--- a/src/HOL/MicroJava/BV/Opt.thy Fri Dec 04 11:44:57 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,295 +0,0 @@
-(* Title: HOL/MicroJava/BV/Opt.thy
- ID: $Id$
- Author: Tobias Nipkow
- Copyright 2000 TUM
-
-More about options
-*)
-
-header {* \isaheader{More about Options} *}
-
-theory Opt
-imports Err
-begin
-
-constdefs
- le :: "'a ord \<Rightarrow> 'a option ord"
-"le r o1 o2 == case o2 of None \<Rightarrow> o1=None |
- Some y \<Rightarrow> (case o1 of None \<Rightarrow> True
- | Some x \<Rightarrow> x <=_r y)"
-
- opt :: "'a set \<Rightarrow> 'a option set"
-"opt A == insert None {x . ? y:A. x = Some y}"
-
- sup :: "'a ebinop \<Rightarrow> 'a option ebinop"
-"sup f o1 o2 ==
- case o1 of None \<Rightarrow> OK o2 | Some x \<Rightarrow> (case o2 of None \<Rightarrow> OK o1
- | Some y \<Rightarrow> (case f x y of Err \<Rightarrow> Err | OK z \<Rightarrow> OK (Some z)))"
-
- esl :: "'a esl \<Rightarrow> '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 \<Rightarrow> o1=None |
- Some y \<Rightarrow> (case o1 of None \<Rightarrow> True | Some x \<Rightarrow> x <=_r y))"
-apply (unfold lesub_def le_def)
-apply (rule refl)
-done
-
-lemma le_opt_refl:
- "order r \<Longrightarrow> o1 <=_(le r) o1"
-by (simp add: unfold_le_opt split: option.split)
-
-lemma le_opt_trans [rule_format]:
- "order r \<Longrightarrow>
- o1 <=_(le r) o2 \<longrightarrow> o2 <=_(le r) o3 \<longrightarrow> 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 \<Longrightarrow> o1 <=_(le r) o2 \<longrightarrow> o2 <=_(le r) o1 \<longrightarrow> 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 \<Longrightarrow> 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]:
- "\<And>L. err_semilat L \<Longrightarrow> 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: "\<forall>x\<in>?A0. \<forall>y\<in>?A0. x <=_?r0 x +_?f0 y" and
- ub2: "\<forall>x\<in>?A0. \<forall>y\<in>?A0. y <=_?r0 x +_?f0 y" and
- lub: "\<forall>x\<in>?A0. \<forall>y\<in>?A0. \<forall>z\<in>?A0. x <=_?r0 z \<and> y <=_?r0 z \<longrightarrow> 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: "\<And>c. a = Some c \<Longrightarrow> c : A"
- by (clarsimp simp add: opt_def)
-
- from ab y
- have b: "\<And>d. b = Some d \<Longrightarrow> 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 \<in> opt A" "b \<in> opt A" "a +_(sup f) b = OK c"
- moreover
- from ord have "order r" by simp
- moreover
- { fix x y z
- assume "x \<in> A" "y \<in> A"
- hence "OK x \<in> err A \<and> OK y \<in> err A" by simp
- with ub1 ub2
- have "(OK x) <=_(Err.le r) (OK x) +_(lift2 f) (OK y) \<and>
- (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 \<and> y <=_r z"
- by (auto simp add: plussub_def lift2_def Err.le_def lesub_def)
- }
- ultimately
- have "a <=_(le r) c \<and> 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 "(\<forall>x\<in>?A. \<forall>y\<in>?A. x <=_?r x +_?f y) \<and> (\<forall>x\<in>?A. \<forall>y\<in>?A. y <=_?r x +_?f y)"
- by (auto simp add: lesub_def plussub_def Err.le_def lift2_def split: err.split)
-
- moreover
-
- have "\<forall>x\<in>?A. \<forall>y\<in>?A. \<forall>z\<in>?A. x <=_?r z \<and> y <=_?r z \<longrightarrow> 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 "\<lbrakk> (OK d) <=_(Err.le r) (OK g); (OK e) <=_(Err.le r) (OK g) \<rbrakk>
- \<Longrightarrow> (OK d) +_(lift2 f) (OK e) <=_(Err.le r) (OK g)"
- by blast
- hence "\<lbrakk> d <=_r g; e <=_r g \<rbrakk> \<Longrightarrow> \<exists>y. d +_f e = OK y \<and> 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:
- "\<lbrakk> order r; top r T \<rbrakk> \<Longrightarrow> (T <=_r x) = (x = T)"
-apply (unfold top_def)
-apply (blast intro: order_antisym)
-done
-
-
-lemma acc_le_optI [intro!]:
- "acc r \<Longrightarrow> 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:
- "\<lbrakk> ox : opt S; !x:S. ox = Some x \<longrightarrow> f x : S \<rbrakk>
- \<Longrightarrow> Option.map f ox : opt S";
-apply (unfold Option.map_def)
-apply (simp split: option.split)
-apply blast
-done
-
-end