author  nipkow 
Thu, 15 Feb 1996 08:10:36 +0100  
changeset 1496  c443b2adaf52 
parent 1465  5d7a7e439cec 
child 1552  6f71b5d46700 
permissions  rwrr 
1465  1 
(* Title: HOL/trancl 
923  2 
ID: $Id$ 
1465  3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
923  4 
Copyright 1992 University of Cambridge 
5 

6 
For trancl.thy. Theorems about the transitive closure of a relation 

7 
*) 

8 

9 
open Trancl; 

10 

11 
(** The relation rtrancl **) 

12 

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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)); 

16 
qed "rtrancl_fun_mono"; 

17 

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val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski); 

19 

20 
(*Reflexivity of rtrancl*) 

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goal Trancl.thy "(a,a) : r^*"; 
923  22 
by (stac rtrancl_unfold 1); 
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by (fast_tac rel_cs 1); 
923  24 
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^*"; 
923  29 
by (stac rtrancl_unfold 1); 
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by (fast_tac (rel_cs addIs prems) 1); 
923  31 
qed "rtrancl_into_rtrancl"; 
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(*rtrancl of r contains r*) 

1301  34 
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); 

923  37 
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))"; 
923  51 
by (rtac ([rtrancl_def, rtrancl_fun_mono, major] MRS def_induct) 1); 
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by (fast_tac (rel_cs addIs prems) 1); 
923  53 
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); \ 
1465  59 
\ !!y z.[ (a,y) : r^*; (y,z) : r; P(y) ] ==> P(z) ] \ 
923  60 
\ ==> 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); 
923  63 
(*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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(*transitivity of transitive closure!!  by induction.*) 

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goal Trancl.thy "!!r. [ (a,b):r^*; (b,c):r^* ] ==> (a,c):r^*"; 
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by (eres_inst_tac [("b","c")] rtrancl_induct 1); 
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by(ALLGOALS(fast_tac (HOL_cs addIs [rtrancl_into_rtrancl]))); 
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qed "rtrancl_trans"; 
923  76 

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(*elimination of rtrancl  by induction on a special formula*) 

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val major::prems = goal Trancl.thy 

1465  79 
"[ (a::'a,b) : r^*; (a = b) ==> P; \ 
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\ !!y.[ (a,y) : r^*; (y,b) : r ] ==> P \ 

923  81 
\ ] ==> P"; 
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by (subgoal_tac "(a::'a) = b  (? y. (a,y) : r^* & (y,b) : r)" 1); 
923  83 
by (rtac (major RS rtrancl_induct) 2); 
84 
by (fast_tac (set_cs addIs prems) 2); 

85 
by (fast_tac (set_cs addIs prems) 2); 

86 
by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1)); 

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qed "rtranclE"; 

88 

1496  89 
goal Trancl.thy "!!R. (y,z):R^* ==> !x. (x,y):R > (x,z):R^*"; 
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be rtrancl_induct 1; 

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by(fast_tac (HOL_cs addIs [r_into_rtrancl]) 1); 

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by(fast_tac (HOL_cs addEs [rtrancl_into_rtrancl]) 1); 

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val lemma = result(); 

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goal Trancl.thy "!!R. [ (x,y) : R; (y,z) : R^* ] ==> (x,z) : R^*"; 

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by(fast_tac (HOL_cs addDs [lemma]) 1); 

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qed "rtrancl_into_rtrancl2"; 

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923  99 

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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 (rtrancl_trans 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 rel_cs 1); 
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by (fast_tac (rel_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 (rtrancl_trans 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"; 

161 

1130  162 
(** More about r^* **) 
1121  163 

164 
goal Trancl.thy "(r^*)^* = r^*"; 

1465  165 
by (rtac set_ext 1); 
1121  166 
by(res_inst_tac [("p","x")] PairE 1); 
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by(hyp_subst_tac 1); 

1465  168 
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); 
1121  173 
qed "rtrancl_idemp"; 
1301  174 
Addsimps [rtrancl_idemp]; 
923  175 

1301  176 
goal Trancl.thy "!!R. [ R <= S; S <= R^* ] ==> S^* = R^*"; 
1465  177 
by (dtac rtrancl_mono 1); 
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by (dtac rtrancl_mono 1); 

1301  179 
by(Asm_full_simp_tac 1); 
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by(fast_tac eq_cs 1); 

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qed "rtrancl_subset"; 

182 

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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, 

185 
rtrancl_mono RS subsetD]) 1); 

186 
qed "trancl_Un_trancl"; 

187 

188 
goal Trancl.thy "(R^=)^* = R^*"; 

189 
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]; 

1130  192 

193 
goal Trancl.thy "!!r. (x,y) : (converse r)^* ==> (x,y) : converse(r^*)"; 

1465  194 
by (rtac converseI 1); 
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by (etac rtrancl_induct 1); 

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by (rtac rtrancl_refl 1); 

1130  197 
by(fast_tac (rel_cs addIs [r_into_rtrancl,rtrancl_trans]) 1); 
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qed "rtrancl_converseD"; 

199 

200 
goal Trancl.thy "!!r. (x,y) : converse(r^*) ==> (x,y) : (converse r)^*"; 

1465  201 
by (dtac converseD 1); 
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by (etac rtrancl_induct 1); 

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by (rtac rtrancl_refl 1); 

1130  204 
by(fast_tac (rel_cs addIs [r_into_rtrancl,rtrancl_trans]) 1); 
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qed "rtrancl_converseI"; 

206 

207 
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); 

1465  211 
by (etac rtrancl_converseD 1); 
1130  212 
qed "rtrancl_converse"; 
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214 

923  215 
val major::prems = goal Trancl.thy 
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"[ (a,b) : r^*; r <= Sigma A (%x.A) ] ==> a=b  a:A"; 
923  217 
by (cut_facts_tac prems 1); 
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by (rtac (major RS rtrancl_induct) 1); 

219 
by (rtac (refl RS disjI1) 1); 

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by (fast_tac (rel_cs addSEs [SigmaE2]) 1); 
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qed "trancl_subset_Sigma_lemma"; 
222 

223 
goalw Trancl.thy [trancl_def] 

224 
"!!r. r <= Sigma A (%x.A) ==> trancl(r) <= Sigma A (%x.A)"; 

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by (fast_tac (rel_cs addSDs [trancl_subset_Sigma_lemma]) 1); 
923  226 
qed "trancl_subset_Sigma"; 
1130  227 

1301  228 
(* Don't add r_into_rtrancl: it messes up the proofs in Lambda *) 
1130  229 
val trancl_cs = rel_cs addIs [rtrancl_refl]; 