src/HOL/Transitive_Closure.thy
author haftmann
Mon Aug 14 13:46:06 2006 +0200 (2006-08-14)
changeset 20380 14f9f2a1caa6
parent 19656 09be06943252
child 20716 a6686a8e1b68
permissions -rw-r--r--
simplified code generator setup
     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
    10 imports Inductive
    11 uses ("../Provers/trancl.ML")
    12 begin
    13 
    14 text {*
    15   @{text rtrancl} is reflexive/transitive closure,
    16   @{text trancl} is transitive closure,
    17   @{text reflcl} is reflexive closure.
    18 
    19   These postfix operators have \emph{maximum priority}, forcing their
    20   operands to be atomic.
    21 *}
    22 
    23 consts
    24   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^*)" [1000] 999)
    25 
    26 inductive "r^*"
    27   intros
    28     rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
    29     rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
    30 
    31 consts
    32   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^+)" [1000] 999)
    33 
    34 inductive "r^+"
    35   intros
    36     r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
    37     trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
    38 
    39 abbreviation
    40   reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^=)" [1000] 999)
    41   "r^= == r \<union> Id"
    42 
    43 const_syntax (xsymbols)
    44   rtrancl  ("(_\<^sup>*)" [1000] 999)
    45   trancl  ("(_\<^sup>+)" [1000] 999)
    46   reflcl  ("(_\<^sup>=)" [1000] 999)
    47 
    48 const_syntax (HTML output)
    49   rtrancl  ("(_\<^sup>*)" [1000] 999)
    50   trancl  ("(_\<^sup>+)" [1000] 999)
    51   reflcl  ("(_\<^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) (iprover intro: cases)+
    77   thus ?thesis by iprover
    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 reflexive_rtrancl: "reflexive (r^*)"
    85   by (unfold refl_def) fast
    86 
    87 lemma trans_rtrancl: "trans(r^*)"
    88   -- {* transitivity of transitive closure!! -- by induction *}
    89 proof (rule transI)
    90   fix x y z
    91   assume "(x, y) \<in> r\<^sup>*"
    92   assume "(y, z) \<in> r\<^sup>*"
    93   thus "(x, z) \<in> r\<^sup>*" by induct (iprover!)+
    94 qed
    95 
    96 lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
    97 
    98 lemma rtranclE:
    99   assumes major: "(a::'a,b) : r^*"
   100     and cases: "(a = b) ==> P"
   101       "!!y. [| (a,y) : r^*; (y,b) : r |] ==> P"
   102   shows P
   103   -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
   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 cases)+
   109   done
   110 
   111 lemma converse_rtrancl_into_rtrancl:
   112   "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
   113   by (rule rtrancl_trans) iprover+
   114 
   115 text {*
   116   \medskip More @{term "r^*"} equations and inclusions.
   117 *}
   118 
   119 lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
   120   apply auto
   121   apply (erule rtrancl_induct)
   122    apply (rule rtrancl_refl)
   123   apply (blast intro: rtrancl_trans)
   124   done
   125 
   126 lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
   127   apply (rule set_ext)
   128   apply (simp only: split_tupled_all)
   129   apply (blast intro: rtrancl_trans)
   130   done
   131 
   132 lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
   133 by (drule rtrancl_mono, simp)
   134 
   135 lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
   136   apply (drule rtrancl_mono)
   137   apply (drule rtrancl_mono, simp)
   138   done
   139 
   140 lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
   141   by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
   142 
   143 lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
   144   by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
   145 
   146 lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
   147   apply (rule sym)
   148   apply (rule rtrancl_subset, blast, clarify)
   149   apply (rename_tac a b)
   150   apply (case_tac "a = b", blast)
   151   apply (blast intro!: r_into_rtrancl)
   152   done
   153 
   154 theorem rtrancl_converseD:
   155   assumes r: "(x, y) \<in> (r^-1)^*"
   156   shows "(y, x) \<in> r^*"
   157 proof -
   158   from r show ?thesis
   159     by induct (iprover intro: rtrancl_trans dest!: converseD)+
   160 qed
   161 
   162 theorem rtrancl_converseI:
   163   assumes r: "(y, x) \<in> r^*"
   164   shows "(x, y) \<in> (r^-1)^*"
   165 proof -
   166   from r show ?thesis
   167     by induct (iprover intro: rtrancl_trans converseI)+
   168 qed
   169 
   170 lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
   171   by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
   172 
   173 lemma sym_rtrancl: "sym r ==> sym (r^*)"
   174   by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
   175 
   176 theorem converse_rtrancl_induct[consumes 1]:
   177   assumes major: "(a, b) : r^*"
   178     and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
   179   shows "P a"
   180 proof -
   181   from rtrancl_converseI [OF major]
   182   show ?thesis
   183     by induct (iprover intro: cases dest!: converseD rtrancl_converseD)+
   184 qed
   185 
   186 lemmas converse_rtrancl_induct2 =
   187   converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
