(* Title: HOL/MicroJava/BV/Opt.thy ID: $Id: Opt.thy,v 1.9 2007/02/07 16:48:51 berghofe Exp $ Author: Tobias Nipkow Copyright 2000 TUM More about options *) header {* \isaheader{More about Options} *} theory Opt imports Err begin constdefs le :: "'a ord => 'a option ord" "le r o1 o2 == case o2 of None => o1=None | Some y => (case o1 of None => True | Some x => x <=_r y)" opt :: "'a set => 'a option set" "opt A == insert None {x . ? y:A. x = Some y}" sup :: "'a ebinop => 'a option ebinop" "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)))" esl :: "'a esl => 'a option esl" "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" by (simp add: opt_def) 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" by (clarsimp simp add: opt_def) from ab y have b: "!!d. b = Some d ==> d : A" by (clarsimp simp add: opt_def) { 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: wfP_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 |] ==> option_map f ox : opt S"; apply (unfold option_map_def) apply (simp split: option.split) apply blast done end
lemma unfold_le_opt:
(o1.0 <=_(Opt.le r) o2.0) =
(case o2.0 of None => o1.0 = None
| Some y => case o1.0 of None => True | Some x => x <=_r y)
lemma le_opt_refl:
Semilat.order r ==> o1.0 <=_(Opt.le r) o1.0
lemma le_opt_trans:
[| Semilat.order r; o1.0 <=_(Opt.le r) o2.0; o2.0 <=_(Opt.le r) o3.0 |]
==> o1.0 <=_(Opt.le r) o3.0
lemma le_opt_antisym:
[| Semilat.order r; o1.0 <=_(Opt.le r) o2.0; o2.0 <=_(Opt.le r) o1.0 |]
==> o1.0 = o2.0
lemma order_le_opt:
Semilat.order r ==> Semilat.order (Opt.le r)
lemma None_bot:
None <=_(Opt.le r) ox
lemma Some_le:
(Some x <=_(Opt.le r) ox) = (∃y. ox = Some y ∧ x <=_r y)
lemma le_None:
(ox <=_(Opt.le r) None) = (ox = None)
lemma OK_None_bot:
OK None <=_(Err.le (Opt.le r)) x
lemma sup_None1:
x +_(Opt.sup f) None = OK x
lemma sup_None2:
None +_(Opt.sup f) x = OK x
lemma None_in_opt:
None ∈ opt A
lemma Some_in_opt:
(Some x ∈ opt A) = (x ∈ A)
lemma semilat_opt:
semilat (sl L) ==> semilat (sl (Opt.esl L))
lemma top_le_opt_Some:
Semilat.top (Opt.le r) (Some T) = Semilat.top r T
lemma Top_le_conv:
[| Semilat.order r; Semilat.top r T |] ==> (T <=_r x) = (x = T)
lemma acc_le_optI:
Semilat.acc r ==> Semilat.acc (Opt.le r)
lemma option_map_in_optionI:
[| ox ∈ opt S; ∀x∈S. ox = Some x --> f x ∈ S |] ==> option_map f ox ∈ opt S