src/HOL/Predicate.thy
 author wenzelm Wed Nov 01 20:46:23 2017 +0100 (21 months ago) changeset 66983 df83b66f1d94 parent 66251 cd935b7cb3fb child 67091 1393c2340eec permissions -rw-r--r--
proper merge (amending fb46c031c841);
```     1 (*  Title:      HOL/Predicate.thy
```
```     2     Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Predicates as enumerations\<close>
```
```     6
```
```     7 theory Predicate
```
```     8 imports String
```
```     9 begin
```
```    10
```
```    11 subsection \<open>The type of predicate enumerations (a monad)\<close>
```
```    12
```
```    13 datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
```
```    14
```
```    15 lemma pred_eqI:
```
```    16   "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
```
```    17   by (cases P, cases Q) (auto simp add: fun_eq_iff)
```
```    18
```
```    19 lemma pred_eq_iff:
```
```    20   "P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)"
```
```    21   by (simp add: pred_eqI)
```
```    22
```
```    23 instantiation pred :: (type) complete_lattice
```
```    24 begin
```
```    25
```
```    26 definition
```
```    27   "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
```
```    28
```
```    29 definition
```
```    30   "P < Q \<longleftrightarrow> eval P < eval Q"
```
```    31
```
```    32 definition
```
```    33   "\<bottom> = Pred \<bottom>"
```
```    34
```
```    35 lemma eval_bot [simp]:
```
```    36   "eval \<bottom>  = \<bottom>"
```
```    37   by (simp add: bot_pred_def)
```
```    38
```
```    39 definition
```
```    40   "\<top> = Pred \<top>"
```
```    41
```
```    42 lemma eval_top [simp]:
```
```    43   "eval \<top>  = \<top>"
```
```    44   by (simp add: top_pred_def)
```
```    45
```
```    46 definition
```
```    47   "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
```
```    48
```
```    49 lemma eval_inf [simp]:
```
```    50   "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
```
```    51   by (simp add: inf_pred_def)
```
```    52
```
```    53 definition
```
```    54   "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
```
```    55
```
```    56 lemma eval_sup [simp]:
```
```    57   "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
```
```    58   by (simp add: sup_pred_def)
```
```    59
```
```    60 definition
```
```    61   "\<Sqinter>A = Pred (INFIMUM A eval)"
```
```    62
```
```    63 lemma eval_Inf [simp]:
```
```    64   "eval (\<Sqinter>A) = INFIMUM A eval"
```
```    65   by (simp add: Inf_pred_def)
```
```    66
```
```    67 definition
```
```    68   "\<Squnion>A = Pred (SUPREMUM A eval)"
```
```    69
```
```    70 lemma eval_Sup [simp]:
```
```    71   "eval (\<Squnion>A) = SUPREMUM A eval"
```
```    72   by (simp add: Sup_pred_def)
```
```    73
```
```    74 instance proof
```
```    75 qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
```
```    76
```
```    77 end
```
```    78
```
```    79 lemma eval_INF [simp]:
```
```    80   "eval (INFIMUM A f) = INFIMUM A (eval \<circ> f)"
```
```    81   using eval_Inf [of "f ` A"] by simp
```
```    82
```
```    83 lemma eval_SUP [simp]:
```
```    84   "eval (SUPREMUM A f) = SUPREMUM A (eval \<circ> f)"
```
```    85   using eval_Sup [of "f ` A"] by simp
```
```    86
```
```    87 instantiation pred :: (type) complete_boolean_algebra
```
```    88 begin
```
```    89
```
```    90 definition
```
```    91   "- P = Pred (- eval P)"
```
```    92
```
```    93 lemma eval_compl [simp]:
```
```    94   "eval (- P) = - eval P"
```
```    95   by (simp add: uminus_pred_def)
```
```    96
```
```    97 definition
```
```    98   "P - Q = Pred (eval P - eval Q)"
```
```    99
```
```   100 lemma eval_minus [simp]:
```
```   101   "eval (P - Q) = eval P - eval Q"
```
```   102   by (simp add: minus_pred_def)
```
```   103
```
```   104 instance proof
```
```   105 qed (auto intro!: pred_eqI)
```
```   106
```
```   107 end
```
```   108
```
```   109 definition single :: "'a \<Rightarrow> 'a pred" where
```
```   110   "single x = Pred ((op =) x)"
```
```   111
```
```   112 lemma eval_single [simp]:
```
```   113   "eval (single x) = (op =) x"
```
```   114   by (simp add: single_def)
```
```   115
```
