(*  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 (blast_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 (blast_tac (claset() addIs prems) 1);

 by (blast_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;

 by (eres_inst_tac [("b","z")] rtrancl_induct 1);

 by (ALLGOALS(blast_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 (blast_tac (claset() addIs prems) 2);

 by (blast_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 (blast_tac (claset() addIs [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 (blast_tac (claset() addSIs [rtrancl_subset]

                        addIs [r_into_rtrancl, rtrancl_mono RS subsetD]) 1);

 qed "rtrancl_Un_rtrancl";

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

 by (blast_tac (claset() addSIs [rtrancl_subset]

                        addIs  [rtrancl_refl, r_into_rtrancl]) 1);

 qed "rtrancl_reflcl";

 Addsimps [rtrancl_reflcl];

 goal Trancl.thy "!!r. (x,y) : (r^-1)^* ==> (x,y) : (r^*)^-1";

 by (rtac converseI 1);

 by (etac rtrancl_induct 1);

 by (rtac rtrancl_refl 1);

 by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);

 qed "rtrancl_converseD";

 goal Trancl.thy "!!r. (x,y) : (r^*)^-1 ==> (x,y) : (r^-1)^*";

 by (dtac converseD 1);

 by (etac rtrancl_induct 1);

 by (rtac rtrancl_refl 1);

 by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);

 qed "rtrancl_converseI";

 goal Trancl.thy "(r^-1)^* = (r^*)^-1";

 by (safe_tac (claset() addSDs [rtrancl_converseD] addSIs [rtrancl_converseI]

 			addSaltern ("split_all_tac", split_all_tac)));

 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 (blast_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";

 val major::prems = goal Trancl.thy

  "[| (x,z):r^*; \

 \    x=z ==> P; \

 \    !!y. [| (x,y):r; (y,z):r^* |] ==> P \

 \ |] ==> P";

 by (subgoal_tac "x = z  | (? y. (x,y) : r & (y,z) : r^*)" 1);

 by (rtac (major RS converse_rtrancl_induct) 2);

 by (blast_tac (claset() addIs prems) 2);

 by (blast_tac (claset() addIs prems) 2);

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

 qed "rtranclE2";

 goal Trancl.thy "r O r^* = r^* O r";

 by (blast_tac (claset() addEs [rtranclE, rtranclE2] 

 	               addIs [rtrancl_into_rtrancl, rtrancl_into_rtrancl2]) 1);

 qed "r_comp_rtrancl_eq";

 (**** The relation trancl ****)

 goalw Trancl.thy [trancl_def] "!!p.[| p:r^+; r <= s |] ==> p:s^+";

 by (blast_tac (claset() addIs [rtrancl_mono RS subsetD]) 1);

 qed "trancl_mono";

 (** Conversions between trancl and rtrancl **)

 goalw Trancl.thy [trancl_def]

     "!!p. p : r^+ ==> p : r^*";

 by (split_all_tac 1);

 by (etac compEpair 1);

 by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));

 qed "trancl_into_rtrancl";

 (*r^+ contains r*)

 goalw Trancl.thy [trancl_def]

    "!!p. p : r ==> p : r^+";

 by (split_all_tac 1);

 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 (blast_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 (blast_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);

 goalw Trancl.thy [trancl_def]

   "!!r. [| (x,y):r^*; (y,z):r^+ |] ==> (x,z):r^+";

 by (blast_tac (claset() addIs [rtrancl_trans,r_into_rtrancl]) 1);

 qed "rtrancl_trancl_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";

 (* primitive recursion for trancl over finite relations: *)

 goal Trancl.thy "(insert (y,x) r)^+ = r^+ Un {(a,b). (a,y):r^* & (x,b):r^*}";

 by (rtac equalityI 1);

  by (rtac subsetI 1);

  by (split_all_tac 1);

  by (etac trancl_induct 1);

   by (blast_tac (claset() addIs [r_into_trancl]) 1);

  by (blast_tac (claset() addIs

      [rtrancl_into_trancl1,trancl_into_rtrancl,r_into_trancl,trancl_trans]) 1);

 by (rtac subsetI 1);

 by (blast_tac (claset() addIs

      [rtrancl_into_trancl2, rtrancl_trancl_trancl,

       impOfSubs rtrancl_mono, trancl_mono]) 1);

 qed "trancl_insert";

 goalw Trancl.thy [trancl_def] "(r^-1)^+ = (r^+)^-1";

 by (simp_tac (simpset() addsimps [rtrancl_converse,converse_comp]) 1);

 by (simp_tac (simpset() addsimps [rtrancl_converse RS sym,r_comp_rtrancl_eq])1);

 qed "trancl_converse";

 val irrefl_tranclI = prove_goal Trancl.thy 

 	"!!r. r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+" (K [

 	rtac allI 1,

 	subgoal_tac "!y. (x, y) : r^+ --> x~=y" 1,

 	 Fast_tac 1,

 	strip_tac 1,

 	etac trancl_induct 1,

 	 auto_tac (claset() addEs [equals0D, r_into_trancl], simpset())]);

 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";

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

 by (safe_tac (claset() addSaltern ("split_all_tac", split_all_tac)));

 by  (etac trancl_into_rtrancl 1);

 by (etac rtranclE 1);

 by  (Auto_tac );

 by (etac rtrancl_into_trancl1 1);

 ba 1;

 qed "reflcl_trancl";

 Addsimps[reflcl_trancl];

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

 by (safe_tac (claset() addSaltern ("split_all_tac", split_all_tac)));

 by  (dtac trancl_into_rtrancl 1);

 by  (Asm_full_simp_tac 1);

 by (etac rtranclE 1);

 by  Safe_tac;

 by  (rtac r_into_trancl 1);

 by  (Simp_tac 1);

 by (rtac rtrancl_into_trancl1 1);

 by (etac (rtrancl_reflcl RS equalityD2 RS subsetD) 1);

 by (Fast_tac 1);

 qed "trancl_reflcl";

 Addsimps[trancl_reflcl];

 qed_goal "trancl_empty" Trancl.thy "{}^+ = {}" 

   (K [auto_tac (claset() addEs [trancl_induct], simpset())]);

 Addsimps[trancl_empty];

 qed_goal "rtrancl_empty" Trancl.thy "{}^* = id" 

   (K [rtac (reflcl_trancl RS subst) 1, Simp_tac 1]);

 Addsimps[rtrancl_empty];