src/HOL/Predicate.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 51143 0a2371e7ced3
child 53374 a14d2a854c02
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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(*  Title:      HOL/Predicate.thy
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    Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
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*)
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header {* Predicates as enumerations *}
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theory Predicate
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imports List
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begin
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subsection {* The type of predicate enumerations (a monad) *}
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datatype 'a pred = Pred "'a \<Rightarrow> bool"
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primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
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  eval_pred: "eval (Pred f) = f"
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lemma Pred_eval [simp]:
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  "Pred (eval x) = x"
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  by (cases x) simp
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lemma pred_eqI:
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  "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
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  by (cases P, cases Q) (auto simp add: fun_eq_iff)
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lemma pred_eq_iff:
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  "P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)"
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  by (simp add: pred_eqI)
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instantiation pred :: (type) complete_lattice
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begin
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definition
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  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
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definition
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  "P < Q \<longleftrightarrow> eval P < eval Q"
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definition
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  "\<bottom> = Pred \<bottom>"
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lemma eval_bot [simp]:
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  "eval \<bottom>  = \<bottom>"
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  by (simp add: bot_pred_def)
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definition
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  "\<top> = Pred \<top>"
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lemma eval_top [simp]:
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  "eval \<top>  = \<top>"
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  by (simp add: top_pred_def)
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definition
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  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
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lemma eval_inf [simp]:
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  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
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  by (simp add: inf_pred_def)
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definition
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  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
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lemma eval_sup [simp]:
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  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
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  by (simp add: sup_pred_def)
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definition
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  "\<Sqinter>A = Pred (INFI A eval)"
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lemma eval_Inf [simp]:
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  "eval (\<Sqinter>A) = INFI A eval"
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  by (simp add: Inf_pred_def)
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definition
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  "\<Squnion>A = Pred (SUPR A eval)"
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lemma eval_Sup [simp]:
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  "eval (\<Squnion>A) = SUPR A eval"
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  by (simp add: Sup_pred_def)
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instance proof
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qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
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end
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lemma eval_INFI [simp]:
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  "eval (INFI A f) = INFI A (eval \<circ> f)"
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  by (simp only: INF_def eval_Inf image_compose)
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lemma eval_SUPR [simp]:
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  "eval (SUPR A f) = SUPR A (eval \<circ> f)"
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  by (simp only: SUP_def eval_Sup image_compose)
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instantiation pred :: (type) complete_boolean_algebra
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begin
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definition
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  "- P = Pred (- eval P)"
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lemma eval_compl [simp]:
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  "eval (- P) = - eval P"
