src/HOL/Predicate.thy
 author nipkow Fri Aug 28 18:52:41 2009 +0200 (2009-08-28) changeset 32436 10cd49e0c067 parent 32372 b0d2b49bfaed child 32578 22117a76f943 permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
```     1 (*  Title:      HOL/Predicate.thy
```
```     2     Author:     Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Predicates as relations and enumerations *}
```
```     6
```
```     7 theory Predicate
```
```     8 imports Inductive Relation
```
```     9 begin
```
```    10
```
```    11 notation
```
```    12   inf (infixl "\<sqinter>" 70) and
```
```    13   sup (infixl "\<squnion>" 65) and
```
```    14   Inf ("\<Sqinter>_" [900] 900) and
```
```    15   Sup ("\<Squnion>_" [900] 900) and
```
```    16   top ("\<top>") and
```
```    17   bot ("\<bottom>")
```
```    18
```
```    19
```
```    20 subsection {* Predicates as (complete) lattices *}
```
```    21
```
```    22 subsubsection {* @{const sup} on @{typ bool} *}
```
```    23
```
```    24 lemma sup_boolI1:
```
```    25   "P \<Longrightarrow> P \<squnion> Q"
```
```    26   by (simp add: sup_bool_eq)
```
```    27
```
```    28 lemma sup_boolI2:
```
```    29   "Q \<Longrightarrow> P \<squnion> Q"
```
```    30   by (simp add: sup_bool_eq)
```
```    31
```
```    32 lemma sup_boolE:
```
```    33   "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```    34   by (auto simp add: sup_bool_eq)
```
```    35
```
```    36
```
```    37 subsubsection {* Equality and Subsets *}
```
```    38
```
```    39 lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)"
```
```    40   by (simp add: mem_def)
```
```    41
```
```    42 lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
```
```    43   by (simp add: expand_fun_eq mem_def)
```
```    44
```
```    45 lemma pred_subset_eq: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)"
```
```    46   by (simp add: mem_def)
```
```    47
```
```    48 lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) <= (\<lambda>x y. (x, y) \<in> S)) = (R <= S)"
```
```    49   by fast
```
```    50
```
```    51
```
```    52 subsubsection {* Top and bottom elements *}
```
```    53
```
```    54 lemma top1I [intro!]: "top x"
```
```    55   by (simp add: top_fun_eq top_bool_eq)
```
```    56
```
```    57 lemma top2I [intro!]: "top x y"
```
```    58   by (simp add: top_fun_eq top_bool_eq)
```
```    59
```
```    60 lemma bot1E [elim!]: "bot x \<Longrightarrow> P"
```
```    61   by (simp add: bot_fun_eq bot_bool_eq)
```
```    62
```
```    63 lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
```
```    64   by (simp add: bot_fun_eq bot_bool_eq)
```
```    65
```
```    66
```
```    67 subsubsection {* The empty set *}
```
```    68
```
```    69 lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
```
```    70   by (auto simp add: expand_fun_eq)
```
```    71
```
```    72 lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
```
```    73   by (auto simp add: expand_fun_eq)
```
```    74
```
```    75
```
```    76 subsubsection {* Binary union *}
```
```    77
```
```    78 lemma sup1_iff [simp]: "sup A B x \<longleftrightarrow> A x | B x"
```
```    79   by (simp add: sup_fun_eq sup_bool_eq)
```
```    80
```
```    81 lemma sup2_iff [simp]: "sup A B x y \<longleftrightarrow> A x y | B x y"
```
```    82   by (simp add: sup_fun_eq sup_bool_eq)
```
```    83
```
```    84 lemma sup_Un_eq [pred_set_conv]: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
```
```    85   by (simp add: expand_fun_eq)
```
```    86
```
```    87 lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
```
```    88   by (simp add: expand_fun_eq)
```
```    89
```
```    90 lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
```
```    91   by simp
```
```    92
```
```    93 lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
```
```    94   by simp
```
```    95
```
```    96 lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
```
```    97   by simp
```
```    98
```
```    99 lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
```
```   100   by simp
```
```   101
```
```   102 text {*
```
```   103   \medskip Classical introduction rule: no commitment to @{text A} vs
```
```   104   @{text B}.
