# Theory Predicate

theory Predicate
imports String
```(*  Title:      HOL/Predicate.thy
Author:     Lukas Bulwahn and Florian Haftmann, TU Muenchen
*)

section ‹Predicates as enumerations›

theory Predicate
imports String
begin

subsection ‹The type of predicate enumerations (a monad)›

datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a ⇒ bool")

lemma pred_eqI:
"(⋀w. eval P w ⟷ eval Q w) ⟹ P = Q"
by (cases P, cases Q) (auto simp add: fun_eq_iff)

lemma pred_eq_iff:
"P = Q ⟹ (⋀w. eval P w ⟷ eval Q w)"

instantiation pred :: (type) complete_lattice
begin

definition
"P ≤ Q ⟷ eval P ≤ eval Q"

definition
"P < Q ⟷ eval P < eval Q"

definition
"⊥ = Pred ⊥"

lemma eval_bot [simp]:
"eval ⊥  = ⊥"

definition
"⊤ = Pred ⊤"

lemma eval_top [simp]:
"eval ⊤  = ⊤"

definition
"P ⊓ Q = Pred (eval P ⊓ eval Q)"

lemma eval_inf [simp]:
"eval (P ⊓ Q) = eval P ⊓ eval Q"

definition
"P ⊔ Q = Pred (eval P ⊔ eval Q)"

lemma eval_sup [simp]:
"eval (P ⊔ Q) = eval P ⊔ eval Q"

definition
"⨅A = Pred (INFIMUM A eval)"

lemma eval_Inf [simp]:
"eval (⨅A) = INFIMUM A eval"

definition
"⨆A = Pred (SUPREMUM A eval)"

lemma eval_Sup [simp]:
"eval (⨆A) = SUPREMUM A eval"

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 ∘ f)"
using eval_Inf [of "f ` A"] by simp

lemma eval_SUP [simp]:
"eval (SUPREMUM A f) = SUPREMUM A (eval ∘ 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"

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

lemma eval_minus [simp]:
"eval (P - Q) = eval P - eval Q"

instance proof
qed (auto intro!: pred_eqI)

end

definition single :: "'a ⇒ 'a pred" where
"single x = Pred ((op =) x)"

lemma eval_single [simp]:
"eval (single x) = (op =) x"

definition bind :: "'a pred ⇒ ('a ⇒ 'b pred) ⇒ 'b pred" (infixl "⤜" 70) where
"P ⤜ f = (SUPREMUM {x. eval P x} f)"

lemma eval_bind [simp]:
"eval (P ⤜ f) = eval (SUPREMUM {x. eval P x} f)"

lemma bind_bind:
"(P ⤜ Q) ⤜ R = P ⤜ (λx. Q x ⤜ R)"
by (rule pred_eqI) auto

lemma bind_single:
"P ⤜ single = P"
by (rule pred_eqI) auto

lemma single_bind:
"single x ⤜ P = P x"
by (rule pred_eqI) auto

lemma bottom_bind:
"⊥ ⤜ P = ⊥"
by (rule pred_eqI) auto

lemma sup_bind:
"(P ⊔ Q) ⤜ R = P ⤜ R ⊔ Q ⤜ R"
by (rule pred_eqI) auto

lemma Sup_bind:
"(⨆A ⤜ f) = ⨆((λx. x ⤜ f) ` A)"
by (rule pred_eqI) auto

lemma pred_iffI:
assumes "⋀x. eval A x ⟹ eval B x"
and "⋀x. eval B x ⟹ 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 ⟹ (y = x ⟹ P) ⟹ P"
by simp

lemma singleE': "eval (single x) y ⟹ (x = y ⟹ P) ⟹ P"
by simp

lemma bindI: "eval P x ⟹ eval (Q x) y ⟹ eval (P ⤜ Q) y"
by auto

lemma bindE: "eval (R ⤜ Q) y ⟹ (⋀x. eval R x ⟹ eval (Q x) y ⟹ P) ⟹ P"
by auto

lemma botE: "eval ⊥ x ⟹ P"
by auto

lemma supI1: "eval A x ⟹ eval (A ⊔ B) x"
by auto

lemma supI2: "eval B x ⟹ eval (A ⊔ B) x"
by auto

lemma supE: "eval (A ⊔ B) x ⟹ (eval A x ⟹ P) ⟹ (eval B x ⟹ P) ⟹ P"
by auto

lemma single_not_bot [simp]:
"single x ≠ ⊥"
by (auto simp add: single_def bot_pred_def fun_eq_iff)

lemma not_bot:
assumes "A ≠ ⊥"
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 ⇒ bool" where
"is_empty A ⟷ A = ⊥"

lemma is_empty_bot:
"is_empty ⊥"

lemma not_is_empty_single:
"¬ 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 ⊔ B) ⟷ is_empty A ∧ is_empty B"

definition singleton :: "(unit ⇒ 'a) ⇒ 'a pred ⇒ 'a" where
"singleton default A = (if ∃!x. eval A x then THE x. eval A x else default ())" for default