   188                  consumes 1, case_names refl step]
   189 
   190 lemma converse_rtranclE:
   191   assumes major: "(x,z):r^*"
   192     and cases: "x=z ==> P"
   193       "!!y. [| (x,y):r; (y,z):r^* |] ==> P"
   194   shows P
   195   apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
   196    apply (rule_tac [2] major [THEN converse_rtrancl_induct])
   197     prefer 2 apply iprover
   198    prefer 2 apply iprover
   199   apply (erule asm_rl exE disjE conjE cases)+
   200   done
   201 
   202 ML_setup {*
   203   bind_thm ("converse_rtranclE2", split_rule
   204     (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
   205 *}
   206 
   207 lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
   208   by (blast elim: rtranclE converse_rtranclE
   209     intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
   210 
   211 lemma rtrancl_unfold: "r^* = Id Un (r O r^*)"
   212   by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
   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 (iprover 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) iprover+
   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 iprover+
   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, iprover)
   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) (iprover intro: cases)+
   253   thus ?thesis by iprover
   254 qed
   255 
   256 lemma trancl_trans_induct:
   257   assumes major: "(x,y) : r^+"
   258     and cases: "!!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   shows "P x y"
   261   -- {* Another induction rule for trancl, incorporating transitivity *}
   262   by (iprover intro: r_into_trancl major [THEN trancl_induct] cases)
   263 
   264 inductive_cases tranclE: "(a, b) : r^+"
   265 
   266 lemma trancl_unfold: "r^+ = r Un (r O r^+)"
   267   by (auto intro: trancl_into_trancl elim: tranclE)
   268 
   269 lemma trans_trancl[simp]: "trans(r^+)"
   270   -- {* Transitivity of @{term "r^+"} *}
   271 proof (rule transI)
   272   fix x y z
   273   assume xy: "(x, y) \<in> r^+"
   274   assume "(y, z) \<in> r^+"
   275   thus "(x, z) \<in> r^+" by induct (insert xy, iprover)+
   276 qed
   277 
   278 lemmas trancl_trans = trans_trancl [THEN transD, standard]
   279 
   280 lemma trancl_id[simp]: "trans r \<Longrightarrow> r^+ = r"
   281 apply(auto)
   282 apply(erule trancl_induct)
   283 apply assumption
   284 apply(unfold trans_def)
   285 apply(blast)
   286 done
   287 
   288 lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
   289   shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
   290   by induct (iprover intro: trancl_trans)+
   291 
   292 lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
   293   by (erule transD [OF trans_trancl r_into_trancl])
   294 
   295 lemma trancl_insert:
   296   "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
   297   -- {* primitive recursion for @{text trancl} over finite relations *}
   298   apply (rule equalityI)
   299    apply (rule subsetI)
   300    apply (simp only: split_tupled_all)
   301    apply (erule trancl_induct, blast)
   302    apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
   303   apply (rule subsetI)
   304   apply (blast intro: trancl_mono rtrancl_mono
   305     [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
   306   done
   307 
   308 lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
   309   apply (drule converseD)
   310   apply (erule trancl.induct)
   311   apply (iprover intro: converseI trancl_trans)+
   312   done
   313 
   314 lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
   315   apply (rule converseI)
   316   apply (erule trancl.induct)
   317   apply (iprover dest: converseD intro: trancl_trans)+
   318   done
   319 
   320 lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
   321   by (fastsimp simp add: split_tupled_all
   322     intro!: trancl_converseI trancl_converseD)
   323 
   324 lemma sym_trancl: "sym r ==> sym (r^+)"
   325   by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
   326 
   327 lemma converse_trancl_induct:
   328   assumes major: "(a,b) : r^+"
   329     and cases: "!!y. (y,b) : r ==> P(y)"
   330       "!!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y)"
   331   shows "P a"
   332   apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
   333    apply (rule cases)
   334    apply (erule converseD)
   335   apply (blast intro: prems dest!: trancl_converseD)
   336   done
   337 
   338 lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
   339   apply (erule converse_trancl_induct, auto)
   340   apply (blast intro: rtrancl_trans)
   341   done
   342 
   343 lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
   344   by (blast elim: tranclE dest: trancl_into_rtrancl)
   345 
   346 lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
   347   by (blast dest: r_into_trancl)
   348 
   349 lemma trancl_subset_Sigma_aux:
   350     "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
   351   by (induct rule: rtrancl_induct) auto
   352 
   353 lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
   354   apply (rule subsetI)
   355   apply (simp only: split_tupled_all)
   356   apply (erule tranclE)
   357   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
   358   done
