src/HOL/Transitive_Closure.thy
author berghofe
Sun Jan 10 18:06:30 2010 +0100 (2010-01-10)
changeset 34909 a799687944af
parent 33878 85102f57b4a8
child 34970 4c316d777461
permissions -rw-r--r--
Tuned some proofs; nicer case names for some of the induction / cases rules.
nipkow@10213
     1
(*  Title:      HOL/Transitive_Closure.thy
nipkow@10213
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213
     3
    Copyright   1992  University of Cambridge
nipkow@10213
     4
*)
nipkow@10213
     5
wenzelm@12691
     6
header {* Reflexive and Transitive closure of a relation *}
wenzelm@12691
     7
nipkow@15131
     8
theory Transitive_Closure
berghofe@22262
     9
imports Predicate
wenzelm@21589
    10
uses "~~/src/Provers/trancl.ML"
nipkow@15131
    11
begin
wenzelm@12691
    12
wenzelm@12691
    13
text {*
wenzelm@12691
    14
  @{text rtrancl} is reflexive/transitive closure,
wenzelm@12691
    15
  @{text trancl} is transitive closure,
wenzelm@12691
    16
  @{text reflcl} is reflexive closure.
wenzelm@12691
    17
wenzelm@12691
    18
  These postfix operators have \emph{maximum priority}, forcing their
wenzelm@12691
    19
  operands to be atomic.
wenzelm@12691
    20
*}
nipkow@10213
    21
berghofe@23743
    22
inductive_set
berghofe@23743
    23
  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)
berghofe@23743
    24
  for r :: "('a \<times> 'a) set"
berghofe@22262
    25
where
berghofe@23743
    26
    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
berghofe@23743
    27
  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
berghofe@11327
    28
berghofe@23743
    29
inductive_set
berghofe@23743
    30
  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)
berghofe@23743
    31
  for r :: "('a \<times> 'a) set"
berghofe@22262
    32
where
berghofe@23743
    33
    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
berghofe@23743
    34
  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
berghofe@13704
    35
blanchet@33878
    36
declare rtrancl_def [nitpick_def del]
blanchet@33878
    37
        rtranclp_def [nitpick_def del]
blanchet@33878
    38
        trancl_def [nitpick_def del]
blanchet@33878
    39
        tranclp_def [nitpick_def del]
blanchet@33878
    40
berghofe@23743
    41
notation
berghofe@23743
    42
  rtranclp  ("(_^**)" [1000] 1000) and
berghofe@23743
    43
  tranclp  ("(_^++)" [1000] 1000)
nipkow@10213
    44
wenzelm@19656
    45
abbreviation
berghofe@23743
    46
  reflclp :: "('a => 'a => bool) => 'a => 'a => bool"  ("(_^==)" [1000] 1000) where
haftmann@22422
    47
  "r^== == sup r op ="
berghofe@22262
    48
berghofe@22262
    49
abbreviation
berghofe@23743
    50
  reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"  ("(_^=)" [1000] 999) where
wenzelm@19656
    51
  "r^= == r \<union> Id"
nipkow@10213
    52
wenzelm@21210
    53
notation (xsymbols)
berghofe@23743
    54
  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
berghofe@23743
    55
  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
berghofe@23743
    56
  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
berghofe@23743
    57
  rtrancl  ("(_\<^sup>*)" [1000] 999) and
berghofe@23743
    58
  trancl  ("(_\<^sup>+)" [1000] 999) and
berghofe@23743
    59
  reflcl  ("(_\<^sup>=)" [1000] 999)
wenzelm@12691
    60
wenzelm@21210
    61
notation (HTML output)
berghofe@23743
    62
  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
berghofe@23743
    63
  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
berghofe@23743
    64
  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
berghofe@23743
    65
  rtrancl  ("(_\<^sup>*)" [1000] 999) and
berghofe@23743
    66
  trancl  ("(_\<^sup>+)" [1000] 999) and
berghofe@23743
    67
  reflcl  ("(_\<^sup>=)" [1000] 999)
kleing@14565
    68
wenzelm@12691
    69
nipkow@26271
    70
subsection {* Reflexive closure *}
nipkow@26271
    71
nipkow@30198
    72
lemma refl_reflcl[simp]: "refl(r^=)"
nipkow@30198
    73
by(simp add:refl_on_def)
nipkow@26271
    74
nipkow@26271
    75
lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
nipkow@26271
    76
by(simp add:antisym_def)
nipkow@26271
    77
nipkow@26271
    78
lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
nipkow@26271
    79
unfolding trans_def by blast
nipkow@26271
    80
nipkow@26271
    81
wenzelm@12691
    82
subsection {* Reflexive-transitive closure *}
wenzelm@12691
    83
haftmann@32883
    84
lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
haftmann@32901
    85
  by (auto simp add: expand_fun_eq)
berghofe@22262
    86
wenzelm@12691
    87
lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
wenzelm@12691
    88
  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
wenzelm@12691
    89
  apply (simp only: split_tupled_all)
wenzelm@12691
    90
  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
wenzelm@12691
    91
  done
wenzelm@12691
    92
berghofe@23743
    93
lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
berghofe@22262
    94
  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
berghofe@23743
    95
  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
berghofe@22262
    96
berghofe@23743
    97
lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
wenzelm@12691
    98
  -- {* monotonicity of @{text rtrancl} *}
berghofe@22262
    99
  apply (rule predicate2I)
berghofe@23743
   100
  apply (erule rtranclp.induct)
berghofe@23743
   101
   apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
wenzelm@12691
   102
  done
wenzelm@12691
   103
berghofe@23743
   104
lemmas rtrancl_mono = rtranclp_mono [to_set]
berghofe@22262
   105
wenzelm@26179
   106
theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
berghofe@22262
   107
  assumes a: "r^** a b"
berghofe@22262
   108
    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
berghofe@34909
   109
  shows "P b" using a
berghofe@34909
   110
  by (induct x\<equiv>a b) (rule cases)+
wenzelm@12691
   111
berghofe@25425
   112
lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
berghofe@22262
   113
berghofe@23743
   114
lemmas rtranclp_induct2 =
berghofe@23743
   115
  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
berghofe@22262
   116
                 consumes 1, case_names refl step]
berghofe@22262
   117
nipkow@14404
   118
lemmas rtrancl_induct2 =
nipkow@14404
   119
  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
nipkow@14404
   120
                 consumes 1, case_names refl step]
wenzelm@18372
   121
nipkow@30198
   122
lemma refl_rtrancl: "refl (r^*)"
nipkow@30198
   123
by (unfold refl_on_def) fast
huffman@19228
   124
wenzelm@26179
   125
text {* Transitivity of transitive closure. *}
wenzelm@26179
   126
lemma trans_rtrancl: "trans (r^*)"
berghofe@12823
   127
proof (rule transI)
berghofe@12823
   128
  fix x y z
berghofe@12823
   129
  assume "(x, y) \<in> r\<^sup>*"
berghofe@12823
   130
  assume "(y, z) \<in> r\<^sup>*"
wenzelm@26179
   131
  then show "(x, z) \<in> r\<^sup>*"
wenzelm@26179
   132
  proof induct
wenzelm@26179
   133
    case base
wenzelm@26179
   134
    show "(x, y) \<in> r\<^sup>*" by fact
wenzelm@26179
   135
  next
wenzelm@26179
   136
    case (step u v)
wenzelm@26179
   137
    from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`
wenzelm@26179
   138
    show "(x, v) \<in> r\<^sup>*" ..
wenzelm@26179
   139
  qed
berghofe@12823
   140
qed
wenzelm@12691
   141
wenzelm@12691
   142
lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
wenzelm@12691
   143
berghofe@23743
   144
lemma rtranclp_trans:
berghofe@22262
   145
  assumes xy: "r^** x y"
berghofe@22262
   146
  and yz: "r^** y z"
berghofe@22262
   147
  shows "r^** x z" using yz xy
berghofe@22262
   148
  by induct iprover+
berghofe@22262
   149
wenzelm@26174
   150
lemma rtranclE [cases set: rtrancl]:
wenzelm@26174
   151
  assumes major: "(a::'a, b) : r^*"
wenzelm@26174
   152
  obtains
wenzelm@26174
   153
    (base) "a = b"
wenzelm@26174
   154
  | (step) y where "(a, y) : r^*" and "(y, b) : r"
wenzelm@12691
   155
  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
wenzelm@18372
   156
  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
wenzelm@18372
   157
   apply (rule_tac [2] major [THEN rtrancl_induct])
wenzelm@18372
   158
    prefer 2 apply blast
wenzelm@18372
   159
   prefer 2 apply blast
wenzelm@26174
   160
  apply (erule asm_rl exE disjE conjE base step)+
wenzelm@18372
   161
  done
wenzelm@12691
   162
krauss@32235
   163
lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
paulson@22080
   164
  apply (rule subsetI)
paulson@22080
   165
  apply (rule_tac p="x" in PairE, clarify)
paulson@22080
   166
  apply (erule rtrancl_induct, auto) 
paulson@22080
   167
  done
paulson@22080
   168
berghofe@23743
   169
lemma converse_rtranclp_into_rtranclp:
berghofe@22262
   170
  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
berghofe@23743
   171
  by (rule rtranclp_trans) iprover+
berghofe@22262
   172
berghofe@23743
   173
lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
wenzelm@12691
   174
wenzelm@12691
   175
text {*
wenzelm@12691
   176
  \medskip More @{term "r^*"} equations and inclusions.
