src/HOL/Predicate.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 56154 f0a927235162 child 56212 3253aaf73a01 permissions -rw-r--r--
normalising simp rules for compound operators
```     1 (*  Title:      HOL/Predicate.thy
```
```     2     Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Predicates as enumerations *}
```
```     6
```
```     7 theory Predicate
```
```     8 imports String
```
```     9 begin
```
```    10
```
```    11 subsection {* The type of predicate enumerations (a monad) *}
```
```    12
```
```    13 datatype 'a pred = Pred "'a \<Rightarrow> bool"
```
```    14
```
```    15 primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
```
```    16   eval_pred: "eval (Pred f) = f"
```
```    17
```
```    18 lemma Pred_eval [simp]:
```
```    19   "Pred (eval x) = x"
```
```    20   by (cases x) simp
```
```    21
```
```    22 lemma pred_eqI:
```
```    23   "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
```
```    24   by (cases P, cases Q) (auto simp add: fun_eq_iff)
```
```    25
```
```    26 lemma pred_eq_iff:
```
```    27   "P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)"
```
```    28   by (simp add: pred_eqI)
```
```    29
```
```    30 instantiation pred :: (type) complete_lattice
```
```    31 begin
```
```    32
```
```    33 definition
```
```    34   "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
```
```    35
```
```    36 definition
```
```    37   "P < Q \<longleftrightarrow> eval P < eval Q"
```
```    38
```
```    39 definition
```
```    40   "\<bottom> = Pred \<bottom>"
```
```    41
```
```    42 lemma eval_bot [simp]:
```
```    43   "eval \<bottom>  = \<bottom>"
```
```    44   by (simp add: bot_pred_def)
```
```    45
```
```    46 definition
```
```    47   "\<top> = Pred \<top>"
```
```    48
```
```    49 lemma eval_top [simp]:
```
```    50   "eval \<top>  = \<top>"
```
```    51   by (simp add: top_pred_def)
```
```    52
```
```    53 definition
```
```    54   "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
```
```    55
```
```    56 lemma eval_inf [simp]:
```
```    57   "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
```
```    58   by (simp add: inf_pred_def)
```
```    59
```
```    60 definition
```
```    61   "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
```
```    62
```
```    63 lemma eval_sup [simp]:
```
```    64   "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
```
```    65   by (simp add: sup_pred_def)
```
```    66
```
```    67 definition
```
```    68   "\<Sqinter>A = Pred (INFI A eval)"
```
```    69
```
```    70 lemma eval_Inf [simp]:
```
```    71   "eval (\<Sqinter>A) = INFI A eval"
```
```    72   by (simp add: Inf_pred_def)
```
```    73
```
```    74 definition
```
```    75   "\<Squnion>A = Pred (SUPR A eval)"
```
```    76
```
```    77 lemma eval_Sup [simp]:
```
```    78   "eval (\<Squnion>A) = SUPR A eval"
```
```    79   by (simp add: Sup_pred_def)
```
```    80
```
```    81 instance proof
```
```    82 qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
```
```    83
```
```    84 end
```
```    85
```
```    86 lemma eval_INFI [simp]:
```
```    87   "eval (INFI A f) = INFI A (eval \<circ> f)"
```
```    88   using eval_Inf [of "f ` A"] by simp
```
```    89
```
```    90 lemma eval_SUPR [simp]:
```
```    91   "eval (SUPR A f) = SUPR A (eval \<circ> f)"
```
```    92   using eval_Sup [of "f ` A"] by simp
```
```    93
```
```    94 instantiation pred :: (type) complete_boolean_algebra
```
```    95 begin
```
```    96
```
```    97 definition
```
```    98   "- P = Pred (- eval P)"
```
```    99
```
```   100 lemma eval_compl [simp]:
```
```   101   "eval (- P) = - eval P"
```
```   102   by (simp add: uminus_pred_def)
```
```   103
```
```   104 definition
```
```   105   "P - Q = Pred (eval P - eval Q)"
```
```   106
```
```   107 lemma eval_minus [simp]:
```
```   108   "eval (P - Q) = eval P - eval Q"
```
```   109   by (simp add: minus_pred_def)
```
```   110
```
```   111 instance proof
```
```   112 qed (auto intro!: pred_eqI)
```
```   113
```
```   114 end
```
```   115
```
```   116 definition single :: "'a \<Rightarrow> 'a pred" where
```
```   117   "single x = Pred ((op =) x)"
```
```   118
```
