(* Title: ZF/trancl.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
For trancl.thy. Transitive closure of a relation
*)
open Trancl;
goal Trancl.thy "bnd_mono(field(r)*field(r), %s. id(field(r)) Un (r O s))";
by (rtac bnd_monoI 1);
by (REPEAT (ares_tac [subset_refl, Un_mono, comp_mono] 2));
by (Fast_tac 1);
qed "rtrancl_bnd_mono";
val [prem] = goalw Trancl.thy [rtrancl_def] "r<=s ==> r^* <= s^*";
by (rtac lfp_mono 1);
by (REPEAT (resolve_tac [rtrancl_bnd_mono, prem, subset_refl, id_mono,
comp_mono, Un_mono, field_mono, Sigma_mono] 1));
qed "rtrancl_mono";
(* r^* = id(field(r)) Un ( r O r^* ) *)
val rtrancl_unfold = rtrancl_bnd_mono RS (rtrancl_def RS def_lfp_Tarski);
(** The relation rtrancl **)
bind_thm ("rtrancl_type", (rtrancl_def RS def_lfp_subset));
(*Reflexivity of rtrancl*)
val [prem] = goal Trancl.thy "[| a: field(r) |] ==> <a,a> : r^*";
by (resolve_tac [rtrancl_unfold RS ssubst] 1);
by (rtac (prem RS idI RS UnI1) 1);
qed "rtrancl_refl";
(*Closure under composition with r *)
val prems = goal Trancl.thy
"[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^*";
by (resolve_tac [rtrancl_unfold RS ssubst] 1);
by (rtac (compI RS UnI2) 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
qed "rtrancl_into_rtrancl";
(*rtrancl of r contains all pairs in r *)
val prems = goal Trancl.thy "<a,b> : r ==> <a,b> : r^*";
by (resolve_tac [rtrancl_refl RS rtrancl_into_rtrancl] 1);
by (REPEAT (resolve_tac (prems@[fieldI1]) 1));
qed "r_into_rtrancl";
(*The premise ensures that r consists entirely of pairs*)
val prems = goal Trancl.thy "r <= Sigma(A,B) ==> r <= r^*";
by (cut_facts_tac prems 1);
by (fast_tac (!claset addIs [r_into_rtrancl]) 1);
qed "r_subset_rtrancl";
goal Trancl.thy "field(r^*) = field(r)";
by (fast_tac (!claset addIs [r_into_rtrancl]
addSDs [rtrancl_type RS subsetD]) 1);
qed "rtrancl_field";
(** standard induction rule **)
val major::prems = goal Trancl.thy
"[| <a,b> : r^*; \
\ !!x. x: field(r) ==> P(<x,x>); \
\ !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |] ==> P(<x,z>) |] \
\ ==> P(<a,b>)";
by (rtac ([rtrancl_def, rtrancl_bnd_mono, major] MRS def_induct) 1);
by (fast_tac (!claset addIs prems) 1);
qed "rtrancl_full_induct";
(*nice induction rule.
