author | lcp |
Thu, 06 Apr 1995 11:49:42 +0200 | |
changeset 246 | 0f9230a24164 |
parent 184 | d8a5435732cf |
permissions | -rw-r--r-- |
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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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qed "transI"; |
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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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qed "transD"; |
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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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qed "idI"; |
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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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qed "idE"; |
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goalw Trancl.thy [id_def] "<a,b>:id = (a=b)"; |
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by(fast_tac prod_cs 1); |
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qed "pair_in_id_conv"; |
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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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qed "compI"; |
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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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qed "compE"; |
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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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qed "compEpair"; |
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val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE]; |
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goal Trancl.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"; |
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by (fast_tac comp_cs 1); |
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qed "comp_mono"; |
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goal Trancl.thy |
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"!!r s. [| s <= Sigma(A,%x.B); r <= Sigma(B,%x.C) |] ==> \ |
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\ (r O s) <= Sigma(A,%x.C)"; |
82c4117aff7f
HOL/Trancl: comp_cs is based upon prod_cs; tidied proofs
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parents:
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changeset
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by (fast_tac comp_cs 1); |
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qed "comp_subset_Sigma"; |
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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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qed "rtrancl_fun_mono"; |
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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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qed "rtrancl_refl"; |
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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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qed "rtrancl_into_rtrancl"; |
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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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qed "r_into_rtrancl"; |
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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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qed "rtrancl_mono"; |
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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 ([rtrancl_def, rtrancl_fun_mono, major] MRS def_induct) 1); |
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by (fast_tac (comp_cs addIs prems) 1); |
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qed "rtrancl_full_induct"; |
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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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qed "rtrancl_induct"; |
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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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qed "trans_rtrancl"; |
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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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qed "rtranclE"; |
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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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qed "trancl_into_rtrancl"; |
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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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qed "r_into_trancl"; |
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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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qed "rtrancl_into_trancl1"; |
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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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qed "rtrancl_into_trancl2"; |
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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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qed "tranclE"; |
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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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qed "trans_trancl"; |
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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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qed "trancl_into_trancl2"; |
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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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qed "trancl_subset_Sigma_lemma"; |
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82c4117aff7f
HOL/Trancl: comp_cs is based upon prod_cs; tidied proofs
lcp
parents:
90
diff
changeset
|
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goalw Trancl.thy [trancl_def] |
82c4117aff7f
HOL/Trancl: comp_cs is based upon prod_cs; tidied proofs
lcp
parents:
90
diff
changeset
|
233 |
"!!r. r <= Sigma(A,%x.A) ==> trancl(r) <= Sigma(A,%x.A)"; |
82c4117aff7f
HOL/Trancl: comp_cs is based upon prod_cs; tidied proofs
lcp
parents:
90
diff
changeset
|
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by (fast_tac (comp_cs addSDs [trancl_subset_Sigma_lemma]) 1); |
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qed "trancl_subset_Sigma"; |
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val prod_ss = prod_ss addsimps [pair_in_id_conv]; |