```   116 definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<bind>" 70) where
```
```   117   "P \<bind> f = (SUPREMUM {x. eval P x} f)"
```
```   118
```
```   119 lemma eval_bind [simp]:
```
```   120   "eval (P \<bind> f) = eval (SUPREMUM {x. eval P x} f)"
```
```   121   by (simp add: bind_def)
```
```   122
```
```   123 lemma bind_bind:
```
```   124   "(P \<bind> Q) \<bind> R = P \<bind> (\<lambda>x. Q x \<bind> R)"
```
```   125   by (rule pred_eqI) auto
```
```   126
```
```   127 lemma bind_single:
```
```   128   "P \<bind> single = P"
```
```   129   by (rule pred_eqI) auto
```
```   130
```
```   131 lemma single_bind:
```
```   132   "single x \<bind> P = P x"
```
```   133   by (rule pred_eqI) auto
```
```   134
```
```   135 lemma bottom_bind:
```
```   136   "\<bottom> \<bind> P = \<bottom>"
```
```   137   by (rule pred_eqI) auto
```
```   138
```
```   139 lemma sup_bind:
```
```   140   "(P \<squnion> Q) \<bind> R = P \<bind> R \<squnion> Q \<bind> R"
```
```   141   by (rule pred_eqI) auto
```
```   142
```
```   143 lemma Sup_bind:
```
```   144   "(\<Squnion>A \<bind> f) = \<Squnion>((\<lambda>x. x \<bind> f) ` A)"
```
```   145   by (rule pred_eqI) auto
```
```   146
```
```   147 lemma pred_iffI:
```
```   148   assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
```
```   149   and "\<And>x. eval B x \<Longrightarrow> eval A x"
```
```   150   shows "A = B"
```
```   151   using assms by (auto intro: pred_eqI)
```
```   152
```
```   153 lemma singleI: "eval (single x) x"
```
```   154   by simp
```
```   155
```
```   156 lemma singleI_unit: "eval (single ()) x"
```
```   157   by simp
```
```   158
```
```   159 lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   160   by simp
```
```   161
```
```   162 lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   163   by simp
```
```   164
```
```   165 lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<bind> Q) y"
```
```   166   by auto
```
```   167
```
```   168 lemma bindE: "eval (R \<bind> Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   169   by auto
```
```   170
```
```   171 lemma botE: "eval \<bottom> x \<Longrightarrow> P"
```
```   172   by auto
```
```   173
```
```   174 lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   175   by auto
```
```   176
```
```   177 lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   178   by auto
```
```   179
```
```   180 lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   181   by auto
```
```   182
```
```   183 lemma single_not_bot [simp]:
```
```   184   "single x \<noteq> \<bottom>"
```
```   185   by (auto simp add: single_def bot_pred_def fun_eq_iff)
```
```   186
```
```   187 lemma not_bot:
```
```   188   assumes "A \<noteq> \<bottom>"
```
```   189   obtains x where "eval A x"
```
```   190   using assms by (cases A) (auto simp add: bot_pred_def)
```
```   191
```
```   192
```
```   193 subsection \<open>Emptiness check and definite choice\<close>
```
```   194
```
```   195 definition is_empty :: "'a pred \<Rightarrow> bool" where
```
```   196   "is_empty A \<longleftrightarrow> A = \<bottom>"
```
```   197
```
```   198 lemma is_empty_bot:
```
```   199   "is_empty \<bottom>"
```
```   200   by (simp add: is_empty_def)
```
```   201
```
```   202 lemma not_is_empty_single:
```
```   203   "\<not> is_empty (single x)"
```
```   204   by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
```
```   205
```
```   206 lemma is_empty_sup:
```
```   207   "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
```
```   208   by (auto simp add: is_empty_def)
```
```   209
```
```   210 definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
```
```   211   "singleton default A = (if \<exists>!x. eval A x then THE x. eval A x else default ())" for default
```
```   212
```
```   213 lemma singleton_eqI:
```
```   214   "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton default A = x" for default
```
```   215   by (auto simp add: singleton_def)
```
```   216
```
```   217 lemma eval_singletonI:
```
```   218   "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton default A)" for default
```
```   219 proof -
```
```   220   assume assm: "\<exists>!x. eval A x"
```
```   221   then obtain x where x: "eval A x" ..