Tried adding the typing hypotheses y,z:field(r), but these
caused expensive case splits!*)
val major::prems = goal Trancl.thy
"[| <a,b> : r^*; \
\ P(a); \
\ !!y z.[| <a,y> : r^*; <y,z> : r; P(y) |] ==> P(z) \
\ |] ==> P(b)";
(*by induction on this formula*)
by (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)" 1);
(*now solve first subgoal: this formula is sufficient*)
by (EVERY1 [etac (spec RS mp), rtac refl]);
(*now do the induction*)
by (resolve_tac [major RS rtrancl_full_induct] 1);
by (ALLGOALS (fast_tac (!claset addIs prems)));
qed "rtrancl_induct";
(*transitivity of transitive closure!! -- by induction.*)
goalw Trancl.thy [trans_def] "trans(r^*)";
by (REPEAT (resolve_tac [allI,impI] 1));
by (eres_inst_tac [("b","z")] rtrancl_induct 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, rtrancl_into_rtrancl] 1));
qed "trans_rtrancl";
(*elimination of rtrancl -- by induction on a special formula*)
val major::prems = goal Trancl.thy
"[| <a,b> : r^*; (a=b) ==> P; \
\ !!y.[| <a,y> : r^*; <y,b> : r |] ==> P |] \
\ ==> P";
by (subgoal_tac "a = b | (EX y. <a,y> : r^* & <y,b> : r)" 1);
(*see HOL/trancl*)
by (rtac (major RS rtrancl_induct) 2);
by (ALLGOALS (fast_tac (!claset addSEs prems)));
qed "rtranclE";
(**** The relation trancl ****)
(*Transitivity of r^+ is proved by transitivity of r^* *)
goalw Trancl.thy [trans_def,trancl_def] "trans(r^+)";
by (safe_tac (!claset));
by (rtac (rtrancl_into_rtrancl RS (trans_rtrancl RS transD RS compI)) 1);
by (REPEAT (assume_tac 1));
qed "trans_trancl";
(** Conversions between trancl and rtrancl **)
val [major] = goalw Trancl.thy [trancl_def] "<a,b> : r^+ ==> <a,b> : r^*";
by (resolve_tac [major RS compEpair] 1);
by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));
qed "trancl_into_rtrancl";
(*r^+ contains all pairs in r *)
val [prem] = goalw Trancl.thy [trancl_def] "<a,b> : r ==> <a,b> : r^+";
by (REPEAT (ares_tac [prem,compI,rtrancl_refl,fieldI1] 1));
qed "r_into_trancl";
(*The premise ensures that r consists entirely of pairs*)
val prems = goal Trancl.thy "r <= Sigma(A,B) ==> r <= r^+";
by (cut_facts_tac prems 1);
by (fast_tac (!claset addIs [r_into_trancl]) 1);
qed "r_subset_trancl";
(*intro rule by definition: from r^* and r *)
val prems = goalw Trancl.thy [trancl_def]
"[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^+";
by (REPEAT (resolve_tac ([compI]@prems) 1));
qed "rtrancl_into_trancl1";
(*intro rule from r and r^* *)
val prems = goal Trancl.thy
"[| <a,b> : r; <b,c> : r^* |] ==> <a,c> : r^+";
by (resolve_tac (prems RL [rtrancl_induct]) 1);
by (resolve_tac (prems RL [r_into_trancl]) 1);
by (etac (trans_trancl RS transD) 1);
by (etac r_into_trancl 1);
qed "rtrancl_into_trancl2";
(*Nice induction rule for trancl*)
val major::prems = goal Trancl.thy
"[| <a,b> : r^+; \
\ !!y. [| <a,y> : r |] ==> P(y); \
\ !!y z.[| <a,y> : r^+; <y,z> : r; P(y) |] ==> P(z) \
\ |] ==> P(b)";
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
(*by induction on this formula*)
by (subgoal_tac "ALL z. <y,z> : r --> P(z)" 1);
(*now solve first subgoal: this formula is sufficient*)
by (Fast_tac 1);
by (etac rtrancl_induct 1);
by (ALLGOALS (fast_tac (!claset addIs (rtrancl_into_trancl1::prems))));
qed "trancl_induct";
(*elimination of r^+ -- NOT an induction rule*)
val major::prems = goal Trancl.thy
"[| <a,b> : r^+; \
\ <a,b> : r ==> P; \
\ !!y.[| <a,y> : r^+; <y,b> : r |] ==> P \
\ |] ==> P";
by (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ & <y,b> : r)" 1);
by (fast_tac (!claset addIs prems) 1);
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
by (etac rtranclE 1);
by (ALLGOALS (fast_tac (!claset addIs [rtrancl_into_trancl1])));
qed "tranclE";
goalw Trancl.thy [trancl_def] "r^+ <= field(r)*field(r)";
by (fast_tac (!claset addEs [rtrancl_type RS subsetD RS SigmaE2]) 1);
qed "trancl_type";
val [prem] = goalw Trancl.thy [trancl_def] "r<=s ==> r^+ <= s^+";
by (REPEAT (resolve_tac [prem, comp_mono, rtrancl_mono] 1));
qed "trancl_mono";