src/HOL/Predicate.thy
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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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notation
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  bot ("\<bottom>") and
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  top ("\<top>") and
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  inf (infixl "\<sqinter>" 70) and
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  sup (infixl "\<squnion>" 65) and
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  Inf ("\<Sqinter>_" [900] 900) and
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  Sup ("\<Squnion>_" [900] 900)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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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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   305
  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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   308
    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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22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   314
    with True rhs show ?thesis by simp
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   315
  next
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   316
    case False then show ?thesis by auto
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   317
  qed
22117a76f943 added emptiness check predicate and singleton projection
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parents: 32372
diff changeset
   318
qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   319
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   320
lemma singleton_sup:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   321
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   322
    else if B = \<bottom> then singleton dfault A
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   323
    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   324
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   325
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   326
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   327
subsection {* Derived operations *}
30328
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haftmann
parents: 26797
diff changeset
   328
ab47f43f7581 added enumeration of predicates
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parents: 26797
diff changeset
   329
definition if_pred :: "bool \<Rightarrow> unit pred" where
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haftmann
parents: 26797
diff changeset
   330
  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
ab47f43f7581 added enumeration of predicates
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parents: 26797
diff changeset
   331
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
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parents: 33622
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   332
definition holds :: "unit pred \<Rightarrow> bool" where
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   333
  holds_eq: "holds P = eval P ()"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   334
30328
ab47f43f7581 added enumeration of predicates
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   335
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   336
  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
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parents: 26797
diff changeset
   337
ab47f43f7581 added enumeration of predicates
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diff changeset
   338
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   339
  unfolding if_pred_eq by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
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parents: 26797
diff changeset
   340
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   341
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   342
  unfolding if_pred_eq by (cases b) (auto elim: botE)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   343
ab47f43f7581 added enumeration of predicates
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diff changeset
   344
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
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haftmann
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diff changeset
   345
  unfolding not_pred_eq eval_pred by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
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parents: 26797
diff changeset
   346
ab47f43f7581 added enumeration of predicates
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diff changeset
   347
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   348
  unfolding not_pred_eq by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   349
ab47f43f7581 added enumeration of predicates
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diff changeset
   350
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   351
  unfolding not_pred_eq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   352
  by (auto split: split_if_asm elim: botE)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   353
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   354
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   355
  unfolding not_pred_eq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   356
  by (auto split: split_if_asm elim: botE)
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   357
lemma "f () = False \<or> f () = True"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   358
by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   359
37549
a62f742f1d58 yields ill-typed ATP/metis proofs -- raus!
blanchet
parents: 36531
diff changeset
   360
lemma closure_of_bool_cases [no_atp]:
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   361
  fixes f :: "unit \<Rightarrow> bool"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   362
  assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   363
  assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   364
  shows "P f"
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   365
proof -
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   366
  have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   367
    apply (cases "f ()")
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   368
    apply (rule disjI2)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   369
    apply (rule ext)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   370
    apply (simp add: unit_eq)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   371
    apply (rule disjI1)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   372
    apply (rule ext)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   373
    apply (simp add: unit_eq)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   374
    done
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41505
diff changeset
   375
  from this assms show ?thesis by blast
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   376
qed
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   377
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   378
lemma unit_pred_cases:
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   379
  assumes "P \<bottom>"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   380
  assumes "P (single ())"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   381
  shows "P Q"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   382
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   383
  fix f
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   384
  assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   385
  then have "P (Pred f)" 
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   386
    by (cases _ f rule: closure_of_bool_cases) simp_all
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   387
  moreover assume "Q = Pred f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   388
  ultimately show "P Q" by simp
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   389
qed
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   390
  
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   391
lemma holds_if_pred:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   392
  "holds (if_pred b) = b"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   393
unfolding if_pred_eq holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   394
by (cases b) (auto intro: singleI elim: botE)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   395
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   396
lemma if_pred_holds:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   397
  "if_pred (holds P) = P"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   398
unfolding if_pred_eq holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   399
by (rule unit_pred_cases) (auto intro: singleI elim: botE)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   400
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   401
lemma is_empty_holds:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   402