lemma singleton_eqI:
"∃!x. eval A x ⟹ eval A x ⟹ singleton default A = x" for default

lemma eval_singletonI:
"∃!x. eval A x ⟹ eval A (singleton default A)" for default
proof -
assume assm: "∃!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:
"∃!x. eval A x ⟹ single (singleton default A) = A" for default
proof -
assume assm: "∃!x. eval A x"
then have "eval A (singleton default A)"
by (rule eval_singletonI)
moreover from assm have "⋀x. eval A x ⟹ 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 "⋀x. eval (single (singleton default A)) x = eval A x"
by simp
then show ?thesis by (rule pred_eqI)
qed

lemma singleton_undefinedI:
"¬ (∃!x. eval A x) ⟹ singleton default A = default ()" for default

lemma singleton_bot:
"singleton default ⊥ = default ()" for default
by (auto simp add: bot_pred_def intro: singleton_undefinedI)

lemma singleton_single:
"singleton default (single x) = x" for default
by (auto simp add: intro: singleton_eqI singleI elim: singleE)

lemma singleton_sup_single_single:
"singleton default (single x ⊔ single y) = (if x = y then x else default ())" for default
proof (cases "x = y")
case True then show ?thesis by (simp add: singleton_single)
next
case False
have "eval (single x ⊔ single y) x"
and "eval (single x ⊔ single y) y"
by (auto intro: supI1 supI2 singleI)
with False have "¬ (∃!z. eval (single x ⊔ single y) z)"
by blast
then have "singleton default (single x ⊔ single y) = default ()"
by (rule singleton_undefinedI)
with False show ?thesis by simp
qed

lemma singleton_sup_aux:
"singleton default (A ⊔ B) = (if A = ⊥ then singleton default B
else if B = ⊥ then singleton default A
else singleton default
(single (singleton default A) ⊔ single (singleton default B)))" for default
proof (cases "(∃!x. eval A x) ∧ (∃!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 () ∨ singleton default B = default ()"
by (auto intro!: singleton_undefinedI)
then have rhs: "singleton default
(single (singleton default A) ⊔ single (singleton default B)) = default ()"
by (auto simp add: singleton_sup_single_single singleton_single)
from False have not_unique:
"¬ (∃!x. eval A x) ∨ ¬ (∃!y. eval B y)" by simp
show ?thesis proof (cases "A ≠ ⊥ ∧ B ≠ ⊥")
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 "¬ (∃!x. eval (A ⊔ B) x)"
by (auto simp add: sup_pred_def bot_pred_def)
then have "singleton default (A ⊔ 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:
"singleton default (A ⊔ B) = (if A = ⊥ then singleton default B
else if B = ⊥ then singleton default A
else if singleton default A = singleton default B then singleton default A else default ())" for default
using singleton_sup_aux [of default A B] by (simp only: singleton_sup_single_single)

subsection ‹Derived operations›

definition if_pred :: "bool ⇒ unit pred" where
if_pred_eq: "if_pred b = (if b then single () else ⊥)"

definition holds :: "unit pred ⇒ bool" where
holds_eq: "holds P = eval P ()"

definition not_pred :: "unit pred ⇒ unit pred" where
not_pred_eq: "not_pred P = (if eval P () then ⊥ else single ())"

lemma if_predI: "P ⟹ eval (if_pred P) ()"
unfolding if_pred_eq by (auto intro: singleI)

lemma if_predE: "eval (if_pred b) x ⟹ (b ⟹ x = () ⟹ P) ⟹ P"
unfolding if_pred_eq by (cases b) (auto elim: botE)

lemma not_predI: "¬ P ⟹ eval (not_pred (Pred (λu. P))) ()"
