# HG changeset patch
# User wenzelm
# Date 980464536 -3600
# Node ID 0a45f2efaaecc9142c5bf0bdc11eac4f70d6a5de
# Parent  3da4543034e7f53e0bfadf5a0e7a5fd1f25a2e76
Transitive_Closure turned into new-style theory;

diff -r 3da4543034e7 -r 0a45f2efaaec src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Jan 26 00:14:25 2001 +0100
+++ b/src/HOL/IsaMakefile	Fri Jan 26 00:15:36 2001 +0100
@@ -97,7 +97,7 @@
   Tools/inductive_package.ML Tools/meson.ML Tools/numeral_syntax.ML \
   Tools/primrec_package.ML Tools/recdef_package.ML \
   Tools/record_package.ML Tools/svc_funcs.ML Tools/typedef_package.ML \
-  Transitive_Closure.ML Transitive_Closure.thy Wellfounded_Recursion.ML \
+  Transitive_Closure.thy Transitive_Closure_lemmas.ML Wellfounded_Recursion.ML \
   Wellfounded_Recursion.thy Wellfounded_Relations.ML \
   Wellfounded_Relations.thy arith_data.ML blastdata.ML cladata.ML \
   equalities.ML equalities.thy hologic.ML meson_lemmas.ML mono.ML \
diff -r 3da4543034e7 -r 0a45f2efaaec src/HOL/Transitive_Closure.thy
--- a/src/HOL/Transitive_Closure.thy	Fri Jan 26 00:14:25 2001 +0100
+++ b/src/HOL/Transitive_Closure.thy	Fri Jan 26 00:15:36 2001 +0100
@@ -13,7 +13,8 @@
 to be atomic.
 *)
 
-Transitive_Closure = Lfp + Relation + 
+theory Transitive_Closure = Lfp + Relation
+files ("Transitive_Closure_lemmas.ML"):
 
 constdefs
   rtrancl :: "('a * 'a) set => ('a * 'a) set"    ("(_^*)" [1000] 999)
@@ -32,4 +33,6 @@
   trancl :: "('a * 'a) set => ('a * 'a) set"    ("(_\\<^sup>+)" [1000] 999)
   "_reflcl" :: "('a * 'a) set => ('a * 'a) set"    ("(_\\<^sup>=)" [1000] 999)
 
+use "Transitive_Closure_lemmas.ML"
+
 end
diff -r 3da4543034e7 -r 0a45f2efaaec src/HOL/Transitive_Closure_lemmas.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Transitive_Closure_lemmas.ML	Fri Jan 26 00:15:36 2001 +0100
@@ -0,0 +1,406 @@
+(*  Title:      HOL/Transitive_Closure_lemmas.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Theorems about the transitive closure of a relation
+*)
+
+val rtrancl_def = thm "rtrancl_def";
+val trancl_def = thm "trancl_def";
+
+
+(** The relation rtrancl **)
+
+section "^*";
+
+Goal "mono(%s. Id Un (r O s))";
+by (rtac monoI 1);
+by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1));
+qed "rtrancl_fun_mono";
+
+bind_thm ("rtrancl_unfold", rtrancl_fun_mono RS (rtrancl_def RS def_lfp_unfold));
+
+(*Reflexivity of rtrancl*)
+Goal "(a,a) : r^*";
+by (stac rtrancl_unfold 1);
+by (Blast_tac 1);
+qed "rtrancl_refl";
+
+Addsimps [rtrancl_refl];
+AddSIs   [rtrancl_refl];
+
+
+(*Closure under composition with r*)
+Goal "[| (a,b) : r^*;  (b,c) : r |] ==> (a,c) : r^*";
+by (stac rtrancl_unfold 1);
+by (Blast_tac 1);
+qed "rtrancl_into_rtrancl";
+
+(*rtrancl of r contains r*)
+Goal "!!p. p : r ==> p : r^*";
+by (split_all_tac 1);
+by (etac (rtrancl_refl RS rtrancl_into_rtrancl) 1);
+qed "r_into_rtrancl";
+
+AddIs [r_into_rtrancl];
+
+(*monotonicity of rtrancl*)
+Goalw [rtrancl_def] "r <= s ==> r^* <= s^*";
+by (REPEAT(ares_tac [lfp_mono,Un_mono,comp_mono,subset_refl] 1));
+qed "rtrancl_mono";
+
+(** standard induction rule **)
+
+val major::prems = Goal 
+  "[| (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 ([rtrancl_def, rtrancl_fun_mono, major] MRS def_lfp_induct) 1);
+by (blast_tac (claset() addIs prems) 1);
