--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Trancl.ML Thu Sep 16 12:21:07 1993 +0200
@@ -0,0 +1,240 @@
+(* Title: HOL/trancl
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+For trancl.thy. Theorems about the transitive closure of a relation
+*)
+
+open Trancl;
+
+(** Natural deduction for trans(r) **)
+
+val prems = goalw Trancl.thy [trans_def]
+ "(!! x y z. [| <x,y>:r; <y,z>:r |] ==> <x,z>:r) ==> trans(r)";
+by (REPEAT (ares_tac (prems@[allI,impI]) 1));
+val transI = result();
+
+val major::prems = goalw Trancl.thy [trans_def]
+ "[| trans(r); <a,b>:r; <b,c>:r |] ==> <a,c>:r";
+by (cut_facts_tac [major] 1);
+by (fast_tac (HOL_cs addIs prems) 1);
+val transD = result();
+
+(** Identity relation **)
+
+goalw Trancl.thy [id_def] "<a,a> : id";
+by (rtac CollectI 1);
+by (rtac exI 1);
+by (rtac refl 1);
+val idI = result();
+
+val major::prems = goalw Trancl.thy [id_def]
+ "[| p: id; !!x.[| p = <x,x> |] ==> P \
+\ |] ==> P";
+by (rtac (major RS CollectE) 1);
+by (etac exE 1);
+by (eresolve_tac prems 1);
+val idE = result();
+
+(** Composition of two relations **)
+
+val prems = goalw Trancl.thy [comp_def]
+ "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s";
+by (fast_tac (set_cs addIs prems) 1);
+val compI = result();
+
+(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
+val prems = goalw Trancl.thy [comp_def]
+ "[| xz : r O s; \
+\ !!x y z. [| xz = <x,z>; <x,y>:s; <y,z>:r |] ==> P \
+\ |] ==> P";
+by (cut_facts_tac prems 1);
+by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1));
+val compE = result();
+
+val prems = goal Trancl.thy
+ "[| <a,c> : r O s; \
+\ !!y. [| <a,y>:s; <y,c>:r |] ==> P \
+\ |] ==> P";
+by (rtac compE 1);
+by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
+val compEpair = result();
+
+val comp_cs = set_cs addIs [compI,idI]
+ addSEs [compE,idE,Pair_inject];
+
+val prems = goal Trancl.thy
+ "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
+by (cut_facts_tac prems 1);
+by (fast_tac comp_cs 1);
+val comp_mono = result();
+
+val prems = goal Trancl.thy
+ "[| s <= Sigma(A,%x.B); r <= Sigma(B,%x.C) |] ==> \
+\ (r O s) <= Sigma(A,%x.C)";
+by (cut_facts_tac prems 1);
+by (fast_tac (comp_cs addIs [SigmaI] addSEs [SigmaE2]) 1);
+val comp_subset_Sigma = result();
+
+
+(** The relation rtrancl **)
+
+goal Trancl.thy "mono(%s. id Un (r O s))";
+by (rtac monoI 1);
+by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1));
+val rtrancl_fun_mono = result();
+
+val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski);
+
+(*Reflexivity of rtrancl*)
+goal Trancl.thy "<a,a> : r^*";
+by (stac rtrancl_unfold 1);
+by (fast_tac comp_cs 1);
+val rtrancl_refl = result();
+
+(*Closure under composition with r*)
+val prems = goal Trancl.thy
+ "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^*";
+by (stac rtrancl_unfold 1);
+by (fast_tac (comp_cs addIs prems) 1);
+val rtrancl_into_rtrancl = result();
+
+(*rtrancl of r contains r*)
+val [prem] = goal Trancl.thy "[| <a,b> : r |] ==> <a,b> : r^*";
+by (rtac (rtrancl_refl RS rtrancl_into_rtrancl) 1);
+by (rtac prem 1);
+val r_into_rtrancl = result();
+
+(*monotonicity of rtrancl*)
+goalw Trancl.thy [rtrancl_def] "!!r s. r <= s ==> r^* <= s^*";
+by(REPEAT(ares_tac [lfp_mono,Un_mono,comp_mono,subset_refl] 1));
+val rtrancl_mono = result();
+
+(** standard induction rule **)
+
+val major::prems = goal Trancl.thy
+ "[| <a,b> : r^*; \
+\ !!x. P(<x,x>); \
+\ !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |] ==> P(<x,z>) |] \
+\ ==> P(<a,b>)";
+by (rtac (major RS (rtrancl_def RS def_induct)) 1);
+by (rtac rtrancl_fun_mono 1);
+by (fast_tac (comp_cs addIs prems) 1);
+val rtrancl_full_induct = result();
+
+(*nice induction rule*)
+val major::prems = goal Trancl.thy
+ "[| <a::'a,b> : r^*; \
+\ P(a); \
+\ !!y z.[| <a,y> : r^*; <y,z> : r; P(y) |] ==> P(z) |] \
+\ ==> P(b)";
