author | paulson |
Wed, 09 Oct 1996 13:36:17 +0200 | |
changeset 2074 | 30a65172e003 |
parent 2065 | b696f087f052 |
child 2469 | b50b8c0eec01 |
permissions | -rw-r--r-- |
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(* Title: FOL/simpdata |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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Simplification data for FOL |
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*) |
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(*** Rewrite rules ***) |
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fun int_prove_fun s = |
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(writeln s; |
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prove_goal IFOL.thy s |
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(fn prems => [ (cut_facts_tac prems 1), |
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(Int.fast_tac 1) ])); |
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val conj_simps = map int_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", |
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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 int_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", |
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"(P | Q) | R <-> P | (Q | R)"]; |
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val not_simps = map int_prove_fun |
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["~(P|Q) <-> ~P & ~Q", |
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"~ False <-> True", "~ True <-> False"]; |
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val imp_simps = map int_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 int_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 int_prove_fun |
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["(ALL x.P) <-> P", "(EX x.P) <-> P"]; |
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(*These are NOT supplied by default!*) |
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val distrib_simps = map int_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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fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th; |
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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",_) $ p => |
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(case p of |
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Const("op -->",_)$_$_ => atomize(r RS mp) |
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| Const("op &",_)$_$_ => atomize(r RS conjunct1) @ |
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atomize(r RS conjunct2) |
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| Const("All",_)$_ => atomize(r RS spec) |
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| Const("True",_) => [] (*True is DELETED*) |
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| Const("False",_) => [] (*should False do something?*) |
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| _ => [r]) |
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| _ => [r]; |
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val P_iff_F = int_prove_fun "~P ==> (P <-> False)"; |
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val iff_reflection_F = P_iff_F RS iff_reflection; |
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val P_iff_T = int_prove_fun "P ==> (P <-> True)"; |
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val iff_reflection_T = P_iff_T RS iff_reflection; |
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(*Make meta-equalities. The operator below is Trueprop*) |
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fun mk_meta_eq th = case concl_of th of |
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Const("==",_)$_$_ => th |
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| _ $ (Const("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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(*** Classical laws ***) |
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fun prove_fun s = |
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(writeln s; |
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prove_goal FOL.thy s |
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(fn prems => [ (cut_facts_tac prems 1), |
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(Cla.fast_tac FOL_cs 1) ])); |
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(*Avoids duplication of subgoals after expand_if, when the true and false |
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cases boil down to the same thing.*) |
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val cases_simp = prove_fun "(P --> Q) & (~P --> Q) <-> Q"; |
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(*At present, miniscoping is for classical logic only. We do NOT include |
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distribution of ALL over &, or dually that of EX over |.*) |
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(*Miniscoping: pushing in existential quantifiers*) |
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val ex_simps = map prove_fun |
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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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"(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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(*Miniscoping: pushing in universal quantifiers*) |
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val all_simps = map prove_fun |
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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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"(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) <-> (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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fun int_prove nm thm = qed_goal nm IFOL.thy thm |
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(fn prems => [ (cut_facts_tac prems 1), |
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(Int.fast_tac 1) ]); |
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fun prove nm thm = qed_goal nm FOL.thy thm (fn _ => [fast_tac FOL_cs 1]); |
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int_prove "conj_commute" "P&Q <-> Q&P"; |
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int_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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int_prove "disj_commute" "P|Q <-> Q|P"; |
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int_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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int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)"; |
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int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)"; |
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int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)"; |
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int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)"; |
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int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)"; |
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int_prove "imp_conj" "((P&Q)-->R) <-> (P --> (Q --> R))"; |
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int_prove "imp_disj" "(P|Q --> R) <-> (P-->R) & (Q-->R)"; |
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int_prove "de_Morgan_disj" "(~(P | Q)) <-> (~P & ~Q)"; |
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prove "de_Morgan_conj" "(~(P & Q)) <-> (~P | ~Q)"; |
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prove "not_iff" "~(P <-> Q) <-> (P <-> ~Q)"; |
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prove "not_all" "(~ (ALL x.P(x))) <-> (EX x.~P(x))"; |
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prove "imp_all" "((ALL x.P(x)) --> Q) <-> (EX x.P(x) --> Q)"; |
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int_prove "not_ex" "(~ (EX x.P(x))) <-> (ALL x.~P(x))"; |
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int_prove "imp_ex" "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)"; |
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int_prove "ex_disj_distrib" |
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"(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))"; |
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int_prove "all_conj_distrib" |
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"(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))"; |
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(*Used in ZF, perhaps elsewhere?*) |
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val meta_eq_to_obj_eq = prove_goal IFOL.thy "x==y ==> x=y" |
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(fn [prem] => [rewtac prem, rtac refl 1]); |
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(*** case splitting ***) |
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qed_goal "meta_iffD" IFOL.thy "[| P==Q; Q |] ==> P" |
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(fn [prem1,prem2] => [rewtac prem1, rtac prem2 1]); |
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local val mktac = mk_case_split_tac meta_iffD |
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in |
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fun split_tac splits = mktac (map mk_meta_eq splits) |
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end; |
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local val mktac = mk_case_split_inside_tac meta_iffD |
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in |
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fun split_inside_tac splits = mktac (map mk_meta_eq splits) |
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end; |
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(*** Standard simpsets ***) |
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structure Induction = InductionFun(struct val spec=IFOL.spec end); |
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open Simplifier Induction; |
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(*Add congruence rules for = or <-> (instead of ==) *) |
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infix 4 addcongs; |
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fun ss addcongs congs = |
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ss addeqcongs (congs RL [eq_reflection,iff_reflection]); |
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(*Add a simpset to a classical set!*) |
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infix 4 addss; |
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fun cs addss ss = cs addbefore asm_full_simp_tac ss 1; |
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val IFOL_simps = |
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[refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @ |
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imp_simps @ iff_simps @ quant_simps; |
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val notFalseI = int_prove_fun "~False"; |
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val triv_rls = [TrueI,refl,iff_refl,notFalseI]; |
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val IFOL_ss = |
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empty_ss |
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setmksimps (map mk_meta_eq o atomize o gen_all) |
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setsolver (fn prems => resolve_tac (triv_rls@prems) |
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ORELSE' assume_tac |
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ORELSE' etac FalseE) |
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setsubgoaler asm_simp_tac |
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addsimps IFOL_simps |
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addcongs [imp_cong]; |
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val cla_simps = |
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[de_Morgan_conj, de_Morgan_disj, not_all, not_ex, cases_simp] @ |
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map prove_fun |
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["~(P&Q) <-> ~P | ~Q", |
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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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val FOL_ss = IFOL_ss addsimps (cla_simps @ ex_simps @ all_simps); |
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