+qed "rtrancl_full_induct";
+
+(*nice induction rule*)
+val major::prems = Goal
+    "[| (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 (Blast_tac 1);
+(*now do the induction*)
+by (resolve_tac [major RS rtrancl_full_induct] 1);
+by (blast_tac (claset() addIs prems) 1);
+by (blast_tac (claset() addIs prems) 1);
+qed "rtrancl_induct";
+
+bind_thm ("rtrancl_induct2", split_rule
+  (read_instantiate [("a","(ax,ay)"), ("b","(bx,by)")] rtrancl_induct));
+
+(*transitivity of transitive closure!! -- by induction.*)
+Goalw [trans_def] "trans(r^*)";
+by Safe_tac;
+by (eres_inst_tac [("b","z")] rtrancl_induct 1);
+by (ALLGOALS(blast_tac (claset() addIs [rtrancl_into_rtrancl])));
+qed "trans_rtrancl";
+
+bind_thm ("rtrancl_trans", trans_rtrancl RS transD);
+
+
+(*elimination of rtrancl -- by induction on a special formula*)
+val major::prems = Goal
+    "[| (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 (blast_tac (claset() addIs prems) 2);
+by (blast_tac (claset() addIs prems) 2);
+by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
+qed "rtranclE";
+
+bind_thm ("rtrancl_into_rtrancl2", r_into_rtrancl RS rtrancl_trans);
+
+(*** More r^* equations and inclusions ***)
+
+Goal "(r^*)^* = r^*";
+by Auto_tac;
+by (etac rtrancl_induct 1);
+by (rtac rtrancl_refl 1);
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "rtrancl_idemp";
+Addsimps [rtrancl_idemp];
+
+Goal "R^* O R^* = R^*";
+by (rtac set_ext 1);
+by (split_all_tac 1);
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "rtrancl_idemp_self_comp";
+Addsimps [rtrancl_idemp_self_comp];
+
+Goal "r <= s^* ==> r^* <= s^*";
+by (dtac rtrancl_mono 1);
+by (Asm_full_simp_tac 1);
+qed "rtrancl_subset_rtrancl";
+
+Goal "[| R <= S; S <= R^* |] ==> S^* = R^*";
+by (dtac rtrancl_mono 1);
+by (dtac rtrancl_mono 1);
+by (Asm_full_simp_tac 1);
+by (Blast_tac 1);
+qed "rtrancl_subset";
+
+Goal "(R^* Un S^*)^* = (R Un S)^*";
+by (blast_tac (claset() addSIs [rtrancl_subset]
+                        addIs [r_into_rtrancl, rtrancl_mono RS subsetD]) 1);
+qed "rtrancl_Un_rtrancl";
+
+Goal "(R^=)^* = R^*";
+by (blast_tac (claset() addSIs [rtrancl_subset] addIs [r_into_rtrancl]) 1);
+qed "rtrancl_reflcl";
+Addsimps [rtrancl_reflcl];
+
+Goal "(r - Id)^* = r^*";
+by (rtac sym 1);
+by (rtac rtrancl_subset 1);
+ by (Blast_tac 1);
+by (Clarify_tac 1);
+by (rename_tac "a b" 1);
+by (case_tac "a=b" 1);
+ by (Blast_tac 1);
+by (blast_tac (claset() addSIs [r_into_rtrancl]) 1);
+qed "rtrancl_r_diff_Id";
+
+Goal "(x,y) : (r^-1)^* ==> (y,x) : r^*";
+by (etac rtrancl_induct 1);
+by (rtac rtrancl_refl 1);
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "rtrancl_converseD";
+
+Goal "(y,x) : r^* ==> (x,y) : (r^-1)^*";
+by (etac rtrancl_induct 1);
+by (rtac rtrancl_refl 1);
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "rtrancl_converseI";
+
+Goal "(r^-1)^* = (r^*)^-1";
+(*blast_tac fails: the split_all_tac wrapper must be called to convert
+  the set element to a pair*)
+by (safe_tac (claset() addSDs [rtrancl_converseD] addSIs [rtrancl_converseI]));
+qed "rtrancl_converse";
+
+val major::prems = Goal
+    "[| (a,b) : r^*; P(b); \
+\       !!y z.[| (y,z) : r;  (z,b) : r^*;  P(z) |] ==> P(y) |]  \
+\     ==> P(a)";
+by (rtac (major RS rtrancl_converseI RS rtrancl_induct) 1);
+by (resolve_tac prems 1);