```   119 lemma eval_single [simp]:
```
```   120   "eval (single x) = (op =) x"
```
```   121   by (simp add: single_def)
```
```   122
```
```   123 definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
```
```   124   "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
```
```   125
```
```   126 lemma eval_bind [simp]:
```
```   127   "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
```
```   128   by (simp add: bind_def)
```
```   129
```
```   130 lemma bind_bind:
```
```   131   "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
```
```   132   by (rule pred_eqI) auto
```
```   133
```
```   134 lemma bind_single:
```
```   135   "P \<guillemotright>= single = P"
```
```   136   by (rule pred_eqI) auto
```
```   137
```
```   138 lemma single_bind:
```
```   139   "single x \<guillemotright>= P = P x"
```
```   140   by (rule pred_eqI) auto
```
```   141
```
```   142 lemma bottom_bind:
```
```   143   "\<bottom> \<guillemotright>= P = \<bottom>"
```
```   144   by (rule pred_eqI) auto
```
```   145
```
```   146 lemma sup_bind:
```
```   147   "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
```
```   148   by (rule pred_eqI) auto
```
```   149
```
```   150 lemma Sup_bind:
```
```   151   "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
```
```   152   by (rule pred_eqI) auto
```
```   153
```
```   154 lemma pred_iffI:
```
```   155   assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
```
```   156   and "\<And>x. eval B x \<Longrightarrow> eval A x"
```
```   157   shows "A = B"
```
```   158   using assms by (auto intro: pred_eqI)
```
```   159
```
```   160 lemma singleI: "eval (single x) x"
```
```   161   by simp
```
```   162
```
```   163 lemma singleI_unit: "eval (single ()) x"
```
```   164   by simp
```
```   165
```
```   166 lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   167   by simp
```
```   168
```
```   169 lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   170   by simp
```
```   171
```
```   172 lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
```
```   173   by auto
```
```   174
```
```   175 lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
```
```   176   by auto
```
```   177
```
```   178 lemma botE: "eval \<bottom> x \<Longrightarrow> P"
```
```   179   by auto
```
```   180
```
```   181 lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   182   by auto
```
```   183
```
```   184 lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
```
```   185   by auto
```
```   186
```
```   187 lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   188   by auto
```
```   189
```
```   190 lemma single_not_bot [simp]:
```
```   191   "single x \<noteq> \<bottom>"
```
```   192   by (auto simp add: single_def bot_pred_def fun_eq_iff)
```
```   193
```
```   194 lemma not_bot:
```
```   195   assumes "A \<noteq> \<bottom>"
```
```   196   obtains x where "eval A x"
```
```   197   using assms by (cases A) (auto simp add: bot_pred_def)
```
```   198
```
```   199
```
```   200 subsection {* Emptiness check and definite choice *}
```
```   201
```
```   202 definition is_empty :: "'a pred \<Rightarrow> bool" where
```
```   203   "is_empty A \<longleftrightarrow> A = \<bottom>"
```
```   204
```
```   205 lemma is_empty_bot:
```
```   206   "is_empty \<bottom>"
```
```   207   by (simp add: is_empty_def)
```
```   208
```
```   209 lemma not_is_empty_single:
```
```   210   "\<not> is_empty (single x)"
```
```   211   by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
```
```   212
```
```   213 lemma is_empty_sup:
```
```   214   "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
```
```   215   by (auto simp add: is_empty_def)
```
```   216
```
```   217 definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
```
```   218   "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
```
```   219
```
```   220 lemma singleton_eqI:
```
```   221   "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
```
```   222   by (auto simp add: singleton_def)
```
```   223
```
```   224 lemma eval_singletonI:
```
```   225   "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
```
```   226 proof -
```
```   227   assume assm: "\<exists>!x. eval A x"
```
```   228   then obtain x where x: "eval A x" ..