```
```   105 *}
```
```   106
```
```   107 lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"
```
```   108   by auto
```
```   109
```
```   110 lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
```
```   111   by auto
```
```   112
```
```   113 lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
```
```   114   by simp iprover
```
```   115
```
```   116 lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
```
```   117   by simp iprover
```
```   118
```
```   119
```
```   120 subsubsection {* Binary intersection *}
```
```   121
```
```   122 lemma inf1_iff [simp]: "inf A B x \<longleftrightarrow> A x \<and> B x"
```
```   123   by (simp add: inf_fun_eq inf_bool_eq)
```
```   124
```
```   125 lemma inf2_iff [simp]: "inf A B x y \<longleftrightarrow> A x y \<and> B x y"
```
```   126   by (simp add: inf_fun_eq inf_bool_eq)
```
```   127
```
```   128 lemma inf_Int_eq [pred_set_conv]: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
```
```   129   by (simp add: expand_fun_eq)
```
```   130
```
```   131 lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
```
```   132   by (simp add: expand_fun_eq)
```
```   133
```
```   134 lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
```
```   135   by simp
```
```   136
```
```   137 lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
```
```   138   by simp
```
```   139
```
```   140 lemma inf1D1: "inf A B x ==> A x"
```
```   141   by simp
```
```   142
```
```   143 lemma inf2D1: "inf A B x y ==> A x y"
```
```   144   by simp
```
```   145
```
```   146 lemma inf1D2: "inf A B x ==> B x"
```
```   147   by simp
```
```   148
```
```   149 lemma inf2D2: "inf A B x y ==> B x y"
```
```   150   by simp
```
```   151
```
```   152 lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
```
```   153   by simp
```
```   154
```
```   155 lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
```
```   156   by simp
```
```   157
```
```   158
```
```   159 subsubsection {* Unions of families *}
```
```   160
```
```   161 lemma SUP1_iff [simp]: "(SUP x:A. B x) b = (EX x:A. B x b)"
```
```   162   by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
```
```   163
```
```   164 lemma SUP2_iff [simp]: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
```
```   165   by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
```
```   166
```
```   167 lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
```
```   168   by auto
```
```   169
```
```   170 lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
```
```   171   by auto
```
```   172
```
```   173 lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
```
```   174   by auto
```
```   175
```
```   176 lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
```
```   177   by auto
```
```   178
```
```   179 lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
```
```   180   by (simp add: expand_fun_eq)
```
```   181
```
```   182 lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
```
```   183   by (simp add: expand_fun_eq)
```
```   184
```
```   185
```
```   186 subsubsection {* Intersections of families *}
```
```   187
```
```   188 lemma INF1_iff [simp]: "(INF x:A. B x) b = (ALL x:A. B x b)"
```
```   189   by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
```
```   190
```
```   191 lemma INF2_iff [simp]: "(INF x:A. B x) b c = (ALL x:A. B x b c)"
```
```   192   by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
```
```   193
```
```   194 lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"
```
```   195   by auto
```
```   196
```
```   197 lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"
```
```   198   by auto
```
```   199
```
```   200 lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"
```
```   201   by auto
```
```   202
```
```   203 lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"
```
```   204   by auto
```
```   205
```
```   206 lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"
```
```   207   by auto
```
```   208
```
```   209 lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"
```
```   210   by auto
```
```   211
```
```   212 lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))"
```