```
```   222   with assm have "singleton default A = x" by (rule singleton_eqI)
```
```   223   with x show ?thesis by simp
```
```   224 qed
```
```   225
```
```   226 lemma single_singleton:
```
```   227   "\<exists>!x. eval A x \<Longrightarrow> single (singleton default A) = A" for default
```
```   228 proof -
```
```   229   assume assm: "\<exists>!x. eval A x"
```
```   230   then have "eval A (singleton default A)"
```
```   231     by (rule eval_singletonI)
```
```   232   moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton default A = x"
```
```   233     by (rule singleton_eqI)
```
```   234   ultimately have "eval (single (singleton default A)) = eval A"
```
```   235     by (simp (no_asm_use) add: single_def fun_eq_iff) blast
```
```   236   then have "\<And>x. eval (single (singleton default A)) x = eval A x"
```
```   237     by simp
```
```   238   then show ?thesis by (rule pred_eqI)
```
```   239 qed
```
```   240
```
```   241 lemma singleton_undefinedI:
```
```   242   "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton default A = default ()" for default
```
```   243   by (simp add: singleton_def)
```
```   244
```
```   245 lemma singleton_bot:
```
```   246   "singleton default \<bottom> = default ()" for default
```
```   247   by (auto simp add: bot_pred_def intro: singleton_undefinedI)
```
```   248
```
```   249 lemma singleton_single:
```
```   250   "singleton default (single x) = x" for default
```
```   251   by (auto simp add: intro: singleton_eqI singleI elim: singleE)
```
```   252
```
```   253 lemma singleton_sup_single_single:
```
```   254   "singleton default (single x \<squnion> single y) = (if x = y then x else default ())" for default
```
```   255 proof (cases "x = y")
```
```   256   case True then show ?thesis by (simp add: singleton_single)
```
```   257 next
```
```   258   case False
```
```   259   have "eval (single x \<squnion> single y) x"
```
```   260     and "eval (single x \<squnion> single y) y"
```
```   261   by (auto intro: supI1 supI2 singleI)
```
```   262   with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
```
```   263     by blast
```
```   264   then have "singleton default (single x \<squnion> single y) = default ()"
```
```   265     by (rule singleton_undefinedI)
```
```   266   with False show ?thesis by simp
```
```   267 qed
```
```   268
```
```   269 lemma singleton_sup_aux:
```
```   270   "singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B
```
```   271     else if B = \<bottom> then singleton default A
```
```   272     else singleton default
```
```   273       (single (singleton default A) \<squnion> single (singleton default B)))" for default
```
```   274 proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
```
```   275   case True then show ?thesis by (simp add: single_singleton)
```
```   276 next
```
```   277   case False
```
```   278   from False have A_or_B:
```
```   279     "singleton default A = default () \<or> singleton default B = default ()"
```
```   280     by (auto intro!: singleton_undefinedI)
```
```   281   then have rhs: "singleton default
```
```   282     (single (singleton default A) \<squnion> single (singleton default B)) = default ()"
```
```   283     by (auto simp add: singleton_sup_single_single singleton_single)
```
```   284   from False have not_unique:
```
```   285     "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
```
```   286   show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
```
```   287     case True
```
```   288     then obtain a b where a: "eval A a" and b: "eval B b"
```
```   289       by (blast elim: not_bot)
```
```   290     with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
```
```   291       by (auto simp add: sup_pred_def bot_pred_def)
```
```   292     then have "singleton default (A \<squnion> B) = default ()" by (rule singleton_undefinedI)
```
```   293     with True rhs show ?thesis by simp
```
```   294   next
```
```   295     case False then show ?thesis by auto
```
```   296   qed
```
```   297 qed
```
```   298
```
```   299 lemma singleton_sup:
```
```   300   "singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B
```
```   301     else if B = \<bottom> then singleton default A
```
```   302     else if singleton default A = singleton default B then singleton default A else default ())" for default
```
```   303   using singleton_sup_aux [of default A B] by (simp only: singleton_sup_single_single)
```
```   304
```
```   305
```
```   306 subsection \<open>Derived operations\<close>
```
```   307
```
```   308 definition if_pred :: "bool \<Rightarrow> unit pred" where
