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(* Title: HOL/MicroJava/BV/Opt.thy
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10496
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 2000 TUM
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More about options
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*)
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12911
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header {* \isaheader{More about Options} *}
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10496
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theory Opt = Err:
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constdefs
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le :: "'a ord \<Rightarrow> 'a option ord"
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"le r o1 o2 == case o2 of None \<Rightarrow> o1=None |
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Some y \<Rightarrow> (case o1 of None \<Rightarrow> True
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| Some x \<Rightarrow> x <=_r y)"
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opt :: "'a set \<Rightarrow> 'a option set"
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"opt A == insert None {x . ? y:A. x = Some y}"
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sup :: "'a ebinop \<Rightarrow> 'a option ebinop"
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"sup f o1 o2 ==
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case o1 of None \<Rightarrow> OK o2 | Some x \<Rightarrow> (case o2 of None \<Rightarrow> OK o1
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| Some y \<Rightarrow> (case f x y of Err \<Rightarrow> Err | OK z \<Rightarrow> OK (Some z)))"
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esl :: "'a esl \<Rightarrow> 'a option esl"
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"esl == %(A,r,f). (opt A, le r, sup f)"
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lemma unfold_le_opt:
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"o1 <=_(le r) o2 =
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(case o2 of None \<Rightarrow> o1=None |
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Some y \<Rightarrow> (case o1 of None \<Rightarrow> True | Some x \<Rightarrow> x <=_r y))"
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apply (unfold lesub_def le_def)
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apply (rule refl)
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done
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lemma le_opt_refl:
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"order r \<Longrightarrow> o1 <=_(le r) o1"
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by (simp add: unfold_le_opt split: option.split)
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lemma le_opt_trans [rule_format]:
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"order r \<Longrightarrow>
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o1 <=_(le r) o2 \<longrightarrow> o2 <=_(le r) o3 \<longrightarrow> o1 <=_(le r) o3"
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apply (simp add: unfold_le_opt split: option.split)
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apply (blast intro: order_trans)
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done
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lemma le_opt_antisym [rule_format]:
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"order r \<Longrightarrow> o1 <=_(le r) o2 \<longrightarrow> o2 <=_(le r) o1 \<longrightarrow> o1=o2"
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apply (simp add: unfold_le_opt split: option.split)
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apply (blast intro: order_antisym)
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done
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lemma order_le_opt [intro!,simp]:
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"order r \<Longrightarrow> order(le r)"
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apply (subst order_def)
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apply (blast intro: le_opt_refl le_opt_trans le_opt_antisym)
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done
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lemma None_bot [iff]:
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"None <=_(le r) ox"
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apply (unfold lesub_def le_def)
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apply (simp split: option.split)
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done
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lemma Some_le [iff]:
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"(Some x <=_(le r) ox) = (? y. ox = Some y & x <=_r y)"
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apply (unfold lesub_def le_def)
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apply (simp split: option.split)
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done
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lemma le_None [iff]:
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"(ox <=_(le r) None) = (ox = None)";
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apply (unfold lesub_def le_def)
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apply (simp split: option.split)
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done
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lemma OK_None_bot [iff]:
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"OK None <=_(Err.le (le r)) x"
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by (simp add: lesub_def Err.le_def le_def split: option.split err.split)
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lemma sup_None1 [iff]:
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"x +_(sup f) None = OK x"
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by (simp add: plussub_def sup_def split: option.split)
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lemma sup_None2 [iff]:
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"None +_(sup f) x = OK x"
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by (simp add: plussub_def sup_def split: option.split)
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lemma None_in_opt [iff]:
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"None : opt A"
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by (simp add: opt_def)
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lemma Some_in_opt [iff]:
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"(Some x : opt A) = (x:A)"
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apply (unfold opt_def)
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apply auto
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done
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lemma semilat_opt [intro, simp]:
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"\<And>L. err_semilat L \<Longrightarrow> err_semilat (Opt.esl L)"
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proof (unfold Opt.esl_def Err.sl_def, simp add: split_tupled_all)
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fix A r f
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assume s: "semilat (err A, Err.le r, lift2 f)"
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let ?A0 = "err A"
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let ?r0 = "Err.le r"
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let ?f0 = "lift2 f"
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from s
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obtain
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ord: "order ?r0" and
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clo: "closed ?A0 ?f0" and
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ub1: "\<forall>x\<in>?A0. \<forall>y\<in>?A0. x <=_?r0 x +_?f0 y" and
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ub2: "\<forall>x\<in>?A0. \<forall>y\<in>?A0. y <=_?r0 x +_?f0 y" and
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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"
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by (unfold semilat_def) simp
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let ?A = "err (opt A)"
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let ?r = "Err.le (Opt.le r)"
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let ?f = "lift2 (Opt.sup f)"
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from ord
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have "order ?r"
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by simp
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moreover
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have "closed ?A ?f"
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proof (unfold closed_def, intro strip)
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fix x y
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assume x: "x : ?A"
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assume y: "y : ?A"
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{ fix a b
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assume ab: "x = OK a" "y = OK b"
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with x
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have a: "\<And>c. a = Some c \<Longrightarrow> c : A"
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by (clarsimp simp add: opt_def)
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from ab y
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have b: "\<And>d. b = Some d \<Longrightarrow> d : A"
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by (clarsimp simp add: opt_def)
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{ fix c d assume "a = Some c" "b = Some d"
