(* Title: HOL/trancl
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
For trancl.thy. Theorems about the transitive closure of a relation
*)
open Trancl;
(** Natural deduction for trans(r) **)
val prems = goalw Trancl.thy [trans_def]
"(!! x y z. [| (x,y):r; (y,z):r |] ==> (x,z):r) ==> trans(r)";
by (REPEAT (ares_tac (prems@[allI,impI]) 1));
qed "transI";
val major::prems = goalw Trancl.thy [trans_def]
"[| trans(r); (a,b):r; (b,c):r |] ==> (a,c):r";
by (cut_facts_tac [major] 1);
by (fast_tac (HOL_cs addIs prems) 1);
qed "transD";
(** Identity relation **)
goalw Trancl.thy [id_def] "(a,a) : id";
by (rtac CollectI 1);
by (rtac exI 1);
by (rtac refl 1);
qed "idI";
val major::prems = goalw Trancl.thy [id_def]
"[| p: id; !!x.[| p = (x,x) |] ==> P \
\ |] ==> P";
by (rtac (major RS CollectE) 1);
by (etac exE 1);
by (eresolve_tac prems 1);
qed "idE";
goalw Trancl.thy [id_def] "(a,b):id = (a=b)";
by(fast_tac prod_cs 1);
qed "pair_in_id_conv";
(** Composition of two relations **)
val prems = goalw Trancl.thy [comp_def]
"[| (a,b):s; (b,c):r |] ==> (a,c) : r O s";
by (fast_tac (set_cs addIs prems) 1);
qed "compI";
(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
val prems = goalw Trancl.thy [comp_def]
"[| xz : r O s; \
\ !!x y z. [| xz = (x,z); (x,y):s; (y,z):r |] ==> P \
\ |] ==> P";
by (cut_facts_tac prems 1);
by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1));
qed "compE";
val prems = goal Trancl.thy
"[| (a,c) : r O s; \
\ !!y. [| (a,y):s; (y,c):r |] ==> P \
\ |] ==> P";
by (rtac compE 1);
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
qed "compEpair";
val comp_cs = prod_cs addIs [compI, idI] addSEs [compE, idE];
goal Trancl.thy "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
by (fast_tac comp_cs 1);
qed "comp_mono";
goal Trancl.thy
"!!r s. [| s <= Sigma A (%x.B); r <= Sigma B (%x.C) |] ==> \
\ (r O s) <= Sigma A (%x.C)";
by (fast_tac comp_cs 1);
qed "comp_subset_Sigma";
(** The relation rtrancl **)
goal Trancl.thy "mono(%s. id Un (r O s))";
by (rtac monoI 1);
by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1));
qed "rtrancl_fun_mono";
val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski);
(*Reflexivity of rtrancl*)
goal Trancl.thy "(a,a) : r^*";
by (stac rtrancl_unfold 1);
by (fast_tac comp_cs 1);
qed "rtrancl_refl";
(*Closure under composition with r*)
val prems = goal Trancl.thy
"[| (a,b) : r^*; (b,c) : r |] ==> (a,c) : r^*";
by (stac rtrancl_unfold 1);
by (fast_tac (comp_cs addIs prems) 1);
qed "rtrancl_into_rtrancl";
(*rtrancl of r contains r*)
val [prem] = goal Trancl.thy "[| (a,b) : r |] ==> (a,b) : r^*";
by (rtac (rtrancl_refl RS rtrancl_into_rtrancl) 1);
by (rtac prem 1);
qed "r_into_rtrancl";
(*monotonicity of rtrancl*)
goalw Trancl.thy [rtrancl_def] "!!r s. r <= s ==> r^* <= s^*";
by(REPEAT(ares_tac [lfp_mono,Un_mono,comp_mono,subset_refl] 1));
qed "rtrancl_mono";
(** standard induction rule **)
val major::prems = goal Trancl.thy
"[| (a,b) : r^*; \
\ !!x. 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_fun_mono, major] MRS def_induct) 1);
by (fast_tac (comp_cs addIs prems) 1);
qed "rtrancl_full_induct";