```
```   229   with assm have "singleton dfault A = x" by (rule singleton_eqI)
```
```   230   with x show ?thesis by simp
```
```   231 qed
```
```   232
```
```   233 lemma single_singleton:
```
```   234   "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
```
```   235 proof -
```
```   236   assume assm: "\<exists>!x. eval A x"
```
```   237   then have "eval A (singleton dfault A)"
```
```   238     by (rule eval_singletonI)
```
```   239   moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
```
```   240     by (rule singleton_eqI)
```
```   241   ultimately have "eval (single (singleton dfault A)) = eval A"
```
```   242     by (simp (no_asm_use) add: single_def fun_eq_iff) blast
```
```   243   then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"
```
```   244     by simp
```
```   245   then show ?thesis by (rule pred_eqI)
```
```   246 qed
```
```   247
```
```   248 lemma singleton_undefinedI:
```
```   249   "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
```
```   250   by (simp add: singleton_def)
```
```   251
```
```   252 lemma singleton_bot:
```
```   253   "singleton dfault \<bottom> = dfault ()"
```
```   254   by (auto simp add: bot_pred_def intro: singleton_undefinedI)
```
```   255
```
```   256 lemma singleton_single:
```
```   257   "singleton dfault (single x) = x"
```
```   258   by (auto simp add: intro: singleton_eqI singleI elim: singleE)
```
```   259
```
```   260 lemma singleton_sup_single_single:
```
```   261   "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
```
```   262 proof (cases "x = y")
```
```   263   case True then show ?thesis by (simp add: singleton_single)
```
```   264 next
```
```   265   case False
```
```   266   have "eval (single x \<squnion> single y) x"
```
```   267     and "eval (single x \<squnion> single y) y"
```
```   268   by (auto intro: supI1 supI2 singleI)
```
```   269   with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
```
```   270     by blast
```
```   271   then have "singleton dfault (single x \<squnion> single y) = dfault ()"
```
```   272     by (rule singleton_undefinedI)
```
```   273   with False show ?thesis by simp
```
```   274 qed
```
```   275
```
```   276 lemma singleton_sup_aux:
```
```   277   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
```
```   278     else if B = \<bottom> then singleton dfault A
```
```   279     else singleton dfault
```
```   280       (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
```
```   281 proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
```
```   282   case True then show ?thesis by (simp add: single_singleton)
```
```   283 next
```
```   284   case False
```
```   285   from False have A_or_B:
```
```   286     "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
```
```   287     by (auto intro!: singleton_undefinedI)
```
```   288   then have rhs: "singleton dfault
```
```   289     (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
```
```   290     by (auto simp add: singleton_sup_single_single singleton_single)
```
```   291   from False have not_unique:
```
```   292     "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
```
```   293   show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
```
```   294     case True
```
```   295     then obtain a b where a: "eval A a" and b: "eval B b"
```
```   296       by (blast elim: not_bot)
```
```   297     with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
```
```   298       by (auto simp add: sup_pred_def bot_pred_def)
```
```   299     then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
```
```   300     with True rhs show ?thesis by simp
```
```   301   next
```
```   302     case False then show ?thesis by auto
```
```   303   qed
```
```   304 qed
```
```   305
```
```   306 lemma singleton_sup:
```
```   307   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
```
```   308     else if B = \<bottom> then singleton dfault A
```
```   309     else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
```
```   310 using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
```
```   311
```
```   312
```
```   313 subsection {* Derived operations *}
```
```   314
```
```   315 definition if_pred :: "bool \<Rightarrow> unit pred" where
```
```   316   if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
```
```   317
```
```   318 definition holds :: "unit pred \<Rightarrow> bool" where
```
```   319   holds_eq: "holds P = eval P ()"
```
```   320
```
