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(* Title: HOL/MicroJava/BV/Product.thy
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 2000 TUM
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Products as semilattices
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*)
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12911
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header {* \isaheader{Products as Semilattices} *}
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16417
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theory Product imports Err begin
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constdefs
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le :: "'a ord \<Rightarrow> 'b ord \<Rightarrow> ('a * 'b) ord"
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"le rA rB == %(a,b) (a',b'). a <=_rA a' & b <=_rB b'"
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sup :: "'a ebinop \<Rightarrow> 'b ebinop \<Rightarrow> ('a * 'b)ebinop"
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"sup f g == %(a1,b1)(a2,b2). Err.sup Pair (a1 +_f a2) (b1 +_g b2)"
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esl :: "'a esl \<Rightarrow> 'b esl \<Rightarrow> ('a * 'b ) esl"
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"esl == %(A,rA,fA) (B,rB,fB). (A <*> B, le rA rB, sup fA fB)"
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syntax "@lesubprod" :: "'a*'b \<Rightarrow> 'a ord \<Rightarrow> 'b ord \<Rightarrow> 'b \<Rightarrow> bool"
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("(_ /<='(_,_') _)" [50, 0, 0, 51] 50)
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translations "p <=(rA,rB) q" == "p <=_(Product.le rA rB) q"
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lemma unfold_lesub_prod:
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"p <=(rA,rB) q == le rA rB p q"
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by (simp add: lesub_def)
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lemma le_prod_Pair_conv [iff]:
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"((a1,b1) <=(rA,rB) (a2,b2)) = (a1 <=_rA a2 & b1 <=_rB b2)"
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by (simp add: lesub_def le_def)
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lemma less_prod_Pair_conv:
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"((a1,b1) <_(Product.le rA rB) (a2,b2)) =
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(a1 <_rA a2 & b1 <=_rB b2 | a1 <=_rA a2 & b1 <_rB b2)"
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apply (unfold lesssub_def)
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apply simp
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apply blast
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done
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lemma order_le_prod [iff]:
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"order(Product.le rA rB) = (order rA & order rB)"
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apply (unfold order_def)
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apply simp
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15613
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apply blast
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done
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lemma acc_le_prodI [intro!]:
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"\<lbrakk> acc rA; acc rB \<rbrakk> \<Longrightarrow> acc(Product.le rA rB)"
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apply (unfold acc_def)
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apply (rule wf_subset)
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apply (erule wf_lex_prod)
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apply assumption
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apply (auto simp add: lesssub_def less_prod_Pair_conv lex_prod_def)
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done
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lemma closed_lift2_sup:
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"\<lbrakk> closed (err A) (lift2 f); closed (err B) (lift2 g) \<rbrakk> \<Longrightarrow>
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closed (err(A<*>B)) (lift2(sup f g))";
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apply (unfold closed_def plussub_def lift2_def err_def sup_def)
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apply (simp split: err.split)
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apply blast
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done
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lemma unfold_plussub_lift2:
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"e1 +_(lift2 f) e2 == lift2 f e1 e2"
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by (simp add: plussub_def)
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lemma plus_eq_Err_conv [simp]:
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assumes "x:A" and "y:A"
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and "semilat(err A, Err.le r, lift2 f)"
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shows "(x +_f y = Err) = (~(? z:A. x <=_r z & y <=_r z))"
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proof -
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have plus_le_conv2:
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"\<And>r f z. \<lbrakk> z : err A; semilat (err A, r, f); OK x : err A; OK y : err A;
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OK x +_f OK y <=_r z\<rbrakk> \<Longrightarrow> OK x <=_r z \<and> OK y <=_r z"
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by (rule semilat.plus_le_conv [THEN iffD1])
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from prems show ?thesis
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apply (rule_tac iffI)
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apply clarify
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apply (drule OK_le_err_OK [THEN iffD2])
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apply (drule OK_le_err_OK [THEN iffD2])
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apply (drule semilat.lub[of _ _ _ "OK x" _ "OK y"])
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apply assumption
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apply assumption
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apply simp
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apply simp
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apply simp
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apply simp
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apply (case_tac "x +_f y")
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apply assumption
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apply (rename_tac "z")
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apply (subgoal_tac "OK z: err A")
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apply (frule plus_le_conv2)
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apply assumption
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apply simp
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apply blast
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apply simp
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apply (blast dest: semilat.orderI order_refl)
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apply blast
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apply (erule subst)
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apply (unfold semilat_def err_def closed_def)
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apply simp
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done
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qed
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lemma err_semilat_Product_esl:
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"\<And>L1 L2. \<lbrakk> err_semilat L1; err_semilat L2 \<rbrakk> \<Longrightarrow> err_semilat(Product.esl L1 L2)"
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apply (unfold esl_def Err.sl_def)
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apply (simp (no_asm_simp) only: split_tupled_all)
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apply simp
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apply (simp (no_asm) only: semilat_Def)
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apply (simp (no_asm_simp) only: semilat.closedI closed_lift2_sup)
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apply (simp (no_asm) only: unfold_lesub_err Err.le_def unfold_plussub_lift2 sup_def)
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apply (auto elim: semilat_le_err_OK1 semilat_le_err_OK2
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simp add: lift2_def split: err.split)
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apply (blast dest: semilat.orderI)
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apply (blast dest: semilat.orderI)
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apply (rule OK_le_err_OK [THEN iffD1])
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apply (erule subst, subst OK_lift2_OK [symmetric], rule semilat.lub)
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apply simp
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apply simp
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apply simp
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apply simp
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apply simp
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apply simp
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apply (rule OK_le_err_OK [THEN iffD1])
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apply (erule subst, subst OK_lift2_OK [symmetric], rule semilat.lub)
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apply simp
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apply simp
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apply simp
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apply simp
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apply simp
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apply simp
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done
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end
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