src/ZF/Trancl.ML
author paulson
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permissions -rw-r--r--
ZF arith
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(*  Title:      ZF/trancl.ML
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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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Transitive closure of a relation
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*)
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Goal "bnd_mono(field(r)*field(r), %s. id(field(r)) Un (r O s))";
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by (rtac bnd_monoI 1);
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by (REPEAT (ares_tac [subset_refl, Un_mono, comp_mono] 2));
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by (Blast_tac 1);
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qed "rtrancl_bnd_mono";
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Goalw [rtrancl_def] "r<=s ==> r^* <= s^*";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac [rtrancl_bnd_mono, subset_refl, id_mono,
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		      comp_mono, Un_mono, field_mono, Sigma_mono] 1));
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qed "rtrancl_mono";
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(* r^* = id(field(r)) Un ( r O r^* )    *)
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val rtrancl_unfold = rtrancl_bnd_mono RS (rtrancl_def RS def_lfp_Tarski);
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(** The relation rtrancl **)
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(*  r^* <= field(r) * field(r)  *)
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bind_thm ("rtrancl_type", rtrancl_def RS def_lfp_subset);
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(*Reflexivity of rtrancl*)
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Goal "[| a: field(r) |] ==> <a,a> : r^*";
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by (resolve_tac [rtrancl_unfold RS ssubst] 1);
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by (etac (idI RS UnI1) 1);
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qed "rtrancl_refl";
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(*Closure under composition with r  *)
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Goal "[| <a,b> : r^*;  <b,c> : r |] ==> <a,c> : r^*";
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by (resolve_tac [rtrancl_unfold RS ssubst] 1);
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by (rtac (compI RS UnI2) 1);
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by (assume_tac 1);
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by (assume_tac 1);
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qed "rtrancl_into_rtrancl";
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(*rtrancl of r contains all pairs in r  *)
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Goal "<a,b> : r ==> <a,b> : r^*";
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by (resolve_tac [rtrancl_refl RS rtrancl_into_rtrancl] 1);
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by (REPEAT (ares_tac [fieldI1] 1));
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qed "r_into_rtrancl";
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(*The premise ensures that r consists entirely of pairs*)
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Goal "r <= Sigma(A,B) ==> r <= r^*";
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by (blast_tac (claset() addIs [r_into_rtrancl]) 1);
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qed "r_subset_rtrancl";
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Goal "field(r^*) = field(r)";
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by (blast_tac (claset() addIs [r_into_rtrancl] 
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                    addSDs [rtrancl_type RS subsetD]) 1);
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qed "rtrancl_field";
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(** standard induction rule **)
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val major::prems = Goal
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  "[| <a,b> : r^*; \
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\     !!x. x: field(r) ==> 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_bnd_mono, major] MRS def_induct) 1);
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by (blast_tac (claset() addIs prems) 1);
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qed "rtrancl_full_induct";
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(*nice induction rule.
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  Tried adding the typing hypotheses y,z:field(r), but these
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  caused expensive case splits!*)
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val major::prems = Goal
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  "[| <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 "ALL y. <a,b> = <a,y> --> P(y)" 1);
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(*now solve first subgoal: this formula is sufficient*)
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by (EVERY1 [etac (spec RS mp), rtac refl]);
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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 (ALLGOALS (blast_tac (claset() addIs prems)));
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qed "rtrancl_induct";
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(*transitivity of transitive closure!! -- by induction.*)
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Goalw [trans_def] "trans(r^*)";
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by (REPEAT (resolve_tac [allI,impI] 1));
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by (eres_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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bind_thm ("rtrancl_trans", trans_rtrancl RS transD);
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(*elimination of rtrancl -- by induction on a special formula*)
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val major::prems = Goal
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    "[| <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 = b  | (EX y. <a,y> : r^* & <y,b> : r)" 1);
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(*see HOL/trancl*)
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by (rtac (major RS rtrancl_induct) 2);
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by (ALLGOALS (fast_tac (claset() addSEs prems)));
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qed "rtranclE";
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(**** The relation trancl ****)
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(*Transitivity of r^+ is proved by transitivity of r^*  *)
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Goalw [trans_def,trancl_def] "trans(r^+)";
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by (blast_tac (claset() addIs [rtrancl_into_rtrancl RS 
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			      (trans_rtrancl RS transD RS compI)]) 1);
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qed "trans_trancl";
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bind_thm ("trancl_trans", trans_trancl RS transD);
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(** Conversions between trancl and rtrancl **)
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Goalw [trancl_def] "<a,b> : r^+ ==> <a,b> : r^*";
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by (blast_tac (claset() addIs [rtrancl_into_rtrancl]) 1);
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qed "trancl_into_rtrancl";
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(*r^+ contains all pairs in r  *)
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Goalw [trancl_def] "<a,b> : r ==> <a,b> : r^+";
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by (blast_tac (claset() addSIs [rtrancl_refl]) 1);
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qed "r_into_trancl";
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(*The premise ensures that r consists entirely of pairs*)
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Goal "r <= Sigma(A,B) ==> r <= r^+";
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by (blast_tac (claset() addIs [r_into_trancl]) 1);
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qed "r_subset_trancl";
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(*intro rule by definition: from r^* and r  *)
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Goalw [trancl_def] "[| <a,b> : r^*;  <b,c> : r |]   ==>  <a,c> : r^+";
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by (Blast_tac 1);
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qed "rtrancl_into_trancl1";
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(*intro rule from r and 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 (resolve_tac (prems RL [rtrancl_induct]) 1);
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by (resolve_tac (prems RL [r_into_trancl]) 1);
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by (etac trancl_trans 1);
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by (etac r_into_trancl 1);
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qed "rtrancl_into_trancl2";
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(*Nice induction rule for trancl*)
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val major::prems = Goal