```   321 definition not_pred :: "unit pred \<Rightarrow> unit pred" where
```
```   322   not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
```
```   323
```
```   324 lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
```
```   325   unfolding if_pred_eq by (auto intro: singleI)
```
```   326
```
```   327 lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
```
```   328   unfolding if_pred_eq by (cases b) (auto elim: botE)
```
```   329
```
```   330 lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
```
```   331   unfolding not_pred_eq eval_pred by (auto intro: singleI)
```
```   332
```
```   333 lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
```
```   334   unfolding not_pred_eq by (auto intro: singleI)
```
```   335
```
```   336 lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   337   unfolding not_pred_eq
```
```   338   by (auto split: split_if_asm elim: botE)
```
```   339
```
```   340 lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
```
```   341   unfolding not_pred_eq
```
```   342   by (auto split: split_if_asm elim: botE)
```
```   343 lemma "f () = False \<or> f () = True"
```
```   344 by simp
```
```   345
```
```   346 lemma closure_of_bool_cases [no_atp]:
```
```   347   fixes f :: "unit \<Rightarrow> bool"
```
```   348   assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
```
```   349   assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
```
```   350   shows "P f"
```
```   351 proof -
```
```   352   have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
```
```   353     apply (cases "f ()")
```
```   354     apply (rule disjI2)
```
```   355     apply (rule ext)
```
```   356     apply (simp add: unit_eq)
```
```   357     apply (rule disjI1)
```
```   358     apply (rule ext)
```
```   359     apply (simp add: unit_eq)
```
```   360     done
```
```   361   from this assms show ?thesis by blast
```
```   362 qed
```
```   363
```
```   364 lemma unit_pred_cases:
```
```   365   assumes "P \<bottom>"
```
```   366   assumes "P (single ())"
```
```   367   shows "P Q"
```
```   368 using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
```
```   369   fix f
```
```   370   assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
```
```   371   then have "P (Pred f)"
```
```   372     by (cases _ f rule: closure_of_bool_cases) simp_all
```
```   373   moreover assume "Q = Pred f"
```
```   374   ultimately show "P Q" by simp
```
```   375 qed
```
```   376
```
```   377 lemma holds_if_pred:
```
```   378   "holds (if_pred b) = b"
```
```   379 unfolding if_pred_eq holds_eq
```
```   380 by (cases b) (auto intro: singleI elim: botE)
```
```   381
```
```   382 lemma if_pred_holds:
```
```   383   "if_pred (holds P) = P"
```
```   384 unfolding if_pred_eq holds_eq
```
```   385 by (rule unit_pred_cases) (auto intro: singleI elim: botE)
```
```   386
```
```   387 lemma is_empty_holds:
```
```   388   "is_empty P \<longleftrightarrow> \<not> holds P"
```
```   389 unfolding is_empty_def holds_eq
```
```   390 by (rule unit_pred_cases) (auto elim: botE intro: singleI)
```
```   391
```
```   392 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
```
```   393   "map f P = P \<guillemotright>= (single o f)"
```
```   394
```
```   395 lemma eval_map [simp]:
```
```   396   "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
```
```   397   by (auto simp add: map_def comp_def)
```
```   398
```
```   399 functor map: map
```
```   400   by (rule ext, rule pred_eqI, auto)+
```
```   401
```
```   402
```
```   403 subsection {* Implementation *}
```
```   404
```
```   405 datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
```
```   406
```
```   407 primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
```
```   408   "pred_of_seq Empty = \<bottom>"
```
```   409 | "pred_of_seq (Insert x P) = single x \<squnion> P"
```
```   410 | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
```
```   411
```
```   412 definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
```
```   413   "Seq f = pred_of_seq (f ())"
```
```   414
```
```   415 code_datatype Seq
```
```   416
```
```   417 primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
```
```   418   "member Empty x \<longleftrightarrow> False"
```
```   419 | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
```
```   420 | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
```
```   421
```
```   422 lemma eval_member:
```
```   423   "member xq = eval (pred_of_seq xq)"