wenzelm@12691
   177
*}
wenzelm@12691
   178
berghofe@23743
   179
lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"
berghofe@22262
   180
  apply (auto intro!: order_antisym)
berghofe@23743
   181
  apply (erule rtranclp_induct)
berghofe@23743
   182
   apply (rule rtranclp.rtrancl_refl)
berghofe@23743
   183
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   184
  done
wenzelm@12691
   185
berghofe@23743
   186
lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
berghofe@22262
   187
wenzelm@12691
   188
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
wenzelm@12691
   189
  apply (rule set_ext)
wenzelm@12691
   190
  apply (simp only: split_tupled_all)
wenzelm@12691
   191
  apply (blast intro: rtrancl_trans)
wenzelm@12691
   192
  done
wenzelm@12691
   193
wenzelm@12691
   194
lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
wenzelm@26179
   195
  apply (drule rtrancl_mono)
wenzelm@26179
   196
  apply simp
wenzelm@26179
   197
  done
wenzelm@12691
   198
berghofe@23743
   199
lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
berghofe@23743
   200
  apply (drule rtranclp_mono)
wenzelm@26179
   201
  apply (drule rtranclp_mono)
wenzelm@26179
   202
  apply simp
wenzelm@12691
   203
  done
wenzelm@12691
   204
berghofe@23743
   205
lemmas rtrancl_subset = rtranclp_subset [to_set]
berghofe@22262
   206
berghofe@23743
   207
lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"
berghofe@23743
   208
  by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
wenzelm@12691
   209
berghofe@23743
   210
lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
berghofe@22262
   211
berghofe@23743
   212
lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"
berghofe@23743
   213
  by (blast intro!: rtranclp_subset)
berghofe@22262
   214
berghofe@23743
   215
lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]
wenzelm@12691
   216
wenzelm@12691
   217
lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
wenzelm@12691
   218
  apply (rule sym)
paulson@14208
   219
  apply (rule rtrancl_subset, blast, clarify)
wenzelm@12691
   220
  apply (rename_tac a b)
wenzelm@26179
   221
  apply (case_tac "a = b")
wenzelm@26179
   222
   apply blast
wenzelm@12691
   223
  apply (blast intro!: r_into_rtrancl)
wenzelm@12691
   224
  done
wenzelm@12691
   225
berghofe@23743
   226
lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
berghofe@22262
   227
  apply (rule sym)
berghofe@23743
   228
  apply (rule rtranclp_subset)
wenzelm@26179
   229
   apply blast+
berghofe@22262
   230
  done
berghofe@22262
   231
berghofe@23743
   232
theorem rtranclp_converseD:
berghofe@22262
   233
  assumes r: "(r^--1)^** x y"
berghofe@22262
   234
  shows "r^** y x"
berghofe@12823
   235
proof -
berghofe@12823
   236
  from r show ?thesis
berghofe@23743
   237
    by induct (iprover intro: rtranclp_trans dest!: conversepD)+
berghofe@12823
   238
qed
wenzelm@12691
   239
berghofe@23743
   240
lemmas rtrancl_converseD = rtranclp_converseD [to_set]
berghofe@22262
   241
berghofe@23743
   242
theorem rtranclp_converseI:
wenzelm@26179
   243
  assumes "r^** y x"
berghofe@22262
   244
  shows "(r^--1)^** x y"
wenzelm@26179
   245
  using assms
wenzelm@26179
   246
  by induct (iprover intro: rtranclp_trans conversepI)+
wenzelm@12691
   247
berghofe@23743
   248
lemmas rtrancl_converseI = rtranclp_converseI [to_set]
berghofe@22262
   249
wenzelm@12691
   250
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
wenzelm@12691
   251
  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
wenzelm@12691
   252
huffman@19228
   253
lemma sym_rtrancl: "sym r ==> sym (r^*)"
huffman@19228
   254
  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
huffman@19228
   255
berghofe@34909
   256
theorem converse_rtranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   257
  assumes major: "r^** a b"
berghofe@22262
   258
    and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
wenzelm@12937
   259
  shows "P a"
wenzelm@26179
   260
  using rtranclp_converseI [OF major]
wenzelm@26179
   261
  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
wenzelm@12691
   262
berghofe@25425
   263
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
berghofe@22262
   264
berghofe@23743
   265
lemmas converse_rtranclp_induct2 =
wenzelm@26179
   266
  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
berghofe@22262
   267
                 consumes 1, case_names refl step]
berghofe@22262
   268
nipkow@14404
   269
lemmas converse_rtrancl_induct2 =
wenzelm@26179
   270
  converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
nipkow@14404
   271
                 consumes 1, case_names refl step]
wenzelm@12691
   272
berghofe@34909
   273
lemma converse_rtranclpE [consumes 1, case_names base step]:
berghofe@22262
   274
  assumes major: "r^** x z"
wenzelm@18372
   275
    and cases: "x=z ==> P"
berghofe@22262
   276
      "!!y. [| r x y; r^** y z |] ==> P"
wenzelm@18372
   277
  shows P
berghofe@22262
   278
  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
berghofe@23743
   279
   apply (rule_tac [2] major [THEN converse_rtranclp_induct])
wenzelm@18372
   280
    prefer 2 apply iprover
wenzelm@18372
   281
   prefer 2 apply iprover
wenzelm@18372
   282
  apply (erule asm_rl exE disjE conjE cases)+
wenzelm@18372
   283
  done
wenzelm@12691
   284
berghofe@23743
   285
lemmas converse_rtranclE = converse_rtranclpE [to_set]
berghofe@22262
   286
berghofe@23743
   287
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
berghofe@22262
   288
berghofe@22262
   289
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
wenzelm@12691
   290
wenzelm@12691
   291
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
wenzelm@12691
   292
