src/HOL/Transitive_Closure.thy
author ballarin
Mon Aug 02 10:12:02 2004 +0200 (2004-08-02)
changeset 15096 be1d3b8cfbd5
parent 15076 4b3d280ef06a
child 15131 c69542757a4d
permissions -rw-r--r--
Documentation added; minor improvements.
     1 (*  Title:      HOL/Transitive_Closure.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 *)
     6 
     7 header {* Reflexive and Transitive closure of a relation *}
     8 
     9 theory Transitive_Closure = Inductive
    10 
    11 files ("../Provers/trancl.ML"):
    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 consts
    23   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^*)" [1000] 999)
    24 
    25 inductive "r^*"
    26   intros
    27     rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*"
    28     rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
    29 
    30 consts
    31   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^+)" [1000] 999)
    32 
    33 inductive "r^+"
    34   intros
    35     r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+"
    36     trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
    37 
    38 syntax
    39   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^=)" [1000] 999)
    40 translations
    41   "r^=" == "r \<union> Id"
    42 
    43 syntax (xsymbols)
    44   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
    45   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
    46   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
    47 
    48 syntax (HTML output)
    49   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
    50   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
    51   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
    52 
    53 
    54 subsection {* Reflexive-transitive closure *}
    55 
    56 lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
    57   -- {* @{text rtrancl} of @{text r} contains @{text r} *}
    58   apply (simp only: split_tupled_all)
    59   apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
    60   done
    61 
    62 lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
    63   -- {* monotonicity of @{text rtrancl} *}
    64   apply (rule subsetI)
    65   apply (simp only: split_tupled_all)
    66   apply (erule rtrancl.induct)
    67    apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
    68   done
    69 
    70 theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
    71   assumes a: "(a, b) : r^*"
    72     and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
    73   shows "P b"
    74 proof -
    75   from a have "a = a --> P b"
    76     by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
    77   thus ?thesis by rules
    78 qed
    79 
    80 lemmas rtrancl_induct2 =
    81   rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
    82                  consumes 1, case_names refl step]
    83  
    84 lemma trans_rtrancl: "trans(r^*)"
    85   -- {* transitivity of transitive closure!! -- by induction *}
    86 proof (rule transI)
    87   fix x y z
    88   assume "(x, y) \<in> r\<^sup>*"
    89   assume "(y, z) \<in> r\<^sup>*"
    90   thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+
    91 qed
    92 
    93 lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
    94 
    95 lemma rtranclE:
    96   "[| (a::'a,b) : r^*;  (a = b) ==> P;
    97       !!y.[| (a,y) : r^*; (y,b) : r |] ==> P
    98    |] ==> P"
    99   -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
   100 proof -
   101   assume major: "(a::'a,b) : r^*"
   102   case rule_context
   103   show ?thesis
   104     apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
   105      apply (rule_tac [2] major [THEN rtrancl_induct])
   106       prefer 2 apply (blast!)
   107       prefer 2 apply (blast!)
   108     apply (erule asm_rl exE disjE conjE prems)+
   109     done
   110 qed
   111 
   112 lemma converse_rtrancl_into_rtrancl:
   113   "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
   114   by (rule rtrancl_trans) rules+
   115 
   116 text {*
   117   \medskip More @{term "r^*"} equations and inclusions.
   118 *}
   119 
   120 lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
   121   apply auto
   122   apply (erule rtrancl_induct)
   123    apply (rule rtrancl_refl)
   124   apply (blast intro: rtrancl_trans)
   125   done
   126 
   127 lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
   128   apply (rule set_ext)
   129   apply (simp only: split_tupled_all)
   130   apply (blast intro: rtrancl_trans)
   131   done
   132 
   133 lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
   134 by (drule rtrancl_mono, simp)
   135 
   136 lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
   137   apply (drule rtrancl_mono)
   138   apply (drule rtrancl_mono, simp)
   139   done
   140 
   141 lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
   142   by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
   143 
   144 lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
   145   by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
   146 
   147 lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
   148   apply (rule sym)
   149   apply (rule rtrancl_subset, blast, clarify)
   150   apply (rename_tac a b)
   151   apply (case_tac "a = b", blast)
   152   apply (blast intro!: r_into_rtrancl)
   153   done
   154 
   155 theorem rtrancl_converseD:
   156   assumes r: "(x, y) \<in> (r^-1)^*"
   157   shows "(y, x) \<in> r^*"
   158 proof -
   159   from r show ?thesis
   160     by induct (rules intro: rtrancl_trans dest!: converseD)+
   161 qed
   162 
   163 theorem rtrancl_converseI:
   164   assumes r: "(y, x) \<in> r^*"
   165   shows "(x, y) \<in> (r^-1)^*"
   166 proof -
   167   from r show ?thesis
   168     by induct (rules intro: rtrancl_trans converseI)+
   169 qed
   170 
   171 lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