```   213   by (simp add: expand_fun_eq)
```
```   214
```
```   215 lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))"
```
```   216   by (simp add: expand_fun_eq)
```
```   217
```
```   218
```
```   219 subsection {* Predicates as relations *}
```
```   220
```
```   221 subsubsection {* Composition  *}
```
```   222
```
```   223 inductive
```
```   224   pred_comp  :: "['a => 'b => bool, 'b => 'c => bool] => 'a => 'c => bool"
```
```   225     (infixr "OO" 75)
```
```   226   for r :: "'a => 'b => bool" and s :: "'b => 'c => bool"
```
```   227 where
```
```   228   pred_compI [intro]: "r a b ==> s b c ==> (r OO s) a c"
```
```   229
```
```   230 inductive_cases pred_compE [elim!]: "(r OO s) a c"
```
```   231
```
```   232 lemma pred_comp_rel_comp_eq [pred_set_conv]:
```
```   233   "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
```
```   234   by (auto simp add: expand_fun_eq elim: pred_compE)
```
```   235
```
```   236
```
```   237 subsubsection {* Converse *}
```
```   238
```
```   239 inductive
```
```   240   conversep :: "('a => 'b => bool) => 'b => 'a => bool"
```
```   241     ("(_^--1)" [1000] 1000)
```
```   242   for r :: "'a => 'b => bool"
```
```   243 where
```
```   244   conversepI: "r a b ==> r^--1 b a"
```
```   245
```
```   246 notation (xsymbols)
```
```   247   conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
```
```   248
```
```   249 lemma conversepD:
```
```   250   assumes ab: "r^--1 a b"
```
```   251   shows "r b a" using ab
```
```   252   by cases simp
```
```   253
```
```   254 lemma conversep_iff [iff]: "r^--1 a b = r b a"
```
```   255   by (iprover intro: conversepI dest: conversepD)
```
```   256
```
```   257 lemma conversep_converse_eq [pred_set_conv]:
```
```   258   "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
```
```   259   by (auto simp add: expand_fun_eq)
```
```   260
```
```   261 lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
```
```   262   by (iprover intro: order_antisym conversepI dest: conversepD)
```
```   263
```
```   264 lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
```
```   265   by (iprover intro: order_antisym conversepI pred_compI
```
```   266     elim: pred_compE dest: conversepD)
```
```   267
```
```   268 lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"
```
```   269   by (simp add: inf_fun_eq inf_bool_eq)
```
```   270     (iprover intro: conversepI ext dest: conversepD)
```
```   271
```
```   272 lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
```
```   273   by (simp add: sup_fun_eq sup_bool_eq)
```
```   274     (iprover intro: conversepI ext dest: conversepD)
```
```   275
```
```   276 lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
```
```   277   by (auto simp add: expand_fun_eq)
```
```   278
```
```   279 lemma conversep_eq [simp]: "(op =)^--1 = op ="
```
```   280   by (auto simp add: expand_fun_eq)
```
```   281
```
```   282
```
```   283 subsubsection {* Domain *}
```
```   284
```
```   285 inductive
```
```   286   DomainP :: "('a => 'b => bool) => 'a => bool"
```
```   287   for r :: "'a => 'b => bool"
```
```   288 where
```
```   289   DomainPI [intro]: "r a b ==> DomainP r a"
```
```   290
```
```   291 inductive_cases DomainPE [elim!]: "DomainP r a"
```
```   292
```
```   293 lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
```
```   294   by (blast intro!: Orderings.order_antisym predicate1I)
```
```   295
```
```   296
```
```   297 subsubsection {* Range *}
```
```   298
```
```   299 inductive
```
```   300   RangeP :: "('a => 'b => bool) => 'b => bool"
```
```   301   for r :: "'a => 'b => bool"
```
```   302 where
```
```   303   RangePI [intro]: "r a b ==> RangeP r b"
```
```   304
```
```   305 inductive_cases RangePE [elim!]: "RangeP r b"
```
```   306
```
```   307 lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
```
```   308   by (blast intro!: Orderings.order_antisym predicate1I)
```
```   309
```
```   310
```
```   311 subsubsection {* Inverse image *}
```
```   312
```
```   313 definition
```
```   314   inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
```
```   315   "inv_imagep r f == %x y. r (f x) (f y)"