  by (blast elim: rtranclE converse_rtranclE
wenzelm@12691
   293
    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
wenzelm@12691
   294
krauss@32235
   295
lemma rtrancl_unfold: "r^* = Id Un r^* O r"
paulson@15551
   296
  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
paulson@15551
   297
nipkow@31690
   298
lemma rtrancl_Un_separatorE:
nipkow@31690
   299
  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
nipkow@31690
   300
apply (induct rule:rtrancl.induct)
nipkow@31690
   301
 apply blast
nipkow@31690
   302
apply (blast intro:rtrancl_trans)
nipkow@31690
   303
done
nipkow@31690
   304
nipkow@31690
   305
lemma rtrancl_Un_separator_converseE:
nipkow@31690
   306
  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
nipkow@31690
   307
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   308
 apply blast
nipkow@31690
   309
apply (blast intro:rtrancl_trans)
nipkow@31690
   310
done
nipkow@31690
   311
wenzelm@12691
   312
wenzelm@12691
   313
subsection {* Transitive closure *}
wenzelm@10331
   314
berghofe@13704
   315
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   316
  apply (simp add: split_tupled_all)
berghofe@13704
   317
  apply (erule trancl.induct)
wenzelm@26179
   318
   apply (iprover dest: subsetD)+
wenzelm@12691
   319
  done
wenzelm@12691
   320
berghofe@13704
   321
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   322
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   323
wenzelm@12691
   324
text {*
wenzelm@12691
   325
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@12691
   326
*}
wenzelm@12691
   327
berghofe@23743
   328
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   329
  by (erule tranclp.induct) iprover+
wenzelm@12691
   330
berghofe@23743
   331
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   332
berghofe@23743
   333
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   334
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   335
  by induct iprover+
wenzelm@12691
   336
berghofe@23743
   337
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   338
berghofe@23743
   339
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@12691
   340
  -- {* intro rule from @{text r} and @{text rtrancl} *}
wenzelm@26179
   341
  apply (erule rtranclp.cases)
wenzelm@26179
   342
   apply iprover
berghofe@23743
   343
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   344
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   345
  done
wenzelm@12691
   346
berghofe@23743
   347
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   348
wenzelm@26179
   349
text {* Nice induction rule for @{text trancl} *}
wenzelm@26179
   350
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
berghofe@34909
   351
  assumes a: "r^++ a b"
berghofe@22262
   352
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   353
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@34909
   354
  shows "P b" using a
berghofe@34909
   355
  by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   356
berghofe@25425
   357
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   358
berghofe@23743
   359
lemmas tranclp_induct2 =
wenzelm@26179
   360
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   361
    consumes 1, case_names base step]
berghofe@22262
   362
paulson@22172
   363
lemmas trancl_induct2 =
wenzelm@26179
   364
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   365
    consumes 1, case_names base step]
paulson@22172
   366
berghofe@23743
   367
lemma tranclp_trans_induct:
berghofe@22262
   368
  assumes major: "r^++ x y"
berghofe@22262
   369
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   370
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   371
  shows "P x y"
wenzelm@12691
   372
  -- {* Another induction rule for trancl, incorporating transitivity *}
berghofe@23743
   373
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   374
berghofe@23743
   375
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   376
wenzelm@26174
   377
lemma tranclE [cases set: trancl]:
wenzelm@26174
   378
  assumes "(a, b) : r^+"
wenzelm@26174
   379
  obtains
wenzelm@26174
   380
    (base) "(a, b) : r"
wenzelm@26174
   381
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   382
  using assms by cases simp_all
wenzelm@10980
   383
krauss@32235
   384
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   385
  apply (rule subsetI)
wenzelm@26179
   386
  apply (rule_tac p = x in PairE)
wenzelm@26179
   387
  apply clarify
wenzelm@26179
   388
  apply (erule trancl_induct)
wenzelm@26179
   389
   apply auto
paulson@22080
   390
  done
paulson@22080
   391
krauss@32235
   392
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   393
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   394
wenzelm@26179
   395
text {* Transitivity of @{term "r^+"} *}
wenzelm@26179
   396
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   397
proof (rule transI)
berghofe@13704
   398
  fix x y z
wenzelm@26179
   399
  assume "(x, y) \<in> r^+"
berghofe@13704
   400
  assume "(y, z) \<in> r^+"
wenzelm@26179
   401
  then show "(x, z) \<in> r^+"
wenzelm@26179
   402
  proof induct
wenzelm@26179
   403
    case (base u)
wenzelm@26179
   404
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
wenzelm@26179
   405
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   406
  next
wenzelm@26179
   407
    case (step u v)
wenzelm@26179
   408
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
wenzelm@26179
   409
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   410
  qed
berghofe@13704
   411
qed
wenzelm@12691
   412
wenzelm@12691
   413
lemmas trancl_trans = trans_trancl [THEN transD, standard]
wenzelm@12691
   414
berghofe@23743
   415
lemma tranclp_trans:
berghofe@22262
   416
  assumes xy: "r^++ x y"
berghofe@22262
   417
  and yz: "r^++ y z"
berghofe@22262
   418
  shows "r^++ x z" using yz xy
berghofe@22262
   419
  by induct iprover+
berghofe@22262
   420
wenzelm@26179
   421
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   422
  apply auto
wenzelm@26179
   423
  apply (erule trancl_induct)
wenzelm@26179
   424
   apply assumption
wenzelm@26179
   425
  apply (unfold trans_def)
wenzelm@26179
   426
  apply blast
wenzelm@26179
   427
  done
nipkow@19623
   428
wenzelm@26179
   429
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   430
  assumes "r^** x y"
wenzelm@26179
   431
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   432
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   433
berghofe@23743
   434
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   435
berghofe@23743
   436
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   437
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   438
berghofe@23743
   439
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   440
wenzelm@12691
   441
lemma trancl_insert:
wenzelm@12691
   442
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@12691
   443
  -- {* primitive recursion for @{text trancl} over finite relations *}
wenzelm@12691
   444
  apply (rule equalityI)
wenzelm@12691
   445
   apply (rule subsetI)
wenzelm@12691
   446
   apply (simp only: split_tupled_all)
paulson@14208
   447
   apply (erule trancl_induct, blast)
wenzelm@12691
   448
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
wenzelm@12691
   449
  apply (rule subsetI)
wenzelm@12691
   450
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@12691
   451
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
wenzelm@12691
   452
  done
wenzelm@12691
   453
berghofe@23743
   454
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   455
  apply (drule conversepD)
berghofe@23743
   456
  apply (erule tranclp_induct)
berghofe@23743
   457
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   458
  done
wenzelm@12691
   459
berghofe@23743
   460
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   461
berghofe@23743
   462
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   463
  apply (rule conversepI)
berghofe@23743
   464
  apply (erule tranclp_induct)
berghofe@23743
   465
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   466
  done
wenzelm@12691
   467
berghofe@23743
   468
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   469
berghofe@23743
   470
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
berghofe@22262
   471
  by (fastsimp simp add: expand_fun_eq
berghofe@23743
   472
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   473
berghofe@23743
   474
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   475
huffman@19228
   476
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   477
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   478
berghofe@34909
   479
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   480
  assumes major: "r^++ a b"
berghofe@22262
   481
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   482
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   483
  shows "P a"
berghofe@23743
   484
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   485
   apply (rule cases)
berghofe@22262
   486
   apply (erule conversepD)
berghofe@23743
   487
  apply (blast intro: prems dest!: tranclp_converseD conversepD)
wenzelm@18372
   488
  done
wenzelm@12691
   489
berghofe@23743
   490
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   491
berghofe@23743
   492
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   493
  apply (erule converse_tranclp_induct)
wenzelm@26179
   494
   apply auto
berghofe@23743
   495
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   496
  done
wenzelm@12691
   497
berghofe@23743
   498
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   499
bulwahn@31577
   500
lemma converse_tranclpE:
bulwahn@31577
   501
  assumes major: "tranclp r x z"
bulwahn@31577
   502
  assumes base: "r x z ==> P"
bulwahn@31577
   503
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   504
  shows P
bulwahn@31577
   505
proof -
bulwahn@31577
   506
  from tranclpD[OF major]
bulwahn@31577
   507
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   508
  from this(2) show P
bulwahn@31577
   509
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   510
    case rtrancl_refl
bulwahn@31577
   511
    with `r x y` base show P by iprover
bulwahn@31577
   512
  next
bulwahn@31577
   513
    case rtrancl_into_rtrancl
bulwahn@31577
   514
    from this have "tranclp r y z"
bulwahn@31577
   515
      by (iprover intro: rtranclp_into_tranclp1)
bulwahn@31577
   516
    with `r x y` step show P by iprover
bulwahn@31577
   517
  qed
bulwahn@31577
   518
qed
bulwahn@31577
   519
bulwahn@31577
   520
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   521
kleing@25295
   522
lemma tranclD2:
kleing@25295
   523
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   524
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   525
nipkow@13867
   526
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   527
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   528
wenzelm@12691
   529
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   530
  by (blast dest: r_into_trancl)
wenzelm@12691
   531
wenzelm@12691
   532
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   533
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   534
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   535
wenzelm@12691
   536
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   537
  apply (rule subsetI)
berghofe@13704
   538
