--- a/src/HOL/Induct/PropLog.ML Tue Apr 02 13:47:01 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,209 +0,0 @@
-(* Title: HOL/ex/pl.ML
- ID: $Id$
- Author: Tobias Nipkow & Lawrence C Paulson
- Copyright 1994 TU Muenchen & University of Cambridge
-
-Soundness and completeness of propositional logic w.r.t. truth-tables.
-
-Prove: If H|=p then G|=p where G:Fin(H)
-*)
-
-(** Monotonicity **)
-Goalw thms.defs "G<=H ==> thms(G) <= thms(H)";
-by (rtac lfp_mono 1);
-by (REPEAT (ares_tac basic_monos 1));
-qed "thms_mono";
-
-(*Rule is called I for Identity Combinator, not for Introduction*)
-Goal "H |- p->p";
-by (best_tac (claset() addIs [thms.K,thms.S,thms.MP]) 1);
-qed "thms_I";
-
-(** Weakening, left and right **)
-
-(* "[| G<=H; G |- p |] ==> H |- p"
- This order of premises is convenient with RS
-*)
-bind_thm ("weaken_left", (thms_mono RS subsetD));
-
-(* H |- p ==> insert(a,H) |- p *)
-val weaken_left_insert = subset_insertI RS weaken_left;
-
-val weaken_left_Un1 = Un_upper1 RS weaken_left;
-val weaken_left_Un2 = Un_upper2 RS weaken_left;
-
-Goal "H |- q ==> H |- p->q";
-by (fast_tac (claset() addIs [thms.K,thms.MP]) 1);
-qed "weaken_right";
-
-(*The deduction theorem*)
-Goal "insert p H |- q ==> H |- p->q";
-by (etac thms.induct 1);
-by (ALLGOALS
- (fast_tac (claset() addIs [thms_I, thms.H, thms.K, thms.S, thms.DN,
- thms.S RS thms.MP RS thms.MP, weaken_right])));
-qed "deduction";
-
-
-(* "[| insert p H |- q; H |- p |] ==> H |- q"
- The cut rule - not used
-*)
-val cut = deduction RS thms.MP;
-
-(* H |- false ==> H |- p *)
-val thms_falseE = weaken_right RS (thms.DN RS thms.MP);
-
-(* [| H |- p->false; H |- p; q: pl |] ==> H |- q *)
-bind_thm ("thms_notE", (thms.MP RS thms_falseE));
-
-
-(*Soundness of the rules wrt truth-table semantics*)
-Goalw [sat_def] "H |- p ==> H |= p";
-by (etac thms.induct 1);
-by (fast_tac (claset() addSDs [eval_imp RS iffD1 RS mp]) 5);
-by (ALLGOALS Asm_simp_tac);
-qed "soundness";
-
-(*** Towards the completeness proof ***)
-
-Goal "H |- p->false ==> H |- p->q";
-by (rtac deduction 1);
-by (etac (weaken_left_insert RS thms_notE) 1);
-by (rtac thms.H 1);
-by (rtac insertI1 1);
-qed "false_imp";
-
-val [premp,premq] = goal PropLog.thy
- "[| H |- p; H |- q->false |] ==> H |- (p->q)->false";
-by (rtac deduction 1);
-by (rtac (premq RS weaken_left_insert RS thms.MP) 1);
-by (rtac (thms.H RS thms.MP) 1);
-by (rtac insertI1 1);
-by (rtac (premp RS weaken_left_insert) 1);
-qed "imp_false";
-
-(*This formulation is required for strong induction hypotheses*)
-Goal "hyps p tt |- (if tt[[p]] then p else p->false)";
-by (rtac (split_if RS iffD2) 1);
-by (induct_tac "p" 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [thms_I, thms.H])));
-by (fast_tac (claset() addIs [weaken_left_Un1, weaken_left_Un2,
- weaken_right, imp_false]
- addSEs [false_imp]) 1);
-qed "hyps_thms_if";
-
-(*Key lemma for completeness; yields a set of assumptions satisfying p*)
-val [sat] = goalw PropLog.thy [sat_def] "{} |= p ==> hyps p tt |- p";
-by (rtac (sat RS spec RS mp RS if_P RS subst) 1 THEN
- rtac hyps_thms_if 2);
-by (Simp_tac 1);
-qed "sat_thms_p";
-
-(*For proving certain theorems in our new propositional logic*)
-
-AddSIs [deduction];
-AddIs [thms.H, thms.H RS thms.MP];
-
-(*The excluded middle in the form of an elimination rule*)
-Goal "H |- (p->q) -> ((p->false)->q) -> q";
