src/HOL/Transitive_Closure.thy
 author berghofe Thu Feb 26 16:32:46 2009 +0100 (2009-02-26) changeset 30107 f3b3b0e3d184 parent 29609 a010aab5bed0 child 30198 922f944f03b2 permissions -rw-r--r--
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
in case assumptions are not of the form (Trueprop $...).  1 (* Title: HOL/Transitive_Closure.thy   2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory   3 Copyright 1992 University of Cambridge   4 *)   5   6 header {* Reflexive and Transitive closure of a relation *}   7   8 theory Transitive_Closure   9 imports Predicate   10 uses "~~/src/Provers/trancl.ML"   11 begin   12   13 text {*   14 @{text rtrancl} is reflexive/transitive closure,   15 @{text trancl} is transitive closure,   16 @{text reflcl} is reflexive closure.   17   18 These postfix operators have \emph{maximum priority}, forcing their   19 operands to be atomic.   20 *}   21   22 inductive_set   23 rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^*)" [1000] 999)   24 for r :: "('a \<times> 'a) set"   25 where   26 rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"   27 | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"   28   29 inductive_set   30 trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^+)" [1000] 999)   31 for r :: "('a \<times> 'a) set"   32 where   33 r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"   34 | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"   35   36 notation   37 rtranclp ("(_^**)" [1000] 1000) and   38 tranclp ("(_^++)" [1000] 1000)   39   40 abbreviation   41 reflclp :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where   42 "r^== == sup r op ="   43   44 abbreviation   45 reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where   46 "r^= == r \<union> Id"   47   48 notation (xsymbols)   49 rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and   50 tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and   51 reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and   52 rtrancl ("(_\<^sup>*)" [1000] 999) and   53 trancl ("(_\<^sup>+)" [1000] 999) and   54 reflcl ("(_\<^sup>=)" [1000] 999)   55   56 notation (HTML output)   57 rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and   58 tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and   59 reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and   60 rtrancl ("(_\<^sup>*)" [1000] 999) and   61 trancl ("(_\<^sup>+)" [1000] 999) and   62 reflcl ("(_\<^sup>=)" [1000] 999)   63   64   65 subsection {* Reflexive closure *}   66   67 lemma reflexive_reflcl[simp]: "reflexive(r^=)"   68 by(simp add:refl_def)   69   70 lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"   71 by(simp add:antisym_def)   72   73 lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"   74 unfolding trans_def by blast   75   76   77 subsection {* Reflexive-transitive closure *}   78   79 lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r Un Id)"   80 by (simp add: expand_fun_eq)   81   82 lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"   83 -- {* @{text rtrancl} of @{text r} contains @{text r} *}   84 apply (simp only: split_tupled_all)   85 apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])   86 done   87   88 lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"   89 -- {* @{text rtrancl} of @{text r} contains @{text r} *}   90 by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])   91   92 lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"   93 -- {* monotonicity of @{text rtrancl} *}   94 apply (rule predicate2I)   95 apply (erule rtranclp.induct)   96 apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)   97 done   98   99 lemmas rtrancl_mono = rtranclp_mono [to_set]   100   101 theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:   102 assumes a: "r^** a b"   103 and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"   104 shows "P b"   105 proof -   106 from a have "a = a --> P b"   107 by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+   108 then show ?thesis by iprover   109 qed   110   111 lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]   112   113 lemmas rtranclp_induct2 =   114 rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,   115 consumes 1, case_names refl step]   116   117 lemmas rtrancl_induct2 =   118 rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),   119 consumes 1, case_names refl step]   120   121 lemma reflexive_rtrancl: "reflexive (r^*)"   122 by (unfold refl_def) fast   123   124 text {* Transitivity of transitive closure. *}   125 lemma trans_rtrancl: "trans (r^*)"   126 proof (rule transI)   127 fix x y z   128 assume "(x, y) \<in> r\<^sup>*"   129 assume "(y, z) \<in> r\<^sup>*"   130 then show "(x, z) \<in> r\<^sup>*"   131 proof induct   132 case base   133 show "(x, y) \<in> r\<^sup>*" by fact   134 next   135 case (step u v)   136 from (x, u) \<in> r\<^sup>* and (u, v) \<in> r   137 show "(x, v) \<in> r\<^sup>*" ..   138 qed   139 qed   140   141 lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]   142   143 lemma rtranclp_trans:   144 assumes xy: "r^** x y"   145 and yz: "r^** y z"   146 shows "r^** x z" using yz xy   147 by induct iprover+   148   149 lemma rtranclE [cases set: rtrancl]:   150 assumes major: "(a::'a, b) : r^*"   151 obtains   152 (base) "a = b"   153 | (step) y where "(a, y) : r^*" and "(y, b) : r"   154 -- {* elimination of @{text rtrancl} -- by induction on a special formula *}   155 apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")   156 apply (rule_tac [2] major [THEN rtrancl_induct])   157 prefer 2 apply blast   158 prefer 2 apply blast   159 apply (erule asm_rl exE disjE conjE base step)+   160 done   161   162 lemma rtrancl_Int_subset: "[| Id \<subseteq> s; r O (r^* \<inter> s) \<subseteq> s|] ==> r^* \<subseteq> s"   163 apply (rule subsetI)   164 apply (rule_tac p="x" in PairE, clarify)   165 apply (erule rtrancl_induct, auto)   166 done   167   168 lemma converse_rtranclp_into_rtranclp:   169 "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"   170 by (rule rtranclp_trans) iprover+   171   172 lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]   173   174 text {*   175 \medskip More @{term "r^*"} equations and inclusions.   