unfolding not_pred_eq by (auto intro: singleI)

lemma not_predI': "¬ eval P () ⟹ eval (not_pred P) ()"
unfolding not_pred_eq by (auto intro: singleI)

lemma not_predE: "eval (not_pred (Pred (λu. P))) x ⟹ (¬ P ⟹ thesis) ⟹ thesis"
unfolding not_pred_eq
by (auto split: if_split_asm elim: botE)

lemma not_predE': "eval (not_pred P) x ⟹ (¬ eval P x ⟹ thesis) ⟹ thesis"
unfolding not_pred_eq
by (auto split: if_split_asm elim: botE)
lemma "f () = False ∨ f () = True"
by simp

lemma closure_of_bool_cases [no_atp]:
fixes f :: "unit ⇒ bool"
assumes "f = (λu. False) ⟹ P f"
assumes "f = (λu. True) ⟹ P f"
shows "P f"
proof -
have "f = (λu. False) ∨ f = (λu. True)"
apply (cases "f ()")
apply (rule disjI2)
apply (rule ext)
apply (rule disjI1)
apply (rule ext)
done
from this assms show ?thesis by blast
qed

lemma unit_pred_cases:
assumes "P ⊥"
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 (λu. False))" "P (Pred (λu. () = 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 ⟷ ¬ holds P"
unfolding is_empty_def holds_eq
by (rule unit_pred_cases) (auto elim: botE intro: singleI)

definition map :: "('a ⇒ 'b) ⇒ 'a pred ⇒ 'b pred" where
"map f P = P ⤜ (single o f)"

lemma eval_map [simp]:
"eval (map f P) = (⨆x∈{x. eval P x}. (λy. f x = y))"
by (auto simp add: map_def comp_def)

functor map: map
by (rule ext, rule pred_eqI, auto)+

subsection ‹Implementation›

datatype (plugins only: code extraction) (dead 'a) seq =
Empty
| Insert "'a" "'a pred"
| Join "'a pred" "'a seq"

primrec pred_of_seq :: "'a seq ⇒ 'a pred" where
"pred_of_seq Empty = ⊥"
| "pred_of_seq (Insert x P) = single x ⊔ P"
| "pred_of_seq (Join P xq) = P ⊔ pred_of_seq xq"

definition Seq :: "(unit ⇒ 'a seq) ⇒ 'a pred" where
"Seq f = pred_of_seq (f ())"

code_datatype Seq

primrec member :: "'a seq ⇒ 'a ⇒ bool"  where
"member Empty x ⟷ False"
| "member (Insert y P) x ⟷ x = y ∨ eval P x"
| "member (Join P xq) x ⟷ eval P x ∨ 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 (λu. Insert x ⊥)"
unfolding Seq_def by simp

primrec "apply" :: "('a ⇒ 'b pred) ⇒ 'a seq ⇒ 'b seq" where
"apply f Empty = Empty"
| "apply f (Insert x P) = Join (f x) (Join (P ⤜ f) Empty)"
| "apply f (Join P xq) = Join (P ⤜ f) (apply f xq)"

lemma apply_bind:
"pred_of_seq (apply f xq) = pred_of_seq xq ⤜ f"
proof (induct xq)
case Empty show ?case
next
case Insert show ?case
next
case Join then show ?case
qed

lemma bind_code [code]:
"Seq g ⤜ f = Seq (λu. apply f (g ()))"
unfolding Seq_def by (rule sym, rule apply_bind)

lemma bot_set_code [code]:
"⊥ = Seq (λu. Empty)"
unfolding Seq_def by simp

primrec adjunct :: "'a pred ⇒ 'a seq ⇒ 'a seq" where
"adjunct P Empty = Join P Empty"
| "adjunct P (Insert x Q) = Insert x (Q ⊔ P)"
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"

"pred_of_seq (adjunct P xq) = P ⊔ pred_of_seq xq"
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)

lemma sup_code [code]:
"Seq f ⊔ Seq g = Seq (λu. case f ()
of Empty ⇒ g ()
| Insert x P ⇒ Insert x (P ⊔ Seq g)
| Join P xq ⇒ adjunct (Seq g) (Join P xq))"
proof (cases "f ()")
case Empty
thus ?thesis
unfolding Seq_def by (simp add: sup_commute [of "⊥"])
next
case Insert
thus ?thesis
unfolding Seq_def by (simp add: sup_assoc)
next
case Join
thus ?thesis
unfolding Seq_def
qed

primrec contained :: "'a seq ⇒ 'a pred ⇒ bool" where
"contained Empty Q ⟷ True"
| "contained (Insert x P) Q ⟷ eval Q x ∧ P ≤ Q"
