
berghofe@22259: (*  Title:      HOL/Predicate.thy
haftmann@46664:     Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
berghofe@22259: *)
berghofe@22259: 
haftmann@46664: header {* Predicates as enumerations *}
berghofe@22259: 
berghofe@22259: theory Predicate
Andreas@53943: imports String
berghofe@22259: begin
berghofe@22259: 
haftmann@46664: subsection {* The type of predicate enumerations (a monad) *}
haftmann@30328: 
haftmann@30328: datatype 'a pred = Pred "'a \<Rightarrow> bool"
haftmann@30328: 
haftmann@30328: primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
haftmann@30328:   eval_pred: "eval (Pred f) = f"
haftmann@30328: 
haftmann@30328: lemma Pred_eval [simp]:
haftmann@30328:   "Pred (eval x) = x"
haftmann@30328:   by (cases x) simp
haftmann@30328: 
haftmann@40616: lemma pred_eqI:
haftmann@40616:   "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
haftmann@40616:   by (cases P, cases Q) (auto simp add: fun_eq_iff)
haftmann@30328: 
haftmann@46038: lemma pred_eq_iff:
haftmann@46038:   "P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)"
haftmann@46038:   by (simp add: pred_eqI)
haftmann@46038: 
haftmann@44033: instantiation pred :: (type) complete_lattice
haftmann@30328: begin
haftmann@30328: 
haftmann@30328: definition
haftmann@30328:   "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
haftmann@30328: 
haftmann@30328: definition
haftmann@30328:   "P < Q \<longleftrightarrow> eval P < eval Q"
haftmann@30328: 
haftmann@30328: definition
haftmann@30328:   "\<bottom> = Pred \<bottom>"
haftmann@30328: 
haftmann@40616: lemma eval_bot [simp]:
haftmann@40616:   "eval \<bottom>  = \<bottom>"
haftmann@40616:   by (simp add: bot_pred_def)
haftmann@40616: 
haftmann@30328: definition
haftmann@30328:   "\<top> = Pred \<top>"
haftmann@30328: 
haftmann@40616: lemma eval_top [simp]:
haftmann@40616:   "eval \<top>  = \<top>"
haftmann@40616:   by (simp add: top_pred_def)
haftmann@40616: 
haftmann@30328: definition
haftmann@30328:   "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
haftmann@30328: 
haftmann@40616: lemma eval_inf [simp]:
haftmann@40616:   "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
haftmann@40616:   by (simp add: inf_pred_def)
haftmann@40616: 
haftmann@30328: definition
haftmann@30328:   "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
haftmann@30328: 
haftmann@40616: lemma eval_sup [simp]:
haftmann@40616:   "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
haftmann@40616:   by (simp add: sup_pred_def)
haftmann@40616: 
haftmann@30328: definition
haftmann@37767:   "\<Sqinter>A = Pred (INFI A eval)"
haftmann@30328: 
haftmann@40616: lemma eval_Inf [simp]:
haftmann@40616:   "eval (\<Sqinter>A) = INFI A eval"
haftmann@40616:   by (simp add: Inf_pred_def)
haftmann@40616: 
haftmann@30328: definition
haftmann@37767:   "\<Squnion>A = Pred (SUPR A eval)"
haftmann@30328: 
haftmann@40616: lemma eval_Sup [simp]:
haftmann@40616:   "eval (\<Squnion>A) = SUPR A eval"
haftmann@40616:   by (simp add: Sup_pred_def)
haftmann@40616: 
haftmann@44033: instance proof
haftmann@44415: qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
haftmann@44033: 
haftmann@44033: end
haftmann@44033: 
haftmann@44033: lemma eval_INFI [simp]:
haftmann@44033:   "eval (INFI A f) = INFI A (eval \<circ> f)"
hoelzl@44928:   by (simp only: INF_def eval_Inf image_compose)
haftmann@44033: 
haftmann@44033: lemma eval_SUPR [simp]:
haftmann@44033:   "eval (SUPR A f) = SUPR A (eval \<circ> f)"
hoelzl@44928:   by (simp only: SUP_def eval_Sup image_compose)
haftmann@44033: 
haftmann@44033: instantiation pred :: (type) complete_boolean_algebra
haftmann@44033: begin
haftmann@44033: 
haftmann@32578: definition
haftmann@32578:   "- P = Pred (- eval P)"
haftmann@32578: 
haftmann@40616: lemma eval_compl [simp]:
haftmann@40616:   "eval (- P) = - eval P"
haftmann@40616:   by (simp add: uminus_pred_def)
haftmann@40616: 
haftmann@32578: definition
haftmann@32578:   "P - Q = Pred (eval P - eval Q)"
haftmann@32578: 
haftmann@40616: lemma eval_minus [simp]:
haftmann@40616:   "eval (P - Q) = eval P - eval Q"
haftmann@40616:   by (simp add: minus_pred_def)
haftmann@40616: 
haftmann@32578: instance proof
noschinl@46884: qed (auto intro!: pred_eqI)
haftmann@30328: 
berghofe@22259: end
haftmann@30328: 
haftmann@40616: definition single :: "'a \<Rightarrow> 'a pred" where
haftmann@40616:   "single x = Pred ((op =) x)"
haftmann@40616: 
haftmann@40616: lemma eval_single [simp]:
haftmann@40616:   "eval (single x) = (op =) x"
haftmann@40616:   by (simp add: single_def)
haftmann@40616: 
haftmann@40616: definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
haftmann@41080:   "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
haftmann@40616: 
haftmann@40616: lemma eval_bind [simp]:
haftmann@41080:   "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
haftmann@40616:   by (simp add: bind_def)
haftmann@40616: 
haftmann@30328: lemma bind_bind:
haftmann@30328:   "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
noschinl@46884:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@30328: lemma bind_single:
haftmann@30328:   "P \<guillemotright>= single = P"
haftmann@40616:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@30328: lemma single_bind:
haftmann@30328:   "single x \<guillemotright>= P = P x"
haftmann@40616:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@30328: lemma bottom_bind:
haftmann@30328:   "\<bottom> \<guillemotright>= P = \<bottom>"
haftmann@40674:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@30328: lemma sup_bind:
haftmann@30328:   "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
haftmann@40674:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@40616: lemma Sup_bind:
haftmann@40616:   "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
noschinl@46884:   by (rule pred_eqI) auto
haftmann@30328: 
haftmann@30328: lemma pred_iffI:
haftmann@30328:   assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
haftmann@30328:   and "\<And>x. eval B x \<Longrightarrow> eval A x"
haftmann@30328:   shows "A = B"
haftmann@40616:   using assms by (auto intro: pred_eqI)
haftmann@30328:   
haftmann@30328: lemma singleI: "eval (single x) x"
haftmann@40616:   by simp
haftmann@30328: 
haftmann@30328: lemma singleI_unit: "eval (single ()) x"
haftmann@40616:   by simp
haftmann@30328: 
haftmann@30328: lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616:   by simp
haftmann@30328: 
haftmann@30328: lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616:   by simp
haftmann@30328: 
haftmann@30328: lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
haftmann@40616:   by auto
haftmann@30328: 
haftmann@30328: lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616:   by auto
haftmann@30328: 
haftmann@30328: lemma botE: "eval \<bottom> x \<Longrightarrow> P"
haftmann@40616:   by auto
haftmann@30328: 
haftmann@30328: lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
haftmann@40616:   by auto
haftmann@30328: 
haftmann@30328: lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 
haftmann@40616:   by auto
haftmann@30328: 
haftmann@30328: lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616:   by auto
haftmann@30328: 
haftmann@32578: lemma single_not_bot [simp]:
haftmann@32578:   "single x \<noteq> \<bottom>"
nipkow@39302:   by (auto simp add: single_def bot_pred_def fun_eq_iff)
haftmann@32578: 
haftmann@32578: lemma not_bot:
haftmann@32578:   assumes "A \<noteq> \<bottom>"
haftmann@32578:   obtains x where "eval A x"
haftmann@45970:   using assms by (cases A) (auto simp add: bot_pred_def)
haftmann@45970: 
haftmann@32578: 
haftmann@46664: subsection {* Emptiness check and definite choice *}
haftmann@32578: 
haftmann@32578: definition is_empty :: "'a pred \<Rightarrow> bool" where
haftmann@32578:   "is_empty A \<longleftrightarrow> A = \<bottom>"
haftmann@32578: 
haftmann@32578: lemma is_empty_bot:
haftmann@32578:   "is_empty \<bottom>"
haftmann@32578:   by (simp add: is_empty_def)
haftmann@32578: 
haftmann@32578: lemma not_is_empty_single:
haftmann@32578:   "\<not> is_empty (single x)"