```
```   316
```
```   317 lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
```
```   318   by (simp add: inv_image_def inv_imagep_def)
```
```   319
```
```   320 lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
```
```   321   by (simp add: inv_imagep_def)
```
```   322
```
```   323
```
```   324 subsubsection {* Powerset *}
```
```   325
```
```   326 definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
```
```   327   "Powp A == \<lambda>B. \<forall>x \<in> B. A x"
```
```   328
```
```   329 lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
```
```   330   by (auto simp add: Powp_def expand_fun_eq)
```
```   331
```
```   332 lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]
```
```   333
```
```   334
```
```   335 subsubsection {* Properties of relations *}
```
```   336
```
```   337 abbreviation antisymP :: "('a => 'a => bool) => bool" where
```
```   338   "antisymP r == antisym {(x, y). r x y}"
```
```   339
```
```   340 abbreviation transP :: "('a => 'a => bool) => bool" where
```
```   341   "transP r == trans {(x, y). r x y}"
```
```   342
```
```   343 abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
```
```   344   "single_valuedP r == single_valued {(x, y). r x y}"
```
```   345
```
```   346
```
```   347 subsection {* Predicates as enumerations *}
```
```   348
```
```   349 subsubsection {* The type of predicate enumerations (a monad) *}
```
```   350
```
```   351 datatype 'a pred = Pred "'a \<Rightarrow> bool"
```
```   352
```
```   353 primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
```
```   354   eval_pred: "eval (Pred f) = f"
```
```   355
```
```   356 lemma Pred_eval [simp]:
```
```   357   "Pred (eval x) = x"
```
```   358   by (cases x) simp
```
```   359
```
```   360 lemma eval_inject: "eval x = eval y \<longleftrightarrow> x = y"
```
```   361   by (cases x) auto
```
```   362
```
```   363 definition single :: "'a \<Rightarrow> 'a pred" where
```
```   364   "single x = Pred ((op =) x)"
```
```   365
```
```   366 definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
```
```   367   "P \<guillemotright>= f = Pred (\<lambda>x. (\<exists>y. eval P y \<and> eval (f y) x))"
```
```   368
```
```   369 instantiation pred :: (type) complete_lattice
```
```   370 begin
```
```   371
```
```   372 definition
```
```   373   "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
```
```   374
```
```   375 definition
```
```   376   "P < Q \<longleftrightarrow> eval P < eval Q"
```
```   377
```
```   378 definition
```
```   379   "\<bottom> = Pred \<bottom>"
```
```   380
```
```   381 definition
```
```   382   "\<top> = Pred \<top>"
```
```   383
```
```   384 definition
```
```   385   "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
```
```   386
```
```   387 definition
```
```   388   "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
```
```   389
```
```   390 definition
```
```   391   [code del]: "\<Sqinter>A = Pred (INFI A eval)"
```
```   392
```
```   393 definition
```
```   394   [code del]: "\<Squnion>A = Pred (SUPR A eval)"
```
```   395
```
```   396 instance by default
```
```   397   (auto simp add: less_eq_pred_def less_pred_def
```
```   398     inf_pred_def sup_pred_def bot_pred_def top_pred_def
```
```   399     Inf_pred_def Sup_pred_def,
```
```   400     auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def
```
```   401     eval_inject mem_def)
```
```   402
```
```   403 end
```
```   404
```
```   405 lemma bind_bind:
```
```   406   "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
```
```   407   by (auto simp add: bind_def expand_fun_eq)
```
```   408
```
```   409 lemma bind_single:
```
```   410   "P \<guillemotright>= single = P"
```
```   411   by (simp add: bind_def single_def)
```
```   412
```
```   413 lemma single_bind:
```
```   414   "single x \<guillemotright>= P = P x"
```
```   415   by (simp add: bind_def single_def)
```
```   416
```
```   417 lemma bottom_bind:
```
```   418   "\<bottom> \<guillemotright>= P = \<bottom>"
```
```   419   by (auto simp add: bot_pred_def bind_def expand_fun_eq)
```
```   420
```
```   421 lemma sup_bind:
```
