author | nipkow |
Tue, 30 Apr 1996 17:30:54 +0200 | |
changeset 1706 | 4e0d5c7f57fa |
parent 1642 | 21db0cf9a1a4 |
child 1746 | f0c6aabc6c02 |
permissions | -rw-r--r-- |
1465 | 1 |
(* 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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(** 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 rel_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 (rel_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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goal Trancl.thy "!!p. p : r ==> p : r^*"; |
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by (split_all_tac 1); |
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by (etac (rtrancl_refl RS rtrancl_into_rtrancl) 1); |
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qed "r_into_rtrancl"; |
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||
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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 (rel_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 (rel_cs addIs prems) 1); |
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by (fast_tac (rel_cs addIs prems) 1); |
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qed "rtrancl_induct"; |
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val prems = goal Trancl.thy |
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"[| ((a,b),(c,d)) : r^*; P a b; \ |
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\ !!x y z u.[| ((a,b),(x,y)) : r^*; ((x,y),(z,u)) : r; P x y |] ==> P z u\ |
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\ |] ==> P c d"; |
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by(res_inst_tac[("R","P")]splitD 1); |
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by(res_inst_tac[("P","split P")]rtrancl_induct 1); |
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brs prems 1; |
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by(Simp_tac 1); |
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brs prems 1; |
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by(split_all_tac 1); |
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by(Asm_full_simp_tac 1); |
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by(REPEAT(ares_tac prems 1)); |
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qed "rtrancl_induct2"; |
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(*transitivity of transitive closure!! -- by induction.*) |
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goalw Trancl.thy [trans_def] "trans(r^*)"; |
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by (safe_tac HOL_cs); |
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by (eres_inst_tac [("b","z")] rtrancl_induct 1); |
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by (ALLGOALS(fast_tac (HOL_cs addIs [rtrancl_into_rtrancl]))); |
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qed "trans_rtrancl"; |
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bind_thm ("rtrancl_trans", trans_rtrancl RS transD); |
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||
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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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bind_thm ("rtrancl_into_rtrancl2", r_into_rtrancl RS rtrancl_trans); |
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(*** More r^* equations and inclusions ***) |
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goal Trancl.thy "(r^*)^* = r^*"; |
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by (rtac set_ext 1); |
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by (res_inst_tac [("p","x")] PairE 1); |
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by (hyp_subst_tac 1); |
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by (rtac iffI 1); |
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by (etac rtrancl_induct 1); |
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by (rtac rtrancl_refl 1); |
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by (fast_tac (HOL_cs addEs [rtrancl_trans]) 1); |
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by (etac r_into_rtrancl 1); |
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qed "rtrancl_idemp"; |
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Addsimps [rtrancl_idemp]; |
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goal Trancl.thy "!!r s. r <= s^* ==> r^* <= s^*"; |
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bd rtrancl_mono 1; |
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by (Asm_full_simp_tac 1); |
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qed "rtrancl_subset_rtrancl"; |
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goal Trancl.thy "!!R. [| R <= S; S <= R^* |] ==> S^* = R^*"; |
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by (dtac rtrancl_mono 1); |
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by (dtac rtrancl_mono 1); |