```
```   309   if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
```
```   310
```
```   311 definition holds :: "unit pred \<Rightarrow> bool" where
```
```   312   holds_eq: "holds P = eval P ()"
```
```   313
```
```   314 definition not_pred :: "unit pred \<Rightarrow> unit pred" where
```
```   315   not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
```
```   316
```
```   317 lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
```
```   318   unfolding if_pred_eq by (auto intro: singleI)
```
```   319
```
```   320 lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
```
```   321   unfolding if_pred_eq by (cases b) (auto elim: botE)
```
```   322
```
```   323 lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
```
```   324   unfolding not_pred_eq by (auto intro: singleI)
```
```   325
```
```   326 lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
```
```   327   unfolding not_pred_eq by (auto intro: singleI)
```
```   328
```
```   329 lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   330   unfolding not_pred_eq
```
```   331   by (auto split: if_split_asm elim: botE)
```
```   332
```
```   333 lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   334   unfolding not_pred_eq
```
```   335   by (auto split: if_split_asm elim: botE)
```
```   336 lemma "f () = False \<or> f () = True"
```
```   337 by simp
```
```   338
```
```   339 lemma closure_of_bool_cases [no_atp]:
```
```   340   fixes f :: "unit \<Rightarrow> bool"
```
```   341   assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
```
```   342   assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
```
```   343   shows "P f"
```
```   344 proof -
```
```   345   have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
```
```   346     apply (cases "f ()")
```
```   347     apply (rule disjI2)
```
```   348     apply (rule ext)
```
```   349     apply (simp add: unit_eq)
```
```   350     apply (rule disjI1)
```
```   351     apply (rule ext)
```
```   352     apply (simp add: unit_eq)
```
```   353     done
```
```   354   from this assms show ?thesis by blast
```
```   355 qed
```
```   356
```
```   357 lemma unit_pred_cases:
```
```   358   assumes "P \<bottom>"
```
```   359   assumes "P (single ())"
```
```   360   shows "P Q"
```
```   361 using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
```
```   362   fix f
```
```   363   assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
```
```   364   then have "P (Pred f)"
```
```   365     by (cases _ f rule: closure_of_bool_cases) simp_all
```
```   366   moreover assume "Q = Pred f"
```
```   367   ultimately show "P Q" by simp
```
```   368 qed
```
```   369
```
```   370 lemma holds_if_pred:
```
```   371   "holds (if_pred b) = b"
```
```   372 unfolding if_pred_eq holds_eq
```
```   373 by (cases b) (auto intro: singleI elim: botE)
```
```   374
```
```   375 lemma if_pred_holds:
```
```   376   "if_pred (holds P) = P"
```
```   377 unfolding if_pred_eq holds_eq
```
```   378 by (rule unit_pred_cases) (auto intro: singleI elim: botE)
```
```   379
```
```   380 lemma is_empty_holds:
```
```   381   "is_empty P \<longleftrightarrow> \<not> holds P"
```
```   382 unfolding is_empty_def holds_eq
```
```   383 by (rule unit_pred_cases) (auto elim: botE intro: singleI)
```
```   384
```
```   385 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
```
```   386   "map f P = P \<bind> (single o f)"
```
```   387
```
```   388 lemma eval_map [simp]:
```
```   389   "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
```
```   390   by (auto simp add: map_def comp_def)
```
```   391
```
```   392 functor map: map
```
```   393   by (rule ext, rule pred_eqI, auto)+
```
```   394
```
```   395
```
```   396 subsection \<open>Implementation\<close>
```
```   397
```
```   398 datatype (plugins only: code extraction) (dead 'a) seq =
```
```   399   Empty
```
```   400 | Insert "'a" "'a pred"
```
```   401 | Join "'a pred" "'a seq"
```
```   402
```
```   403 primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
```
```   404   "pred_of_seq Empty = \<bottom>"
```
```   405 | "pred_of_seq (Insert x P) = single x \<squnion> P"
```
```   406 | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
```
```   407
```
```   408 definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
```
```   409   "Seq f = pred_of_seq (f ())"
```
```   410
```
```   411 code_datatype Seq
```
```   412
```
```   413 primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
```
```   414   "member Empty x \<longleftrightarrow> False"
```
```   415 | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
```