   172   by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
   173 
   174 theorem converse_rtrancl_induct[consumes 1]:
   175   assumes major: "(a, b) : r^*"
   176     and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
   177   shows "P a"
   178 proof -
   179   from rtrancl_converseI [OF major]
   180   show ?thesis
   181     by induct (rules intro: cases dest!: converseD rtrancl_converseD)+
   182 qed
   183 
   184 lemmas converse_rtrancl_induct2 =
   185   converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
   186                  consumes 1, case_names refl step]
   187 
   188 lemma converse_rtranclE:
   189   "[| (x,z):r^*;
   190       x=z ==> P;
   191       !!y. [| (x,y):r; (y,z):r^* |] ==> P
   192    |] ==> P"
   193 proof -
   194   assume major: "(x,z):r^*"
   195   case rule_context
   196   show ?thesis
   197     apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
   198      apply (rule_tac [2] major [THEN converse_rtrancl_induct])
   199       prefer 2 apply rules
   200      prefer 2 apply rules
   201     apply (erule asm_rl exE disjE conjE prems)+
   202     done
   203 qed
   204 
   205 ML_setup {*
   206   bind_thm ("converse_rtranclE2", split_rule
   207     (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
   208 *}
   209 
   210 lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
   211   by (blast elim: rtranclE converse_rtranclE
   212     intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
   213 
   214 
   215 subsection {* Transitive closure *}
   216 
   217 lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
   218   apply (simp only: split_tupled_all)
   219   apply (erule trancl.induct)
   220   apply (rules dest: subsetD)+
   221   done
   222 
   223 lemma r_into_trancl': "!!p. p : r ==> p : r^+"
   224   by (simp only: split_tupled_all) (erule r_into_trancl)
   225 
   226 text {*
   227   \medskip Conversions between @{text trancl} and @{text rtrancl}.
   228 *}
   229 
   230 lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
   231   by (erule trancl.induct) rules+
   232 
   233 lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
   234   shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
   235   by induct rules+
   236 
   237 lemma rtrancl_into_trancl2: "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+"
   238   -- {* intro rule from @{text r} and @{text rtrancl} *}
   239   apply (erule rtranclE, rules)
   240   apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
   241    apply (assumption | rule r_into_rtrancl)+
   242   done
   243 
   244 lemma trancl_induct [consumes 1, induct set: trancl]:
   245   assumes a: "(a,b) : r^+"
   246   and cases: "!!y. (a, y) : r ==> P y"
   247     "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
   248   shows "P b"
   249   -- {* Nice induction rule for @{text trancl} *}
   250 proof -
   251   from a have "a = a --> P b"
   252     by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
   253   thus ?thesis by rules
   254 qed
   255 
   256 lemma trancl_trans_induct:
   257   "[| (x,y) : r^+;
   258       !!x y. (x,y) : r ==> P x y;
   259       !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z
   260    |] ==> P x y"
   261   -- {* Another induction rule for trancl, incorporating transitivity *}
   262 proof -
   263   assume major: "(x,y) : r^+"
   264   case rule_context
   265   show ?thesis
   266     by (rules intro: r_into_trancl major [THEN trancl_induct] prems)
   267 qed
   268 
   269 inductive_cases tranclE: "(a, b) : r^+"
   270 
   271 lemma trans_trancl: "trans(r^+)"
   272   -- {* Transitivity of @{term "r^+"} *}
   273 proof (rule transI)
   274   fix x y z
   275   assume "(x, y) \<in> r^+"
   276   assume "(y, z) \<in> r^+"
   277   thus "(x, z) \<in> r^+" by induct (rules!)+
   278 qed
   279 
   280 lemmas trancl_trans = trans_trancl [THEN transD, standard]
   281 
   282 lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
   283   shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
   284   by induct (rules intro: trancl_trans)+
   285 
   286 lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
   287   by (erule transD [OF trans_trancl r_into_trancl])
   288 
   289 lemma trancl_insert:
   290   "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
   291   -- {* primitive recursion for @{text trancl} over finite relations *}
   292   apply (rule equalityI)
   293    apply (rule subsetI)
   294    apply (simp only: split_tupled_all)
   295    apply (erule trancl_induct, blast)
   296    apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
   297   apply (rule subsetI)
   298   apply (blast intro: trancl_mono rtrancl_mono
   299     [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
   300   done
   301 
   302 lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
   303   apply (drule converseD)
   304   apply (erule trancl.induct)
   305   apply (rules intro: converseI trancl_trans)+
   306   done
   307 
   308 lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
   309   apply (rule converseI)
   310   apply (erule trancl.induct)
   311   apply (rules dest: converseD intro: trancl_trans)+
   312   done
   313 
   314 lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
   315   by (fastsimp simp add: split_tupled_all
   316     intro!: trancl_converseI trancl_converseD)
   317 
   318 lemma converse_trancl_induct:
   319   "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y);
   320       !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]
   321     ==> P(a)"
   322 proof -
   323   assume major: "(a,b) : r^+"
   324   case rule_context
   325   show ?thesis
   326     apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
   327      apply (rule prems)
   328      apply (erule converseD)
   329     apply (blast intro: prems dest!: trancl_converseD)
   330     done
   331 qed
   332 