  "is_empty P \<longleftrightarrow> \<not> holds P"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   403
unfolding is_empty_def holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   404
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   405
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   406
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   407
  "map f P = P \<guillemotright>= (single o f)"
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   408
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   409
lemma eval_map [simp]:
44363
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
   410
  "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   411
  by (auto simp add: map_def comp_def)
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   412
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41372
diff changeset
   413
enriched_type map: map
44363
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
   414
  by (rule ext, rule pred_eqI, auto)+
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   415
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   416
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   417
subsection {* Implementation *}
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   418
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   419
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   420
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   421
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   422
  "pred_of_seq Empty = \<bottom>"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   423
| "pred_of_seq (Insert x P) = single x \<squnion> P"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   424
| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   425
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   426
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   427
  "Seq f = pred_of_seq (f ())"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   428
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   429
code_datatype Seq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   430
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   431
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   432
  "member Empty x \<longleftrightarrow> False"
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   433
| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   434
| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   435
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   436
lemma eval_member:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   437
  "member xq = eval (pred_of_seq xq)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   438
proof (induct xq)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   439
  case Empty show ?case
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   440
  by (auto simp add: fun_eq_iff elim: botE)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   441
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   442
  case Insert show ?case
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   443
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   444
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   445
  case Join then show ?case
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   446
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   447
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   448
46038
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   449
lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   450
  unfolding Seq_def by (rule sym, rule eval_member)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   451
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   452
lemma single_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   453
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   454
  unfolding Seq_def by simp
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   455
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   456
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   457
  "apply f Empty = Empty"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   458
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   459
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   460
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   461
lemma apply_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   462
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   463
proof (induct xq)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   464
  case Empty show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   465
    by (simp add: bottom_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   466
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   467
  case Insert show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   468
    by (simp add: single_bind sup_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   469
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   470
  case Join then show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   471
    by (simp add: sup_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   472
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   473
  
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   474
lemma bind_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   475
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   476
  unfolding Seq_def by (rule sym, rule apply_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   477
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   478
lemma bot_set_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   479
  "\<bottom> = Seq (\<lambda>u. Empty)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   480
  unfolding Seq_def by simp
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   481
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   482
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   483
  "adjunct P Empty = Join P Empty"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   484
| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   485
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   486
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   487
lemma adjunct_sup:
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   488
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   489
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   490
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   491
lemma sup_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   492
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   493
    of Empty \<Rightarrow> g ()
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   494
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   495
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   496
proof (cases "f ()")
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   497
  case Empty
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   498
  thus ?thesis
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 33988
diff changeset
   499
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   500
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   501
  case Insert
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   502
  thus ?thesis
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   503
    unfolding Seq_def by (simp add: sup_assoc)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   504
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   505
  case Join
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   506
  thus ?thesis
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   507
    unfolding Seq_def
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   508
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   509
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   510
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   511
lemma [code]:
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   512
  "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   513
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   514
lemma [code]:
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   515
  "pred_size f P = 0" by (cases P) simp
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   516
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   517
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   518
  "contained Empty Q \<longleftrightarrow> True"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   519
| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   520
| "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   521
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   522
lemma single_less_eq_eval:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   523
  "single x \<le> P \<longleftrightarrow> eval P x"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   524
  by (auto simp add: less_eq_pred_def le_fun_def)
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   525
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   526
lemma contained_less_eq:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   527
  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   528