```
```   424 proof (induct xq)
```
```   425   case Empty show ?case
```
```   426   by (auto simp add: fun_eq_iff elim: botE)
```
```   427 next
```
```   428   case Insert show ?case
```
```   429   by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
```
```   430 next
```
```   431   case Join then show ?case
```
```   432   by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
```
```   433 qed
```
```   434
```
```   435 lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
```
```   436   unfolding Seq_def by (rule sym, rule eval_member)
```
```   437
```
```   438 lemma single_code [code]:
```
```   439   "single x = Seq (\<lambda>u. Insert x \<bottom>)"
```
```   440   unfolding Seq_def by simp
```
```   441
```
```   442 primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
```
```   443   "apply f Empty = Empty"
```
```   444 | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
```
```   445 | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
```
```   446
```
```   447 lemma apply_bind:
```
```   448   "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
```
```   449 proof (induct xq)
```
```   450   case Empty show ?case
```
```   451     by (simp add: bottom_bind)
```
```   452 next
```
```   453   case Insert show ?case
```
```   454     by (simp add: single_bind sup_bind)
```
```   455 next
```
```   456   case Join then show ?case
```
```   457     by (simp add: sup_bind)
```
```   458 qed
```
```   459
```
```   460 lemma bind_code [code]:
```
```   461   "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
```
```   462   unfolding Seq_def by (rule sym, rule apply_bind)
```
```   463
```
```   464 lemma bot_set_code [code]:
```
```   465   "\<bottom> = Seq (\<lambda>u. Empty)"
```
```   466   unfolding Seq_def by simp
```
```   467
```
```   468 primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
```
```   469   "adjunct P Empty = Join P Empty"
```
```   470 | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
```
```   471 | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
```
```   472
```
```   473 lemma adjunct_sup:
```
```   474   "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
```
```   475   by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
```
```   476
```
```   477 lemma sup_code [code]:
```
```   478   "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
```
```   479     of Empty \<Rightarrow> g ()
```
```   480      | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
```
```   481      | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
```
```   482 proof (cases "f ()")
```
```   483   case Empty
```
```   484   thus ?thesis
```
```   485     unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
```
```   486 next
```
```   487   case Insert
```
```   488   thus ?thesis
```
```   489     unfolding Seq_def by (simp add: sup_assoc)
```
```   490 next
```
```   491   case Join
```
```   492   thus ?thesis
```
```   493     unfolding Seq_def
```
```   494     by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
```
```   495 qed
```
```   496
```
```   497 lemma [code]:
```
```   498   "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
```
```   499
```
```   500 lemma [code]:
```
```   501   "pred_size f P = 0" by (cases P) simp
```
```   502
```
```   503 primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
```
```   504   "contained Empty Q \<longleftrightarrow> True"
```
```   505 | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
```
```   506 | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
```
```   507
```
```   508 lemma single_less_eq_eval:
```
```   509   "single x \<le> P \<longleftrightarrow> eval P x"
```
```   510   by (auto simp add: less_eq_pred_def le_fun_def)
```
```   511
```
```   512 lemma contained_less_eq:
```
```   513   "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
```
```   514   by (induct xq) (simp_all add: single_less_eq_eval)
```
```   515
```
```   516 lemma less_eq_pred_code [code]:
```
```   517   "Seq f \<le> Q = (case f ()
```
```   518    of Empty \<Rightarrow> True
```
```   519     | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
```
```   520     | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
```
```   521   by (cases "f ()")
```
```   522     (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
```
```   523
```
```   524 lemma eq_pred_code [code]:
```
```   525   fixes P Q :: "'a pred"
```
```   526   shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
