src/HOL/Transitive_Closure.thy
 author wenzelm Wed Nov 29 15:44:56 2006 +0100 (2006-11-29) changeset 21589 1b02201d7195 parent 21404 eb85850d3eb7 child 22080 7bf8868ab3e4 permissions -rw-r--r--
simplified method 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 "~~/src/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) where

    41   "r^= == r \<union> Id"

    42

    43 notation (xsymbols)

    44   rtrancl  ("(_\<^sup>*)" [1000] 999) and

    45   trancl  ("(_\<^sup>+)" [1000] 999) and

    46   reflcl  ("(_\<^sup>=)" [1000] 999)

    47

    48 notation (HTML output)

    49   rtrancl  ("(_\<^sup>*)" [1000] 999) and

    50   trancl  ("(_\<^sup>+)" [1000] 999) and

    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 ML_setup {*

   463

   464 structure Trancl_Tac = Trancl_Tac_Fun (

   465   struct

   466     val r_into_trancl = thm "r_into_trancl";

   467     val trancl_trans  = thm "trancl_trans";

   468     val rtrancl_refl = thm "rtrancl_refl";

   469     val r_into_rtrancl = thm "r_into_rtrancl";

   470     val trancl_into_rtrancl = thm "trancl_into_rtrancl";

   471     val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl";

   472     val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl";

   473     val rtrancl_trans = thm "rtrancl_trans";

   474

   475   fun decomp (Trueprop $t) =   476 let fun dec (Const ("op :", _)$ (Const ("Pair", _) $a$ b) $rel ) =   477 let fun decr (Const ("Transitive_Closure.rtrancl", _ )$ r) = (r,"r*")

   478               | decr (Const ("Transitive_Closure.trancl", _ ) \$ r)  = (r,"r+")

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

   480             val (rel,r) = decr rel;

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

   482       | dec _ =  NONE

   483     in dec t end;

   484

   485   end);

   486

   487 change_simpset (fn ss => ss

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

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

   490

   491 *}

   492

   493 (* Optional methods *)

   494

   495 method_setup trancl =

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

   497   {* simple transitivity reasoner *}

   498 method_setup rtrancl =

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

   500   {* simple transitivity reasoner *}

   501

   502 end