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by (Asm_full_simp_tac 1); |
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by (fast_tac eq_cs 1); |
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qed "rtrancl_subset"; |
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goal Trancl.thy "!!R. (R^* Un S^*)^* = (R Un S)^*"; |
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by (best_tac (set_cs addIs [rtrancl_subset,r_into_rtrancl, |
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rtrancl_mono RS subsetD]) 1); |
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qed "rtrancl_Un_rtrancl"; |
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goal Trancl.thy "(R^=)^* = R^*"; |
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by (fast_tac (rel_cs addIs [rtrancl_refl,rtrancl_subset,r_into_rtrancl]) 1); |
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qed "rtrancl_reflcl"; |
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Addsimps [rtrancl_reflcl]; |
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goal Trancl.thy "!!r. (x,y) : (converse r)^* ==> (x,y) : converse(r^*)"; |
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by (rtac converseI 1); |
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by (etac rtrancl_induct 1); |
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by (rtac rtrancl_refl 1); |
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by (fast_tac (rel_cs addIs [r_into_rtrancl,rtrancl_trans]) 1); |
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qed "rtrancl_converseD"; |
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goal Trancl.thy "!!r. (x,y) : converse(r^*) ==> (x,y) : (converse r)^*"; |
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by (dtac converseD 1); |
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by (etac rtrancl_induct 1); |
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by (rtac rtrancl_refl 1); |
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by (fast_tac (rel_cs addIs [r_into_rtrancl,rtrancl_trans]) 1); |
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qed "rtrancl_converseI"; |
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goal Trancl.thy "(converse r)^* = converse(r^*)"; |
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by (safe_tac (rel_eq_cs addSIs [rtrancl_converseI])); |
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by (res_inst_tac [("p","x")] PairE 1); |
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by (hyp_subst_tac 1); |
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by (etac rtrancl_converseD 1); |
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qed "rtrancl_converse"; |
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val major::prems = goal Trancl.thy |
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"[| (a,b) : r^*; P(b); \ |
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\ !!y z.[| (y,z) : r; (z,b) : r^*; P(z) |] ==> P(y) |] \ |
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\ ==> P(a)"; |
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br ((major RS converseI RS rtrancl_converseI) RS rtrancl_induct) 1; |
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brs prems 1; |
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by(fast_tac (HOL_cs addIs prems addSEs[converseD]addSDs[rtrancl_converseD])1); |
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qed "converse_rtrancl_induct"; |
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|
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val prems = goal Trancl.thy |
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"[| ((a,b),(c,d)) : r^*; P c d; \ |
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\ !!x y z u.[| ((x,y),(z,u)) : r; ((z,u),(c,d)) : r^*; P z u |] ==> P x y\ |
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\ |] ==> P a b"; |
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by(res_inst_tac[("R","P")]splitD 1); |
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by(res_inst_tac[("P","split P")]converse_rtrancl_induct 1); |
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brs prems 1; |
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by(Simp_tac 1); |
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brs prems 1; |
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by(split_all_tac 1); |
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by(Asm_full_simp_tac 1); |
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by(REPEAT(ares_tac prems 1)); |
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|
188 |
qed "converse_rtrancl_induct2"; |
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|
191 |
(**** The relation trancl ****) |
|
192 |
||
193 |
(** Conversions between trancl and rtrancl **) |
|
194 |
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195 |
val [major] = goalw Trancl.thy [trancl_def] |