-by (rtac (deduction RS deduction) 1);
-by (rtac (thms.DN RS thms.MP) 1);
-by (ALLGOALS (best_tac (claset() addSIs prems)));
-qed "thms_excluded_middle";
-
-(*Hard to prove directly because it requires cuts*)
-val prems = goal PropLog.thy
- "[| insert p H |- q; insert (p->false) H |- q |] ==> H |- q";
-by (rtac (thms_excluded_middle RS thms.MP RS thms.MP) 1);
-by (REPEAT (resolve_tac (prems@[deduction]) 1));
-qed "thms_excluded_middle_rule";
-
-(*** Completeness -- lemmas for reducing the set of assumptions ***)
-
-(*For the case hyps(p,t)-insert(#v,Y) |- p;
- we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
-Goal "hyps p (t-{v}) <= insert (#v->false) ((hyps p t)-{#v})";
-by (induct_tac "p" 1);
-by (ALLGOALS Simp_tac);
-by (Fast_tac 1);
-qed "hyps_Diff";
-
-(*For the case hyps(p,t)-insert(#v -> false,Y) |- p;
- we also have hyps(p,t)-{#v->false} <= hyps(p, insert(v,t)) *)
-Goal "hyps p (insert v t) <= insert (#v) (hyps p t-{#v->false})";
-by (induct_tac "p" 1);
-by (ALLGOALS Simp_tac);
-by (Fast_tac 1);
-qed "hyps_insert";
-
-(** Two lemmas for use with weaken_left **)
-
-goal Set.thy "B-C <= insert a (B-insert a C)";
-by (Fast_tac 1);
-qed "insert_Diff_same";
-
-goal Set.thy "insert a (B-{c}) - D <= insert a (B-insert c D)";
-by (Fast_tac 1);
-qed "insert_Diff_subset2";
-
-Goal "finite(hyps p t)";
-by (induct_tac "p" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "hyps_finite";
-
-Goal "hyps p t <= (UN v. {#v, #v->false})";
-by (induct_tac "p" 1);
-by (ALLGOALS Simp_tac);
-by (Blast_tac 1);
-qed "hyps_subset";
-
-val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
-
-(*Induction on the finite set of assumptions hyps(p,t0).
- We may repeatedly subtract assumptions until none are left!*)
-val [sat] = goal PropLog.thy
- "{} |= p ==> !t. hyps p t - hyps p t0 |- p";
-by (rtac (hyps_subset RS (hyps_finite RS finite_subset_induct)) 1);
-by (simp_tac (simpset() addsimps [sat RS sat_thms_p]) 1);
-by Safe_tac;
-(*Case hyps(p,t)-insert(#v,Y) |- p *)
-by (rtac thms_excluded_middle_rule 1);
-by (rtac (insert_Diff_same RS weaken_left) 1);
-by (etac spec 1);
-by (rtac (insert_Diff_subset2 RS weaken_left) 1);
-by (rtac (hyps_Diff RS Diff_weaken_left) 1);
-by (etac spec 1);
-(*Case hyps(p,t)-insert(#v -> false,Y) |- p *)
-by (rtac thms_excluded_middle_rule 1);
-by (rtac (insert_Diff_same RS weaken_left) 2);
-by (etac spec 2);
-by (rtac (insert_Diff_subset2 RS weaken_left) 1);
-by (rtac (hyps_insert RS Diff_weaken_left) 1);
-by (etac spec 1);
-qed "completeness_0_lemma";
-
-(*The base case for completeness*)
-val [sat] = goal PropLog.thy "{} |= p ==> {} |- p";
-by (rtac (Diff_cancel RS subst) 1);
-by (rtac (sat RS (completeness_0_lemma RS spec)) 1);
-qed "completeness_0";
-
-(*A semantic analogue of the Deduction Theorem*)
-Goalw [sat_def] "insert p H |= q ==> H |= p->q";
-by (Simp_tac 1);
-by (Fast_tac 1);
-qed "sat_imp";
-
-Goal "finite H ==> !p. H |= p --> H |- p";
-by (etac finite_induct 1);
-by (safe_tac ((claset_of Fun.thy) addSIs [completeness_0]));
-by (rtac (weaken_left_insert RS thms.MP) 1);
-by (fast_tac ((claset_of Fun.thy) addSIs [sat_imp]) 1);
-by (Fast_tac 1);
-qed "completeness_lemma";
-
-val completeness = completeness_lemma RS spec RS mp;
-
-Goal "finite H ==> (H |- p) = (H |= p)";
-by (fast_tac (claset() addSEs [soundness, completeness]) 1);
-qed "syntax_iff_semantics";