nipkow@39302:   by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
haftmann@32578: 
haftmann@32578: lemma is_empty_sup:
haftmann@32578:   "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
huffman@36008:   by (auto simp add: is_empty_def)
haftmann@32578: 
haftmann@40616: definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
bulwahn@33111:   "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
haftmann@32578: 
haftmann@32578: lemma singleton_eqI:
bulwahn@33110:   "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578:   by (auto simp add: singleton_def)
haftmann@32578: 
haftmann@32578: lemma eval_singletonI:
bulwahn@33110:   "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
haftmann@32578: proof -
haftmann@32578:   assume assm: "\<exists>!x. eval A x"
wenzelm@53374:   then obtain x where x: "eval A x" ..
wenzelm@53374:   with assm have "singleton dfault A = x" by (rule singleton_eqI)
wenzelm@53374:   with x show ?thesis by simp
haftmann@32578: qed
haftmann@32578: 
haftmann@32578: lemma single_singleton:
bulwahn@33110:   "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
haftmann@32578: proof -
haftmann@32578:   assume assm: "\<exists>!x. eval A x"
bulwahn@33110:   then have "eval A (singleton dfault A)"
haftmann@32578:     by (rule eval_singletonI)
bulwahn@33110:   moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578:     by (rule singleton_eqI)
bulwahn@33110:   ultimately have "eval (single (singleton dfault A)) = eval A"
nipkow@39302:     by (simp (no_asm_use) add: single_def fun_eq_iff) blast
haftmann@40616:   then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"
haftmann@40616:     by simp
haftmann@40616:   then show ?thesis by (rule pred_eqI)
haftmann@32578: qed
haftmann@32578: 
haftmann@32578: lemma singleton_undefinedI:
bulwahn@33111:   "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
haftmann@32578:   by (simp add: singleton_def)
haftmann@32578: 
haftmann@32578: lemma singleton_bot:
bulwahn@33111:   "singleton dfault \<bottom> = dfault ()"
haftmann@32578:   by (auto simp add: bot_pred_def intro: singleton_undefinedI)
haftmann@32578: 
haftmann@32578: lemma singleton_single:
bulwahn@33110:   "singleton dfault (single x) = x"
haftmann@32578:   by (auto simp add: intro: singleton_eqI singleI elim: singleE)
haftmann@32578: 
haftmann@32578: lemma singleton_sup_single_single:
bulwahn@33111:   "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
haftmann@32578: proof (cases "x = y")
haftmann@32578:   case True then show ?thesis by (simp add: singleton_single)
haftmann@32578: next
haftmann@32578:   case False
haftmann@32578:   have "eval (single x \<squnion> single y) x"
haftmann@32578:     and "eval (single x \<squnion> single y) y"
haftmann@32578:   by (auto intro: supI1 supI2 singleI)
haftmann@32578:   with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
haftmann@32578:     by blast
bulwahn@33111:   then have "singleton dfault (single x \<squnion> single y) = dfault ()"
haftmann@32578:     by (rule singleton_undefinedI)
haftmann@32578:   with False show ?thesis by simp
haftmann@32578: qed
haftmann@32578: 
haftmann@32578: lemma singleton_sup_aux:
bulwahn@33110:   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110:     else if B = \<bottom> then singleton dfault A
bulwahn@33110:     else singleton dfault
bulwahn@33110:       (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
haftmann@32578: proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
haftmann@32578:   case True then show ?thesis by (simp add: single_singleton)
haftmann@32578: next
haftmann@32578:   case False
haftmann@32578:   from False have A_or_B:
bulwahn@33111:     "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
haftmann@32578:     by (auto intro!: singleton_undefinedI)
bulwahn@33110:   then have rhs: "singleton dfault
bulwahn@33111:     (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
haftmann@32578:     by (auto simp add: singleton_sup_single_single singleton_single)
haftmann@32578:   from False have not_unique:
haftmann@32578:     "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
haftmann@32578:   show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
haftmann@32578:     case True
haftmann@32578:     then obtain a b where a: "eval A a" and b: "eval B b"
haftmann@32578:       by (blast elim: not_bot)
haftmann@32578:     with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
haftmann@32578:       by (auto simp add: sup_pred_def bot_pred_def)
bulwahn@33111:     then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
haftmann@32578:     with True rhs show ?thesis by simp
haftmann@32578:   next
haftmann@32578:     case False then show ?thesis by auto
haftmann@32578:   qed
haftmann@32578: qed
haftmann@32578: 
haftmann@32578: lemma singleton_sup:
bulwahn@33110:   "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110:     else if B = \<bottom> then singleton dfault A
bulwahn@33111:     else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
bulwahn@33110: using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
haftmann@32578: 
haftmann@30328: 
haftmann@46664: subsection {* Derived operations *}
haftmann@30328: 
haftmann@30328: definition if_pred :: "bool \<Rightarrow> unit pred" where
haftmann@30328:   if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
haftmann@30328: 
bulwahn@33754: definition holds :: "unit pred \<Rightarrow> bool" where
bulwahn@33754:   holds_eq: "holds P = eval P ()"
bulwahn@33754: 
haftmann@30328: definition not_pred :: "unit pred \<Rightarrow> unit pred" where
haftmann@30328:   not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
haftmann@30328: 
haftmann@30328: lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
haftmann@30328:   unfolding if_pred_eq by (auto intro: singleI)
haftmann@30328: 
haftmann@30328: lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30328:   unfolding if_pred_eq by (cases b) (auto elim: botE)
haftmann@30328: 
haftmann@30328: lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
haftmann@30328:   unfolding not_pred_eq eval_pred by (auto intro: singleI)
haftmann@30328: 
haftmann@30328: lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
haftmann@30328:   unfolding not_pred_eq by (auto intro: singleI)
haftmann@30328: 
haftmann@30328: lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328:   unfolding not_pred_eq
haftmann@30328:   by (auto split: split_if_asm elim: botE)
haftmann@30328: 
haftmann@30328: lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328:   unfolding not_pred_eq
haftmann@30328:   by (auto split: split_if_asm elim: botE)
bulwahn@33754: lemma "f () = False \<or> f () = True"
bulwahn@33754: by simp
haftmann@30328: 
blanchet@37549: lemma closure_of_bool_cases [no_atp]:
haftmann@44007:   fixes f :: "unit \<Rightarrow> bool"
haftmann@44007:   assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
haftmann@44007:   assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
haftmann@44007:   shows "P f"
bulwahn@33754: proof -
haftmann@44007:   have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
bulwahn@33754:     apply (cases "f ()")
bulwahn@33754:     apply (rule disjI2)
bulwahn@33754:     apply (rule ext)
bulwahn@33754:     apply (simp add: unit_eq)
bulwahn@33754:     apply (rule disjI1)
bulwahn@33754:     apply (rule ext)
bulwahn@33754:     apply (simp add: unit_eq)
bulwahn@33754:     done
wenzelm@41550:   from this assms show ?thesis by blast
bulwahn@33754: qed
bulwahn@33754: 
bulwahn@33754: lemma unit_pred_cases:
haftmann@44007:   assumes "P \<bottom>"
haftmann@44007:   assumes "P (single ())"
haftmann@44007:   shows "P Q"
haftmann@44415: using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
haftmann@44007:   fix f
haftmann@44007:   assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
haftmann@44007:   then have "P (Pred f)" 
haftmann@44007:     by (cases _ f rule: closure_of_bool_cases) simp_all
haftmann@44007:   moreover assume "Q = Pred f"
haftmann@44007:   ultimately show "P Q" by simp
haftmann@44007: qed
haftmann@44007:   
bulwahn@33754: lemma holds_if_pred:
bulwahn@33754:   "holds (if_pred b) = b"
bulwahn@33754: unfolding if_pred_eq holds_eq
bulwahn@33754: by (cases b) (auto intro: singleI elim: botE)
bulwahn@33754: 
bulwahn@33754: lemma if_pred_holds:
bulwahn@33754:   "if_pred (holds P) = P"
bulwahn@33754: unfolding if_pred_eq holds_eq
bulwahn@33754: by (rule unit_pred_cases) (auto intro: singleI elim: botE)
bulwahn@33754: 
bulwahn@33754: lemma is_empty_holds:
bulwahn@33754:   "is_empty P \<longleftrightarrow> \<not> holds P"
bulwahn@33754: unfolding is_empty_def holds_eq
bulwahn@33754: by (rule unit_pred_cases) (auto elim: botE intro: singleI)
haftmann@30328: 
haftmann@41311: definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
haftmann@41311:   "map f P = P \<guillemotright>= (single o f)"
haftmann@41311: 
haftmann@41311: lemma eval_map [simp]:
haftmann@44363:   "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
haftmann@44415:   by (auto simp add: map_def comp_def)
haftmann@41311: 
haftmann@41505: enriched_type map: map
haftmann@44363:   by (rule ext, rule pred_eqI, auto)+
haftmann@41311: 
haftmann@41311: 
haftmann@46664: subsection {* Implementation *}
haftmann@30328: 
haftmann@30328: datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
haftmann@30328: 
haftmann@30328: primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
haftmann@44414:   "pred_of_seq Empty = \<bottom>"
haftmann@44414: | "pred_of_seq (Insert x P) = single x \<squnion> P"
haftmann@44414: | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
haftmann@30328: 
haftmann@30328: definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
haftmann@30328:   "Seq f = pred_of_seq (f ())"
haftmann@30328: 
haftmann@30328: code_datatype Seq
haftmann@30328: 
haftmann@30328: primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
haftmann@30328:   "member Empty x \<longleftrightarrow> False"
haftmann@44414: | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
haftmann@44414: | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
haftmann@30328: 
haftmann@30328: lemma eval_member:
haftmann@30328:   "member xq = eval (pred_of_seq xq)"
haftmann@30328: proof (induct xq)
haftmann@30328:   case Empty show ?case
nipkow@39302:   by (auto simp add: fun_eq_iff elim: botE)
haftmann@30328: next
haftmann@30328:   case Insert show ?case
nipkow@39302:   by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
haftmann@30328: next
haftmann@30328:   case Join then show ?case
nipkow@39302:   by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
haftmann@30328: qed
haftmann@30328: 
haftmann@46038: lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"
haftmann@30328:   unfolding Seq_def by (rule sym, rule eval_member)
haftmann@30328: 
haftmann@30328: lemma single_code [code]:
haftmann@30328:   "single x = Seq (\<lambda>u. Insert x \<bottom>)"
haftmann@30328:   unfolding Seq_def by simp
haftmann@30328: 
haftmann@41080: primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
haftmann@44415:   "apply f Empty = Empty"
haftmann@44415: | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
haftmann@44415: | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
haftmann@30328: 
haftmann@30328: lemma apply_bind:
haftmann@30328:   "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
haftmann@30328: proof (induct xq)
haftmann@30328:   case Empty show ?case
haftmann@30328:     by (simp add: bottom_bind)
haftmann@30328: next
haftmann@30328:   case Insert show ?case
haftmann@30328:     by (simp add: single_bind sup_bind)
haftmann@30328: next
haftmann@30328:   case Join then show ?case
haftmann@30328:     by (simp add: sup_bind)
haftmann@30328: qed
haftmann@30328:   
haftmann@30328: lemma bind_code [code]:
haftmann@30328:   "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
haftmann@30328:   unfolding Seq_def by (rule sym, rule apply_bind)
haftmann@30328: 
haftmann@30328: lemma bot_set_code [code]:
haftmann@30328:   "\<bottom> = Seq (\<lambda>u. Empty)"
haftmann@30328:   unfolding Seq_def by simp
haftmann@30328: 
