
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qed "split_if"; |
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Goal "|- (if P then x else x) = x"; |
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by (lemma_tac split_if 1); |
|
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by (Fast_tac 1); |
|
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qed "if_cancel"; |
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279 |
||
280 |
Goal "|- (if x=y then y else x) = x"; |
|
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by (lemma_tac split_if 1); |
|
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by (Safe_tac 1); |
|
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by (rtac (thm "symL") 1); |
284 |
by (rtac (thm "basic") 1); |
|
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qed "if_eq_cancel"; |
286 |
||
287 |
(*Putting in automatic case splits seems to require a lot of work.*) |