(* Title: HOL/Predicate.thy
Author: Lukas Bulwahn and Florian Haftmann, TU Muenchen
*)
section \<open>Predicates as enumerations\<close>
theory Predicate
imports String
begin
subsection \<open>The type of predicate enumerations (a monad)\<close>
datatype (plugins only: code extraction) (dead '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 (INFIMUM A eval)"
lemma eval_Inf [simp]:
"eval (\<Sqinter>A) = INFIMUM A eval"
by (simp add: Inf_pred_def)
definition
"\<Squnion>A = Pred (SUPREMUM A eval)"
lemma eval_Sup [simp]:
"eval (\<Squnion>A) = SUPREMUM 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_INF [simp]:
"eval (INFIMUM A f) = INFIMUM A (eval \<circ> f)"
using eval_Inf [of "f ` A"] by simp
lemma eval_SUP [simp]:
"eval (SUPREMUM A f) = SUPREMUM A (eval \<circ> f)"
using eval_Sup [of "f ` A"] by simp
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 "\<bind>" 70) where
"P \<bind> f = (SUPREMUM {x. eval P x} f)"
lemma eval_bind [simp]:
"eval (P \<bind> f) = eval (SUPREMUM {x. eval P x} f)"
by (simp add: bind_def)
lemma bind_bind:
"(P \<bind> Q) \<bind> R = P \<bind> (\<lambda>x. Q x \<bind> R)"
by (rule pred_eqI) auto
lemma bind_single:
"P \<bind> single = P"
by (rule pred_eqI) auto
lemma single_bind:
"single x \<bind> P = P x"
by (rule pred_eqI) auto
lemma bottom_bind:
"\<bottom> \<bind> P = \<bottom>"
by (rule pred_eqI) auto
lemma sup_bind:
"(P \<squnion> Q) \<bind> R = P \<bind> R \<squnion> Q \<bind> R"
by (rule pred_eqI) auto
lemma Sup_bind:
"(\<Squnion>A \<bind> f) = \<Squnion>((\<lambda>x. x \<bind> 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 \<bind> Q) y"
by auto
lemma bindE: "eval (R \<bind> 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 \<open>Emptiness check and definite choice\<close>
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
"\<And>default. singleton default A = (if \<exists>!x. eval A x then THE x. eval A x else default ())"
lemma singleton_eqI:
fixes default
shows "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton default A = x"
by (auto simp add: singleton_def)
lemma eval_singletonI:
fixes default
shows "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton default A)"
proof -
assume assm: "\<exists>!x. eval A x"
then obtain x where x: "eval A x" ..
with assm have "singleton default A = x" by (rule singleton_eqI)
with x show ?thesis by simp
qed
lemma single_singleton:
fixes default
shows "\<exists>!x. eval A x \<Longrightarrow> single (singleton default A) = A"
proof -
assume assm: "\<exists>!x. eval A x"
then have "eval A (singleton default A)"
by (rule eval_singletonI)
moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton default A = x"
by (rule singleton_eqI)
ultimately have "eval (single (singleton default A)) = eval A"
by (simp (no_asm_use) add: single_def fun_eq_iff) blast
then have "\<And>x. eval (single (singleton default A)) x = eval A x"
by simp
then show ?thesis by (rule pred_eqI)
qed
lemma singleton_undefinedI:
fixes default
shows "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton default A = default ()"
by (simp add: singleton_def)
lemma singleton_bot:
fixes default
shows "singleton default \<bottom> = default ()"
by (auto simp add: bot_pred_def intro: singleton_undefinedI)
lemma singleton_single:
fixes default
shows "singleton default (single x) = x"
by (auto simp add: intro: singleton_eqI singleI elim: singleE)
lemma singleton_sup_single_single:
fixes default
shows "singleton default (single x \<squnion> single y) = (if x = y then x else default ())"
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 default (single x \<squnion> single y) = default ()"
by (rule singleton_undefinedI)
with False show ?thesis by simp
qed
lemma singleton_sup_aux:
fixes default
shows
"singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B
else if B = \<bottom> then singleton default A
else singleton default
(single (singleton default A) \<squnion> single (singleton default 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 default A = default () \<or> singleton default B = default ()"
by (auto intro!: singleton_undefinedI)
then have rhs: "singleton default
(single (singleton default A) \<squnion> single (singleton default B)) = default ()"
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 default (A \<squnion> B) = default ()" by (rule singleton_undefinedI)
with True rhs show ?thesis by simp
next
case False then show ?thesis by auto
qed
qed
lemma singleton_sup:
fixes default
shows
"singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B
else if B = \<bottom> then singleton default A
else if singleton default A = singleton default B then singleton default A else default ())"
using singleton_sup_aux [of default A B] by (simp only: singleton_sup_single_single)
subsection \<open>Derived operations\<close>
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: if_split_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: if_split_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 \<bind> (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)
functor map: map
by (rule ext, rule pred_eqI, auto)+
subsection \<open>Implementation\<close>
datatype (plugins only: code extraction) (dead '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 \<bind> f) Empty)"
| "apply f (Join P xq) = Join (P \<bind> f) (apply f xq)"
lemma apply_bind:
"pred_of_seq (apply f xq) = pred_of_seq xq \<bind> 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 \<bind> 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
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]:
"case_pred f P = f (eval P)"
by (cases P) simp
lemma [code]:
"rec_pred 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]: "\<And>default. the_only default Empty = default ()"
| "\<And>default. the_only default (Insert x P) =
(if is_empty P then x else let y = singleton default P in if x = y then x else default ())"
| "\<And>default. the_only default (Join P xq) =
(if is_empty P then the_only default xq else if null xq then singleton default P
else let x = singleton default P; y = the_only default xq in
if x = y then x else default ())"
lemma the_only_singleton:
fixes default
shows "the_only default xq = singleton default (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]:
fixes default
shows
"singleton default (Seq f) =
(case f () of
Empty \<Rightarrow> default ()
| Insert x P \<Rightarrow> if is_empty P then x
else let y = singleton default P in
if x = y then x else default ()
| Join P xq \<Rightarrow> if is_empty P then the_only default xq
else if null xq then singleton default P
else let x = singleton default P; y = the_only default xq in
if x = y then x else default ())"
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)
lemma the_eq [code]: "the A = singleton (\<lambda>x. Code.abort (STR ''not_unique'') (\<lambda>_. the A)) A"
by (rule the_eqI) (simp add: singleton_def the_def)
code_reflect Predicate
datatypes pred = Seq and seq = Empty | Insert | Join
ML \<open>
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;
\<close>
text \<open>Conversion from and to sets\<close>
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 \<open>Lazy Evaluation of an indexed function\<close>
function iterate_upto :: "(natural \<Rightarrow> 'a) \<Rightarrow> natural \<Rightarrow> natural \<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). nat_of_natural (m + 1 - n))")
(auto simp add: less_natural_def)
text \<open>Misc\<close>
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 \<open>finite A\<close> 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 "\<bind>" 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 the
iterate_upto
hide_fact (open) null_def member_def
end