   359 
   360 lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
   361   apply safe
   362    apply (erule trancl_into_rtrancl)
   363   apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
   364   done
   365 
   366 lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
   367   apply safe
   368    apply (drule trancl_into_rtrancl, simp)
   369   apply (erule rtranclE, safe)
   370    apply (rule r_into_trancl, simp)
   371   apply (rule rtrancl_into_trancl1)
   372    apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
   373   done
   374 
   375 lemma trancl_empty [simp]: "{}^+ = {}"
   376   by (auto elim: trancl_induct)
   377 
   378 lemma rtrancl_empty [simp]: "{}^* = Id"
   379   by (rule subst [OF reflcl_trancl]) simp
   380 
   381 lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
   382   by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
   383 
   384 lemma rtrancl_eq_or_trancl:
   385   "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
   386   by (fast elim: trancl_into_rtrancl dest: rtranclD)
   387 
   388 text {* @{text Domain} and @{text Range} *}
   389 
   390 lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
   391   by blast
   392 
   393 lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
   394   by blast
   395 
   396 lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
   397   by (rule rtrancl_Un_rtrancl [THEN subst]) fast
   398 
   399 lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
   400   by (blast intro: subsetD [OF rtrancl_Un_subset])
   401 
   402 lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
   403   by (unfold Domain_def) (blast dest: tranclD)
   404 
   405 lemma trancl_range [simp]: "Range (r^+) = Range r"
   406   by (simp add: Range_def trancl_converse [symmetric])
   407 
   408 lemma Not_Domain_rtrancl:
   409     "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
   410   apply auto
   411   by (erule rev_mp, erule rtrancl_induct, auto)
   412 
   413 
   414 text {* More about converse @{text rtrancl} and @{text trancl}, should
   415   be merged with main body. *}
   416 
   417 lemma single_valued_confluent:
   418   "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
   419   \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
   420 apply(erule rtrancl_induct)
   421  apply simp
   422 apply(erule disjE)
   423  apply(blast elim:converse_rtranclE dest:single_valuedD)
   424 apply(blast intro:rtrancl_trans)
   425 done
   426 
   427 lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
   428   by (fast intro: trancl_trans)
   429 
   430 lemma trancl_into_trancl [rule_format]:
   431     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
   432   apply (erule trancl_induct)
   433    apply (fast intro: r_r_into_trancl)
   434   apply (fast intro: r_r_into_trancl trancl_trans)
   435   done
   436 
   437 lemma trancl_rtrancl_trancl:
   438     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
   439   apply (drule tranclD)
   440   apply (erule exE, erule conjE)
   441   apply (drule rtrancl_trans, assumption)
   442   apply (drule rtrancl_into_trancl2, assumption, assumption)
   443   done
   444 
   445 lemmas transitive_closure_trans [trans] =
   446   r_r_into_trancl trancl_trans rtrancl_trans
   447   trancl_into_trancl trancl_into_trancl2
   448   rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
   449   rtrancl_trancl_trancl trancl_rtrancl_trancl
   450 
   451 declare trancl_into_rtrancl [elim]
   452 
   453 declare rtranclE [cases set: rtrancl]
   454 declare tranclE [cases set: trancl]
   455 
   456 
   457 
   458 
   459 
   460 subsection {* Setup of transitivity reasoner *}
   461 
   462 use "../Provers/trancl.ML";
   463 
   464 ML_setup {*
   465 
   466 structure Trancl_Tac = Trancl_Tac_Fun (
   467   struct
   468     val r_into_trancl = thm "r_into_trancl";
   469     val trancl_trans  = thm "trancl_trans";
   470     val rtrancl_refl = thm "rtrancl_refl";
   471     val r_into_rtrancl = thm "r_into_rtrancl";
   472     val trancl_into_rtrancl = thm "trancl_into_rtrancl";
   473     val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl";
   474     val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl";
   475     val rtrancl_trans = thm "rtrancl_trans";
   476 
   477   fun decomp (Trueprop $ t) =
   478     let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) =
   479         let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
   480               | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
   481               | decr r = (r,"r");
   482             val (rel,r) = decr rel;
   483         in SOME (a,b,rel,r) end
   484       | dec _ =  NONE
   485     in dec t end;
   486 
   487   end); (* struct *)
   488 
   489 change_simpset (fn ss => ss
   490   addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
   491   addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac)));
   492 
   493 *}
   494 
   495 (* Optional methods
   496 
   497 method_setup trancl =
   498   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *}
   499   {* simple transitivity reasoner *}
   500 method_setup rtrancl =
   501   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *}
   502   {* simple transitivity reasoner *}
   503 
   504 *)
   505 
   506 end