```   422   "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
```
```   423   by (auto simp add: bind_def sup_pred_def expand_fun_eq)
```
```   424
```
```   425 lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
```
```   426   by (auto simp add: bind_def Sup_pred_def expand_fun_eq)
```
```   427
```
```   428 lemma pred_iffI:
```
```   429   assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
```
```   430   and "\<And>x. eval B x \<Longrightarrow> eval A x"
```
```   431   shows "A = B"
```
```   432 proof -
```
```   433   from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast
```
```   434   then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq)
```
```   435 qed
```
```   436
```
```   437 lemma singleI: "eval (single x) x"
```
```   438   unfolding single_def by simp
```
```   439
```
```   440 lemma singleI_unit: "eval (single ()) x"
```
```   441   by simp (rule singleI)
```
```   442
```
```   443 lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   444   unfolding single_def by simp
```
```   445
```
```   446 lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   447   by (erule singleE) simp
```
```   448
```
```   449 lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
```
```   450   unfolding bind_def by auto
```
```   451
```
```   452 lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   453   unfolding bind_def by auto
```
```   454
```
```   455 lemma botE: "eval \<bottom> x \<Longrightarrow> P"
```
```   456   unfolding bot_pred_def by auto
```
```   457
```
```   458 lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   459   unfolding sup_pred_def by simp
```
```   460
```
```   461 lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   462   unfolding sup_pred_def by simp
```
```   463
```
```   464 lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   465   unfolding sup_pred_def by auto
```
```   466
```
```   467
```
```   468 subsubsection {* Derived operations *}
```
```   469
```
```   470 definition if_pred :: "bool \<Rightarrow> unit pred" where
```
```   471   if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
```
```   472
```
```   473 definition not_pred :: "unit pred \<Rightarrow> unit pred" where
```
```   474   not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
```
```   475
```
```   476 lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
```
```   477   unfolding if_pred_eq by (auto intro: singleI)
```
```   478
```
```   479 lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
```
```   480   unfolding if_pred_eq by (cases b) (auto elim: botE)
```
```   481
```
```   482 lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
```
```   483   unfolding not_pred_eq eval_pred by (auto intro: singleI)
```
```   484
```
```   485 lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
```
```   486   unfolding not_pred_eq by (auto intro: singleI)
```
```   487
```
```   488 lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   489   unfolding not_pred_eq
```
```   490   by (auto split: split_if_asm elim: botE)
```
```   491
```
```   492 lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   493   unfolding not_pred_eq
```
```   494   by (auto split: split_if_asm elim: botE)
```
```   495
```
```   496
```
```   497 subsubsection {* Implementation *}
```
```   498
```
```   499 datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
```
```   500
```
```   501 primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
```
```   502     "pred_of_seq Empty = \<bottom>"
```
```   503   | "pred_of_seq (Insert x P) = single x \<squnion> P"
```
```   504   | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
```
```   505
```
```   506 definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
```
```   507   "Seq f = pred_of_seq (f ())"
```
```   508
```
```   509 code_datatype Seq
```
```   510
```
```   511 primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
```
```   512   "member Empty x \<longleftrightarrow> False"
```
```   513   | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
```
```   514   | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
```
```   515
```
```   516 lemma eval_member:
