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--
     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