  apply (simp only: split_tupled_all)
berghofe@13704
   539
  apply (erule tranclE)
wenzelm@26179
   540
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   541
  done
nipkow@10996
   542
berghofe@23743
   543
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   544
  apply (safe intro!: order_antisym)
berghofe@23743
   545
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   546
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   547
  done
nipkow@10996
   548
berghofe@23743
   549
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
berghofe@22262
   550
wenzelm@11090
   551
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   552
  apply safe
paulson@14208
   553
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   554
  apply (erule rtranclE, safe)
paulson@14208
   555
   apply (rule r_into_trancl, simp)
wenzelm@11084
   556
  apply (rule rtrancl_into_trancl1)
paulson@14208
   557
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   558
  done
nipkow@10996
   559
wenzelm@11090
   560
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   561
  by (auto elim: trancl_induct)
nipkow@10996
   562
wenzelm@11090
   563
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   564
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   565
berghofe@23743
   566
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
berghofe@23743
   567
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
berghofe@22262
   568
berghofe@23743
   569
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   570
kleing@16514
   571
lemma rtrancl_eq_or_trancl:
kleing@16514
   572
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   573
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   574
krauss@33656
   575
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   576
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   577
krauss@33656
   578
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   579
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   580
krauss@33656
   581
krauss@33656
   582
text {* Simplifying nested closures *}
krauss@33656
   583
krauss@33656
   584
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   585
by (simp add: trans_rtrancl)
krauss@33656
   586
krauss@33656
   587
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   588
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   589
krauss@33656
   590
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   591
by auto
krauss@33656
   592
krauss@33656
   593
wenzelm@12691
   594
text {* @{text Domain} and @{text Range} *}
nipkow@10996
   595
wenzelm@11090
   596
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   597
  by blast
nipkow@10996
   598
wenzelm@11090
   599
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   600
  by blast
nipkow@10996
   601
wenzelm@11090
   602
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   603
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   604
wenzelm@11090
   605
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   606
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   607
wenzelm@11090
   608
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
wenzelm@11084
   609
  by (unfold Domain_def) (blast dest: tranclD)
nipkow@10996
   610
wenzelm@11090
   611
lemma trancl_range [simp]: "Range (r^+) = Range r"
nipkow@26271
   612
unfolding Range_def by(simp add: trancl_converse [symmetric])
nipkow@10996
   613
paulson@11115
   614
lemma Not_Domain_rtrancl:
wenzelm@12691
   615
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   616
  apply auto
wenzelm@26179
   617
  apply (erule rev_mp)
wenzelm@26179
   618
  apply (erule rtrancl_induct)
wenzelm@26179
   619
   apply auto
wenzelm@26179
   620
  done
berghofe@11327
   621
haftmann@29609
   622
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   623
  apply clarify
haftmann@29609
   624
  apply (erule trancl_induct)
haftmann@29609
   625
   apply (auto simp add: Field_def)
haftmann@29609
   626
  done
haftmann@29609
   627
haftmann@29609
   628
lemma finite_trancl: "finite (r^+) = finite r"
haftmann@29609
   629
  apply auto
haftmann@29609
   630
   prefer 2
haftmann@29609
   631
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   632
   apply (rule finite_SigmaI)
haftmann@29609
   633
    prefer 3
haftmann@29609
   634
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   635
   apply (auto simp add: finite_Field)
haftmann@29609
   636
  done
haftmann@29609
   637
wenzelm@12691
   638
text {* More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@12691
   639
  be merged with main body. *}
kleing@12428
   640
nipkow@14337
   641
lemma single_valued_confluent:
nipkow@14337
   642
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   643
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   644
  apply (erule rtrancl_induct)
wenzelm@26179
   645
  apply simp
wenzelm@26179
   646
  apply (erule disjE)
wenzelm@26179
   647
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   648
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   649
  done
nipkow@14337
   650
wenzelm@12691
   651
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   652
  by (fast intro: trancl_trans)
kleing@12428
   653
kleing@12428
   654
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   655
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   656
  apply (erule trancl_induct)
kleing@12428
   657
   apply (fast intro: r_r_into_trancl)
kleing@12428
   658
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   659
  done
kleing@12428
   660
berghofe@23743
   661
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   662
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   663
  apply (drule tranclpD)
wenzelm@26179
   664
  apply (elim exE conjE)
