Trancl.ML

--- a/Trancl.ML	Fri Nov 11 10:35:03 1994 +0100
+++ b/Trancl.ML	Mon Nov 21 17:50:34 1994 +0100
@@ -13,13 +13,13 @@
 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();
+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);
-val transD = result();
+qed "transD";
 
 (** Identity relation **)
 
@@ -27,7 +27,7 @@
 by (rtac CollectI 1);
 by (rtac exI 1);
 by (rtac refl 1);
-val idI = result();
+qed "idI";
 
 val major::prems = goalw Trancl.thy [id_def]
     "[| p: id;  !!x.[| p = <x,x> |] ==> P  \
@@ -35,14 +35,14 @@
 by (rtac (major RS CollectE) 1);
 by (etac exE 1);
 by (eresolve_tac prems 1);
-val idE = result();
+qed "idE";
 
 (** 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();
+qed "compI";
 
 (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
 val prems = goalw Trancl.thy [comp_def]
@@ -51,7 +51,7 @@
 \    |] ==> P";
 by (cut_facts_tac prems 1);
 by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1));
-val compE = result();
+qed "compE";
 
 val prems = goal Trancl.thy
     "[| <a,c> : r O s;  \
@@ -59,19 +59,19 @@
 \    |] ==> P";
 by (rtac compE 1);
 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
-val compEpair = result();
+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);
-val comp_mono = result();
+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);
-val comp_subset_Sigma = result();
+qed "comp_subset_Sigma";
 
 
 (** The relation rtrancl **)
@@ -79,7 +79,7 @@
 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();
+qed "rtrancl_fun_mono";
 
 val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski);
 
@@ -87,25 +87,25 @@
 goal Trancl.thy "<a,a> : r^*";
 by (stac rtrancl_unfold 1);
 by (fast_tac comp_cs 1);
-val rtrancl_refl = result();
+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);
-val rtrancl_into_rtrancl = result();
+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);
-val r_into_rtrancl = result();
+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));
-val rtrancl_mono = result();
+qed "rtrancl_mono";
 
 (** standard induction rule **)
 
@@ -116,7 +116,7 @@
 \  ==>  P(<a,b>)";
 by (rtac ([rtrancl_def, rtrancl_fun_mono, major] MRS def_induct) 1);
 by (fast_tac (comp_cs addIs prems) 1);
-val rtrancl_full_induct = result();
+qed "rtrancl_full_induct";
 
 (*nice induction rule*)
 val major::prems = goal Trancl.thy
@@ -132,14 +132,14 @@
 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();
+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));
-val trans_rtrancl = result();
+qed "trans_rtrancl";
 
 (*elimination of rtrancl -- by induction on a special formula*)
 val major::prems = goal Trancl.thy
@@ -151,7 +151,7 @@
 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();
+qed "rtranclE";
 
 
 (**** The relation trancl ****)
@@ -162,19 +162,19 @@
     "<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();
+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));
-val r_into_trancl = result();
+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));
-val rtrancl_into_trancl1 = result();
+qed "rtrancl_into_trancl1";
 
 (*intro rule from r and rtrancl*)
 val prems = goal Trancl.thy
@@ -184,7 +184,7 @@
 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();
+qed "rtrancl_into_trancl2";
 
 (*elimination of r^+ -- NOT an induction rule*)
 val major::prems = goal Trancl.thy
@@ -198,7 +198,7 @@
 by (etac rtranclE 1);
 by (fast_tac comp_cs 1);
 by (fast_tac (comp_cs addSIs [rtrancl_into_trancl1]) 1);
-val tranclE = result();
+qed "tranclE";
 
 (*Transitivity of r^+.
   Proved by unfolding since it uses transitivity of rtrancl. *)
@@ -207,14 +207,14 @@
 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();
+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);
-val trancl_into_trancl2 = result();
+qed "trancl_into_trancl2";
 
 
 val major::prems = goal Trancl.thy
@@ -223,10 +223,10 @@
 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();
+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);
-val trancl_subset_Sigma = result();
+qed "trancl_subset_Sigma";