(*nice induction rule*)
val major::prems = goal Trancl.thy
"[| (a::'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 "! y. (a::'a,b) = (a,y) --> P(y)" 1);
(*now solve first subgoal: this formula is sufficient*)
by (fast_tac HOL_cs 1);
(*now do the induction*)
by (resolve_tac [major RS rtrancl_full_induct] 1);
by (fast_tac (comp_cs addIs prems) 1);
by (fast_tac (comp_cs addIs prems) 1);
qed "rtrancl_induct";
(*transitivity of transitive closure!! -- by induction.*)
goal Trancl.thy "trans(r^*)";
by (rtac transI 1);
by (res_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::'a,b) : r^*; (a = b) ==> P; \
\ !!y.[| (a,y) : r^*; (y,b) : r |] ==> P \
\ |] ==> P";
by (subgoal_tac "(a::'a) = b | (? y. (a,y) : r^* & (y,b) : r)" 1);
by (rtac (major RS rtrancl_induct) 2);
by (fast_tac (set_cs addIs prems) 2);
by (fast_tac (set_cs addIs prems) 2);
by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
qed "rtranclE";
(**** The relation 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 r*)
val [prem] = goalw Trancl.thy [trancl_def]
"[| (a,b) : r |] ==> (a,b) : r^+";
by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1));
qed "r_into_trancl";
(*intro rule by definition: from rtrancl 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 rtrancl*)
val prems = goal Trancl.thy
"[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+";
by (resolve_tac (prems RL [rtranclE]) 1);
by (etac subst 1);
by (resolve_tac (prems RL [r_into_trancl]) 1);
by (rtac (trans_rtrancl RS transD RS rtrancl_into_trancl1) 1);
by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 1));
qed "rtrancl_into_trancl2";
(*elimination of r^+ -- NOT an induction rule*)
val major::prems = goal Trancl.thy
"[| (a::'a,b) : r^+; \
\ (a,b) : r ==> P; \
\ !!y.[| (a,y) : r^+; (y,b) : r |] ==> P \
\ |] ==> P";
by (subgoal_tac "(a::'a,b) : r | (? y. (a,y) : r^+ & (y,b) : r)" 1);
by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1));
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
by (etac rtranclE 1);
by (fast_tac comp_cs 1);
by (fast_tac (comp_cs addSIs [rtrancl_into_trancl1]) 1);
qed "tranclE";
(*Transitivity of r^+.
Proved by unfolding since it uses transitivity of rtrancl. *)
goalw Trancl.thy [trancl_def] "trans(r^+)";
by (rtac transI 1);
by (REPEAT (etac compEpair 1));
by (rtac (rtrancl_into_rtrancl RS (trans_rtrancl RS transD RS compI)) 1);
by (REPEAT (assume_tac 1));
qed "trans_trancl";
val prems = goal Trancl.thy
"[| (a,b) : r; (b,c) : r^+ |] ==> (a,c) : r^+";
by (rtac (r_into_trancl RS (trans_trancl RS transD)) 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
qed "trancl_into_trancl2";
val major::prems = goal Trancl.thy
"[| (a,b) : r^*; r <= Sigma A (%x.A) |] ==> a=b | a:A";
by (cut_facts_tac prems 1);
by (rtac (major RS rtrancl_induct) 1);
by (rtac (refl RS disjI1) 1);
by (fast_tac (comp_cs addSEs [SigmaE2]) 1);
qed "trancl_subset_Sigma_lemma";
goalw Trancl.thy [trancl_def]
"!!r. r <= Sigma A (%x.A) ==> trancl(r) <= Sigma A (%x.A)";
by (fast_tac (comp_cs addSDs [trancl_subset_Sigma_lemma]) 1);
qed "trancl_subset_Sigma";
val prod_ss = prod_ss addsimps [pair_in_id_conv];