diff -r f04b33ce250f -r a4dc62a46ee4 Trancl.ML
--- a/Trancl.ML	Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,237 +0,0 @@
-(*  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));
-qed "transI";
-
-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);
-qed "transD";
-
-(** Identity relation **)
-
-goalw Trancl.thy [id_def] "<a,a> : id";  
-by (rtac CollectI 1);
-by (rtac exI 1);
-by (rtac refl 1);
-qed "idI";
-
-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);
-qed "idE";
-
-goalw Trancl.thy [id_def] "<a,b>:id = (a=b)";
-by(fast_tac prod_cs 1);
-qed "pair_in_id_conv";
-
-(** 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);
-qed "compI";
-
-(*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));
-qed "compE";
-
-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));
-qed "compEpair";
-
-val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE];
-
-goal Trancl.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
-by (fast_tac comp_cs 1);
-qed "comp_mono";
-
-goal Trancl.thy
-    "!!r s. [| s <= Sigma(A,%x.B);  r <= Sigma(B,%x.C) |] ==> \
-\           (r O s) <= Sigma(A,%x.C)";
-by (fast_tac comp_cs 1);
-qed "comp_subset_Sigma";
-
-
-(** 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));
-qed "rtrancl_fun_mono";
-
-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);
-qed "rtrancl_refl";
-
-(*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);
-qed "rtrancl_into_rtrancl";
-
-(*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);
-qed "r_into_rtrancl";
-
-(*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));
-qed "rtrancl_mono";
-
-(** 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 ([rtrancl_def, rtrancl_fun_mono, major] MRS def_induct) 1);
-by (fast_tac (comp_cs addIs prems) 1);
-qed "rtrancl_full_induct";
-
-(*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);
-qed "rtrancl_induct";
-
-(*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));
-qed "trans_rtrancl";
-
-(*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));
-qed "rtranclE";
-
-
-(**** 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));
-qed "trancl_into_rtrancl";
-
-(*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));
-qed "r_into_trancl";
-
-(*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));
-qed "rtrancl_into_trancl1";
-
-(*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));
-qed "rtrancl_into_trancl2";
-
-(*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);
-qed "tranclE";
-
-(*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));
-qed "trans_trancl";
-
-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);
-qed "trancl_into_trancl2";
-
-
-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);
-qed "trancl_subset_Sigma_lemma";
-
-goalw Trancl.thy [trancl_def]
-    "!!r. r <= Sigma(A,%x.A) ==> trancl(r) <= Sigma(A,%x.A)";
-by (fast_tac (comp_cs addSDs [trancl_subset_Sigma_lemma]) 1);
-qed "trancl_subset_Sigma";
-
-val prod_ss = prod_ss addsimps [pair_in_id_conv];