# Theory Opt

theory Opt
imports Err
(*  Title:      HOL/MicroJava/DFA/Opt.thy
Author:     Tobias Nipkow
*)

theory Opt
imports Err
begin

definition le :: "'a ord ⇒ 'a option ord" where
"le r o1 o2 == case o2 of None ⇒ o1=None |
Some y ⇒ (case o1 of None ⇒ True
| Some x ⇒ x <=_r y)"

definition opt :: "'a set ⇒ 'a option set" where
"opt A == insert None {x . ? y:A. x = Some y}"

definition sup :: "'a ebinop ⇒ 'a option ebinop" where
"sup f o1 o2 ==
case o1 of None ⇒ OK o2 | Some x ⇒ (case o2 of None ⇒ OK o1
| Some y ⇒ (case f x y of Err ⇒ Err | OK z ⇒ OK (Some z)))"

definition esl :: "'a esl ⇒ 'a option esl" where
"esl == %(A,r,f). (opt A, le r, sup f)"

lemma unfold_le_opt:
"o1 <=_(le r) o2 =
(case o2 of None ⇒ o1=None |
Some y ⇒ (case o1 of None ⇒ True | Some x ⇒ x <=_r y))"
apply (unfold lesub_def le_def)
apply (rule refl)
done

lemma le_opt_refl:
"order r ⟹ o1 <=_(le r) o1"
by (simp add: unfold_le_opt split: option.split)

lemma le_opt_trans [rule_format]:
"order r ⟹
o1 <=_(le r) o2 ⟶ o2 <=_(le r) o3 ⟶ o1 <=_(le r) o3"
apply (simp add: unfold_le_opt split: option.split)
apply (blast intro: order_trans)
done

lemma le_opt_antisym [rule_format]:
"order r ⟹ o1 <=_(le r) o2 ⟶ o2 <=_(le r) o1 ⟶ o1=o2"
apply (simp add: unfold_le_opt split: option.split)
apply (blast intro: order_antisym)
done

lemma order_le_opt [intro!,simp]:
"order r ⟹ order(le r)"
apply (subst Semilat.order_def)
apply (blast intro: le_opt_refl le_opt_trans le_opt_antisym)
done

lemma None_bot [iff]:
"None <=_(le r) ox"
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done

lemma Some_le [iff]:
"(Some x <=_(le r) ox) = (? y. ox = Some y & x <=_r y)"
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done

lemma le_None [iff]:
"(ox <=_(le r) None) = (ox = None)"
apply (unfold lesub_def le_def)
apply (simp split: option.split)
done

lemma OK_None_bot [iff]:
"OK None <=_(Err.le (le r)) x"
by (simp add: lesub_def Err.le_def le_def split: option.split err.split)

lemma sup_None1 [iff]:
"x +_(sup f) None = OK x"
by (simp add: plussub_def sup_def split: option.split)

lemma sup_None2 [iff]:
"None +_(sup f) x = OK x"
by (simp add: plussub_def sup_def split: option.split)

lemma None_in_opt [iff]:
"None : opt A"

lemma Some_in_opt [iff]:
"(Some x : opt A) = (x:A)"
apply (unfold opt_def)
apply auto
done

lemma semilat_opt [intro, simp]:
"⋀L. err_semilat L ⟹ err_semilat (Opt.esl L)"
proof (unfold Opt.esl_def Err.sl_def, simp add: split_tupled_all)

fix A r f
assume s: "semilat (err A, Err.le r, lift2 f)"

let ?A0 = "err A"
let ?r0 = "Err.le r"
let ?f0 = "lift2 f"

from s
obtain
ord: "order ?r0" and
clo: "closed ?A0 ?f0" and
ub1: "∀x∈?A0. ∀y∈?A0. x <=_?r0 x +_?f0 y" and
ub2: "∀x∈?A0. ∀y∈?A0. y <=_?r0 x +_?f0 y" and
lub: "∀x∈?A0. ∀y∈?A0. ∀z∈?A0. x <=_?r0 z ∧ y <=_?r0 z ⟶ x +_?f0 y <=_?r0 z"
by (unfold semilat_def) simp

let ?A = "err (opt A)"
let ?r = "Err.le (Opt.le r)"
let ?f = "lift2 (Opt.sup f)"

from ord
have "order ?r"
by simp

moreover

have "closed ?A ?f"
proof (unfold closed_def, intro strip)
fix x y
assume x: "x : ?A"
assume y: "y : ?A"

{ fix a b
assume ab: "x = OK a" "y = OK b"

with x
have a: "⋀c. a = Some c ⟹ c : A"

from ab y
have b: "⋀d. b = Some d ⟹ d : A"