   333 lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
   334   apply (erule converse_trancl_induct, auto)
   335   apply (blast intro: rtrancl_trans)
   336   done
   337 
   338 lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
   339 by(blast elim: tranclE dest: trancl_into_rtrancl)
   340 
   341 lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
   342   by (blast dest: r_into_trancl)
   343 
   344 lemma trancl_subset_Sigma_aux:
   345     "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
   346   apply (erule rtrancl_induct, auto)
   347   done
   348 
   349 lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
   350   apply (rule subsetI)
   351   apply (simp only: split_tupled_all)
   352   apply (erule tranclE)
   353   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
   354   done
   355 
   356 lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
   357   apply safe
   358    apply (erule trancl_into_rtrancl)
   359   apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
   360   done
   361 
   362 lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
   363   apply safe
   364    apply (drule trancl_into_rtrancl, simp)
   365   apply (erule rtranclE, safe)
   366    apply (rule r_into_trancl, simp)
   367   apply (rule rtrancl_into_trancl1)
   368    apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
   369   done
   370 
   371 lemma trancl_empty [simp]: "{}^+ = {}"
   372   by (auto elim: trancl_induct)
   373 
   374 lemma rtrancl_empty [simp]: "{}^* = Id"
   375   by (rule subst [OF reflcl_trancl]) simp
   376 
   377 lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
   378   by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
   379 
   380 
   381 text {* @{text Domain} and @{text Range} *}
   382 
   383 lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
   384   by blast
   385 
   386 lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
   387   by blast
   388 
   389 lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
   390   by (rule rtrancl_Un_rtrancl [THEN subst]) fast
   391 
   392 lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
   393   by (blast intro: subsetD [OF rtrancl_Un_subset])
   394 
   395 lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
   396   by (unfold Domain_def) (blast dest: tranclD)
   397 
   398 lemma trancl_range [simp]: "Range (r^+) = Range r"
   399   by (simp add: Range_def trancl_converse [symmetric])
   400 
   401 lemma Not_Domain_rtrancl:
   402     "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
   403   apply auto
   404   by (erule rev_mp, erule rtrancl_induct, auto)
   405 
   406 
   407 text {* More about converse @{text rtrancl} and @{text trancl}, should
   408   be merged with main body. *}
   409 
   410 lemma single_valued_confluent:
   411   "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
   412   \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
   413 apply(erule rtrancl_induct)
   414  apply simp
   415 apply(erule disjE)
   416  apply(blast elim:converse_rtranclE dest:single_valuedD)
   417 apply(blast intro:rtrancl_trans)
   418 done
   419 
   420 lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
   421   by (fast intro: trancl_trans)
   422 
   423 lemma trancl_into_trancl [rule_format]:
   424     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
   425   apply (erule trancl_induct)
   426    apply (fast intro: r_r_into_trancl)
   427   apply (fast intro: r_r_into_trancl trancl_trans)
   428   done
   429 
   430 lemma trancl_rtrancl_trancl:
   431     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
   432   apply (drule tranclD)
   433   apply (erule exE, erule conjE)
   434   apply (drule rtrancl_trans, assumption)
   435   apply (drule rtrancl_into_trancl2, assumption, assumption)
   436   done
   437 
   438 lemmas transitive_closure_trans [trans] =
   439   r_r_into_trancl trancl_trans rtrancl_trans
   440   trancl_into_trancl trancl_into_trancl2
   441   rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
   442   rtrancl_trancl_trancl trancl_rtrancl_trancl
   443 
   444 declare trancl_into_rtrancl [elim]
   445 
   446 declare rtranclE [cases set: rtrancl]
   447 declare tranclE [cases set: trancl]
   448 
   449 subsection {* Setup of transitivity reasoner *}
   450 
   451 use "../Provers/trancl.ML";
   452 
   453 ML_setup {*
   454 
   455 structure Trancl_Tac = Trancl_Tac_Fun (
   456   struct
   457     val r_into_trancl = thm "r_into_trancl";
   458     val trancl_trans  = thm "trancl_trans";
   459     val rtrancl_refl = thm "rtrancl_refl";
   460     val r_into_rtrancl = thm "r_into_rtrancl";
   461     val trancl_into_rtrancl = thm "trancl_into_rtrancl";
   462     val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl";
   463     val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl";
   464     val rtrancl_trans = thm "rtrancl_trans";
   465 
   466   fun decomp (Trueprop $ t) = 
   467     let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = 
   468 	let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
   469 	      | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
   470 	      | decr r = (r,"r");
   471 	    val (rel,r) = decr rel;
   472 	in Some (a,b,rel,r) end
   473       | dec _ =  None 
   474     in dec t end;
   475   
   476   end); (* struct *)
   477 
   478 simpset_ref() := simpset ()
   479     addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
   480     addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac));
   481 
   482 *}
   483 
   484 (* Optional methods
   485 
   486 method_setup trancl =
   487   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *}
   488   {* simple transitivity reasoner *}	    
   489 method_setup rtrancl =
   490   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *}
   491   {* simple transitivity reasoner *}
   492 
   493 *)
   494 
   495 end