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  by (simp add: uminus_pred_def)
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definition
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  "P - Q = Pred (eval P - eval Q)"
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lemma eval_minus [simp]:
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  "eval (P - Q) = eval P - eval Q"
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  by (simp add: minus_pred_def)
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instance proof
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qed (auto intro!: pred_eqI)
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end
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definition single :: "'a \<Rightarrow> 'a pred" where
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  "single x = Pred ((op =) x)"
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lemma eval_single [simp]:
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  "eval (single x) = (op =) x"
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  by (simp add: single_def)
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definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
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  "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
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lemma eval_bind [simp]:
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  "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
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  by (simp add: bind_def)
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lemma bind_bind:
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  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
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  by (rule pred_eqI) auto
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lemma bind_single:
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  "P \<guillemotright>= single = P"
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  by (rule pred_eqI) auto
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lemma single_bind:
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  "single x \<guillemotright>= P = P x"
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  by (rule pred_eqI) auto
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lemma bottom_bind:
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  "\<bottom> \<guillemotright>= P = \<bottom>"
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  by (rule pred_eqI) auto
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lemma sup_bind:
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  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
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  by (rule pred_eqI) auto
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lemma Sup_bind:
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  "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
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  by (rule pred_eqI) auto
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lemma pred_iffI:
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  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
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  and "\<And>x. eval B x \<Longrightarrow> eval A x"
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  shows "A = B"
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  using assms by (auto intro: pred_eqI)
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lemma singleI: "eval (single x) x"
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  by simp
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lemma singleI_unit: "eval (single ()) x"
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  by simp
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lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
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  by simp
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lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
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  by simp
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lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
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  by auto
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lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
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  by auto
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lemma botE: "eval \<bottom> x \<Longrightarrow> P"
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  by auto
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lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
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  by auto
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lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 
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  by auto
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lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
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  by auto
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lemma single_not_bot [simp]:
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  "single x \<noteq> \<bottom>"
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  by (auto simp add: single_def bot_pred_def fun_eq_iff)
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lemma not_bot:
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  assumes "A \<noteq> \<bottom>"
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  obtains x where "eval A x"
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  using assms by (cases A) (auto simp add: bot_pred_def)
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subsection {* Emptiness check and definite choice *}
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definition is_empty :: "'a pred \<Rightarrow> bool" where