+(*by induction on this formula*)
+by (subgoal_tac "! y. <a::'a,b> = <a,y> --> P(y)" 1);
+(*now solve first subgoal: this formula is sufficient*)
+by (fast_tac HOL_cs 1);
+(*now do the induction*)
+by (resolve_tac [major RS rtrancl_full_induct] 1);
+by (fast_tac (comp_cs addIs prems) 1);
+by (fast_tac (comp_cs addIs prems) 1);
+val rtrancl_induct = result();
+
+(*transitivity of transitive closure!! -- by induction.*)
+goal Trancl.thy "trans(r^*)";
+by (rtac transI 1);
+by (res_inst_tac [("b","z")] rtrancl_induct 1);
+by (DEPTH_SOLVE (eresolve_tac [asm_rl, rtrancl_into_rtrancl] 1));
+val trans_rtrancl = result();
+
+(*elimination of rtrancl -- by induction on a special formula*)
+val major::prems = goal Trancl.thy
+ "[| <a::'a,b> : r^*; (a = b) ==> P; \
+\ !!y.[| <a,y> : r^*; <y,b> : r |] ==> P \
+\ |] ==> P";
+by (subgoal_tac "a::'a = b | (? y. <a,y> : r^* & <y,b> : r)" 1);
+by (rtac (major RS rtrancl_induct) 2);
+by (fast_tac (set_cs addIs prems) 2);
+by (fast_tac (set_cs addIs prems) 2);
+by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
+val rtranclE = result();
+
+
+(**** The relation trancl ****)
+
+(** Conversions between trancl and rtrancl **)
+
+val [major] = goalw Trancl.thy [trancl_def]
+ "<a,b> : r^+ ==> <a,b> : r^*";
+by (resolve_tac [major RS compEpair] 1);
+by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));
+val trancl_into_rtrancl = result();
+
+(*r^+ contains r*)
+val [prem] = goalw Trancl.thy [trancl_def]
+ "[| <a,b> : r |] ==> <a,b> : r^+";
+by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1));
+val r_into_trancl = result();
+
+(*intro rule by definition: from rtrancl and r*)
+val prems = goalw Trancl.thy [trancl_def]
+ "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^+";
+by (REPEAT (resolve_tac ([compI]@prems) 1));
+val rtrancl_into_trancl1 = result();
+
+(*intro rule from r and rtrancl*)
+val prems = goal Trancl.thy
+ "[| <a,b> : r; <b,c> : r^* |] ==> <a,c> : r^+";
+by (resolve_tac (prems RL [rtranclE]) 1);
+by (etac subst 1);
+by (resolve_tac (prems RL [r_into_trancl]) 1);
+by (rtac (trans_rtrancl RS transD RS rtrancl_into_trancl1) 1);
+by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 1));
+val rtrancl_into_trancl2 = result();
+
+(*elimination of r^+ -- NOT an induction rule*)
+val major::prems = goal Trancl.thy
+ "[| <a::'a,b> : r^+; \
+\ <a,b> : r ==> P; \
+\ !!y.[| <a,y> : r^+; <y,b> : r |] ==> P \
+\ |] ==> P";
+by (subgoal_tac "<a::'a,b> : r | (? y. <a,y> : r^+ & <y,b> : r)" 1);
+by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1));
+by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
+by (etac rtranclE 1);
+by (fast_tac comp_cs 1);
+by (fast_tac (comp_cs addSIs [rtrancl_into_trancl1]) 1);
+val tranclE = result();
+
+(*Transitivity of r^+.
+ Proved by unfolding since it uses transitivity of rtrancl. *)
+goalw Trancl.thy [trancl_def] "trans(r^+)";
+by (rtac transI 1);
+by (REPEAT (etac compEpair 1));
+by (rtac (rtrancl_into_rtrancl RS (trans_rtrancl RS transD RS compI)) 1);
+by (REPEAT (assume_tac 1));
+val trans_trancl = result();
+
+val prems = goal Trancl.thy
+ "[| <a,b> : r; <b,c> : r^+ |] ==> <a,c> : r^+";
+by (rtac (r_into_trancl RS (trans_trancl RS transD)) 1);
+by (resolve_tac prems 1);
+by (resolve_tac prems 1);
+val trancl_into_trancl2 = result();
+
+
+val major::prems = goal Trancl.thy
+ "[| <a,b> : r^*; r <= Sigma(A,%x.A) |] ==> a=b | a:A";
+by (cut_facts_tac prems 1);
+by (rtac (major RS rtrancl_induct) 1);
+by (rtac (refl RS disjI1) 1);
+by (fast_tac (comp_cs addSEs [SigmaE2]) 1);
+val trancl_subset_Sigma_lemma = result();
+
+val prems = goalw Trancl.thy [trancl_def]
+ "r <= Sigma(A,%x.A) ==> trancl(r) <= Sigma(A,%x.A)";
+by (cut_facts_tac prems 1);
+by (fast_tac (comp_cs addIs [SigmaI]
+ addSDs [trancl_subset_Sigma_lemma]
+ addSEs [SigmaE2]) 1);
+val trancl_subset_Sigma = result();
+