berghofe@23743
   665
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   666
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   667
  done
kleing@12428
   668
berghofe@23743
   669
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   670
wenzelm@12691
   671
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   672
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   673
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   674
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   675
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   676
berghofe@23743
   677
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   678
  tranclp_trans rtranclp_trans
berghofe@23743
   679
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   680
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   681
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   682
kleing@12428
   683
declare trancl_into_rtrancl [elim]
berghofe@11327
   684
haftmann@30954
   685
subsection {* The power operation on relations *}
haftmann@30954
   686
haftmann@30954
   687
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
haftmann@30954
   688
haftmann@30971
   689
overloading
haftmann@30971
   690
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@30971
   691
begin
haftmann@30954
   692
haftmann@30971
   693
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   694
    "relpow 0 R = Id"
krauss@32235
   695
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   696
haftmann@30971
   697
end
haftmann@30954
   698
haftmann@30954
   699
lemma rel_pow_1 [simp]:
haftmann@30971
   700
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   701
  shows "R ^^ 1 = R"
haftmann@30954
   702
  by simp
haftmann@30954
   703
haftmann@30954
   704
lemma rel_pow_0_I: 
haftmann@30954
   705
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   706
  by simp
haftmann@30954
   707
haftmann@30954
   708
lemma rel_pow_Suc_I:
haftmann@30954
   709
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   710
  by auto
haftmann@30954
   711
haftmann@30954
   712
lemma rel_pow_Suc_I2:
haftmann@30954
   713
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   714
  by (induct n arbitrary: z) (simp, fastsimp)
haftmann@30954
   715
haftmann@30954
   716
lemma rel_pow_0_E:
haftmann@30954
   717
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   718
  by simp
haftmann@30954
   719
haftmann@30954
   720
lemma rel_pow_Suc_E:
haftmann@30954
   721
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   722
  by auto
haftmann@30954
   723
haftmann@30954
   724
lemma rel_pow_E:
haftmann@30954
   725
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   726
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   727
   \<Longrightarrow> P"
haftmann@30954
   728
  by (cases n) auto
haftmann@30954
   729
haftmann@30954
   730
lemma rel_pow_Suc_D2:
haftmann@30954
   731
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   732
  apply (induct n arbitrary: x z)
haftmann@30954
   733
   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   734
  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   735
  done
haftmann@30954
   736
haftmann@30954
   737
lemma rel_pow_Suc_E2:
haftmann@30954
   738
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   739
  by (blast dest: rel_pow_Suc_D2)
haftmann@30954
   740
haftmann@30954
   741
lemma rel_pow_Suc_D2':
haftmann@30954
   742
  "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
haftmann@30954
   743
  by (induct n) (simp_all, blast)
haftmann@30954
   744
haftmann@30954
   745
lemma rel_pow_E2:
haftmann@30954
   746
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   747
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   748
   \<Longrightarrow> P"
haftmann@30954
   749
  apply (cases n, simp)
haftmann@30954
   750
  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
haftmann@30954
   751
  done
haftmann@30954
   752
krauss@32235
   753
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"
nipkow@31351
   754
by(induct n) auto
nipkow@31351
   755
krauss@31970
   756
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"
krauss@32235
   757
by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   758
haftmann@30954
   759
lemma rtrancl_imp_UN_rel_pow:
haftmann@30954
   760
  assumes "p \<in> R^*"
haftmann@30954
   761
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   762
proof (cases p)
haftmann@30954
   763
  case (Pair x y)
haftmann@30954
   764
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   765
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
haftmann@30954
   766
    case base show ?case by (blast intro: rel_pow_0_I)
haftmann@30954
   767
  next
haftmann@30954
   768
    case step then show ?case by (blast intro: rel_pow_Suc_I)
haftmann@30954
   769
  qed
haftmann@30954
   770
  with Pair show ?thesis by simp
haftmann@30954
   771
qed
haftmann@30954
   772
haftmann@30954
   773
lemma rel_pow_imp_rtrancl:
haftmann@30954
   774
  assumes "p \<in> R ^^ n"
haftmann@30954
   775
  shows "p \<in> R^*"
haftmann@30954
   776
proof (cases p)
haftmann@30954
   777
  case (Pair x y)
haftmann@30954
   778
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   779
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   780
    case 0 then show ?case by simp
haftmann@30954
   781
  next
haftmann@30954
   782
    case Suc then show ?case
haftmann@30954
   783
      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   784
  qed
haftmann@30954
   785
  with Pair show ?thesis by simp
haftmann@30954
   786
qed
haftmann@30954
   787
haftmann@30954
   788
lemma rtrancl_is_UN_rel_pow:
haftmann@30954
   789
  "R^* = (\<Union>n. R ^^ n)"
haftmann@30954
   790
