(*  Title:      HOL/Predicate.thy

    Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen

*)

header {* Predicates as enumerations *}

theory Predicate

imports List

begin

subsection {* The type of predicate enumerations (a monad) *}

datatype 'a pred = Pred "'a \<Rightarrow> bool"

primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where

  eval_pred: "eval (Pred f) = f"

lemma Pred_eval [simp]:

  "Pred (eval x) = x"

  by (cases x) simp

lemma pred_eqI:

  "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"

  by (cases P, cases Q) (auto simp add: fun_eq_iff)

lemma pred_eq_iff:

  "P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)"

  by (simp add: pred_eqI)

instantiation pred :: (type) complete_lattice

begin

definition

  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"

definition

  "P < Q \<longleftrightarrow> eval P < eval Q"

definition

  "\<bottom> = Pred \<bottom>"

lemma eval_bot [simp]:

  "eval \<bottom>  = \<bottom>"

  by (simp add: bot_pred_def)

definition

  "\<top> = Pred \<top>"

lemma eval_top [simp]:

  "eval \<top>  = \<top>"

  by (simp add: top_pred_def)

definition

  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"

lemma eval_inf [simp]:

  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"

  by (simp add: inf_pred_def)

definition

  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"

lemma eval_sup [simp]:

  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"

  by (simp add: sup_pred_def)

definition

  "\<Sqinter>A = Pred (INFI A eval)"

lemma eval_Inf [simp]:

  "eval (\<Sqinter>A) = INFI A eval"

  by (simp add: Inf_pred_def)

definition

  "\<Squnion>A = Pred (SUPR A eval)"

lemma eval_Sup [simp]:

  "eval (\<Squnion>A) = SUPR A eval"

  by (simp add: Sup_pred_def)

instance proof

qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)

end

lemma eval_INFI [simp]:

  "eval (INFI A f) = INFI A (eval \<circ> f)"

  by (simp only: INF_def eval_Inf image_compose)

lemma eval_SUPR [simp]:

  "eval (SUPR A f) = SUPR A (eval \<circ> f)"

  by (simp only: SUP_def eval_Sup image_compose)

instantiation pred :: (type) complete_boolean_algebra

begin

definition

  "- P = Pred (- eval P)"

lemma eval_compl [simp]:

  "eval (- P) = - eval P"

  by (simp add: uminus_pred_def)

definition

  "P - Q = Pred (eval P - eval Q)"

lemma eval_minus [simp]:

  "eval (P - Q) = eval P - eval Q"

  by (simp add: minus_pred_def)

instance proof

qed (auto intro!: pred_eqI)

end

definition single :: "'a \<Rightarrow> 'a pred" where

  "single x = Pred ((op =) x)"

lemma eval_single [simp]:

  "eval (single x) = (op =) x"

  by (simp add: single_def)

definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where

  "P \<guillemotright>= f = (SUPR {x. eval P x} f)"

lemma eval_bind [simp]:

  "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"

  by (simp add: bind_def)

lemma bind_bind:

  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"

  by (rule pred_eqI) auto

lemma bind_single:

  "P \<guillemotright>= single = P"

  by (rule pred_eqI) auto

lemma single_bind:

  "single x \<guillemotright>= P = P x"

  by (rule pred_eqI) auto

lemma bottom_bind:

  "\<bottom> \<guillemotright>= P = \<bottom>"

  by (rule pred_eqI) auto

lemma sup_bind:

  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"

  by (rule pred_eqI) auto

lemma Sup_bind:

  "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"

  by (rule pred_eqI) auto

lemma pred_iffI:

  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"

  and "\<And>x. eval B x \<Longrightarrow> eval A x"

  shows "A = B"

  using assms by (auto intro: pred_eqI)

lemma singleI: "eval (single x) x"

  by simp

lemma singleI_unit: "eval (single ()) x"

  by simp

lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"

  by simp

lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"

  by simp

lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"

  by auto

lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"

  by auto

lemma botE: "eval \<bottom> x \<Longrightarrow> P"

  by auto

lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"

  by auto

lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

  by auto

lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"

  by auto

lemma single_not_bot [simp]:

  "single x \<noteq> \<bottom>"

  by (auto simp add: single_def bot_pred_def fun_eq_iff)

lemma not_bot:

  assumes "A \<noteq> \<bottom>"

  obtains x where "eval A x"

  using assms by (cases A) (auto simp add: bot_pred_def)

subsection {* Emptiness check and definite choice *}

definition is_empty :: "'a pred \<Rightarrow> bool" where

  "is_empty A \<longleftrightarrow> A = \<bottom>"

lemma is_empty_bot:

  "is_empty \<bottom>"

  by (simp add: is_empty_def)

lemma not_is_empty_single:

  "\<not> is_empty (single x)"

  by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)

lemma is_empty_sup:

  "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"

  by (auto simp add: is_empty_def)

definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where

  "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"

lemma singleton_eqI:

  "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"

  by (auto simp add: singleton_def)

lemma eval_singletonI:

  "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"

proof -

  assume assm: "\<exists>!x. eval A x"

  then obtain x where "eval A x" ..

  moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)

  ultimately show ?thesis by simp 

qed

lemma single_singleton:

  "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"

proof -

  assume assm: "\<exists>!x. eval A x"

  then have "eval A (singleton dfault A)"

    by (rule eval_singletonI)

  moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"

    by (rule singleton_eqI)

  ultimately have "eval (single (singleton dfault A)) = eval A"

    by (simp (no_asm_use) add: single_def fun_eq_iff) blast

  then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"

    by simp

  then show ?thesis by (rule pred_eqI)

qed

lemma singleton_undefinedI:

  "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"

  by (simp add: singleton_def)

lemma singleton_bot:

  "singleton dfault \<bottom> = dfault ()"

  by (auto simp add: bot_pred_def intro: singleton_undefinedI)

lemma singleton_single:

  "singleton dfault (single x) = x"

  by (auto simp add: intro: singleton_eqI singleI elim: singleE)

lemma singleton_sup_single_single:

  "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"

proof (cases "x = y")

  case True then show ?thesis by (simp add: singleton_single)

next

  case False

  have "eval (single x \<squnion> single y) x"

    and "eval (single x \<squnion> single y) y"

  by (auto intro: supI1 supI2 singleI)

  with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"

    by blast

  then have "singleton dfault (single x \<squnion> single y) = dfault ()"

    by (rule singleton_undefinedI)

  with False show ?thesis by simp

qed

lemma singleton_sup_aux:

  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B

    else if B = \<bottom> then singleton dfault A

    else singleton dfault

      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"

proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")

  case True then show ?thesis by (simp add: single_singleton)

next

  case False

  from False have A_or_B:

    "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"

    by (auto intro!: singleton_undefinedI)

  then have rhs: "singleton dfault

    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"

    by (auto simp add: singleton_sup_single_single singleton_single)

  from False have not_unique:

    "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp

  show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")

    case True

    then obtain a b where a: "eval A a" and b: "eval B b"

      by (blast elim: not_bot)

    with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"

      by (auto simp add: sup_pred_def bot_pred_def)

    then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)

    with True rhs show ?thesis by simp

  next

    case False then show ?thesis by auto

  qed

qed

lemma singleton_sup:

  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B

    else if B = \<bottom> then singleton dfault A

    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"

using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)

subsection {* Derived operations *}

definition if_pred :: "bool \<Rightarrow> unit pred" where

  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"

definition holds :: "unit pred \<Rightarrow> bool" where

  holds_eq: "holds P = eval P ()"

definition not_pred :: "unit pred \<Rightarrow> unit pred" where

  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"

lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"

  unfolding if_pred_eq by (auto intro: singleI)

lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"

  unfolding if_pred_eq by (cases b) (auto elim: botE)

lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"

  unfolding not_pred_eq eval_pred by (auto intro: singleI)

lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"

  unfolding not_pred_eq by (auto intro: singleI)

lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"

  unfolding not_pred_eq

  by (auto split: split_if_asm elim: botE)

lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"

  unfolding not_pred_eq

  by (auto split: split_if_asm elim: botE)

lemma "f () = False \<or> f () = True"

by simp

lemma closure_of_bool_cases [no_atp]:

  fixes f :: "unit \<Rightarrow> bool"

  assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"

  assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"

  shows "P f"

proof -

  have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"

    apply (cases "f ()")

    apply (rule disjI2)

    apply (rule ext)

    apply (simp add: unit_eq)

    apply (rule disjI1)

    apply (rule ext)

    apply (simp add: unit_eq)

    done

  from this assms show ?thesis by blast

qed

lemma unit_pred_cases:

  assumes "P \<bottom>"

  assumes "P (single ())"

  shows "P Q"

using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)

  fix f

  assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"

  then have "P (Pred f)" 

    by (cases _ f rule: closure_of_bool_cases) simp_all

  moreover assume "Q = Pred f"

  ultimately show "P Q" by simp

qed

lemma holds_if_pred:

  "holds (if_pred b) = b"

unfolding if_pred_eq holds_eq

by (cases b) (auto intro: singleI elim: botE)

lemma if_pred_holds:

  "if_pred (holds P) = P"

unfolding if_pred_eq holds_eq

by (rule unit_pred_cases) (auto intro: singleI elim: botE)

lemma is_empty_holds:

  "is_empty P \<longleftrightarrow> \<not> holds P"

unfolding is_empty_def holds_eq

by (rule unit_pred_cases) (auto elim: botE intro: singleI)

definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where

  "map f P = P \<guillemotright>= (single o f)"

lemma eval_map [simp]:

  "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"

  by (auto simp add: map_def comp_def)

enriched_type map: map

  by (rule ext, rule pred_eqI, auto)+

subsection {* Implementation *}

datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"

primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where

  "pred_of_seq Empty = \<bottom>"

| "pred_of_seq (Insert x P) = single x \<squnion> P"

| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"

definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where

  "Seq f = pred_of_seq (f ())"

code_datatype Seq

primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where

  "member Empty x \<longleftrightarrow> False"

| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"

| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"

lemma eval_member:

  "member xq = eval (pred_of_seq xq)"

proof (induct xq)

  case Empty show ?case

  by (auto simp add: fun_eq_iff elim: botE)

next

  case Insert show ?case

  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)

next

  case Join then show ?case

  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)

qed

lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())"

  unfolding Seq_def by (rule sym, rule eval_member)

lemma single_code [code]:

  "single x = Seq (\<lambda>u. Insert x \<bottom>)"

  unfolding Seq_def by simp

primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where

  "apply f Empty = Empty"

| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"

| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"

lemma apply_bind:

  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"

proof (induct xq)

  case Empty show ?case

    by (simp add: bottom_bind)

next

  case Insert show ?case

    by (simp add: single_bind sup_bind)

next

  case Join then show ?case

    by (simp add: sup_bind)

qed

lemma bind_code [code]:

  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"

  unfolding Seq_def by (rule sym, rule apply_bind)

lemma bot_set_code [code]:

  "\<bottom> = Seq (\<lambda>u. Empty)"

  unfolding Seq_def by simp

primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where

  "adjunct P Empty = Join P Empty"

| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"

| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"

lemma adjunct_sup:

  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"

  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)

lemma sup_code [code]:

  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()

    of Empty \<Rightarrow> g ()

     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)

     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"

proof (cases "f ()")

  case Empty

  thus ?thesis

    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])

next

  case Insert

  thus ?thesis

    unfolding Seq_def by (simp add: sup_assoc)

next

  case Join

  thus ?thesis

    unfolding Seq_def

    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)

qed

lemma [code]:

  "size (P :: 'a Predicate.pred) = 0" by (cases P) simp

lemma [code]:

  "pred_size f P = 0" by (cases P) simp

primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where

  "contained Empty Q \<longleftrightarrow> True"

| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"

| "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"

lemma single_less_eq_eval:

  "single x \<le> P \<longleftrightarrow> eval P x"

  by (auto simp add: less_eq_pred_def le_fun_def)

lemma contained_less_eq:

  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"

  by (induct xq) (simp_all add: single_less_eq_eval)

lemma less_eq_pred_code [code]:

  "Seq f \<le> Q = (case f ()

   of Empty \<Rightarrow> True

    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q

    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"

  by (cases "f ()")

    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)

lemma eq_pred_code [code]:

  fixes P Q :: "'a pred"

  shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"

  by (auto simp add: equal)

lemma [code nbe]:

  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"

  by (fact equal_refl)

lemma [code]:

  "pred_case f P = f (eval P)"

  by (cases P) simp

lemma [code]:

  "pred_rec f P = f (eval P)"

  by (cases P) simp

inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"

lemma eq_is_eq: "eq x y \<equiv> (x = y)"

  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)

primrec null :: "'a seq \<Rightarrow> bool" where

  "null Empty \<longleftrightarrow> True"

| "null (Insert x P) \<longleftrightarrow> False"

| "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"

lemma null_is_empty:

  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"

  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)

lemma is_empty_code [code]:

  "is_empty (Seq f) \<longleftrightarrow> null (f ())"

  by (simp add: null_is_empty Seq_def)

primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where

  [code del]: "the_only dfault Empty = dfault ()"

| "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 ())"

| "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P

       else let x = singleton dfault P; y = the_only dfault xq in

       if x = y then x else dfault ())"

lemma the_only_singleton:

  "the_only dfault xq = singleton dfault (pred_of_seq xq)"

  by (induct xq)

    (auto simp add: singleton_bot singleton_single is_empty_def

    null_is_empty Let_def singleton_sup)

lemma singleton_code [code]:

  "singleton dfault (Seq f) = (case f ()

   of Empty \<Rightarrow> dfault ()

    | Insert x P \<Rightarrow> if is_empty P then x

        else let y = singleton dfault P in

          if x = y then x else dfault ()

    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq

        else if null xq then singleton dfault P

        else let x = singleton dfault P; y = the_only dfault xq in

          if x = y then x else dfault ())"

  by (cases "f ()")

   (auto simp add: Seq_def the_only_singleton is_empty_def

      null_is_empty singleton_bot singleton_single singleton_sup Let_def)

definition the :: "'a pred \<Rightarrow> 'a" where

  "the A = (THE x. eval A x)"

lemma the_eqI:

  "(THE x. eval P x) = x \<Longrightarrow> the P = x"

  by (simp add: the_def)

definition not_unique :: "'a pred \<Rightarrow> 'a" where

  [code del]: "not_unique A = (THE x. eval A x)"

code_abort not_unique

lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"

  by (rule the_eqI) (simp add: singleton_def not_unique_def)

code_reflect Predicate

  datatypes pred = Seq and seq = Empty | Insert | Join

ML {*

signature PREDICATE =

sig

  val anamorph: ('a -> ('b * 'a) option) -> int -> 'a -> 'b list * 'a

  datatype 'a pred = Seq of (unit -> 'a seq)

  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq

  val map: ('a -> 'b) -> 'a pred -> 'b pred

  val yield: 'a pred -> ('a * 'a pred) option

  val yieldn: int -> 'a pred -> 'a list * 'a pred

end;

structure Predicate : PREDICATE =

struct

fun anamorph f k x =

 (if k = 0 then ([], x)

  else case f x

   of NONE => ([], x)

    | SOME (v, y) => let

        val k' = k - 1;

        val (vs, z) = anamorph f k' y

      in (v :: vs, z) end);

datatype pred = datatype Predicate.pred

datatype seq = datatype Predicate.seq

fun map f = @{code Predicate.map} f;

fun yield (Seq f) = next (f ())

and next Empty = NONE

  | next (Insert (x, P)) = SOME (x, P)

  | next (Join (P, xq)) = (case yield P

     of NONE => next xq

      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));

fun yieldn k = anamorph yield k;

end;

*}

text {* Conversion from and to sets *}

definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where

  "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"

lemma eval_pred_of_set [simp]:

  "eval (pred_of_set A) x \<longleftrightarrow> x \<in>A"

  by (simp add: pred_of_set_def)

definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where

  "set_of_pred = Collect \<circ> eval"

lemma member_set_of_pred [simp]:

  "x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x"

  by (simp add: set_of_pred_def)

definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where

  "set_of_seq = set_of_pred \<circ> pred_of_seq"

lemma member_set_of_seq [simp]:

  "x \<in> set_of_seq xq = Predicate.member xq x"

  by (simp add: set_of_seq_def eval_member)

lemma of_pred_code [code]:

  "set_of_pred (Predicate.Seq f) = (case f () of

     Predicate.Empty \<Rightarrow> {}

   | Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P)

   | Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"

  by (auto split: seq.split simp add: eval_code)

lemma of_seq_code [code]:

  "set_of_seq Predicate.Empty = {}"

  "set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)"

  "set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq"

  by auto

text {* Lazy Evaluation of an indexed function *}

function iterate_upto :: "(code_numeral \<Rightarrow> 'a) \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> 'a Predicate.pred"

where

  "iterate_upto f n m =

    Predicate.Seq (%u. if n > m then Predicate.Empty

     else Predicate.Insert (f n) (iterate_upto f (n + 1) m))"

by pat_completeness auto

termination by (relation "measure (%(f, n, m). Code_Numeral.nat_of (m + 1 - n))") auto

text {* Misc *}

declare Inf_set_fold [where 'a = "'a Predicate.pred", code]

declare Sup_set_fold [where 'a = "'a Predicate.pred", code]

(* FIXME: better implement conversion by bisection *)

lemma pred_of_set_fold_sup:

  assumes "finite A"

  shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")

proof (rule sym)

  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"

    by (fact comp_fun_idem_sup)

  from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)

qed

lemma pred_of_set_set_fold_sup:

  "pred_of_set (set xs) = fold sup (List.map Predicate.single xs) bot"

proof -

  interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"

    by (fact comp_fun_idem_sup)

  show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])

qed

lemma pred_of_set_set_foldr_sup [code]:

  "pred_of_set (set xs) = foldr sup (List.map Predicate.single xs) bot"

  by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)

no_notation

  bind (infixl "\<guillemotright>=" 70)

hide_type (open) pred seq

hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds

  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the

  iterate_upto

hide_fact (open) null_def member_def

end