176 *}   177   178 lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"   179 apply (auto intro!: order_antisym)   180 apply (erule rtranclp_induct)   181 apply (rule rtranclp.rtrancl_refl)   182 apply (blast intro: rtranclp_trans)   183 done   184   185 lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]   186   187 lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"   188 apply (rule set_ext)   189 apply (simp only: split_tupled_all)   190 apply (blast intro: rtrancl_trans)   191 done   192   193 lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"   194 apply (drule rtrancl_mono)   195 apply simp   196 done   197   198 lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"   199 apply (drule rtranclp_mono)   200 apply (drule rtranclp_mono)   201 apply simp   202 done   203   204 lemmas rtrancl_subset = rtranclp_subset [to_set]   205   206 lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"   207 by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])   208   209 lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]   210   211 lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"   212 by (blast intro!: rtranclp_subset)   213   214 lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]   215   216 lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"   217 apply (rule sym)   218 apply (rule rtrancl_subset, blast, clarify)   219 apply (rename_tac a b)   220 apply (case_tac "a = b")   221 apply blast   222 apply (blast intro!: r_into_rtrancl)   223 done   224   225 lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"   226 apply (rule sym)   227 apply (rule rtranclp_subset)   228 apply blast+   229 done   230   231 theorem rtranclp_converseD:   232 assumes r: "(r^--1)^** x y"   233 shows "r^** y x"   234 proof -   235 from r show ?thesis   236 by induct (iprover intro: rtranclp_trans dest!: conversepD)+   237 qed   238   239 lemmas rtrancl_converseD = rtranclp_converseD [to_set]   240   241 theorem rtranclp_converseI:   242 assumes "r^** y x"   243 shows "(r^--1)^** x y"   244 using assms   245 by induct (iprover intro: rtranclp_trans conversepI)+   246   247 lemmas rtrancl_converseI = rtranclp_converseI [to_set]   248   249 lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"   250 by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)   251   252 lemma sym_rtrancl: "sym r ==> sym (r^*)"   253 by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])   254   255 theorem converse_rtranclp_induct[consumes 1]:   256 assumes major: "r^** a b"   257 and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"   258 shows "P a"   259 using rtranclp_converseI [OF major]   260 by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+   261   262 lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]   263   264 lemmas converse_rtranclp_induct2 =   265 converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,   266 consumes 1, case_names refl step]   267   268 lemmas converse_rtrancl_induct2 =   269 converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),   270 consumes 1, case_names refl step]   271   272 lemma converse_rtranclpE:   273 assumes major: "r^** x z"   274 and cases: "x=z ==> P"   275 "!!y. [| r x y; r^** y z |] ==> P"   276 shows P   277 apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")   278 apply (rule_tac [2] major [THEN converse_rtranclp_induct])   279 prefer 2 apply iprover   280 prefer 2 apply iprover   281 apply (erule asm_rl exE disjE conjE cases)+   282 done   283   284 lemmas converse_rtranclE = converse_rtranclpE [to_set]   285   286 lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]   287   288 lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]   289   290 lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"   291 by (blast elim: rtranclE converse_rtranclE   292 intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)   293   294 lemma rtrancl_unfold: "r^* = Id Un r O r^*"   295 by (auto intro: rtrancl_into_rtrancl elim: rtranclE)   296   297   298 subsection {* Transitive closure *}   299   300 lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"   301 apply (simp add: split_tupled_all)   302 apply (erule trancl.induct)   303 apply (iprover dest: subsetD)+   304 done   305   306 lemma r_into_trancl': "!!p. p : r ==> p : r^+"   307 by (simp only: split_tupled_all) (erule r_into_trancl)   308   309 text {*   310 \medskip Conversions between @{text trancl} and @{text rtrancl}.   