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  "is_empty A \<longleftrightarrow> A = \<bottom>"
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lemma is_empty_bot:
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  "is_empty \<bottom>"
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  by (simp add: is_empty_def)
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lemma not_is_empty_single:
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  "\<not> is_empty (single x)"
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  by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
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lemma is_empty_sup:
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  "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
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  by (auto simp add: is_empty_def)
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definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
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  "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
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lemma singleton_eqI:
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  "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
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  by (auto simp add: singleton_def)
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lemma eval_singletonI:
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  "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
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proof -
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  assume assm: "\<exists>!x. eval A x"
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  then obtain x where "eval A x" ..
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  moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)
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  ultimately show ?thesis by simp 
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qed
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lemma single_singleton:
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  "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
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proof -
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  assume assm: "\<exists>!x. eval A x"
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  then have "eval A (singleton dfault A)"
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    by (rule eval_singletonI)
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  moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
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    by (rule singleton_eqI)
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  ultimately have "eval (single (singleton dfault A)) = eval A"
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    by (simp (no_asm_use) add: single_def fun_eq_iff) blast
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  then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"
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    by simp
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  then show ?thesis by (rule pred_eqI)
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qed
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lemma singleton_undefinedI:
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  "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
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  by (simp add: singleton_def)
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lemma singleton_bot:
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  "singleton dfault \<bottom> = dfault ()"
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  by (auto simp add: bot_pred_def intro: singleton_undefinedI)
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lemma singleton_single:
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  "singleton dfault (single x) = x"
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  by (auto simp add: intro: singleton_eqI singleI elim: singleE)
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lemma singleton_sup_single_single:
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  "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
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proof (cases "x = y")
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  case True then show ?thesis by (simp add: singleton_single)
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next
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  case False
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  have "eval (single x \<squnion> single y) x"
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    and "eval (single x \<squnion> single y) y"
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  by (auto intro: supI1 supI2 singleI)
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  with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
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    by blast
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  then have "singleton dfault (single x \<squnion> single y) = dfault ()"
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    by (rule singleton_undefinedI)
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  with False show ?thesis by simp
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qed
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lemma singleton_sup_aux:
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  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
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    else if B = \<bottom> then singleton dfault A
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    else singleton dfault
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      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
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proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