  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
haftmann@30954
   791
haftmann@30954
   792
lemma rtrancl_power:
haftmann@30954
   793
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
haftmann@30954
   794
  by (simp add: rtrancl_is_UN_rel_pow)
haftmann@30954
   795
haftmann@30954
   796
lemma trancl_power:
haftmann@30954
   797
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   798
  apply (cases p)
haftmann@30954
   799
  apply simp
haftmann@30954
   800
  apply (rule iffI)
haftmann@30954
   801
   apply (drule tranclD2)
haftmann@30954
   802
   apply (clarsimp simp: rtrancl_is_UN_rel_pow)
haftmann@30971
   803
   apply (rule_tac x="Suc n" in exI)
haftmann@30954
   804
   apply (clarsimp simp: rel_comp_def)
haftmann@30954
   805
   apply fastsimp
haftmann@30954
   806
  apply clarsimp
haftmann@30954
   807
  apply (case_tac n, simp)
haftmann@30954
   808
  apply clarsimp
haftmann@30954
   809
  apply (drule rel_pow_imp_rtrancl)
haftmann@30954
   810
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   811
  done
haftmann@30954
   812
haftmann@30954
   813
lemma rtrancl_imp_rel_pow:
haftmann@30954
   814
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
haftmann@30954
   815
  by (auto dest: rtrancl_imp_UN_rel_pow)
haftmann@30954
   816
haftmann@30954
   817
lemma single_valued_rel_pow:
haftmann@30954
   818
  fixes R :: "('a * 'a) set"
haftmann@30954
   819
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
haftmann@30954
   820
  apply (induct n arbitrary: R)
haftmann@30954
   821
  apply simp_all
haftmann@30954
   822
  apply (rule single_valuedI)
haftmann@30954
   823
  apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
haftmann@30954
   824
  done
paulson@15551
   825
ballarin@15076
   826
subsection {* Setup of transitivity reasoner *}
ballarin@15076
   827
wenzelm@26340
   828
ML {*
ballarin@15076
   829
wenzelm@32215
   830
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
   831
(
wenzelm@32215
   832
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
   833
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
   834
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
   835
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
   836
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
   837
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
   838
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
   839
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
   840
berghofe@30107
   841
  fun decomp (@{const Trueprop} $ t) =
wenzelm@18372
   842
    let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) =
berghofe@23743
   843
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
berghofe@23743
   844
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
wenzelm@18372
   845
              | decr r = (r,"r");
berghofe@26801
   846
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
   847
        in SOME (a,b,rel,r) end
wenzelm@18372
   848
      | dec _ =  NONE
berghofe@30107
   849
    in dec t end
berghofe@30107
   850
    | decomp _ = NONE;
wenzelm@32215
   851
);
ballarin@15076
   852
wenzelm@32215
   853
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
   854
(
wenzelm@32215
   855
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
   856
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
   857
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
   858
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
   859
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
   860
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
   861
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
   862
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
   863
berghofe@30107
   864
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
   865
    let fun dec (rel $ a $ b) =
berghofe@23743
   866
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
berghofe@23743
   867
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
berghofe@22262
   868
              | decr r = (r,"r");
berghofe@22262
   869
            val (rel,r) = decr rel;
berghofe@26801
   870
        in SOME (a, b, rel, r) end
berghofe@22262
   871
      | dec _ =  NONE
berghofe@30107
   872
    in dec t end
berghofe@30107
   873
    | decomp _ = NONE;
wenzelm@32215
   874
);
wenzelm@26340
   875
*}
berghofe@22262
   876
wenzelm@26340
   877
declaration {* fn _ =>
wenzelm@26340
   878
  Simplifier.map_ss (fn ss => ss
wenzelm@32215
   879
    addSolver (mk_solver' "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   880
    addSolver (mk_solver' "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
wenzelm@32215
   881
    addSolver (mk_solver' "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   882
    addSolver (mk_solver' "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
ballarin@15076
   883
*}
ballarin@15076
   884
wenzelm@32215
   885
wenzelm@32215
   886
text {* Optional methods. *}
ballarin@15076
   887
ballarin@15076
   888
method_setup trancl =
wenzelm@32215
   889
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
wenzelm@18372
   890
  {* simple transitivity reasoner *}
ballarin@15076
   891
method_setup rtrancl =
wenzelm@32215
   892
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
ballarin@15076
   893
  {* simple transitivity reasoner *}
berghofe@22262
   894
method_setup tranclp =
wenzelm@32215
   895
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
berghofe@22262
   896
  {* simple transitivity reasoner (predicate version) *}
berghofe@22262
   897
method_setup rtranclp =
wenzelm@32215
   898
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
berghofe@22262
   899
  {* simple transitivity reasoner (predicate version) *}
ballarin@15076
   900
nipkow@10213
   901
end