311 *}   312   313 lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"   314 by (erule tranclp.induct) iprover+   315   316 lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]   317   318 lemma rtranclp_into_tranclp1: assumes r: "r^** a b"   319 shows "!!c. r b c ==> r^++ a c" using r   320 by induct iprover+   321   322 lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]   323   324 lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"   325 -- {* intro rule from @{text r} and @{text rtrancl} *}   326 apply (erule rtranclp.cases)   327 apply iprover   328 apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])   329 apply (simp | rule r_into_rtranclp)+   330 done   331   332 lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]   333   334 text {* Nice induction rule for @{text trancl} *}   335 lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:   336 assumes "r^++ a b"   337 and cases: "!!y. r a y ==> P y"   338 "!!y z. r^++ a y ==> r y z ==> P y ==> P z"   339 shows "P b"   340 proof -   341 from r^++ a b have "a = a --> P b"   342 by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+   343 then show ?thesis by iprover   344 qed   345   346 lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]   347   348 lemmas tranclp_induct2 =   349 tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,   350 consumes 1, case_names base step]   351   352 lemmas trancl_induct2 =   353 trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),   354 consumes 1, case_names base step]   355   356 lemma tranclp_trans_induct:   357 assumes major: "r^++ x y"   358 and cases: "!!x y. r x y ==> P x y"   359 "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"   360 shows "P x y"   361 -- {* Another induction rule for trancl, incorporating transitivity *}   362 by (iprover intro: major [THEN tranclp_induct] cases)   363   364 lemmas trancl_trans_induct = tranclp_trans_induct [to_set]   365   366 lemma tranclE [cases set: trancl]:   367 assumes "(a, b) : r^+"   368 obtains   369 (base) "(a, b) : r"   370 | (step) c where "(a, c) : r^+" and "(c, b) : r"   371 using assms by cases simp_all   372   373 lemma trancl_Int_subset: "[| r \<subseteq> s; r O (r^+ \<inter> s) \<subseteq> s|] ==> r^+ \<subseteq> s"   374 apply (rule subsetI)   375 apply (rule_tac p = x in PairE)   376 apply clarify   377 apply (erule trancl_induct)   378 apply auto   379 done   380   381 lemma trancl_unfold: "r^+ = r Un r O r^+"   382 by (auto intro: trancl_into_trancl elim: tranclE)   383   384 text {* Transitivity of @{term "r^+"} *}   385 lemma trans_trancl [simp]: "trans (r^+)"   386 proof (rule transI)   387 fix x y z   388 assume "(x, y) \<in> r^+"   389 assume "(y, z) \<in> r^+"   390 then show "(x, z) \<in> r^+"   391 proof induct   392 case (base u)   393 from (x, y) \<in> r^+ and (y, u) \<in> r   394 show "(x, u) \<in> r^+" ..   395 next   396 case (step u v)   397 from (x, u) \<in> r^+ and (u, v) \<in> r   398 show "(x, v) \<in> r^+" ..   399 qed   400 qed   401   402 lemmas trancl_trans = trans_trancl [THEN transD, standard]   403   404 lemma tranclp_trans:   405 assumes xy: "r^++ x y"   406 and yz: "r^++ y z"   407 shows "r^++ x z" using yz xy   408 by induct iprover+   409   410 lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"   411 apply auto   412 apply (erule trancl_induct)   413 apply assumption   414 apply (unfold trans_def)   415 apply blast   416 done   417   418 lemma rtranclp_tranclp_tranclp:   419 assumes "r^** x y"   420 shows "!!z. r^++ y z ==> r^++ x z" using assms   421 by induct (iprover intro: tranclp_trans)+   422   423 lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]   424   425 lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"   426 by (erule tranclp_trans [OF tranclp.r_into_trancl])   427   428 lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]   429   430 lemma trancl_insert:   431 "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"   432 -- {* primitive recursion for @{text trancl} over finite relations *}   433 apply (rule equalityI)   434 apply (rule subsetI)   435 apply (simp only: split_tupled_all)   436 apply (erule trancl_induct, blast)   437 apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)   438 apply (rule subsetI)   439 apply (blast intro: trancl_mono rtrancl_mono   440 [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)   441 done   442   443 lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"   444 apply (drule conversepD)   445 apply (erule tranclp_induct)   446 apply (iprover intro: conversepI tranclp_trans)+   447 done   448   449 lemmas trancl_converseI = tranclp_converseI [to_set]   450   451 lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"   452 apply (rule conversepI)   453 apply (erule tranclp_induct)   454 apply (iprover dest: conversepD intro: tranclp_trans)+   455 done   456   457 lemmas trancl_converseD = tranclp_converseD [to_set]   458   459 lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"   460 by (fastsimp simp add: expand_fun_eq   461 intro!: tranclp_converseI dest!: tranclp_converseD)   462   463 lemmas trancl_converse = tranclp_converse [to_set]   464   465 lemma sym_trancl: "sym r ==> sym (r^+)"   466 by (simp only: sym_conv_converse_eq trancl_converse [symmetric])   467   468 lemma converse_tranclp_induct:   469 assumes major: "r^++ a b"   470 and cases: "!!y. r y b ==> P(y)"   471 "!!y z.