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196 |
"(a,b) : r^+ ==> (a,b) : r^*"; |
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by (resolve_tac [major RS compEpair] 1); |
198 |
by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1)); |
|
199 |
qed "trancl_into_rtrancl"; |
|
200 |
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201 |
(*r^+ contains r*) |
|
202 |
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)); |
205 |
qed "r_into_trancl"; |
|
206 |
||
207 |
(*intro rule by definition: from rtrancl and r*) |
|
208 |
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)); |
211 |
qed "rtrancl_into_trancl1"; |
|
212 |
||
213 |
(*intro rule from r and rtrancl*) |
|
214 |
val prems = goal Trancl.thy |
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215 |
"[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+"; |
923 | 216 |
by (resolve_tac (prems RL [rtranclE]) 1); |
217 |
by (etac subst 1); |
|
218 |
by (resolve_tac (prems RL [r_into_trancl]) 1); |
|
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|
219 |
by (rtac (rtrancl_trans RS rtrancl_into_trancl1) 1); |
923 | 220 |
by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 1)); |
221 |
qed "rtrancl_into_trancl2"; |
|
222 |
||
1642 | 223 |
(*Nice induction rule for trancl*) |
224 |
val major::prems = goal Trancl.thy |
|
225 |
"[| (a,b) : r^+; \ |
|
226 |
\ !!y. [| (a,y) : r |] ==> P(y); \ |
|
227 |
\ !!y z.[| (a,y) : r^+; (y,z) : r; P(y) |] ==> P(z) \ |
|
228 |
\ |] ==> P(b)"; |
|
229 |
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1); |
|
230 |
(*by induction on this formula*) |
|
231 |
by (subgoal_tac "ALL z. (y,z) : r --> P(z)" 1); |
|
232 |
(*now solve first subgoal: this formula is sufficient*) |
|
233 |
by (fast_tac HOL_cs 1); |
|
234 |
by (etac rtrancl_induct 1); |
|
235 |
by (ALLGOALS (fast_tac (set_cs addIs (rtrancl_into_trancl1::prems)))); |
|
236 |
qed "trancl_induct"; |
|
237 |
||
923 | 238 |
(*elimination of r^+ -- NOT an induction rule*) |
239 |
val major::prems = goal Trancl.thy |
|
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diff
changeset
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240 |
"[| (a::'a,b) : r^+; \ |
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241 |
\ (a,b) : r ==> P; \ |
1465 | 242 |
\ !!y.[| (a,y) : r^+; (y,b) : r |] ==> P \ |
923 | 243 |
\ |] ==> P"; |
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diff
changeset
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244 |
by (subgoal_tac "(a::'a,b) : r | (? y. (a,y) : r^+ & (y,b) : r)" 1); |
923 | 245 |
by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1)); |
246 |
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1); |
|
247 |
by (etac rtranclE 1); |
|
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
1122
diff
changeset
|
248 |
by (fast_tac rel_cs 1); |
64b30e3cc6d4
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nipkow
parents:
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diff
changeset
|
249 |
by (fast_tac (rel_cs addSIs [rtrancl_into_trancl1]) 1); |
923 | 250 |
qed "tranclE"; |
251 |
||
252 |
(*Transitivity of r^+. |
|
253 |
Proved by unfolding since it uses transitivity of rtrancl. *) |
|
254 |
goalw Trancl.thy [trancl_def] "trans(r^+)"; |
|
255 |
by (rtac transI 1); |
|
256 |
by (REPEAT (etac compEpair 1)); |
|
1122
20b708827030
renamed trans_rtrancl to rtrancl_trans and modified it by expanding trans.
nipkow
parents:
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diff
changeset
|
257 |
by (rtac (rtrancl_into_rtrancl RS (rtrancl_trans RS compI)) 1); |
923 | 258 |
by (REPEAT (assume_tac 1)); |
259 |
qed "trans_trancl"; |
|
260 |
||
1642 | 261 |
bind_thm ("trancl_trans", trans_trancl RS transD); |
262 |
||
923 | 263 |
val prems = goal Trancl.thy |
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clasohm
parents:
923
diff
changeset
|
264 |
"[| (a,b) : r; (b,c) : r^+ |] ==> (a,c) : r^+"; |
923 | 265 |
by (rtac (r_into_trancl RS (trans_trancl RS transD)) 1); |
266 |
by (resolve_tac prems 1); |
|
267 |
by (resolve_tac prems 1); |
|
268 |
qed "trancl_into_trancl2"; |
|
269 |
||
1130 | 270 |
|
923 | 271 |
val major::prems = goal Trancl.thy |
1642 | 272 |
"[| (a,b) : r^*; r <= A Times A |] ==> a=b | a:A"; |
923 | 273 |
by (cut_facts_tac prems 1); |
274 |
by (rtac (major RS rtrancl_induct) 1); |
|
275 |
by (rtac (refl RS disjI1) 1); |
|
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
1122
diff
changeset
|
276 |
by (fast_tac (rel_cs addSEs [SigmaE2]) 1); |
1642 | 277 |
val lemma = result(); |
923 | 278 |
|
279 |
goalw Trancl.thy [trancl_def] |
|
1642 | 280 |
"!!r. r <= A Times A ==> r^+ <= A Times A"; |
281 |
by (fast_tac (rel_cs addSDs [lemma]) 1); |
|
923 | 282 |
qed "trancl_subset_Sigma"; |
1130 | 283 |
|
1301 | 284 |
(* Don't add r_into_rtrancl: it messes up the proofs in Lambda *) |
1130 | 285 |
val trancl_cs = rel_cs addIs [rtrancl_refl]; |
1642 | 286 |
|
287 |