```
```   527   by (auto simp add: equal)
```
```   528
```
```   529 lemma [code nbe]:
```
```   530   "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
```
```   531   by (fact equal_refl)
```
```   532
```
```   533 lemma [code]:
```
```   534   "case_pred f P = f (eval P)"
```
```   535   by (cases P) simp
```
```   536
```
```   537 lemma [code]:
```
```   538   "rec_pred f P = f (eval P)"
```
```   539   by (cases P) simp
```
```   540
```
```   541 inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
```
```   542
```
```   543 lemma eq_is_eq: "eq x y \<equiv> (x = y)"
```
```   544   by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
```
```   545
```
```   546 primrec null :: "'a seq \<Rightarrow> bool" where
```
```   547   "null Empty \<longleftrightarrow> True"
```
```   548 | "null (Insert x P) \<longleftrightarrow> False"
```
```   549 | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
```
```   550
```
```   551 lemma null_is_empty:
```
```   552   "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
```
```   553   by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
```
```   554
```
```   555 lemma is_empty_code [code]:
```
```   556   "is_empty (Seq f) \<longleftrightarrow> null (f ())"
```
```   557   by (simp add: null_is_empty Seq_def)
```
```   558
```
```   559 primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
```
```   560   [code del]: "the_only dfault Empty = dfault ()"
```
```   561 | "the_only dfault (Insert x P) = (if is_empty P then x else let y = singleton dfault P in if x = y then x else dfault ())"
```
```   562 | "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P
```
```   563        else let x = singleton dfault P; y = the_only dfault xq in
```
```   564        if x = y then x else dfault ())"
```
```   565
```
```   566 lemma the_only_singleton:
```
```   567   "the_only dfault xq = singleton dfault (pred_of_seq xq)"
```
```   568   by (induct xq)
```
```   569     (auto simp add: singleton_bot singleton_single is_empty_def
```
```   570     null_is_empty Let_def singleton_sup)
```
```   571
```
```   572 lemma singleton_code [code]:
```
```   573   "singleton dfault (Seq f) = (case f ()
```
```   574    of Empty \<Rightarrow> dfault ()
```
```   575     | Insert x P \<Rightarrow> if is_empty P then x
```
```   576         else let y = singleton dfault P in
```
```   577           if x = y then x else dfault ()
```
```   578     | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
```
```   579         else if null xq then singleton dfault P
```
```   580         else let x = singleton dfault P; y = the_only dfault xq in
```
```   581           if x = y then x else dfault ())"
```
```   582   by (cases "f ()")
```
```   583    (auto simp add: Seq_def the_only_singleton is_empty_def
```
```   584       null_is_empty singleton_bot singleton_single singleton_sup Let_def)
```
```   585
```
```   586 definition the :: "'a pred \<Rightarrow> 'a" where
```
```   587   "the A = (THE x. eval A x)"
```
```   588
```
```   589 lemma the_eqI:
```
```   590   "(THE x. eval P x) = x \<Longrightarrow> the P = x"
```
```   591   by (simp add: the_def)
```
```   592
```
```   593 lemma the_eq [code]: "the A = singleton (\<lambda>x. Code.abort (STR ''not_unique'') (\<lambda>_. the A)) A"
```
```   594   by (rule the_eqI) (simp add: singleton_def the_def)
```
```   595
```
```   596 code_reflect Predicate
```
```   597   datatypes pred = Seq and seq = Empty | Insert | Join
```
```   598
```
```   599 ML {*
```
```   600 signature PREDICATE =
```
```   601 sig
```
```   602   val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a
```
```   603   datatype 'a pred = Seq of (unit -> 'a seq)
```
```   604   and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
```
```   605   val map: ('a -> 'b) -> 'a pred -> 'b pred
```
```   606   val yield: 'a pred -> ('a * 'a pred) option
```
```   607   val yieldn: int -> 'a pred -> 'a list * 'a pred
```
```   608 end;
```
```   609
```
```   610 structure Predicate : PREDICATE =
```
```   611 struct
```
```   612
```
```   613 fun anamorph f k x =
```
```   614  (if k = 0 then ([], x)
```
```   615   else case f x
```
```   616    of NONE => ([], x)
```
```   617     | SOME (v, y) => let
```
```   618         val k' = k - 1;
```
```   619         val (vs, z) = anamorph f k' y
```
```   620       in (v :: vs, z) end);
```
```   621
```
```   622 datatype pred = datatype Predicate.pred
```