```
```   517   "member xq = eval (pred_of_seq xq)"
```
```   518 proof (induct xq)
```
```   519   case Empty show ?case
```
```   520   by (auto simp add: expand_fun_eq elim: botE)
```
```   521 next
```
```   522   case Insert show ?case
```
```   523   by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI)
```
```   524 next
```
```   525   case Join then show ?case
```
```   526   by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2)
```
```   527 qed
```
```   528
```
```   529 lemma eval_code [code]: "eval (Seq f) = member (f ())"
```
```   530   unfolding Seq_def by (rule sym, rule eval_member)
```
```   531
```
```   532 lemma single_code [code]:
```
```   533   "single x = Seq (\<lambda>u. Insert x \<bottom>)"
```
```   534   unfolding Seq_def by simp
```
```   535
```
```   536 primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
```
```   537     "apply f Empty = Empty"
```
```   538   | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
```
```   539   | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
```
```   540
```
```   541 lemma apply_bind:
```
```   542   "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
```
```   543 proof (induct xq)
```
```   544   case Empty show ?case
```
```   545     by (simp add: bottom_bind)
```
```   546 next
```
```   547   case Insert show ?case
```
```   548     by (simp add: single_bind sup_bind)
```
```   549 next
```
```   550   case Join then show ?case
```
```   551     by (simp add: sup_bind)
```
```   552 qed
```
```   553
```
```   554 lemma bind_code [code]:
```
```   555   "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
```
```   556   unfolding Seq_def by (rule sym, rule apply_bind)
```
```   557
```
```   558 lemma bot_set_code [code]:
```
```   559   "\<bottom> = Seq (\<lambda>u. Empty)"
```
```   560   unfolding Seq_def by simp
```
```   561
```
```   562 primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
```
```   563     "adjunct P Empty = Join P Empty"
```
```   564   | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
```
```   565   | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
```
```   566
```
```   567 lemma adjunct_sup:
```
```   568   "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
```
```   569   by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
```
```   570
```
```   571 lemma sup_code [code]:
```
```   572   "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
```
```   573     of Empty \<Rightarrow> g ()
```
```   574      | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
```
```   575      | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
```
```   576 proof (cases "f ()")
```
```   577   case Empty
```
```   578   thus ?thesis
```
```   579     unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"]  sup_bot)
```
```   580 next
```
```   581   case Insert
```
```   582   thus ?thesis
```
```   583     unfolding Seq_def by (simp add: sup_assoc)
```
```   584 next
```
```   585   case Join
```
```   586   thus ?thesis
```
```   587     unfolding Seq_def
```
```   588     by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
```
```   589 qed
```
```   590
```
```   591 primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
```
```   592     "contained Empty Q \<longleftrightarrow> True"
```
```   593   | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
```
```   594   | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
```
```   595
```
```   596 lemma single_less_eq_eval:
```
```   597   "single x \<le> P \<longleftrightarrow> eval P x"
```
```   598   by (auto simp add: single_def less_eq_pred_def mem_def)
```
```   599
```
```   600 lemma contained_less_eq:
```
```   601   "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
```
```   602   by (induct xq) (simp_all add: single_less_eq_eval)
```
```   603
```
```   604 lemma less_eq_pred_code [code]:
```
```   605   "Seq f \<le> Q = (case f ()
```
```   606    of Empty \<Rightarrow> True
```
```   607     | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
```
```   608     | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
```
```   609   by (cases "f ()")
```