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  case True then show ?thesis by (simp add: single_singleton)
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next
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  case False
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  from False have A_or_B:
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    "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
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    by (auto intro!: singleton_undefinedI)
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  then have rhs: "singleton dfault
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    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
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    by (auto simp add: singleton_sup_single_single singleton_single)
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  from False have not_unique:
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    "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
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  show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
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    case True
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    then obtain a b where a: "eval A a" and b: "eval B b"
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      by (blast elim: not_bot)
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    with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
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      by (auto simp add: sup_pred_def bot_pred_def)
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    then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
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    with True rhs show ?thesis by simp
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  next
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    case False then show ?thesis by auto
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  qed
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qed
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lemma singleton_sup:
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  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
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    else if B = \<bottom> then singleton dfault A
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    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
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using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
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subsection {* Derived operations *}
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definition if_pred :: "bool \<Rightarrow> unit pred" where
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  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
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definition holds :: "unit pred \<Rightarrow> bool" where
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  holds_eq: "holds P = eval P ()"
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definition not_pred :: "unit pred \<Rightarrow> unit pred" where
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  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
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lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
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  unfolding if_pred_eq by (auto intro: singleI)
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lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding if_pred_eq by (cases b) (auto elim: botE)
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lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
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  unfolding not_pred_eq eval_pred by (auto intro: singleI)
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lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
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  unfolding not_pred_eq by (auto intro: singleI)
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lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
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   337
  unfolding not_pred_eq
haftmann@30328
   338
  by (auto split: split_if_asm elim: botE)
haftmann@30328
   339
haftmann@30328
   340
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   341
  unfolding not_pred_eq
haftmann@30328
   342
  by (auto split: split_if_asm elim: botE)
bulwahn@33754
   343
lemma "f () = False \<or> f () = True"
bulwahn@33754
   344
by simp
haftmann@30328
   345
blanchet@37549
   346
lemma closure_of_bool_cases [no_atp]:
haftmann@44007
   347
  fixes f :: "unit \<Rightarrow> bool"
haftmann@44007
   348
  assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
haftmann@44007
   349
  assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
haftmann@44007
   350
  shows "P f"
bulwahn@33754
   351
proof -
haftmann@44007
   352
  have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
bulwahn@33754
   353
    apply (cases "f ()")
bulwahn@33754
   354
    apply (rule disjI2)
bulwahn@33754
   355
    apply (rule ext)
bulwahn@33754
   356
    apply (simp add: unit_eq)
bulwahn@33754
   357
    apply (rule disjI1)
bulwahn@33754
   358
    apply (rule ext)
bulwahn@33754
   359
    apply (simp add: unit_eq)
bulwahn@33754
   360
    done
wenzelm@41550
   361
  from this assms show ?thesis by blast
bulwahn@33754
   362
qed
bulwahn@33754
   363
bulwahn@33754
   364
lemma unit_pred_cases:
haftmann@44007
   365
  assumes "P \<bottom>"
haftmann@44007
   366
  assumes "P (single ())"
haftmann@44007
   367
  shows "P Q"
haftmann@44415
   368
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
haftmann@44007
   369
  fix f
haftmann@44007
   370
  assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
haftmann@44007
   371
  then have "P (Pred f)" 
haftmann@44007
   372
    by (cases _ f rule: closure_of_bool_cases) simp_all
haftmann@44007
   373
  moreover assume "Q = Pred f"