haftmann@30376: primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
haftmann@44415:   "adjunct P Empty = Join P Empty"
haftmann@44415: | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
haftmann@44415: | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
haftmann@30376: 
haftmann@30376: lemma adjunct_sup:
haftmann@30376:   "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
haftmann@30376:   by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
haftmann@30376: 
haftmann@30328: lemma sup_code [code]:
haftmann@30328:   "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
haftmann@30328:     of Empty \<Rightarrow> g ()
haftmann@30328:      | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
haftmann@30376:      | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
haftmann@30328: proof (cases "f ()")
haftmann@30328:   case Empty
haftmann@30328:   thus ?thesis
haftmann@34007:     unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
haftmann@30328: next
haftmann@30328:   case Insert
haftmann@30328:   thus ?thesis
haftmann@30328:     unfolding Seq_def by (simp add: sup_assoc)
haftmann@30328: next
haftmann@30328:   case Join
haftmann@30328:   thus ?thesis
haftmann@30376:     unfolding Seq_def
haftmann@30376:     by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
haftmann@30328: qed
haftmann@30328: 
haftmann@46664: lemma [code]:
haftmann@46664:   "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
haftmann@46664: 
haftmann@46664: lemma [code]:
haftmann@46664:   "pred_size f P = 0" by (cases P) simp
haftmann@46664: 
haftmann@30430: primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
haftmann@44415:   "contained Empty Q \<longleftrightarrow> True"
haftmann@44415: | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
haftmann@44415: | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
haftmann@30430: 
haftmann@30430: lemma single_less_eq_eval:
haftmann@30430:   "single x \<le> P \<longleftrightarrow> eval P x"
haftmann@44415:   by (auto simp add: less_eq_pred_def le_fun_def)
haftmann@30430: 
haftmann@30430: lemma contained_less_eq:
haftmann@30430:   "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
haftmann@30430:   by (induct xq) (simp_all add: single_less_eq_eval)
haftmann@30430: 
haftmann@30430: lemma less_eq_pred_code [code]:
haftmann@30430:   "Seq f \<le> Q = (case f ()
haftmann@30430:    of Empty \<Rightarrow> True
haftmann@30430:     | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
haftmann@30430:     | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
haftmann@30430:   by (cases "f ()")
haftmann@30430:     (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
haftmann@30430: 
haftmann@30430: lemma eq_pred_code [code]:
haftmann@31133:   fixes P Q :: "'a pred"
haftmann@38857:   shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
haftmann@38857:   by (auto simp add: equal)
haftmann@38857: 
haftmann@38857: lemma [code nbe]:
haftmann@38857:   "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
haftmann@38857:   by (fact equal_refl)
haftmann@30430: 
haftmann@30430: lemma [code]:
haftmann@30430:   "pred_case f P = f (eval P)"
haftmann@30430:   by (cases P) simp
haftmann@30430: 
haftmann@30430: lemma [code]:
haftmann@30430:   "pred_rec f P = f (eval P)"
haftmann@30430:   by (cases P) simp
haftmann@30328: 
bulwahn@31105: inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
bulwahn@31105: 
bulwahn@31105: lemma eq_is_eq: "eq x y \<equiv> (x = y)"
haftmann@31108:   by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
haftmann@30948: 
haftmann@32578: primrec null :: "'a seq \<Rightarrow> bool" where
haftmann@44415:   "null Empty \<longleftrightarrow> True"
haftmann@44415: | "null (Insert x P) \<longleftrightarrow> False"
haftmann@44415: | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
haftmann@32578: 
haftmann@32578: lemma null_is_empty:
haftmann@32578:   "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
haftmann@32578:   by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
haftmann@32578: 
haftmann@32578: lemma is_empty_code [code]:
haftmann@32578:   "is_empty (Seq f) \<longleftrightarrow> null (f ())"
haftmann@32578:   by (simp add: null_is_empty Seq_def)
haftmann@32578: 
bulwahn@33111: primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
bulwahn@33111:   [code del]: "the_only dfault Empty = dfault ()"
haftmann@44415: | "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: | "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:        else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111:        if x = y then x else dfault ())"
haftmann@32578: 
haftmann@32578: lemma the_only_singleton:
bulwahn@33110:   "the_only dfault xq = singleton dfault (pred_of_seq xq)"
haftmann@32578:   by (induct xq)
haftmann@32578:     (auto simp add: singleton_bot singleton_single is_empty_def
haftmann@32578:     null_is_empty Let_def singleton_sup)
haftmann@32578: 
haftmann@32578: lemma singleton_code [code]:
bulwahn@33110:   "singleton dfault (Seq f) = (case f ()
bulwahn@33111:    of Empty \<Rightarrow> dfault ()
haftmann@32578:     | Insert x P \<Rightarrow> if is_empty P then x
bulwahn@33110:         else let y = singleton dfault P in
bulwahn@33111:           if x = y then x else dfault ()
bulwahn@33110:     | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
bulwahn@33110:         else if null xq then singleton dfault P
bulwahn@33110:         else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111:           if x = y then x else dfault ())"
haftmann@32578:   by (cases "f ()")
haftmann@32578:    (auto simp add: Seq_def the_only_singleton is_empty_def
haftmann@32578:       null_is_empty singleton_bot singleton_single singleton_sup Let_def)
haftmann@32578: 
haftmann@44414: definition the :: "'a pred \<Rightarrow> 'a" where
haftmann@37767:   "the A = (THE x. eval A x)"
bulwahn@33111: 
haftmann@40674: lemma the_eqI:
haftmann@41080:   "(THE x. eval P x) = x \<Longrightarrow> the P = x"
haftmann@40674:   by (simp add: the_def)
haftmann@40674: 
Andreas@53943: lemma the_eq [code]: "the A = singleton (\<lambda>x. Code.abort (STR ''not_unique'') (\<lambda>_. the A)) A"
Andreas@53943:   by (rule the_eqI) (simp add: singleton_def the_def)
bulwahn@33110: 
haftmann@36531: code_reflect Predicate
haftmann@36513:   datatypes pred = Seq and seq = Empty | Insert | Join
haftmann@36513: 
haftmann@30948: ML {*
haftmann@30948: signature PREDICATE =
haftmann@30948: sig
haftmann@51126:   val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a
haftmann@30948:   datatype 'a pred = Seq of (unit -> 'a seq)
haftmann@30948:   and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
haftmann@51126:   val map: ('a -> 'b) -> 'a pred -> 'b pred
haftmann@30959:   val yield: 'a pred -> ('a * 'a pred) option
haftmann@30959:   val yieldn: int -> 'a pred -> 'a list * 'a pred
haftmann@30948: end;
haftmann@30948: 
haftmann@30948: structure Predicate : PREDICATE =
haftmann@30948: struct
haftmann@30948: 
haftmann@51126: fun anamorph f k x =
haftmann@51126:  (if k = 0 then ([], x)
haftmann@51126:   else case f x
haftmann@51126:    of NONE => ([], x)
haftmann@51126:     | SOME (v, y) => let
haftmann@51126:         val k' = k - 1;
haftmann@51126:         val (vs, z) = anamorph f k' y
haftmann@51126:       in (v :: vs, z) end);
haftmann@51126: 
haftmann@36513: datatype pred = datatype Predicate.pred
haftmann@36513: datatype seq = datatype Predicate.seq
haftmann@36513: 
haftmann@51126: fun map f = @{code Predicate.map} f;
haftmann@30959: 
haftmann@36513: fun yield (Seq f) = next (f ())
haftmann@36513: and next Empty = NONE
haftmann@36513:   | next (Insert (x, P)) = SOME (x, P)
haftmann@36513:   | next (Join (P, xq)) = (case yield P
haftmann@30959:      of NONE => next xq
haftmann@36513:       | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
haftmann@30959: 
haftmann@51126: fun yieldn k = anamorph yield k;
haftmann@30948: 
haftmann@30948: end;
haftmann@30948: *}
haftmann@30948: 
haftmann@46038: text {* Conversion from and to sets *}
haftmann@46038: 
haftmann@46038: definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where