[| r y z; r^++ z b; P(z) |] ==> P(y)"   472 shows "P a"   473 apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])   474 apply (rule cases)   475 apply (erule conversepD)   476 apply (blast intro: prems dest!: tranclp_converseD conversepD)   477 done   478   479 lemmas converse_trancl_induct = converse_tranclp_induct [to_set]   480   481 lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"   482 apply (erule converse_tranclp_induct)   483 apply auto   484 apply (blast intro: rtranclp_trans)   485 done   486   487 lemmas tranclD = tranclpD [to_set]   488   489 lemma tranclD2:   490 "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"   491 by (blast elim: tranclE intro: trancl_into_rtrancl)   492   493 lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"   494 by (blast elim: tranclE dest: trancl_into_rtrancl)   495   496 lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"   497 by (blast dest: r_into_trancl)   498   499 lemma trancl_subset_Sigma_aux:   500 "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"   501 by (induct rule: rtrancl_induct) auto   502   503 lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"   504 apply (rule subsetI)   505 apply (simp only: split_tupled_all)   506 apply (erule tranclE)   507 apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+   508 done   509   510 lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"   511 apply (safe intro!: order_antisym)   512 apply (erule tranclp_into_rtranclp)   513 apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)   514 done   515   516 lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]   517   518 lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"   519 apply safe   520 apply (drule trancl_into_rtrancl, simp)   521 apply (erule rtranclE, safe)   522 apply (rule r_into_trancl, simp)   523 apply (rule rtrancl_into_trancl1)   524 apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)   525 done   526   527 lemma trancl_empty [simp]: "{}^+ = {}"   528 by (auto elim: trancl_induct)   529   530 lemma rtrancl_empty [simp]: "{}^* = Id"   531 by (rule subst [OF reflcl_trancl]) simp   532   533 lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"   534 by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)   535   536 lemmas rtranclD = rtranclpD [to_set]   537   538 lemma rtrancl_eq_or_trancl:   539 "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"   540 by (fast elim: trancl_into_rtrancl dest: rtranclD)   541   542 text {* @{text Domain} and @{text Range} *}   543   544 lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"   545 by blast   546   547 lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"   548 by blast   549   550 lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"   551 by (rule rtrancl_Un_rtrancl [THEN subst]) fast   552   553 lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"   554 by (blast intro: subsetD [OF rtrancl_Un_subset])   555   556 lemma trancl_domain [simp]: "Domain (r^+) = Domain r"   557 by (unfold Domain_def) (blast dest: tranclD)   558   559 lemma trancl_range [simp]: "Range (r^+) = Range r"   560 unfolding Range_def by(simp add: trancl_converse [symmetric])   561   562 lemma Not_Domain_rtrancl:   563 "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"   564 apply auto   565 apply (erule rev_mp)   566 apply (erule rtrancl_induct)   567 apply auto   568 done   569   570 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"   571 apply clarify   572 apply (erule trancl_induct)   573 apply (auto simp add: Field_def)   574 done   575   576 lemma finite_trancl: "finite (r^+) = finite r"   577 apply auto   578 prefer 2   579 apply (rule trancl_subset_Field2 [THEN finite_subset])   580 apply (rule finite_SigmaI)   581 prefer 3   582 apply (blast intro: r_into_trancl' finite_subset)   583 apply (auto simp add: finite_Field)   584 done   585   586 text {* More about converse @{text rtrancl} and @{text trancl}, should   587 be merged with main body. *}   588   589 lemma single_valued_confluent:   590 "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>   591 \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"   592 apply (erule rtrancl_induct)   593 apply simp   594 apply (erule disjE)   595 apply (blast elim:converse_rtranclE dest:single_valuedD)   596 apply(blast intro:rtrancl_trans)   597 done   598   599 lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"   600 by (fast intro: trancl_trans)   601   602 lemma trancl_into_trancl [rule_format]:   603 "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"   604 apply (erule trancl_induct)   605 apply (fast intro: r_r_into_trancl)   606 apply (fast intro: r_r_into_trancl trancl_trans)   607 done   608   609 lemma tranclp_rtranclp_tranclp:   610 "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"   611 apply (drule tranclpD)   612 apply (elim exE conjE)   613 apply (drule rtranclp_trans, assumption)   614 apply (drule rtranclp_into_tranclp2, assumption, assumption)   615 done   616   617 lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]   618   619 lemmas transitive_closure_trans [trans] =   620 r_r_into_trancl trancl_trans rtrancl_trans   621 trancl.trancl_into_trancl trancl_into_trancl2   622 rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl   623 rtrancl_trancl_trancl trancl_rtrancl_trancl   624   625 lemmas transitive_closurep_trans' [trans] =   626 tranclp_trans rtranclp_trans   627 tranclp.trancl_into_trancl tranclp_into_tranclp2   628 rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp   629 rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp   630   631 declare trancl_into_rtrancl [elim]   632   633   634 subsection {* Setup of transitivity reasoner *}   635   636 ML {*   637   638 structure Trancl_Tac = Trancl_Tac_Fun (   639 struct   640 val r_into_trancl = @{thm trancl.r_into_trancl};   641 val trancl_trans = @{thm trancl_trans};   642 val rtrancl_refl = @{thm rtrancl.rtrancl_refl};   643 val r_into_rtrancl = @{thm r_into_rtrancl};   644 val trancl_into_rtrancl = @{thm trancl_into_rtrancl};   645 val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};   646 val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};   647 val rtrancl_trans = @{thm rtrancl_trans};   648   649 fun decomp (@{const Trueprop}$ t) =