  by (induct xq) (simp_all add: single_less_eq_eval)
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   529
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   530
lemma less_eq_pred_code [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   531
  "Seq f \<le> Q = (case f ()
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   532
   of Empty \<Rightarrow> True
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   533
    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   534
    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   535
  by (cases "f ()")
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   536
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   537
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   538
lemma eq_pred_code [code]:
31133
a9f728dc5c8e dropped sort constraint on predicate equality
haftmann
parents: 31122
diff changeset
   539
  fixes P Q :: "'a pred"
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   540
  shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   541
  by (auto simp add: equal)
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   542
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   543
lemma [code nbe]:
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   544
  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   545
  by (fact equal_refl)
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   546
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   547
lemma [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   548
  "pred_case f P = f (eval P)"
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   549
  by (cases P) simp
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   550
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   551
lemma [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   552
  "pred_rec f P = f (eval P)"
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   553
  by (cases P) simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   554
31105
95f66b234086 added general preprocessing of equality in predicates for code generation
bulwahn
parents: 30430
diff changeset
   555
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
95f66b234086 added general preprocessing of equality in predicates for code generation
bulwahn
parents: 30430
diff changeset
   556
95f66b234086 added general preprocessing of equality in predicates for code generation
bulwahn
parents: 30430
diff changeset
   557
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
31108
haftmann
parents: 31106 30959
diff changeset
   558
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   559
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   560
primrec null :: "'a seq \<Rightarrow> bool" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   561
  "null Empty \<longleftrightarrow> True"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   562
| "null (Insert x P) \<longleftrightarrow> False"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   563
| "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   564
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   565
lemma null_is_empty:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   566
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   567
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   568
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   569
lemma is_empty_code [code]:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   570
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   571
  by (simp add: null_is_empty Seq_def)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   572
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   573
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   574
  [code del]: "the_only dfault Empty = dfault ()"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   575
| "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 ())"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   576
| "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   577
       else let x = singleton dfault P; y = the_only dfault xq in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   578
       if x = y then x else dfault ())"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   579
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   580
lemma the_only_singleton:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   581
  "the_only dfault xq = singleton dfault (pred_of_seq xq)"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   582
  by (induct xq)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   583
    (auto simp add: singleton_bot singleton_single is_empty_def
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   584
    null_is_empty Let_def singleton_sup)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   585
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   586
lemma singleton_code [code]:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   587
  "singleton dfault (Seq f) = (case f ()
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   588
   of Empty \<Rightarrow> dfault ()
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   589
    | Insert x P \<Rightarrow> if is_empty P then x
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   590
        else let y = singleton dfault P in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   591
          if x = y then x else dfault ()
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   592
    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   593
        else if null xq then singleton dfault P
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   594
        else let x = singleton dfault P; y = the_only dfault xq in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   595
          if x = y then x else dfault ())"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   596
  by (cases "f ()")
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   597
   (auto simp add: Seq_def the_only_singleton is_empty_def
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   598
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   599
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   600
definition the :: "'a pred \<Rightarrow> 'a" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37549
diff changeset
   601
  "the A = (THE x. eval A x)"
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   602
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   603
lemma the_eqI:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   604
  "(THE x. eval P x) = x \<Longrightarrow> the P = x"
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   605
  by (simp add: the_def)
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   606
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   607
definition not_unique :: "'a pred \<Rightarrow> 'a" where
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   608
  [code del]: "not_unique A = (THE x. eval A x)"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   609
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   610
code_abort not_unique
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   611
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   612
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   613
  by (rule the_eqI) (simp add: singleton_def not_unique_def)
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   614
36531
19f6e3b0d9b6 code_reflect: specify module name directly after keyword
haftmann
parents: 36513
diff changeset
   615
code_reflect Predicate
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   616
  datatypes pred = Seq and seq = Empty | Insert | Join
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   617
  functions map
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   618
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   619
ML {*
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   620
signature PREDICATE =
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   621
sig
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   622
  datatype 'a pred = Seq of (unit -> 'a seq)
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   623
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   624
  val yield: 'a pred -> ('a * 'a pred) option
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   625
  val yieldn: int -> 'a pred -> 'a list * 'a pred
31222
4a84ae57b65f added Predicate.map in SML environment
haftmann
parents: 31216
diff changeset
   626
  val map: ('a -> 'b) -> 'a pred -> 'b pred
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   627
end;
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   628
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   629
structure Predicate : PREDICATE =
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   630
struct
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   631
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   632
datatype pred = datatype Predicate.pred
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   633
datatype seq = datatype Predicate.seq
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   634
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   635
fun map f = Predicate.map f;