```   623 datatype seq = datatype Predicate.seq
```
```   624
```
```   625 fun map f = @{code Predicate.map} f;
```
```   626
```
```   627 fun yield (Seq f) = next (f ())
```
```   628 and next Empty = NONE
```
```   629   | next (Insert (x, P)) = SOME (x, P)
```
```   630   | next (Join (P, xq)) = (case yield P
```
```   631      of NONE => next xq
```
```   632       | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
```
```   633
```
```   634 fun yieldn k = anamorph yield k;
```
```   635
```
```   636 end;
```
```   637 *}
```
```   638
```
```   639 text {* Conversion from and to sets *}
```
```   640
```
```   641 definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where
```
```   642   "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
```
```   643
```
```   644 lemma eval_pred_of_set [simp]:
```
```   645   "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"
```
```   646   by (simp add: pred_of_set_def)
```
```   647
```
```   648 definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where
```
```   649   "set_of_pred = Collect \<circ> eval"
```
```   650
```
```   651 lemma member_set_of_pred [simp]:
```
```   652   "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"
```
```   653   by (simp add: set_of_pred_def)
```
```   654
```
```   655 definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where
```
```   656   "set_of_seq = set_of_pred \<circ> pred_of_seq"
```
```   657
```
```   658 lemma member_set_of_seq [simp]:
```
```   659   "x \<in> set_of_seq xq = Predicate.member xq x"
```
```   660   by (simp add: set_of_seq_def eval_member)
```
```   661
```
```   662 lemma of_pred_code [code]:
```
```   663   "set_of_pred (Predicate.Seq f) = (case f () of
```
```   664      Predicate.Empty \<Rightarrow> {}
```
```   665    | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)
```
```   666    | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
```
```   667   by (auto split: seq.split simp add: eval_code)
```
```   668
```
```   669 lemma of_seq_code [code]:
```
```   670   "set_of_seq Predicate.Empty = {}"
```
```   671   "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
```
```   672   "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"
```
```   673   by auto
```
```   674
```
```   675 text {* Lazy Evaluation of an indexed function *}
```
```   676
```
```   677 function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a Predicate.pred"
```
```   678 where
```
```   679   "iterate_upto f n m =
```
```   680     Predicate.Seq (%u. if n > m then Predicate.Empty
```
```   681      else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
```
```   682 by pat_completeness auto
```
```   683
```
```   684 termination by (relation "measure (%(f, n, m). nat_of_natural (m + 1 - n))")
```
```   685   (auto simp add: less_natural_def)
```
```   686
```
```   687 text {* Misc *}
```
```   688
```
```   689 declare Inf_set_fold [where 'a = "'a Predicate.pred", code]
```
```   690 declare Sup_set_fold [where 'a = "'a Predicate.pred", code]
```
```   691
```
```   692 (* FIXME: better implement conversion by bisection *)
```
```   693
```
```   694 lemma pred_of_set_fold_sup:
```
```   695   assumes "finite A"
```
```   696   shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
```
```   697 proof (rule sym)
```
```   698   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
```
```   699     by (fact comp_fun_idem_sup)
```
```   700   from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
```
```   701 qed
```
```   702
```
```   703 lemma pred_of_set_set_fold_sup:
```
```   704   "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
```
```   705 proof -
```
```   706   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
```
```   707     by (fact comp_fun_idem_sup)
```
```   708   show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
```
```   709 qed
```
```   710
```
```   711 lemma pred_of_set_set_foldr_sup [code]:
```
```   712   "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
```
```   713   by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
```
```   714
```
```   715 no_notation
```
```   716   bind (infixl "\<guillemotright>=" 70)
```
```   717
```
```   718 hide_type (open) pred seq
```
```   719 hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
```
```   720   Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map the
```
```   721   iterate_upto
```
```   722 hide_fact (open) null_def member_def
```
```   723
```
```   724 end
```