+by (blast_tac (claset() addIs prems addSDs[rtrancl_converseD])1);
+qed "converse_rtrancl_induct";
+
+bind_thm ("converse_rtrancl_induct2", split_rule
+  (read_instantiate [("a","(ax,ay)"),("b","(bx,by)")]converse_rtrancl_induct));
+
+val major::prems = Goal
+ "[| (x,z):r^*; \
+\    x=z ==> P; \
+\    !!y. [| (x,y):r; (y,z):r^* |] ==> P \
+\ |] ==> P";
+by (subgoal_tac "x = z  | (? y. (x,y) : r & (y,z) : r^*)" 1);
+by (rtac (major RS converse_rtrancl_induct) 2);
+by (blast_tac (claset() addIs prems) 2);
+by (blast_tac (claset() addIs prems) 2);
+by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
+qed "converse_rtranclE";
+
+bind_thm ("converse_rtranclE2", split_rule
+  (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] converse_rtranclE));
+
+Goal "r O r^* = r^* O r";
+by (blast_tac (claset() addEs [rtranclE, converse_rtranclE] 
+	               addIs [rtrancl_into_rtrancl, rtrancl_into_rtrancl2]) 1);
+qed "r_comp_rtrancl_eq";
+
+
+(**** The relation trancl ****)
+
+section "^+";
+
+Goalw [trancl_def] "[| p:r^+; r <= s |] ==> p:s^+";
+by (blast_tac (claset() addIs [rtrancl_mono RS subsetD]) 1);
+qed "trancl_mono";
+
+(** Conversions between trancl and rtrancl **)
+
+Goalw [trancl_def]
+    "!!p. p : r^+ ==> p : r^*";
+by (split_all_tac 1);
+by (etac compEpair 1);
+by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));
+qed "trancl_into_rtrancl";
+
+(*r^+ contains r*)
+Goalw [trancl_def]
+   "!!p. p : r ==> p : r^+";
+by (split_all_tac 1);
+by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1));
+qed "r_into_trancl";
+AddIs [r_into_trancl];
+
+(*intro rule by definition: from rtrancl and r*)
+Goalw [trancl_def] "[| (a,b) : r^*;  (b,c) : r |]   ==>  (a,c) : r^+";
+by Auto_tac;
+qed "rtrancl_into_trancl1";
+
+(*intro rule from r and rtrancl*)
+Goal "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+";
+by (etac rtranclE 1);
+by (blast_tac (claset() addIs [r_into_trancl]) 1);
+by (rtac (rtrancl_trans RS rtrancl_into_trancl1) 1);
+by (REPEAT (ares_tac [r_into_rtrancl] 1));
+qed "rtrancl_into_trancl2";
+
+(*Nice induction rule for trancl*)
+val major::prems = Goal
+  "[| (a,b) : r^+;                                      \
+\     !!y.  [| (a,y) : r |] ==> P(y);                   \
+\     !!y z.[| (a,y) : r^+;  (y,z) : r;  P(y) |] ==> P(z)       \
+\  |] ==> P(b)";
+by (rtac (rewrite_rule [trancl_def] major  RS  compEpair) 1);
+(*by induction on this formula*)
+by (subgoal_tac "ALL z. (y,z) : r --> P(z)" 1);
+(*now solve first subgoal: this formula is sufficient*)
+by (Blast_tac 1);
+by (etac rtrancl_induct 1);
+by (ALLGOALS (blast_tac (claset() addIs (rtrancl_into_trancl1::prems))));
+qed "trancl_induct";
+
+(*Another induction rule for trancl, incorporating transitivity.*)
+val major::prems = Goal
+ "[| (x,y) : r^+; \
+\    !!x y. (x,y) : r ==> P x y; \
+\    !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z \
+\ |] ==> P x y";
+by (blast_tac (claset() addIs ([r_into_trancl,major RS trancl_induct]@prems))1);
+qed "trancl_trans_induct";
+
+(*elimination of r^+ -- NOT an induction rule*)
+val major::prems = Goal
+    "[| (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 (Blast_tac 1);
+by (blast_tac (claset() addSIs [rtrancl_into_trancl1]) 1);
+qed "tranclE";
+
+(*Transitivity of r^+.
