--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/ex/Commutation.ML Sat Feb 03 12:41:38 2001 +0100
@@ -0,0 +1,131 @@
+(* Title: HOL/Lambda/Commutation.thy
+ ID: $Id$
+ Author: Tobias Nipkow & Sidi Ould Ehmety
+ Copyright 1995 TU Muenchen
+
+Commutation theory for proving the Church Rosser theorem
+ ported from Isabelle/HOL by Sidi Ould Ehmety
+*)
+
+Goalw [square_def] "square(r,s,t,u) ==> square(s,r,u,t)";
+by (Blast_tac 1);
+qed "square_sym";
+
+
+Goalw [square_def] "[| square(r,s,t,u); t <= t' |] ==> square(r,s,t',u)";
+by (Blast_tac 1);
+qed "square_subset";
+
+
+Goalw [square_def]
+ "field(s)<=field(t)==> square(r,s,s,t) --> square(r^*,s,s,t^*)";
+by (Clarify_tac 1);
+by (etac rtrancl_induct 1);
+by (blast_tac (claset() addIs [rtrancl_refl]) 1);
+by (blast_tac (claset() addIs [rtrancl_into_rtrancl]) 1);
+qed_spec_mp "square_rtrancl";
+
+(* A special case of square_rtrancl_on *)
+Goalw [diamond_def, commute_def, strip_def]
+ "diamond(r) ==> strip(r)";
+by (resolve_tac [square_rtrancl] 1);
+by (ALLGOALS(Asm_simp_tac));
+qed "diamond_strip";
+
+(*** commute ***)
+
+Goalw [commute_def]
+ "commute(r,s) ==> commute(s,r)";
+by (blast_tac (claset() addIs [square_sym]) 1);
+qed "commute_sym";
+
+Goalw [commute_def]
+"commute(r,s) ==> field(r)=field(s) --> commute(r^*,s^*)";
+by (Clarify_tac 1);
+by (rtac square_rtrancl 1);
+by (rtac square_sym 2);
+by (rtac square_rtrancl 2);
+by (rtac square_sym 3);
+by (ALLGOALS(asm_simp_tac
+ (simpset() addsimps [rtrancl_field])));
+qed_spec_mp "commute_rtrancl";
+
+
+Goalw [strip_def,confluent_def, diamond_def]
+"strip(r) ==> confluent(r)";
+by (dtac commute_rtrancl 1);
+by (ALLGOALS(asm_full_simp_tac (simpset()
+ addsimps [rtrancl_field])));
+qed "strip_confluent";
+
+
+Goalw [commute_def,square_def]
+ "[| commute(r,t); commute(s,t) |] ==> commute(r Un s, t)";
+by (Blast_tac 1);
+qed "commute_Un";
+
+
+Goalw [diamond_def]
+ "[| diamond(r); diamond(s); commute(r, s) |] \
+\ ==> diamond(r Un s)";
+by (REPEAT(ares_tac [commute_Un,commute_sym] 1));
+qed "diamond_Un";
+
+
+Goalw [diamond_def,confluent_def]
+ "diamond(r) ==> confluent(r)";
+by (etac commute_rtrancl 1);
+by (Simp_tac 1);
+qed "diamond_confluent";
+
+
+Goalw [confluent_def]
+ "[| confluent(r); confluent(s); commute(r^*, s^*); \
+\ r<=Sigma(A,B); s<=Sigma(C,D) |] ==> confluent(r Un s)";
+by (rtac (rtrancl_Un_rtrancl RS subst) 1);
+by (blast_tac (claset() addDs [diamond_Un]
+ addIs [rewrite_rule [confluent_def] diamond_confluent]) 3);
+by Auto_tac;
+qed "confluent_Un";
+
+
+Goal
+ "[| diamond(r); s<=r; r<= s^* |] ==> confluent(s)";
+by (dresolve_tac [rtrancl_subset RS sym] 1);
+by (assume_tac 1);
+by (ALLGOALS(asm_simp_tac (simpset() addsimps[confluent_def])));
+by (resolve_tac [rewrite_rule [confluent_def] diamond_confluent] 1);
+by (Asm_simp_tac 1);
+qed "diamond_to_confluence";
+
+(*** Church_Rosser ***)
+
+Goalw [confluent_def, Church_Rosser_def, square_def,commute_def,diamond_def]
+ "Church_Rosser(r) ==> confluent(r)";
+by Auto_tac;
+by (dtac converseI 1);
+by (full_simp_tac (simpset()
+ addsimps [rtrancl_converse RS sym]) 1);
+by (dres_inst_tac [("x", "b")] spec 1);
+by (dres_inst_tac [("x1", "c")] (spec RS mp) 1);
+by (res_inst_tac [("b", "a")] rtrancl_trans 1);
+by (REPEAT(blast_tac (claset() addIs [rtrancl_mono RS subsetD]) 1));
+qed "Church_Rosser1";
+
+
+Goalw [confluent_def, Church_Rosser_def, square_def,commute_def,diamond_def]
+"confluent(r) ==> Church_Rosser(r)";
+by Auto_tac;
+by (forward_tac [fieldI1] 1);
+by (full_simp_tac (simpset() addsimps [rtrancl_field]) 1);
+by (etac rtrancl_induct 1);
+by (ALLGOALS(Clarify_tac));
+by (blast_tac (claset() addIs [rtrancl_refl]) 1);
+by (blast_tac (claset() delrules [rtrancl_refl]
+ addIs [r_into_rtrancl, rtrancl_trans]) 1);
+qed "Church_Rosser2";
+
+
+Goal "Church_Rosser(r) <-> confluent(r)";
+by (blast_tac(claset() addIs [Church_Rosser1,Church_Rosser2]) 1);
+qed "Church_Rosser";
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