(* 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;
(** 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 (Blast_tac 1);
qed "rtrancl_refl";
Addsimps [rtrancl_refl];
AddSIs [rtrancl_refl];
(*Closure under composition with r*)
goal Trancl.thy "!!r. [| (a,b) : r^*; (b,c) : r |] ==> (a,c) : r^*";
by (stac rtrancl_unfold 1);
by (Blast_tac 1);
qed "rtrancl_into_rtrancl";
(*rtrancl of r contains r*)
goal Trancl.thy "!!p. p : r ==> p : r^*";
by (split_all_tac 1);
by (etac (rtrancl_refl RS rtrancl_into_rtrancl) 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 (!claset 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 (Blast_tac 1);
(*now do the induction*)
by (resolve_tac [major RS rtrancl_full_induct] 1);
by (fast_tac (!claset addIs prems) 1);
by (fast_tac (!claset addIs prems) 1);
qed "rtrancl_induct";
bind_thm
("rtrancl_induct2",
Prod_Syntax.split_rule
(read_instantiate [("a","(ax,ay)"), ("b","(bx,by)")] rtrancl_induct));
(*transitivity of transitive closure!! -- by induction.*)
goalw Trancl.thy [trans_def] "trans(r^*)";
by (safe_tac (!claset));
by (eres_inst_tac [("b","z")] rtrancl_induct 1);
by (ALLGOALS(fast_tac (!claset addIs [rtrancl_into_rtrancl])));
qed "trans_rtrancl";
bind_thm ("rtrancl_trans", trans_rtrancl RS transD);
(*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 (!claset addIs prems) 2);
by (fast_tac (!claset addIs prems) 2);
by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
qed "rtranclE";
bind_thm ("rtrancl_into_rtrancl2", r_into_rtrancl RS rtrancl_trans);
(*** More r^* equations and inclusions ***)
goal Trancl.thy "(r^*)^* = r^*";
by (rtac set_ext 1);
by (res_inst_tac [("p","x")] PairE 1);
by (hyp_subst_tac 1);
by (rtac iffI 1);
by (etac rtrancl_induct 1);
by (rtac rtrancl_refl 1);
by (fast_tac (!claset addEs [rtrancl_trans]) 1);
by (etac r_into_rtrancl 1);
qed "rtrancl_idemp";
Addsimps [rtrancl_idemp];
goal Trancl.thy "!!r s. r <= s^* ==> r^* <= s^*";
by (dtac rtrancl_mono 1);
by (Asm_full_simp_tac 1);
qed "rtrancl_subset_rtrancl";
goal Trancl.thy "!!R. [| R <= S; S <= R^* |] ==> S^* = R^*";
by (dtac rtrancl_mono 1);
by (dtac rtrancl_mono 1);
by (Asm_full_simp_tac 1);
by (Blast_tac 1);
qed "rtrancl_subset";
goal Trancl.thy "!!R. (R^* Un S^*)^* = (R Un S)^*";
by (best_tac (!claset addSIs [rtrancl_subset]
addIs [r_into_rtrancl, rtrancl_mono RS subsetD]) 1);
qed "rtrancl_Un_rtrancl";
goal Trancl.thy "(R^=)^* = R^*";
by (fast_tac (!claset addSIs [rtrancl_refl,rtrancl_subset]
addIs [r_into_rtrancl]) 1);
qed "rtrancl_reflcl";
Addsimps [rtrancl_reflcl];
goal Trancl.thy "!!r. (x,y) : (converse r)^* ==> (x,y) : converse(r^*)";
by (rtac converseI 1);
by (etac rtrancl_induct 1);
by (rtac rtrancl_refl 1);
by (deepen_tac (!claset addIs [r_into_rtrancl,rtrancl_trans]) 0 1);
qed "rtrancl_converseD";
goal Trancl.thy "!!r. (x,y) : converse(r^*) ==> (x,y) : (converse r)^*";
by (dtac converseD 1);
by (etac rtrancl_induct 1);
by (rtac rtrancl_refl 1);
by (deepen_tac (!claset addIs [r_into_rtrancl,rtrancl_trans]) 0 1);
qed "rtrancl_converseI";
goal Trancl.thy "(converse r)^* = converse(r^*)";
by (safe_tac (!claset addSIs [rtrancl_converseI]));
by (res_inst_tac [("p","x")] PairE 1);
by (hyp_subst_tac 1);
by (etac rtrancl_converseD 1);
qed "rtrancl_converse";
val major::prems = goal Trancl.thy
"[| (a,b) : r^*; P(b); \
\ !!y z.[| (y,z) : r; (z,b) : r^*; P(z) |] ==> P(y) |] \
\ ==> P(a)";
by (rtac ((major RS converseI RS rtrancl_converseI) RS rtrancl_induct) 1);
by (resolve_tac prems 1);
by (fast_tac (!claset addIs prems addSDs[rtrancl_converseD])1);
qed "converse_rtrancl_induct";
val prems = goal Trancl.thy
"[| ((a,b),(c,d)) : r^*; P c d; \
\ !!x y z u.[| ((x,y),(z,u)) : r; ((z,u),(c,d)) : r^*; P z u |] ==> P x y\
\ |] ==> P a b";
by (res_inst_tac[("R","P")]splitD 1);
by (res_inst_tac[("P","split P")]converse_rtrancl_induct 1);
by (resolve_tac prems 1);
by (Simp_tac 1);
by (resolve_tac prems 1);
by (split_all_tac 1);
by (Asm_full_simp_tac 1);
by (REPEAT(ares_tac prems 1));
qed "converse_rtrancl_induct2";
(**** 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 (rtrancl_trans RS rtrancl_into_trancl1) 1);
by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 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 (Blast_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::'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 (Blast_tac 1);
by (fast_tac (!claset 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 (rtrancl_trans RS compI)) 1);
by (REPEAT (assume_tac 1));
qed "trans_trancl";
bind_thm ("trancl_trans", trans_trancl RS transD);
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 <= A Times 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 (Blast_tac 1);
val lemma = result();
goalw Trancl.thy [trancl_def]
"!!r. r <= A Times A ==> r^+ <= A Times A";
by (blast_tac (!claset addSDs [lemma]) 1);
qed "trancl_subset_Sigma";