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