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  "[| <a,b> : r^+;                                      \
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\     !!y.  [| <a,y> : r |] ==> P(y);                   \
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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 (rtac (rewrite_rule [trancl_def] major  RS  compEpair) 1);
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(*by induction on this formula*)
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by (subgoal_tac "ALL z. <y,z> : r --> P(z)" 1);
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parents:
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   158
(*now solve first subgoal: this formula is sufficient*)
2929
4eefc6c22d41 Changed some fast_tac to blast_tac
paulson
parents: 2493
diff changeset
   159
by (Blast_tac 1);
0
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clasohm
parents:
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   160
by (etac rtrancl_induct 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3016
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   161
by (ALLGOALS (fast_tac (claset() addIs (rtrancl_into_trancl1::prems))));
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clasohm
parents: 435
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   162
qed "trancl_induct";
0
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clasohm
parents:
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   163
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   164
(*elimination of r^+ -- NOT an induction rule*)
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   165
val major::prems = Goal
0
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clasohm
parents:
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   166
    "[| <a,b> : r^+;  \
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clasohm
parents:
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   167
\       <a,b> : r ==> P; \
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   168
\       !!y.[| <a,y> : r^+; <y,b> : r |] ==> P  \
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clasohm
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   169
\    |] ==> P";
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clasohm
parents:
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   170
by (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+  &  <y,b> : r)" 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3016
diff changeset
   171
by (fast_tac (claset() addIs prems) 1);
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   172
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
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clasohm
parents:
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   173
by (etac rtranclE 1);
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3016
diff changeset
   174
by (ALLGOALS (blast_tac (claset() addIs [rtrancl_into_trancl1])));
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clasohm
parents: 435
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   175
qed "tranclE";
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parents:
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   176
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62b6288e6005 isatool fixgoal;
wenzelm
parents: 4091
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   177
Goalw [trancl_def] "r^+ <= field(r)*field(r)";
4091
771b1f6422a8 isatool fixclasimp;
wenzelm
parents: 3016
diff changeset
   178
by (blast_tac (claset() addEs [rtrancl_type RS subsetD RS SigmaE2]) 1);
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clasohm
parents: 435
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   179
qed "trancl_type";
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   180
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parents: 5137
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   181
Goalw [trancl_def] "r<=s ==> r^+ <= s^+";
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paulson
parents: 5137
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   182
by (REPEAT (ares_tac [comp_mono, rtrancl_mono] 1));
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clasohm
parents: 435
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   183
qed "trancl_mono";
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parents:
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   184
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   185
(** Suggested by Sidi Ould Ehmety **)
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   186
54d69141a17f new theorems from Sidi Ould Ehmety
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   187
Goal "(r^*)^* = r^*";
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paulson
parents: 5321
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   188
by (rtac equalityI 1);
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paulson
parents: 5321
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   189
by Auto_tac;
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paulson
parents: 5321
diff changeset
   190
by (ALLGOALS (forward_tac [impOfSubs rtrancl_type]));
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   191
by (ALLGOALS Clarify_tac);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   192
by (etac r_into_rtrancl 2);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   193
by (etac rtrancl_induct 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   194
by (asm_full_simp_tac (simpset() addsimps [rtrancl_refl, rtrancl_field]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   195
by (blast_tac (claset() addIs [rtrancl_trans]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   196
qed "rtrancl_idemp";
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   197
Addsimps [rtrancl_idemp];
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paulson
parents: 5321
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   198
54d69141a17f new theorems from Sidi Ould Ehmety
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   199
Goal "[| R <= S; S <= R^* |] ==> S^* = R^*";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   200
by (dtac rtrancl_mono 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   201
by (dtac rtrancl_mono 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   202
by (ALLGOALS Asm_full_simp_tac);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   203
by (Blast_tac 1);
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paulson
parents: 5321
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   204
qed "rtrancl_subset";
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paulson
parents: 5321
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   205
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   206
Goal "[| r<= Sigma(A,B); s<=Sigma(C,D) |] ==> (r^* Un s^*)^* = (r Un s)^*";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   207
by (rtac rtrancl_subset 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   208
by (blast_tac (claset() addDs [r_subset_rtrancl]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   209
by (blast_tac (claset() addIs [rtrancl_mono RS subsetD]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   210
qed "rtrancl_Un_rtrancl";
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paulson
parents: 5321
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   211
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   212
(** "converse" laws by Sidi Ould Ehmety **)
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paulson
parents: 5321
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   213
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   214
Goal "<x,y>:converse(r)^* ==> <x,y>:converse(r^*)";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   215
by (rtac converseI 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   216
by (forward_tac [rtrancl_type RS subsetD] 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   217
by (etac rtrancl_induct 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   218
by (blast_tac (claset() addIs [rtrancl_refl]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   219
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   220
qed "rtrancl_converseD";
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paulson
parents: 5321
diff changeset
   221
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   222
Goal "<x,y>:converse(r^*) ==> <x,y>:converse(r)^*";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   223
by (dtac converseD 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   224
by (forward_tac [rtrancl_type RS subsetD] 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   225
by (etac rtrancl_induct 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   226
by (blast_tac (claset() addIs [rtrancl_refl]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   227
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
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   228
qed "rtrancl_converseI";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   229
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   230
Goal "converse(r)^* = converse(r^*)";
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   231
by (safe_tac (claset() addSIs [equalityI]));
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   232
by (forward_tac [rtrancl_type RS subsetD] 1);
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   233
by (safe_tac (claset() addSDs [rtrancl_converseD] addSIs [rtrancl_converseI]));
54d69141a17f new theorems from Sidi Ould Ehmety
paulson
parents: 5321
diff changeset
   234
qed "rtrancl_converse";