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theory Product = Err:(* Title: HOL/MicroJava/BV/Product.thy
ID: $Id: Product.html,v 1.1 2002/11/28 16:11:18 kleing Exp $
Author: Tobias Nipkow
Copyright 2000 TUM
Products as semilattices
*)
header {* \isaheader{Products as Semilattices} *}
theory Product = Err:
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 order_def)
apply simp
apply blast
done
lemma acc_le_prodI [intro!]:
"\<lbrakk> acc rA A; acc rB B \<rbrakk> \<Longrightarrow> acc (Product.le rA rB) (A <*> B)"
apply (unfold acc_def)
apply (rule wf_subset)
apply (erule wf_lex_prod)
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]:
"\<lbrakk> x:A; y:A; semilat(err A, Err.le r, lift2 f) \<rbrakk>
\<Longrightarrow> (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 [THEN iffD1])
case rule_context
thus ?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 _ _ _ "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 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 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)
apply (blast dest: semilat.orderI)
apply (rule OK_le_err_OK [THEN iffD1])
apply (erule subst, subst OK_lift2_OK [symmetric], rule semilat.lub)
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)
apply simp
apply simp
apply simp
apply simp
apply simp
apply simp
done
end
lemma unfold_lesub_prod:
p <=(rA,rB) q == Product.le rA rB p q
lemma le_prod_Pair_conv:
((a1, b1) <=(rA,rB) (a2, b2)) = (a1 <=_rA a2 & b1 <=_rB b2)
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)
lemma order_le_prod:
order (Product.le rA rB) = (order rA & order rB)
lemma acc_le_prodI:
[| acc rA A; acc rB B |] ==> acc (Product.le rA rB) (A <*> B)
lemma closed_lift2_sup:
[| closed (err A) (lift2 f); closed (err B) (lift2 g) |] ==> closed (err (A <*> B)) (lift2 (Product.sup f g))
lemma unfold_plussub_lift2:
e1 +_(lift2 f) e2 == lift2 f e1 e2
lemma plus_eq_Err_conv:
[| x : A; y : A; semilat (err A, Err.le r, lift2 f) |] ==> (x +_f y = Err) = (¬ (EX z:A. x <=_r z & y <=_r z))
lemma err_semilat_Product_esl:
[| semilat (sl L1); semilat (sl L2) |] ==> semilat (sl (Product.esl L1 L2))