```   610     (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
```
```   611
```
```   612 lemma eq_pred_code [code]:
```
```   613   fixes P Q :: "'a pred"
```
```   614   shows "eq_class.eq P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
```
```   615   unfolding eq by auto
```
```   616
```
```   617 lemma [code]:
```
```   618   "pred_case f P = f (eval P)"
```
```   619   by (cases P) simp
```
```   620
```
```   621 lemma [code]:
```
```   622   "pred_rec f P = f (eval P)"
```
```   623   by (cases P) simp
```
```   624
```
```   625 inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
```
```   626
```
```   627 lemma eq_is_eq: "eq x y \<equiv> (x = y)"
```
```   628   by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
```
```   629
```
```   630 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
```
```   631   "map f P = P \<guillemotright>= (single o f)"
```
```   632
```
```   633 ML {*
```
```   634 signature PREDICATE =
```
```   635 sig
```
```   636   datatype 'a pred = Seq of (unit -> 'a seq)
```
```   637   and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
```
```   638   val yield: 'a pred -> ('a * 'a pred) option
```
```   639   val yieldn: int -> 'a pred -> 'a list * 'a pred
```
```   640   val map: ('a -> 'b) -> 'a pred -> 'b pred
```
```   641 end;
```
```   642
```
```   643 structure Predicate : PREDICATE =
```
```   644 struct
```
```   645
```
```   646 @{code_datatype pred = Seq};
```
```   647 @{code_datatype seq = Empty | Insert | Join};
```
```   648
```
```   649 fun yield (@{code Seq} f) = next (f ())
```
```   650 and next @{code Empty} = NONE
```
```   651   | next (@{code Insert} (x, P)) = SOME (x, P)
```
```   652   | next (@{code Join} (P, xq)) = (case yield P
```
```   653      of NONE => next xq
```
```   654       | SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq))))
```
```   655
```
```   656 fun anamorph f k x = (if k = 0 then ([], x)
```
```   657   else case f x
```
```   658    of NONE => ([], x)
```
```   659     | SOME (v, y) => let
```
```   660         val (vs, z) = anamorph f (k - 1) y
```
```   661       in (v :: vs, z) end)
```
```   662
```
```   663 fun yieldn P = anamorph yield P;
```
```   664
```
```   665 fun map f = @{code map} f;
```
```   666
```
```   667 end;
```
```   668 *}
```
```   669
```
```   670 code_reserved Eval Predicate
```
```   671
```
```   672 code_type pred and seq
```
```   673   (Eval "_/ Predicate.pred" and "_/ Predicate.seq")
```
```   674
```
```   675 code_const Seq and Empty and Insert and Join
```
```   676   (Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)")
```
```   677
```
```   678 text {* dummy setup for @{text code_pred} and @{text values} keywords *}
```
```   679
```
```   680 ML {*
```
```   681 local
```
```   682
```
```   683 structure P = OuterParse;
```
```   684
```
```   685 val opt_modes = Scan.optional (P.\$\$\$ "(" |-- P.!!! (Scan.repeat1 P.xname --| P.\$\$\$ ")")) [];
```
```   686
```
```   687 in
```
```   688
```
```   689 val _ = OuterSyntax.local_theory_to_proof "code_pred" "sets up goal for cases rule from given introduction rules and compiles predicate"
```
```   690   OuterKeyword.thy_goal (P.term_group >> (K (Proof.theorem_i NONE (K I) [[]])));
```
```   691
```
```   692 val _ = OuterSyntax.improper_command "values" "enumerate and print comprehensions"
```
```   693   OuterKeyword.diag ((opt_modes -- P.term)
```
```   694     >> (fn (modes, t) => Toplevel.no_timing o Toplevel.keep
```
```   695         (K ())));
```
```   696
```
```   697 end
```
```   698 *}
```
```   699
```
```   700 no_notation
```
```   701   inf (infixl "\<sqinter>" 70) and
```
```   702   sup (infixl "\<squnion>" 65) and
```
```   703   Inf ("\<Sqinter>_" [900] 900) and
```
```   704   Sup ("\<Squnion>_" [900] 900) and
```
```   705   top ("\<top>") and
```
```   706   bot ("\<bottom>") and
```
```   707   bind (infixl "\<guillemotright>=" 70)
```
```   708
```
```   709 hide (open) type pred seq
```
```   710 hide (open) const Pred eval single bind if_pred not_pred
```
```   711   Empty Insert Join Seq member pred_of_seq "apply" adjunct eq map
```
```   712
```
```   713 end
```