haftmann@44007
   374
  ultimately show "P Q" by simp
haftmann@44007
   375
qed
haftmann@44007
   376
  
bulwahn@33754
   377
lemma holds_if_pred:
bulwahn@33754
   378
  "holds (if_pred b) = b"
bulwahn@33754
   379
unfolding if_pred_eq holds_eq
bulwahn@33754
   380
by (cases b) (auto intro: singleI elim: botE)
bulwahn@33754
   381
bulwahn@33754
   382
lemma if_pred_holds:
bulwahn@33754
   383
  "if_pred (holds P) = P"
bulwahn@33754
   384
unfolding if_pred_eq holds_eq
bulwahn@33754
   385
by (rule unit_pred_cases) (auto intro: singleI elim: botE)
bulwahn@33754
   386
bulwahn@33754
   387
lemma is_empty_holds:
bulwahn@33754
   388
  "is_empty P \<longleftrightarrow> \<not> holds P"
bulwahn@33754
   389
unfolding is_empty_def holds_eq
bulwahn@33754
   390
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
haftmann@30328
   391
haftmann@41311
   392
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
haftmann@41311
   393
  "map f P = P \<guillemotright>= (single o f)"
haftmann@41311
   394
haftmann@41311
   395
lemma eval_map [simp]:
haftmann@44363
   396
  "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
haftmann@44415
   397
  by (auto simp add: map_def comp_def)
haftmann@41311
   398
haftmann@41505
   399
enriched_type map: map
haftmann@44363
   400
  by (rule ext, rule pred_eqI, auto)+
haftmann@41311
   401
haftmann@41311
   402
haftmann@46664
   403
subsection {* Implementation *}
haftmann@30328
   404
haftmann@30328
   405
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
haftmann@30328
   406
haftmann@30328
   407
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
haftmann@44414
   408
  "pred_of_seq Empty = \<bottom>"
haftmann@44414
   409
| "pred_of_seq (Insert x P) = single x \<squnion> P"
haftmann@44414
   410
| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
haftmann@30328
   411
haftmann@30328
   412
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
haftmann@30328
   413
  "Seq f = pred_of_seq (f ())"
haftmann@30328
   414
haftmann@30328
   415
code_datatype Seq
haftmann@30328
   416
haftmann@30328
   417
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
haftmann@30328
   418
  "member Empty x \<longleftrightarrow> False"
haftmann@44414
   419
| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
haftmann@44414
   420
| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
haftmann@30328
   421
haftmann@30328
   422
lemma eval_member:
haftmann@30328
   423
  "member xq = eval (pred_of_seq xq)"
haftmann@30328
   424
proof (induct xq)
haftmann@30328
   425
  case Empty show ?case
nipkow@39302
   426
  by (auto simp add: fun_eq_iff elim: botE)
haftmann@30328
   427
next
haftmann@30328
   428
  case Insert show ?case
nipkow@39302
   429
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
haftmann@30328
   430
next
haftmann@30328
   431
  case Join then show ?case
nipkow@39302
   432
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
haftmann@30328
   433
qed
haftmann@30328
   434
haftmann@46038
   435
lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
haftmann@30328
   436
  unfolding Seq_def by (rule sym, rule eval_member)
haftmann@30328
   437
haftmann@30328
   438
lemma single_code [code]:
haftmann@30328
   439
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
haftmann@30328
   440
  unfolding Seq_def by simp
haftmann@30328
   441
haftmann@41080
   442
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
haftmann@44415
   443
  "apply f Empty = Empty"
haftmann@44415
   444
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
haftmann@44415
   445
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
haftmann@30328
   446
haftmann@30328
   447
lemma apply_bind:
haftmann@30328
   448
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
haftmann@30328
   449
proof (induct xq)
haftmann@30328
   450
  case Empty show ?case
haftmann@30328
   451
    by (simp add: bottom_bind)
haftmann@30328
   452
next
haftmann@30328
   453
  case Insert show ?case
haftmann@30328
   454
    by (simp add: single_bind sup_bind)
haftmann@30328
   455
next
haftmann@30328
   456
  case Join then show ?case
haftmann@30328
   457
    by (simp add: sup_bind)
haftmann@30328
   458
qed
haftmann@30328
   459
  
haftmann@30328
   460
lemma bind_code [code]:
haftmann@30328
   461
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
haftmann@30328
   462
  unfolding Seq_def by (rule sym, rule apply_bind)
haftmann@30328
   463
haftmann@30328
   464
lemma bot_set_code [code]:
haftmann@30328
   465
  "\<bottom> = Seq (\<lambda>u. Empty)"
haftmann@30328
   466
  unfolding Seq_def by simp
haftmann@30328
   467
haftmann@30376
   468
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
haftmann@44415
   469
  "adjunct P Empty = Join P Empty"
haftmann@44415
   470
| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
haftmann@44415
   471
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
haftmann@30376
   472
haftmann@30376
   473
lemma adjunct_sup:
haftmann@30376
   474
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
haftmann@30376
   475
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
haftmann@30376
   476
haftmann@30328
   477
lemma sup_code [code]:
haftmann@30328
   478
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
haftmann@30328
   479
    of Empty \<Rightarrow> g ()
haftmann@30328
   480
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
haftmann@30376
   481
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
haftmann@30328
   482
proof (cases "f ()")
haftmann@30328
   483
  case Empty
haftmann@30328
   484
  thus ?thesis
haftmann@34007
   485
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
haftmann@30328
   486
next
haftmann@30328
   487
  case Insert
haftmann@30328
   488
  thus ?thesis
haftmann@30328
   489
    unfolding Seq_def by (simp add: sup_assoc)
haftmann@30328
   490
next
haftmann@30328
   491
  case Join
haftmann@30328
   492
  thus ?thesis
haftmann@30376
   493
    unfolding Seq_def
haftmann@30376
   494
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
haftmann@30328
   495
qed
haftmann@30328
   496
haftmann@46664
   497
lemma [code]:
haftmann@46664
   498
  "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
haftmann@46664
   499
haftmann@46664
   500
lemma [code]:
haftmann@46664
   501
  "pred_size f P = 0" by (cases P) simp
haftmann@46664
   502
haftmann@30430
   503
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
haftmann@44415
   504
  "contained Empty Q \<longleftrightarrow> True"
haftmann@44415
   505
| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
haftmann@44415
   506
| "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
haftmann@30430
   507
haftmann@30430
   508
lemma single_less_eq_eval:
haftmann@30430
   509
  "single x \<le> P \<longleftrightarrow> eval P x"
haftmann@44415
   510
  by (auto simp add: less_eq_pred_def le_fun_def)
haftmann@30430
   511
haftmann@30430
   512
lemma contained_less_eq:
haftmann@30430
   513
  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
haftmann@30430
   514
  by (induct xq) (simp_all add: single_less_eq_eval)
haftmann@30430
   515
haftmann@30430
   516
lemma less_eq_pred_code [code]:
haftmann@30430
   517
  "Seq f \<le> Q = (case f ()
haftmann@30430
   518
   of Empty \<Rightarrow> True
haftmann@30430
   519
    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
haftmann@30430
   520
    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
haftmann@30430
   521
  by (cases "f ()")
haftmann@30430
   522
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
haftmann@30430
   523
haftmann@30430
   524
lemma eq_pred_code [code]:
haftmann@31133
   525
  fixes P Q :: "'a pred"
haftmann@38857
   526
  shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
haftmann@38857
   527
  by (auto simp add: equal)
haftmann@38857
   528
haftmann@38857
   529
lemma [code nbe]:
haftmann@38857
   530
  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
haftmann@38857
   531
  by (fact equal_refl)
haftmann@30430
   532
haftmann@30430
   533
lemma [code]:
haftmann@30430
   534
  "pred_case f P = f (eval P)"
haftmann@30430
   535
  by (cases P) simp
haftmann@30430
   536
haftmann@30430
   537
lemma [code]:
haftmann@30430
   538
  "pred_rec f P = f (eval P)"
haftmann@30430
   539
  by (cases P) simp
haftmann@30328
   540
bulwahn@31105
   541
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
bulwahn@31105
   542
bulwahn@31105
   543
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
haftmann@31108
   544
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
haftmann@30948
   545
haftmann@32578
   546
primrec null :: "'a seq \<Rightarrow> bool" where
haftmann@44415
   547
  "null Empty \<longleftrightarrow> True"
haftmann@44415
   548
| "null (Insert x P) \<longleftrightarrow> False"
haftmann@44415
   549
| "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
haftmann@32578
   550
haftmann@32578
   551
lemma null_is_empty:
haftmann@32578
   552
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
haftmann@32578
   553
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
haftmann@32578
   554
haftmann@32578
   555
lemma is_empty_code [code]:
haftmann@32578
   556
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
haftmann@32578
   557
  by (simp add: null_is_empty Seq_def)
haftmann@32578
   558
bulwahn@33111
   559
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
bulwahn@33111
   560
  [code del]: "the_only dfault Empty = dfault ()"
haftmann@44415
   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 ())"
haftmann@44415
   562
| "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P
bulwahn@33110
   563
       else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   564
       if x = y then x else dfault ())"
haftmann@32578
   565
haftmann@32578
   566
lemma the_only_singleton:
bulwahn@33110
   567
  "the_only dfault xq = singleton dfault (pred_of_seq xq)"
haftmann@32578
   568
  by (induct xq)
haftmann@32578
   569
    (auto simp add: singleton_bot singleton_single is_empty_def
haftmann@32578
   570
    null_is_empty Let_def singleton_sup)
haftmann@32578
   571
haftmann@32578
   572
lemma singleton_code [code]:
bulwahn@33110
   573
  "singleton dfault (Seq f) = (case f ()
bulwahn@33111
   574
   of Empty \<Rightarrow> dfault ()
haftmann@32578
   575
    | Insert x P \<Rightarrow> if is_empty P then x
bulwahn@33110
   576
        else let y = singleton dfault P in
bulwahn@33111
   577
          if x = y then x else dfault ()
bulwahn@33110
   578
    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
bulwahn@33110
   579
        else if null xq then singleton dfault P
bulwahn@33110
   580
        else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   581
          if x = y then x else dfault ())"
haftmann@32578
   582
  by (cases "f ()")
haftmann@32578
   583
   (auto simp add: Seq_def the_only_singleton is_empty_def
haftmann@32578
   584
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
haftmann@32578
   585
haftmann@44414
   586
definition the :: "'a pred \<Rightarrow> 'a" where
haftmann@37767
   587
  "the A = (THE x. eval A x)"
bulwahn@33111
   588
haftmann@40674
   589
lemma the_eqI:
haftmann@41080
   590
  "(THE x. eval P x) = x \<Longrightarrow> the P = x"
haftmann@40674
   591
  by (simp add: the_def)
haftmann@40674
   592
haftmann@44414
   593
definition not_unique :: "'a pred \<Rightarrow> 'a" where
haftmann@44414
   594
  [code del]: "not_unique A = (THE x. eval A x)"
haftmann@44414
   595
haftmann@44414
   596
code_abort not_unique
haftmann@44414
   597
haftmann@40674
   598
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"
haftmann@40674
   599
  by (rule the_eqI) (simp add: singleton_def not_unique_def)
bulwahn@33110
   600
haftmann@36531
   601
code_reflect Predicate
haftmann@36513
   602
  datatypes pred = Seq and seq = Empty | Insert | Join
haftmann@36513
   603
haftmann@30948
   604
ML {*
haftmann@30948
   605
signature PREDICATE =
haftmann@30948
   606
sig
haftmann@51126
   607
  val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a
haftmann@30948
   608
  datatype 'a pred = Seq of (unit -> 'a seq)
haftmann@30948
   609
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
haftmann@51126
   610