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   636
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   637
fun yield (Seq f) = next (f ())
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   638
and next Empty = NONE
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   639
  | next (Insert (x, P)) = SOME (x, P)
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   640
  | next (Join (P, xq)) = (case yield P
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   641
     of NONE => next xq
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   642
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   643
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   644
fun anamorph f k x = (if k = 0 then ([], x)
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   645
  else case f x
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   646
   of NONE => ([], x)
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   647
    | SOME (v, y) => let
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   648
        val (vs, z) = anamorph f (k - 1) y
33607
haftmann
parents: 33111
diff changeset
   649
      in (v :: vs, z) end);
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   650
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   651
fun yieldn P = anamorph yield P;
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   652
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   653
end;
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   654
*}
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   655
46038
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   656
text {* Conversion from and to sets *}
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   657
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   658
definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   659
  "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   660
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   661
lemma eval_pred_of_set [simp]:
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   662
  "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   663
  by (simp add: pred_of_set_def)
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   664
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   665
definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   666
  "set_of_pred = Collect \<circ> eval"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   667
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   668
lemma member_set_of_pred [simp]:
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   669
  "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   670
  by (simp add: set_of_pred_def)
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   671
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   672
definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   673
  "set_of_seq = set_of_pred \<circ> pred_of_seq"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   674
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   675
lemma member_set_of_seq [simp]:
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   676
  "x \<in> set_of_seq xq = Predicate.member xq x"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   677
  by (simp add: set_of_seq_def eval_member)
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   678
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   679
lemma of_pred_code [code]:
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   680
  "set_of_pred (Predicate.Seq f) = (case f () of
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   681
     Predicate.Empty \<Rightarrow> {}
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   682
   | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   683
   | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   684
  by (auto split: seq.split simp add: eval_code)
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   685
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   686
lemma of_seq_code [code]:
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   687
  "set_of_seq Predicate.Empty = {}"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   688
  "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   689
  "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   690
  by auto
bb2f7488a0f1 conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents: 45970
diff changeset
   691
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   692
text {* Lazy Evaluation of an indexed function *}
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   693
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   694
function iterate_upto :: "(code_numeral \<Rightarrow> 'a) \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> 'a Predicate.pred"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   695
where
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   696
  "iterate_upto f n m =
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   697
    Predicate.Seq (%u. if n > m then Predicate.Empty
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   698
     else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   699
by pat_completeness auto
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   700
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   701
termination by (relation "measure (%(f, n, m). Code_Numeral.nat_of (m + 1 - n))") auto
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   702
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   703
text {* Misc *}
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   704
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   705
declare Inf_set_foldr [where 'a = "'a Predicate.pred", code] Sup_set_foldr [where 'a = "'a Predicate.pred", code]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   706
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   707
(* FIXME: better implement conversion by bisection *)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   708
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   709
lemma pred_of_set_fold_sup:
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   710
  assumes "finite A"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   711
  shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   712
proof (rule sym)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   713
  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   714
    by (fact comp_fun_idem_sup)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   715
  from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   716
qed
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   717
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   718
lemma pred_of_set_set_fold_sup:
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   719
  "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   720
proof -
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   721
  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   722
    by (fact comp_fun_idem_sup)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   723
  show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   724
qed
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   725
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   726
lemma pred_of_set_set_foldr_sup [code]:
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   727
  "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   728
  by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   729
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   730
no_notation
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   731
  bot ("\<bottom>") and
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   732
  top ("\<top>") and
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   733
  inf (infixl "\<sqinter>" 70) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   734
  sup (infixl "\<squnion>" 65) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   735
  Inf ("\<Sqinter>_" [900] 900) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   736
  Sup ("\<Squnion>_" [900] 900) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   737
  bind (infixl "\<guillemotright>=" 70)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   738
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   739
no_syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   740
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   741
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   742
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   743
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   744
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36008
diff changeset
   745
hide_type (open) pred seq
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36008
diff changeset
   746
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   747
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   748
  iterate_upto
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   749
hide_fact (open) null_def member_def
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   750
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   751
end
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   752