   650     let fun dec (Const ("op :", _) $(Const ("Pair", _)$ a $b)$ rel ) =

   651         let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $r) = (r,"r*")   652 | decr (Const ("Transitive_Closure.trancl", _ )$ r)  = (r,"r+")

   653               | decr r = (r,"r");

   654             val (rel,r) = decr (Envir.beta_eta_contract rel);

   655         in SOME (a,b,rel,r) end

   656       | dec _ =  NONE

   657     in dec t end

   658     | decomp _ = NONE;

   659

   660   end);

   661

   662 structure Tranclp_Tac = Trancl_Tac_Fun (

   663   struct

   664     val r_into_trancl = @{thm tranclp.r_into_trancl};

   665     val trancl_trans  = @{thm tranclp_trans};

   666     val rtrancl_refl = @{thm rtranclp.rtrancl_refl};

   667     val r_into_rtrancl = @{thm r_into_rtranclp};

   668     val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};

   669     val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};

   670     val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};

   671     val rtrancl_trans = @{thm rtranclp_trans};

   672

   673   fun decomp (@{const Trueprop} $t) =   674 let fun dec (rel$ a $b) =   675 let fun decr (Const ("Transitive_Closure.rtranclp", _ )$ r) = (r,"r*")

   676               | decr (Const ("Transitive_Closure.tranclp", _ ) \$ r)  = (r,"r+")

   677               | decr r = (r,"r");

   678             val (rel,r) = decr rel;

   679         in SOME (a, b, rel, r) end

   680       | dec _ =  NONE

   681     in dec t end

   682     | decomp _ = NONE;

   683

   684   end);

   685 *}

   686

   687 declaration {* fn _ =>

   688   Simplifier.map_ss (fn ss => ss

   689     addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))

   690     addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac))

   691     addSolver (mk_solver "Tranclp" (fn _ => Tranclp_Tac.trancl_tac))

   692     addSolver (mk_solver "Rtranclp" (fn _ => Tranclp_Tac.rtrancl_tac)))

   693 *}

   694

   695 (* Optional methods *)

   696

   697 method_setup trancl =

   698   {* Method.no_args (Method.SIMPLE_METHOD' Trancl_Tac.trancl_tac) *}

   699   {* simple transitivity reasoner *}

   700 method_setup rtrancl =

   701   {* Method.no_args (Method.SIMPLE_METHOD' Trancl_Tac.rtrancl_tac) *}

   702   {* simple transitivity reasoner *}

   703 method_setup tranclp =

   704   {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.trancl_tac) *}

   705   {* simple transitivity reasoner (predicate version) *}

   706 method_setup rtranclp =

   707   {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.rtrancl_tac) *}

   708   {* simple transitivity reasoner (predicate version) *}

   709

   710 end