  val map: ('a -> 'b) -> 'a pred -> 'b pred
haftmann@30959
   611
  val yield: 'a pred -> ('a * 'a pred) option
haftmann@30959
   612
  val yieldn: int -> 'a pred -> 'a list * 'a pred
haftmann@30948
   613
end;
haftmann@30948
   614
haftmann@30948
   615
structure Predicate : PREDICATE =
haftmann@30948
   616
struct
haftmann@30948
   617
haftmann@51126
   618
fun anamorph f k x =
haftmann@51126
   619
 (if k = 0 then ([], x)
haftmann@51126
   620
  else case f x
haftmann@51126
   621
   of NONE => ([], x)
haftmann@51126
   622
    | SOME (v, y) => let
haftmann@51126
   623
        val k' = k - 1;
haftmann@51126
   624
        val (vs, z) = anamorph f k' y
haftmann@51126
   625
      in (v :: vs, z) end);
haftmann@51126
   626
haftmann@36513
   627
datatype pred = datatype Predicate.pred
haftmann@36513
   628
datatype seq = datatype Predicate.seq
haftmann@36513
   629
haftmann@51126
   630
fun map f = @{code Predicate.map} f;
haftmann@30959
   631
haftmann@36513
   632
fun yield (Seq f) = next (f ())
haftmann@36513
   633
and next Empty = NONE
haftmann@36513
   634
  | next (Insert (x, P)) = SOME (x, P)
haftmann@36513
   635
  | next (Join (P, xq)) = (case yield P
haftmann@30959
   636
     of NONE => next xq
haftmann@36513
   637
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
haftmann@30959
   638
haftmann@51126
   639
fun yieldn k = anamorph yield k;
haftmann@30948
   640
haftmann@30948
   641
end;
haftmann@30948
   642
*}
haftmann@30948
   643
haftmann@46038
   644
text {* Conversion from and to sets *}
haftmann@46038
   645
haftmann@46038
   646
definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where
haftmann@46038
   647
  "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
haftmann@46038
   648
haftmann@46038
   649
lemma eval_pred_of_set [simp]:
haftmann@46038
   650
  "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"
haftmann@46038
   651
  by (simp add: pred_of_set_def)
haftmann@46038
   652
haftmann@46038
   653
definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where
haftmann@46038
   654
  "set_of_pred = Collect \<circ> eval"
haftmann@46038
   655
haftmann@46038
   656
lemma member_set_of_pred [simp]:
haftmann@46038
   657
  "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"
haftmann@46038
   658
  by (simp add: set_of_pred_def)
haftmann@46038
   659
haftmann@46038
   660
definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where
haftmann@46038
   661
  "set_of_seq = set_of_pred \<circ> pred_of_seq"
haftmann@46038
   662
haftmann@46038
   663
lemma member_set_of_seq [simp]:
haftmann@46038
   664
  "x \<in> set_of_seq xq = Predicate.member xq x"
haftmann@46038
   665
  by (simp add: set_of_seq_def eval_member)
haftmann@46038
   666
haftmann@46038
   667
lemma of_pred_code [code]:
haftmann@46038
   668
  "set_of_pred (Predicate.Seq f) = (case f () of
haftmann@46038
   669
     Predicate.Empty \<Rightarrow> {}
haftmann@46038
   670
   | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)
haftmann@46038
   671
   | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
haftmann@46038
   672
  by (auto split: seq.split simp add: eval_code)
haftmann@46038
   673
haftmann@46038
   674
lemma of_seq_code [code]:
haftmann@46038
   675
  "set_of_seq Predicate.Empty = {}"
haftmann@46038
   676
  "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
haftmann@46038
   677
  "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"
haftmann@46038
   678
  by auto
haftmann@46038
   679
haftmann@46664
   680
text {* Lazy Evaluation of an indexed function *}
haftmann@46664
   681
haftmann@51143
   682
function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a Predicate.pred"
haftmann@46664
   683
where
haftmann@46664
   684
  "iterate_upto f n m =
haftmann@46664
   685
    Predicate.Seq (%u. if n > m then Predicate.Empty
haftmann@46664
   686
     else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
haftmann@46664
   687
by pat_completeness auto
haftmann@46664
   688
haftmann@51143
   689
termination by (relation "measure (%(f, n, m). nat_of_natural (m + 1 - n))")
haftmann@51143
   690
  (auto simp add: less_natural_def)
haftmann@46664
   691
haftmann@46664
   692
text {* Misc *}
haftmann@46664
   693
haftmann@47399
   694
declare Inf_set_fold [where 'a = "'a Predicate.pred", code]
haftmann@47399
   695
declare Sup_set_fold [where 'a = "'a Predicate.pred", code]
haftmann@46664
   696
haftmann@46664
   697
(* FIXME: better implement conversion by bisection *)
haftmann@46664
   698
haftmann@46664
   699
lemma pred_of_set_fold_sup:
haftmann@46664
   700
  assumes "finite A"
haftmann@46664
   701
  shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
haftmann@46664
   702
proof (rule sym)
haftmann@46664
   703
  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
haftmann@46664
   704
    by (fact comp_fun_idem_sup)
haftmann@46664
   705
  from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
haftmann@46664
   706
qed
haftmann@46664
   707
haftmann@46664
   708
lemma pred_of_set_set_fold_sup:
haftmann@46664
   709
  "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
haftmann@46664
   710
proof -
haftmann@46664
   711
  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
haftmann@46664
   712
    by (fact comp_fun_idem_sup)
haftmann@46664
   713
  show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
haftmann@46664
   714
qed
haftmann@46664
   715
haftmann@46664
   716
lemma pred_of_set_set_foldr_sup [code]:
haftmann@46664
   717
  "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
haftmann@46664
   718
  by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
haftmann@46664
   719
haftmann@30328
   720
no_notation
haftmann@30328
   721
  bind (infixl "\<guillemotright>=" 70)
haftmann@30328
   722
wenzelm@36176
   723
hide_type (open) pred seq
wenzelm@36176
   724
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
bulwahn@33111
   725
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
haftmann@46664
   726
  iterate_upto
haftmann@46664
   727
hide_fact (open) null_def member_def
haftmann@30328
   728
haftmann@30328
   729
end
haftmann@46664
   730