+  Proved by unfolding since it uses transitivity of rtrancl. *)
+Goalw [trancl_def] "trans(r^+)";
+by (rtac transI 1);
+by (REPEAT (etac compEpair 1));
+by (rtac (rtrancl_into_rtrancl RS (rtrancl_trans RS compI)) 1);
+by (REPEAT (assume_tac 1));
+qed "trans_trancl";
+
+bind_thm ("trancl_trans", trans_trancl RS transD);
+
+Goalw [trancl_def] "[| (x,y):r^*; (y,z):r^+ |] ==> (x,z):r^+";
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "rtrancl_trancl_trancl";
+
+(* "[| (a,b) : r;  (b,c) : r^+ |]   ==>  (a,c) : r^+" *)
+bind_thm ("trancl_into_trancl2", [trans_trancl, r_into_trancl] MRS transD);
+
+(* primitive recursion for trancl over finite relations: *)
+Goal "(insert (y,x) r)^+ = r^+ Un {(a,b). (a,y):r^* & (x,b):r^*}";
+by (rtac equalityI 1);
+ by (rtac subsetI 1);
+ by (split_all_tac 1);
+ by (etac trancl_induct 1);
+  by (blast_tac (claset() addIs [r_into_trancl]) 1);
+ by (blast_tac (claset() addIs
+     [rtrancl_into_trancl1,trancl_into_rtrancl,r_into_trancl,trancl_trans]) 1);
+by (rtac subsetI 1);
+by (blast_tac (claset() addIs
+     [rtrancl_into_trancl2, rtrancl_trancl_trancl,
+      impOfSubs rtrancl_mono, trancl_mono]) 1);
+qed "trancl_insert";
+
+Goalw [trancl_def] "(r^-1)^+ = (r^+)^-1";
+by (simp_tac (simpset() addsimps [rtrancl_converse,converse_comp]) 1);
+by (simp_tac (simpset() addsimps [rtrancl_converse RS sym,
+				  r_comp_rtrancl_eq]) 1);
+qed "trancl_converse";
+
+Goal "(x,y) : (r^+)^-1 ==> (x,y) : (r^-1)^+";
+by (asm_full_simp_tac (simpset() addsimps [trancl_converse]) 1);
+qed "trancl_converseI";
+
+Goal "(x,y) : (r^-1)^+ ==> (x,y) : (r^+)^-1";
+by (asm_full_simp_tac (simpset() addsimps [trancl_converse]) 1);
+qed "trancl_converseD";
+
+val major::prems = Goal
+    "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y); \
+\       !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]  \
+\     ==> P(a)";
+by (rtac ((major RS converseI RS trancl_converseI) RS trancl_induct) 1);
+ by (resolve_tac prems 1);
+ by (etac converseD 1);
+by (blast_tac (claset() addIs prems addSDs [trancl_converseD])1);
+qed "converse_trancl_induct";
+
+Goal "(x,y):R^+ ==> ? z. (x,z):R & (z,y):R^*";
+be converse_trancl_induct 1;
+by Auto_tac;
+by (blast_tac (claset() addIs [rtrancl_trans]) 1);
+qed "tranclD";
+
+(*Unused*)
+Goal "r^-1 Int r^+ = {} ==> (x, x) ~: r^+";
+by (subgoal_tac "!y. (x, y) : r^+ --> x~=y" 1);
+by (Fast_tac 1);
+by (strip_tac 1);
+by (etac trancl_induct 1);
+by (auto_tac (claset() addIs [r_into_trancl], simpset()));
+qed "irrefl_tranclI";
+
+Goal "!!X. [| !x. (x, x) ~: r^+; (x,y) : r |] ==> x ~= y";
+by (blast_tac (claset() addDs [r_into_trancl]) 1);
+qed "irrefl_trancl_rD";
+
+Goal "[| (a,b) : r^*;  r <= A <*> A |] ==> a=b | a:A";
+by (etac rtrancl_induct 1);
+by Auto_tac;
+val lemma = result();
+
+Goalw [trancl_def] "r <= A <*> A ==> r^+ <= A <*> A";
+by (blast_tac (claset() addSDs [lemma]) 1);
+qed "trancl_subset_Sigma";
+
+
+Goal "(r^+)^= = r^*";
+by Safe_tac;
+by  (etac trancl_into_rtrancl 1);
+by (blast_tac (claset() addEs [rtranclE] addDs [rtrancl_into_trancl1]) 1);
+qed "reflcl_trancl";
+Addsimps[reflcl_trancl];
+
+Goal "(r^=)^+ = r^*";
+by Safe_tac;
+by  (dtac trancl_into_rtrancl 1);
+by  (Asm_full_simp_tac 1);
+by (etac rtranclE 1);
+by  Safe_tac;
+by  (rtac r_into_trancl 1);
+by  (Simp_tac 1);
+by (rtac rtrancl_into_trancl1 1);
+by (etac (rtrancl_reflcl RS equalityD2 RS subsetD) 1);
+by (Fast_tac 1);
+qed "trancl_reflcl";
+Addsimps[trancl_reflcl];
+
+Goal "{}^+ = {}";
+by (auto_tac (claset() addEs [trancl_induct], simpset()));
+qed "trancl_empty";
+Addsimps[trancl_empty];
+
+Goal "{}^* = Id";
+by (rtac (reflcl_trancl RS subst) 1);
+by (Simp_tac 1);
+qed "rtrancl_empty";
+Addsimps[rtrancl_empty];
+
+Goal "(a,b):R^* ==> a=b | a~=b & (a,b):R^+";
+by(force_tac (claset(), simpset() addsimps [reflcl_trancl RS sym] 
+				  delsimps [reflcl_trancl]) 1);
+qed "rtranclD";
+