{ fix c d assume "a = Some c" "b = Some d"
with ab x y
have "c:A & d:A"
by (simp add: err_def opt_def Bex_def)
with clo
have "f c d : err A"
by (simp add: closed_def plussub_def err_def lift2_def)
moreover
fix z assume "f c d = OK z"
ultimately
have "z : A" by simp
} note f_closed = this

have "sup f a b : ?A"
proof (cases a)
case None
thus ?thesis
by (simp add: sup_def opt_def) (cases b, simp, simp add: b Bex_def)
next
case Some
thus ?thesis
by (auto simp add: sup_def opt_def Bex_def a b f_closed split: err.split option.split)
qed
}

thus "x +_?f y : ?A"
by (simp add: plussub_def lift2_def split: err.split)
qed

moreover

{ fix a b c
assume "a ∈ opt A" "b ∈ opt A" "a +_(sup f) b = OK c"
moreover
from ord have "order r" by simp
moreover
{ fix x y z
assume "x ∈ A" "y ∈ A"
hence "OK x ∈ err A ∧ OK y ∈ err A" by simp
with ub1 ub2
have "(OK x) <=_(Err.le r) (OK x) +_(lift2 f) (OK y) ∧
(OK y) <=_(Err.le r) (OK x) +_(lift2 f) (OK y)"
by blast
moreover
assume "x +_f y = OK z"
ultimately
have "x <=_r z ∧ y <=_r z"
by (auto simp add: plussub_def lift2_def Err.le_def lesub_def)
}
ultimately
have "a <=_(le r) c ∧ b <=_(le r) c"
by (auto simp add: sup_def le_def lesub_def plussub_def
dest: order_refl split: option.splits err.splits)
}

hence "(∀x∈?A. ∀y∈?A. x <=_?r x +_?f y) ∧ (∀x∈?A. ∀y∈?A. y <=_?r x +_?f y)"
by (auto simp add: lesub_def plussub_def Err.le_def lift2_def split: err.split)

moreover

have "∀x∈?A. ∀y∈?A. ∀z∈?A. x <=_?r z ∧ y <=_?r z ⟶ x +_?f y <=_?r z"
proof (intro strip, elim conjE)
fix x y z
assume xyz: "x : ?A" "y : ?A" "z : ?A"
assume xz: "x <=_?r z"
assume yz: "y <=_?r z"

{ fix a b c
assume ok: "x = OK a" "y = OK b" "z = OK c"

{ fix d e g
assume some: "a = Some d" "b = Some e" "c = Some g"

with ok xyz
obtain "OK d:err A" "OK e:err A" "OK g:err A"
by simp
with lub
have "⟦ (OK d) <=_(Err.le r) (OK g); (OK e) <=_(Err.le r) (OK g) ⟧
⟹ (OK d) +_(lift2 f) (OK e) <=_(Err.le r) (OK g)"
by blast
hence "⟦ d <=_r g; e <=_r g ⟧ ⟹ ∃y. d +_f e = OK y ∧ y <=_r g"
by simp

with ok some xyz xz yz
have "x +_?f y <=_?r z"
by (auto simp add: sup_def le_def lesub_def lift2_def plussub_def Err.le_def)
} note this [intro!]

from ok xyz xz yz
have "x +_?f y <=_?r z"
by - (cases a, simp, cases b, simp, cases c, simp, blast)
}

with xyz xz yz
show "x +_?f y <=_?r z"
by - (cases x, simp, cases y, simp, cases z, simp+)
qed

ultimately

show "semilat (?A,?r,?f)"
by (unfold semilat_def) simp
qed

lemma top_le_opt_Some [iff]:
"top (le r) (Some T) = top r T"
apply (unfold top_def)
apply (rule iffI)
apply blast
apply (rule allI)
apply (case_tac "x")
apply simp+
done

lemma Top_le_conv:
"⟦ order r; top r T ⟧ ⟹ (T <=_r x) = (x = T)"
apply (unfold top_def)
apply (blast intro: order_antisym)
done

lemma acc_le_optI [intro!]:
"acc r ⟹ acc(le r)"
apply (unfold acc_def lesub_def le_def lesssub_def)
apply (simp add: wf_eq_minimal split: option.split)
apply clarify
apply (case_tac "? a. Some a : Q")
apply (erule_tac x = "{a . Some a : Q}" in allE)
apply blast
apply (case_tac "x")
apply blast
apply blast
done

lemma option_map_in_optionI:
"⟦ ox : opt S; !x:S. ox = Some x ⟶ f x : S ⟧
⟹ map_option f ox : opt S"
apply (unfold map_option_case)
apply (simp split: option.split)
apply blast
done

end