--- a/src/HOL/MicroJava/BV/Product.thy Wed Dec 02 12:04:07 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,146 +0,0 @@
-(* Title: HOL/MicroJava/BV/Product.thy
- ID: $Id$
- Author: Tobias Nipkow
- Copyright 2000 TUM
-
-Products as semilattices
-*)
-
-header {* \isaheader{Products as Semilattices} *}
-
-theory Product
-imports Err
-begin
-
-constdefs
- le :: "'a ord \<Rightarrow> 'b ord \<Rightarrow> ('a * 'b) ord"
-"le rA rB == %(a,b) (a',b'). a <=_rA a' & b <=_rB b'"
-
- sup :: "'a ebinop \<Rightarrow> 'b ebinop \<Rightarrow> ('a * 'b)ebinop"
-"sup f g == %(a1,b1)(a2,b2). Err.sup Pair (a1 +_f a2) (b1 +_g b2)"
-
- esl :: "'a esl \<Rightarrow> 'b esl \<Rightarrow> ('a * 'b ) esl"
-"esl == %(A,rA,fA) (B,rB,fB). (A <*> B, le rA rB, sup fA fB)"
-
-syntax "@lesubprod" :: "'a*'b \<Rightarrow> 'a ord \<Rightarrow> 'b ord \<Rightarrow> 'b \<Rightarrow> bool"
- ("(_ /<='(_,_') _)" [50, 0, 0, 51] 50)
-translations "p <=(rA,rB) q" == "p <=_(Product.le rA rB) q"
-
-lemma unfold_lesub_prod:
- "p <=(rA,rB) q == le rA rB p q"
- by (simp add: lesub_def)
-
-lemma le_prod_Pair_conv [iff]:
- "((a1,b1) <=(rA,rB) (a2,b2)) = (a1 <=_rA a2 & b1 <=_rB b2)"
- by (simp add: lesub_def le_def)
-
-lemma less_prod_Pair_conv:
- "((a1,b1) <_(Product.le rA rB) (a2,b2)) =
- (a1 <_rA a2 & b1 <=_rB b2 | a1 <=_rA a2 & b1 <_rB b2)"
-apply (unfold lesssub_def)
-apply simp
-apply blast
-done
-
-lemma order_le_prod [iff]:
- "order(Product.le rA rB) = (order rA & order rB)"
-apply (unfold Semilat.order_def)
-apply simp
-apply blast
-done
-
-
-lemma acc_le_prodI [intro!]:
- "\<lbrakk> acc rA; acc rB \<rbrakk> \<Longrightarrow> acc(Product.le rA rB)"
-apply (unfold acc_def)
-apply (rule wfP_subset)
- apply (erule wf_lex_prod [to_pred, THEN wfP_wf_eq [THEN iffD2]])
- apply assumption
-apply (auto simp add: lesssub_def less_prod_Pair_conv lex_prod_def)
-done
-
-
-lemma closed_lift2_sup:
- "\<lbrakk> closed (err A) (lift2 f); closed (err B) (lift2 g) \<rbrakk> \<Longrightarrow>
- closed (err(A<*>B)) (lift2(sup f g))";
-apply (unfold closed_def plussub_def lift2_def err_def sup_def)
-apply (simp split: err.split)
-apply blast
-done
-
-lemma unfold_plussub_lift2:
- "e1 +_(lift2 f) e2 == lift2 f e1 e2"
- by (simp add: plussub_def)
-
-
-lemma plus_eq_Err_conv [simp]:
- assumes "x:A" and "y:A"
- and "semilat(err A, Err.le r, lift2 f)"
- shows "(x +_f y = Err) = (~(? z:A. x <=_r z & y <=_r z))"
-proof -
- have plus_le_conv2:
- "\<And>r f z. \<lbrakk> z : err A; semilat (err A, r, f); OK x : err A; OK y : err A;
- OK x +_f OK y <=_r z\<rbrakk> \<Longrightarrow> OK x <=_r z \<and> OK y <=_r z"
- by (rule Semilat.plus_le_conv [OF Semilat.intro, THEN iffD1])
- from prems show ?thesis
- apply (rule_tac iffI)
- apply clarify
- apply (drule OK_le_err_OK [THEN iffD2])
- apply (drule OK_le_err_OK [THEN iffD2])
- apply (drule Semilat.lub [OF Semilat.intro, of _ _ _ "OK x" _ "OK y"])
- apply assumption
- apply assumption
- apply simp
- apply simp
- apply simp
- apply simp
- apply (case_tac "x +_f y")
- apply assumption
- apply (rename_tac "z")
- apply (subgoal_tac "OK z: err A")
- apply (frule plus_le_conv2)
- apply assumption
- apply simp
- apply blast
- apply simp
- apply (blast dest: Semilat.orderI [OF Semilat.intro] order_refl)
- apply blast
- apply (erule subst)
- apply (unfold semilat_def err_def closed_def)
- apply simp
- done
-qed
-
-lemma err_semilat_Product_esl:
- "\<And>L1 L2. \<lbrakk> err_semilat L1; err_semilat L2 \<rbrakk> \<Longrightarrow> err_semilat(Product.esl L1 L2)"
-apply (unfold esl_def Err.sl_def)
-apply (simp (no_asm_simp) only: split_tupled_all)
-apply simp
-apply (simp (no_asm) only: semilat_Def)
-apply (simp (no_asm_simp) only: Semilat.closedI [OF Semilat.intro] closed_lift2_sup)
-apply (simp (no_asm) only: unfold_lesub_err Err.le_def unfold_plussub_lift2 sup_def)
-apply (auto elim: semilat_le_err_OK1 semilat_le_err_OK2
- simp add: lift2_def split: err.split)
-apply (blast dest: Semilat.orderI [OF Semilat.intro])
-apply (blast dest: Semilat.orderI [OF Semilat.intro])
-
-apply (rule OK_le_err_OK [THEN iffD1])
-apply (erule subst, subst OK_lift2_OK [symmetric], rule Semilat.lub [OF Semilat.intro])
-apply simp
-apply simp
-apply simp
-apply simp
-apply simp
-apply simp
-
-apply (rule OK_le_err_OK [THEN iffD1])
-apply (erule subst, subst OK_lift2_OK [symmetric], rule Semilat.lub [OF Semilat.intro])
-apply simp
-apply simp
-apply simp
-apply simp
-apply simp
-apply simp
-done
-
-end