haftmann@46038:   "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
haftmann@46038: 
haftmann@46038: lemma eval_pred_of_set [simp]:
haftmann@46038:   "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"
haftmann@46038:   by (simp add: pred_of_set_def)
haftmann@46038: 
haftmann@46038: definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where
haftmann@46038:   "set_of_pred = Collect \<circ> eval"
haftmann@46038: 
haftmann@46038: lemma member_set_of_pred [simp]:
haftmann@46038:   "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"
haftmann@46038:   by (simp add: set_of_pred_def)
haftmann@46038: 
haftmann@46038: definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where
haftmann@46038:   "set_of_seq = set_of_pred \<circ> pred_of_seq"
haftmann@46038: 
haftmann@46038: lemma member_set_of_seq [simp]:
haftmann@46038:   "x \<in> set_of_seq xq = Predicate.member xq x"
haftmann@46038:   by (simp add: set_of_seq_def eval_member)
haftmann@46038: 
haftmann@46038: lemma of_pred_code [code]:
haftmann@46038:   "set_of_pred (Predicate.Seq f) = (case f () of
haftmann@46038:      Predicate.Empty \<Rightarrow> {}
haftmann@46038:    | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)
haftmann@46038:    | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
haftmann@46038:   by (auto split: seq.split simp add: eval_code)
haftmann@46038: 
haftmann@46038: lemma of_seq_code [code]:
haftmann@46038:   "set_of_seq Predicate.Empty = {}"
haftmann@46038:   "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"
haftmann@46038:   "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"
haftmann@46038:   by auto
haftmann@46038: 
haftmann@46664: text {* Lazy Evaluation of an indexed function *}
haftmann@46664: 
haftmann@51143: function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> 'a Predicate.pred"
haftmann@46664: where
haftmann@46664:   "iterate_upto f n m =
haftmann@46664:     Predicate.Seq (%u. if n > m then Predicate.Empty
haftmann@46664:      else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"
haftmann@46664: by pat_completeness auto
haftmann@46664: 
haftmann@51143: termination by (relation "measure (%(f, n, m). nat_of_natural (m + 1 - n))")
haftmann@51143:   (auto simp add: less_natural_def)
haftmann@46664: 
haftmann@46664: text {* Misc *}
haftmann@46664: 
haftmann@47399: declare Inf_set_fold [where 'a = "'a Predicate.pred", code]
haftmann@47399: declare Sup_set_fold [where 'a = "'a Predicate.pred", code]
haftmann@46664: 
haftmann@46664: (* FIXME: better implement conversion by bisection *)
haftmann@46664: 
haftmann@46664: lemma pred_of_set_fold_sup:
haftmann@46664:   assumes "finite A"
haftmann@46664:   shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
haftmann@46664: proof (rule sym)
haftmann@46664:   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
haftmann@46664:     by (fact comp_fun_idem_sup)
haftmann@46664:   from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
haftmann@46664: qed
haftmann@46664: 
haftmann@46664: lemma pred_of_set_set_fold_sup:
haftmann@46664:   "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"
haftmann@46664: proof -
haftmann@46664:   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
haftmann@46664:     by (fact comp_fun_idem_sup)
haftmann@46664:   show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
haftmann@46664: qed
haftmann@46664: 
haftmann@46664: lemma pred_of_set_set_foldr_sup [code]:
haftmann@46664:   "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"
haftmann@46664:   by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
haftmann@46664: 
haftmann@30328: no_notation
haftmann@30328:   bind (infixl "\<guillemotright>=" 70)
haftmann@30328: 
wenzelm@36176: hide_type (open) pred seq
wenzelm@36176: hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
Andreas@53943:   Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map the
haftmann@46664:   iterate_upto
haftmann@46664: hide_fact (open) null_def member_def
haftmann@30328: 
haftmann@30328: end
haftmann@46664: 