| "contained (Join P xq) Q ⟷ P ≤ Q ∧ contained xq Q"

lemma single_less_eq_eval:
"single x ≤ P ⟷ eval P x"
by (auto simp add: less_eq_pred_def le_fun_def)

lemma contained_less_eq:
"contained xq Q ⟷ pred_of_seq xq ≤ Q"
by (induct xq) (simp_all add: single_less_eq_eval)

lemma less_eq_pred_code [code]:
"Seq f ≤ Q = (case f ()
of Empty ⇒ True
| Insert x P ⇒ eval Q x ∧ P ≤ Q
| Join P xq ⇒ P ≤ Q ∧ contained xq Q)"
by (cases "f ()")

instantiation pred :: (type) equal
begin

definition equal_pred
where [simp]: "HOL.equal P Q ⟷ P = (Q :: 'a pred)"

instance by standard simp

end

lemma [code]:
"HOL.equal P Q ⟷ P ≤ Q ∧ Q ≤ P" for P Q :: "'a pred"
by auto

lemma [code nbe]:
"HOL.equal P P ⟷ True" for P :: "'a pred"
by (fact equal_refl)

lemma [code]:
"case_pred f P = f (eval P)"
by (fact pred.case_eq_if)

lemma [code]:
"rec_pred f P = f (eval P)"
by (cases P) simp

inductive eq :: "'a ⇒ 'a ⇒ bool" where "eq x x"

lemma eq_is_eq: "eq x y ≡ (x = y)"
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)

primrec null :: "'a seq ⇒ bool" where
"null Empty ⟷ True"
| "null (Insert x P) ⟷ False"
| "null (Join P xq) ⟷ is_empty P ∧ null xq"

lemma null_is_empty:
"null xq ⟷ 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) ⟷ null (f ())"

primrec the_only :: "(unit ⇒ 'a) ⇒ 'a seq ⇒ 'a" where
"the_only default Empty = default ()" for 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 ())" for 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 ())" for default

lemma the_only_singleton:
"the_only default xq = singleton default (pred_of_seq xq)" for default
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 default (Seq f) =
(case f () of
Empty ⇒ default ()
| Insert x P ⇒ if is_empty P then x
else let y = singleton default P in
if x = y then x else 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 ())" for 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 ⇒ 'a" where
"the A = (THE x. eval A x)"

lemma the_eqI:
"(THE x. eval P x) = x ⟹ the P = x"

lemma the_eq [code]: "the A = singleton (λx. Code.abort (STR ''not_unique'') (λ_. 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 ‹
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 ⇒ 'a pred" where
"pred_of_set = Pred ∘ (λA x. x ∈ A)"

lemma eval_pred_of_set [simp]:
"eval (pred_of_set A) x ⟷ x ∈A"

definition set_of_pred :: "'a pred ⇒ 'a set" where
"set_of_pred = Collect ∘ eval"

lemma member_set_of_pred [simp]:
"x ∈ set_of_pred P ⟷ Predicate.eval P x"

definition set_of_seq :: "'a seq ⇒ 'a set" where
"set_of_seq = set_of_pred ∘ pred_of_seq"

lemma member_set_of_seq [simp]:
"x ∈ set_of_seq xq = Predicate.member xq x"

lemma of_pred_code [code]:
"set_of_pred (Predicate.Seq f) = (case f () of
Predicate.Empty ⇒ {}
| Predicate.Insert x P ⇒ insert x (set_of_pred P)
| Predicate.Join P xq ⇒ set_of_pred P ∪ 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 ∪ set_of_seq xq"
by auto

text ‹Lazy Evaluation of an indexed function›

function iterate_upto :: "(natural ⇒ 'a) ⇒ natural ⇒ natural ⇒ '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))")

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 ⇒ 'a Predicate.pred ⇒ '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 ⇒ 'a Predicate.pred ⇒ '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 "⤜" 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
```