--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MicroJava/BV/Opt.thy Mon Nov 20 16:37:42 2000 +0100
@@ -0,0 +1,359 @@
+(* Title: HOL/BCV/Opt.thy
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 2000 TUM
+
+More about options
+*)
+
+header "More about Options"
+
+theory Opt = Err:
+
+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 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:
+ "!!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: "\<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: "!!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
+
+ have "\<forall>x\<in>?A. \<forall>y\<in>?A. x <=_?r x +_?f y"
+ proof (intro strip)
+ fix x y
+ assume x: "x : ?A"
+ assume y: "y : ?A"
+
+ show "x <=_?r x +_?f y"
+ proof -
+ from ord
+ have order_r: "order r" by simp
+
+ { fix a b
+ assume ok: "x = OK a" "y = OK b"
+
+ { fix c d
+ assume some: "a = Some c" "b = Some d"
+
+ with x y ok
+ obtain "c:A" "d:A" by simp
+ then
+ obtain "OK c : err A" "OK d : err A" by simp
+ with ub1
+ have OK: "OK c <=_(Err.le r) (OK c) +_(lift2 f) (OK d)"
+ by blast
+
+ { fix e assume "f c d = OK e"
+ with OK
+ have "c <=_r e"
+ by (simp add: lesub_def Err.le_def plussub_def lift2_def)
+ }
+ with ok some
+ have ?thesis
+ by (simp add: plussub_def sup_def lesub_def le_def lift2_def Err.le_def
+ split: err.split)
+ } note this [intro!]
+
+ from ok
+ have ?thesis
+ by - (cases a, simp, cases b, simp add: order_r, blast)
+ }
+ thus ?thesis
+ by - (cases x, simp, cases y, simp+)
+ qed
+ qed
+
+ moreover
+
+ have "\<forall>x\<in>?A. \<forall>y\<in>?A. y <=_?r x +_?f y"
+ proof (intro strip)
+ fix x y
+ assume x: "x : ?A"
+ assume y: "y : ?A"
+
+ show "y <=_?r x +_?f y"
+ proof -
+ from ord
+ have order_r: "order r" by simp
+
+ { fix a b
+ assume ok: "x = OK a" "y = OK b"
+
+ { fix c d
+ assume some: "a = Some c" "b = Some d"
+
+ with x y ok
+ obtain "c:A" "d:A" by simp
+ then
+ obtain "OK c : err A" "OK d : err A" by simp
+ with ub2
+ have OK: "OK d <=_(Err.le r) (OK c) +_(lift2 f) (OK d)"
+ by blast
+
+ { fix e assume "f c d = OK e"
+ with OK
+ have "d <=_r e"
+ by (simp add: lesub_def Err.le_def plussub_def lift2_def)
+ }
+ with ok some
+ have ?thesis
+ by (simp add: plussub_def sup_def lesub_def le_def lift2_def Err.le_def
+ split: err.split)
+ } note this [intro!]
+
+ from ok
+ have ?thesis
+ by - (cases a, simp add: order_r, cases b, simp, blast)
+ }
+ thus ?thesis
+ by - (cases x, simp, cases y, simp+)
+ qed
+ qed
+
+ 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 "[| (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 |] ==> \<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:
+ "[| 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 |]
+ ==> option_map f ox : opt S";
+apply (unfold option_map_def)
+apply (simp split: option.split)
+apply blast
+done
+
+end