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with ab x y
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have "c:A & d:A"
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by (simp add: err_def opt_def Bex_def)
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with clo
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have "f c d : err A"
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by (simp add: closed_def plussub_def err_def lift2_def)
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moreover
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fix z assume "f c d = OK z"
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ultimately
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have "z : A" by simp
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} note f_closed = this
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have "sup f a b : ?A"
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proof (cases a)
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case None
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thus ?thesis
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by (simp add: sup_def opt_def) (cases b, simp, simp add: b Bex_def)
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next
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case Some
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thus ?thesis
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by (auto simp add: sup_def opt_def Bex_def a b f_closed split: err.split option.split)
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qed
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}
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thus "x +_?f y : ?A"
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by (simp add: plussub_def lift2_def split: err.split)
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qed
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moreover
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{ fix a b c
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assume "a \<in> opt A" "b \<in> opt A" "a +_(sup f) b = OK c"
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moreover
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from ord have "order r" by simp
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moreover
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{ fix x y z
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assume "x \<in> A" "y \<in> A"
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hence "OK x \<in> err A \<and> OK y \<in> err A" by simp
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with ub1 ub2
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have "(OK x) <=_(Err.le r) (OK x) +_(lift2 f) (OK y) \<and>
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(OK y) <=_(Err.le r) (OK x) +_(lift2 f) (OK y)"
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by blast
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moreover
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assume "x +_f y = OK z"
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ultimately
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have "x <=_r z \<and> y <=_r z"
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by (auto simp add: plussub_def lift2_def Err.le_def lesub_def)
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}
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ultimately
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have "a <=_(le r) c \<and> b <=_(le r) c"
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by (auto simp add: sup_def le_def lesub_def plussub_def
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dest: order_refl split: option.splits err.splits)
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}
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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)"
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by (auto simp add: lesub_def plussub_def Err.le_def lift2_def split: err.split)
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moreover
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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"
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proof (intro strip, elim conjE)
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fix x y z
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assume xyz: "x : ?A" "y : ?A" "z : ?A"
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assume xz: "x <=_?r z"
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assume yz: "y <=_?r z"
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{ fix a b c
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assume ok: "x = OK a" "y = OK b" "z = OK c"
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{ fix d e g
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assume some: "a = Some d" "b = Some e" "c = Some g"
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with ok xyz
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obtain "OK d:err A" "OK e:err A" "OK g:err A"
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by simp
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with lub
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have "\<lbrakk> (OK d) <=_(Err.le r) (OK g); (OK e) <=_(Err.le r) (OK g) \<rbrakk>
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\<Longrightarrow> (OK d) +_(lift2 f) (OK e) <=_(Err.le r) (OK g)"
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by blast
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hence "\<lbrakk> d <=_r g; e <=_r g \<rbrakk> \<Longrightarrow> \<exists>y. d +_f e = OK y \<and> y <=_r g"
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by simp
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with ok some xyz xz yz
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have "x +_?f y <=_?r z"
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by (auto simp add: sup_def le_def lesub_def lift2_def plussub_def Err.le_def)
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} note this [intro!]
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from ok xyz xz yz
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have "x +_?f y <=_?r z"
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by - (cases a, simp, cases b, simp, cases c, simp, blast)
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}
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with xyz xz yz
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show "x +_?f y <=_?r z"
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by - (cases x, simp, cases y, simp, cases z, simp+)
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qed
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ultimately
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show "semilat (?A,?r,?f)"
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by (unfold semilat_def) simp
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qed
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lemma top_le_opt_Some [iff]:
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"top (le r) (Some T) = top r T"
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apply (unfold top_def)
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apply (rule iffI)
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apply blast
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apply (rule allI)
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apply (case_tac "x")
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apply simp+
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done
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lemma Top_le_conv:
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"\<lbrakk> order r; top r T \<rbrakk> \<Longrightarrow> (T <=_r x) = (x = T)"
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apply (unfold top_def)
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apply (blast intro: order_antisym)
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done
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lemma acc_le_optI [intro!]:
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"acc r \<Longrightarrow> acc(le r)"
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apply (unfold acc_def lesub_def le_def lesssub_def)
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apply (simp add: wf_eq_minimal split: option.split)
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apply clarify
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apply (case_tac "? a. Some a : Q")
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apply (erule_tac x = "{a . Some a : Q}" in allE)
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apply blast
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apply (case_tac "x")
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apply blast
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apply blast
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done
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lemma option_map_in_optionI:
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"\<lbrakk> ox : opt S; !x:S. ox = Some x \<longrightarrow> f x : S \<rbrakk>
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\<Longrightarrow> option_map f ox : opt S";
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apply (unfold option_map_def)
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apply (simp split: option.split)
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apply blast
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done
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end
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