```   416 | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
```
```   417
```
```   418 lemma eval_member:
```
```   419   "member xq = eval (pred_of_seq xq)"
```
```   420 proof (induct xq)
```
```   421   case Empty show ?case
```
```   422   by (auto simp add: fun_eq_iff elim: botE)
```
```   423 next
```
```   424   case Insert show ?case
```
```   425   by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
```
```   426 next
```
```   427   case Join then show ?case
```
```   428   by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
```
```   429 qed
```
```   430
```
```   431 lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
```
```   432   unfolding Seq_def by (rule sym, rule eval_member)
```
```   433
```
```   434 lemma single_code [code]:
```
```   435   "single x = Seq (\<lambda>u. Insert x \<bottom>)"
```
```   436   unfolding Seq_def by simp
```
```   437
```
```   438 primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
```
```   439   "apply f Empty = Empty"
```
```   440 | "apply f (Insert x P) = Join (f x) (Join (P \<bind> f) Empty)"
```
```   441 | "apply f (Join P xq) = Join (P \<bind> f) (apply f xq)"
```
```   442
```
```   443 lemma apply_bind:
```
```   444   "pred_of_seq (apply f xq) = pred_of_seq xq \<bind> f"
```
```   445 proof (induct xq)
```
```   446   case Empty show ?case
```
```   447     by (simp add: bottom_bind)
```
```   448 next
```
```   449   case Insert show ?case
```
```   450     by (simp add: single_bind sup_bind)
```
```   451 next
```
```   452   case Join then show ?case
```
```   453     by (simp add: sup_bind)
```
```   454 qed
```
```   455
```
```   456 lemma bind_code [code]:
```
```   457   "Seq g \<bind> f = Seq (\<lambda>u. apply f (g ()))"
```
```   458   unfolding Seq_def by (rule sym, rule apply_bind)
```
```   459
```
```   460 lemma bot_set_code [code]:
```
```   461   "\<bottom> = Seq (\<lambda>u. Empty)"
```
```   462   unfolding Seq_def by simp
```
```   463
```
```   464 primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
```
```   465   "adjunct P Empty = Join P Empty"
```
```   466 | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
```
```   467 | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
```
```   468
```
```   469 lemma adjunct_sup:
```
```   470   "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
```
```   471   by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
```
```   472
```
```   473 lemma sup_code [code]:
```
```   474   "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
```
```   475     of Empty \<Rightarrow> g ()
```
```   476      | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
```
```   477      | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
```
```   478 proof (cases "f ()")
```
```   479   case Empty
```
```   480   thus ?thesis
```
```   481     unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
```
```   482 next
```
```   483   case Insert
```
```   484   thus ?thesis
```
```   485     unfolding Seq_def by (simp add: sup_assoc)
```
```   486 next
```
```   487   case Join
```
```   488   thus ?thesis
```
```   489     unfolding Seq_def
```
```   490     by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
```
```   491 qed
```
```   492
```
```   493 primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
```
```   494   "contained Empty Q \<longleftrightarrow> True"
```
```   495 | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
```
```   496 | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
```
```   497
```
```   498 lemma single_less_eq_eval:
```
```   499   "single x \<le> P \<longleftrightarrow> eval P x"
```
```   500   by (auto simp add: less_eq_pred_def le_fun_def)
```
```   501
```
```   502 lemma contained_less_eq:
```
```   503   "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
```
```   504   by (induct xq) (simp_all add: single_less_eq_eval)
```
```   505
```
```   506 lemma less_eq_pred_code [code]:
```
```   507   "Seq f \<le> Q = (case f ()
```
```   508    of Empty \<Rightarrow> True
```
```   509     | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
```
```   510     | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
```
```   511   by (cases "f ()")
```
```   512     (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
```
```   513
```
```   514 instantiation pred :: (type) equal
```
```   515 begin
```
```   516
```
```   517 definition equal_pred
```
```   518   where [simp]: "HOL.equal P Q \<longleftrightarrow> P = (Q :: 'a pred)"
```
```   519
```
```   520 instance by standard simp
```
```   521
```
```   522 end
```
```   523
```
```   524 lemma [code]:
```
```   525   "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P" for P Q :: "'a pred"
```
```   526   by auto
```
```   527
```
```   528 lemma [code nbe]:
```
```   529   "HOL.equal P P \<longleftrightarrow> True" for P :: "'a pred"
```
```   530   by (fact equal_refl)
```
```   531
```
```   532 lemma [code]:
```
```   533   "case_pred f P = f (eval P)"
```
```   534   by (fact pred.case_eq_if)
```
```   535
```
```   536 lemma [code]:
```
```   537   "rec_pred f P = f (eval P)"
```
```   538   by (cases P) simp
```
```   539
```
```   540 inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
```
```   541
```
```   542 lemma eq_is_eq: "eq x y \<equiv> (x = y)"
```
```   543   by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
```
```   544
```
```   545 primrec null :: "'a seq \<Rightarrow> bool" where
```
```   546   "null Empty \<longleftrightarrow> True"
```
```   547 | "null (Insert x P) \<longleftrightarrow> False"
```
```   548 | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
```
```   549
```
```   550 lemma null_is_empty:
```
```   551   "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
```
```   552   by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
```
```   553
```
```   554 lemma is_empty_code [code]:
```
```   555   "is_empty (Seq f) \<longleftrightarrow> null (f ())"
```
```   556   by (simp add: null_is_empty Seq_def)
```
```   557
```
```   558 primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
```
```   559   "the_only default Empty = default ()" for default
```
```   560 | "the_only default (Insert x P) =
```
```   561     (if is_empty P then x else let y = singleton default P in if x = y then x else default ())" for default
```
```   562 | "the_only default (Join P xq) =
```
```   563     (if is_empty P then the_only default xq else if null xq then singleton default P
```
```   564        else let x = singleton default P; y = the_only default xq in
```
```   565        if x = y then x else default ())" for default
```
```   566
```
```   567 lemma the_only_singleton:
```
```   568   "the_only default xq = singleton default (pred_of_seq xq)" for default
```
```   569   by (induct xq)
```
```   570     (auto simp add: singleton_bot singleton_single is_empty_def
```
```   571     null_is_empty Let_def singleton_sup)
```
```   572
```
```   573 lemma singleton_code [code]:
```
```   574   "singleton default (Seq f) =
```
```   575     (case f () of
```
```   576       Empty \<Rightarrow> default ()
```
```   577     | Insert x P \<Rightarrow> if is_empty P then x
```
```   578         else let y = singleton default P in
```
```   579           if x = y then x else default ()
```
```   580     | Join P xq \<Rightarrow> if is_empty P then the_only default xq
```
```   581         else if null xq then singleton default P
```
```   582         else let x = singleton default P; y = the_only default xq in
```
```   583           if x = y then x else default ())" for default
```
```   584   by (cases "f ()")
```
```   585    (auto simp add: Seq_def the_only_singleton is_empty_def
```
```   586       null_is_empty singleton_bot singleton_single singleton_sup Let_def)
```
```   587
```
```   588 definition the :: "'a pred \<Rightarrow> 'a" where
```
```   589   "the A = (THE x. eval A x)"
```
```   590
```
```   591 lemma the_eqI:
```
```   592   "(THE x. eval P x) = x \<Longrightarrow> the P = x"
```
```   593   by (simp add: the_def)
```
```   594
```
```   595 lemma the_eq [code]: "the A = singleton (\<lambda>x. Code.abort (STR ''not_unique'') (\<lambda>_. the A)) A"
```
```   596   by (rule the_eqI) (simp add: singleton_def the_def)
```
```   597
```
```   598 code_reflect Predicate
```
```   599   datatypes pred = Seq and seq = Empty | Insert | Join
```
```   600
```
```   601 ML \<open>
```
```   602 signature PREDICATE =
```
```   603 sig
```
```   604   val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a
```
```   605   datatype 'a pred = Seq of (unit -> 'a seq)
```
```   606   and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
```
```   607   val map: ('a -> 'b) -> 'a pred -> 'b pred
```
```   608   val yield: 'a pred -> ('a * 'a pred) option
```
```   609   val yieldn: int -> 'a pred -> 'a list * 'a pred
```
```   610 end;
```
```   611
```
```   612 structure Predicate : PREDICATE =
```
```   613 struct
```
```   614
```
```   615 fun anamorph f k x =
```
```   616  (if k = 0 then ([], x)
```
```   617   else case f x
```
```   618    of NONE => ([], x)
```
```   619     | SOME (v, y) => let
```
```   620         val k' = k - 1;
```
```   621         val (vs, z) = anamorph f k' y
```
```   622       in (v :: vs, z) end);
```
```   623
```
```   624 datatype pred = datatype Predicate.pred
```
```   625 datatype seq = datatype Predicate.seq
```
```   626
```
```   627 fun map f = @{code Predicate.map} f;
```
```   628
```
```   629 fun yield (Seq f) = next (f ())
```
```   630 and next Empty = NONE
```
```   631   | next (Insert (x, P)) = SOME (x, P)
```
```   632   | next (Join (P, xq)) = (case yield P
```
```   633      of NONE => next xq
```
```   634       | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
```
```   635
```
```   636 fun yieldn k = anamorph yield k;
```
```   637
```
```   638 end;
```
```   639 \<close>
```
```   640
```
```   641 text \<open>Conversion from and to sets\<close>
```
```   642
```
```   643 definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where
```
```   644   "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
```
```   645
```
```   646 lemma eval_pred_of_set [simp]:
```
```   647   "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"
```
```   648   by (simp add: pred_of_set_def)
```
```   649
```
```   650 definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where
```
```   651   "set_of_pred = Collect \<circ> eval"
```
```   652
```
```   653 lemma member_set_of_pred [simp]:
```
```   654   "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"
```
```   655   by (simp add: set_of_pred_def)
```
```   656
```
```   657 definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where
```
```   658   "set_of_seq = set_of_pred \<circ> pred_of_seq"
```
```   659
```
```   660 lemma member_set_of_seq [simp]:
```
```   661   "x \<in> set_of_seq xq = Predicate.member xq x"
```
```   662   by (simp add: set_of_seq_def eval_member)
```
```   663
```
```   664 lemma of_pred_code [code]:
```
```   665   "set_of_pred (Predicate.Seq f) = (case f () of
```
```   666      Predicate.Empty \<Rightarrow> {}
```
```   667    | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)
```
```   668    | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
```
```   669   by (auto split: seq.split simp add: eval_code)
```
```   670
```
```   671 lemma of_seq_code [code]:
```
```   672   "set_of_seq Predicate.Empty = {}"
```
```   673   "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
```
```   674   "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"
```
```   675   by auto
```
```   676
```
```   677 text \<open>Lazy Evaluation of an indexed function\<close>
```
```   678
```
```   679 function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a Predicate.pred"
```
```   680 where
```
```   681   "iterate_upto f n m =
```
```   682     Predicate.Seq (%u. if n > m then Predicate.Empty
```
```   683      else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
```
```   684 by pat_completeness auto
```
```   685
```
```   686 termination by (relation "measure (%(f, n, m). nat_of_natural (m + 1 - n))")
```
```   687   (auto simp add: less_natural_def)
```
```   688
```
```   689 text \<open>Misc\<close>
```
```   690
```
```   691 declare Inf_set_fold [where 'a = "'a Predicate.pred", code]
```
```   692 declare Sup_set_fold [where 'a = "'a Predicate.pred", code]
```
```   693
```
```   694 (* FIXME: better implement conversion by bisection *)
```
```   695
```
```   696 lemma pred_of_set_fold_sup:
```
```   697   assumes "finite A"
```
```   698   shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
```
```   699 proof (rule sym)
```
```   700   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
```
```   701     by (fact comp_fun_idem_sup)
```
```   702   from \<open>finite A\<close> show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
```
```   703 qed
```
```   704
```
```   705 lemma pred_of_set_set_fold_sup:
```
```   706   "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
```
```   707 proof -
```
```   708   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
```
```   709     by (fact comp_fun_idem_sup)
```
```   710   show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
```
```   711 qed
```
```   712
```
```   713 lemma pred_of_set_set_foldr_sup [code]:
```
```   714   "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
```
```   715   by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
```
```   716
```
```   717 no_notation
```
```   718   bind (infixl "\<bind>" 70)
```
```   719
```
```   720 hide_type (open) pred seq
```
```   721 hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
```
```   722   Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map the
```
```   723   iterate_upto
```
```   724 hide_fact (open) null_def member_def
```
```   725
```
```   726 end
```