# Theory Product

theory Product
imports Err
```(*  Title:      HOL/MicroJava/DFA/Product.thy
Author:     Tobias Nipkow
*)

section ‹Products as Semilattices›

theory Product
imports Err
begin

definition le :: "'a ord ⇒ 'b ord ⇒ ('a * 'b) ord" where
"le rA rB == %(a,b) (a',b'). a <=_rA a' & b <=_rB b'"

definition sup :: "'a ebinop ⇒ 'b ebinop ⇒ ('a * 'b)ebinop" where
"sup f g == %(a1,b1)(a2,b2). Err.sup Pair (a1 +_f a2) (b1 +_g b2)"

definition esl :: "'a esl ⇒ 'b esl ⇒ ('a * 'b ) esl" where
"esl == %(A,rA,fA) (B,rB,fB). (A × B, le rA rB, sup fA fB)"

abbreviation
lesubprod_sntax :: "'a * 'b ⇒ 'a ord ⇒ 'b ord ⇒ 'a * 'b ⇒ bool"
("(_ /<='(_,_') _)" [50, 0, 0, 51] 50)
where "p <=(rA,rB) q == p <=_(le rA rB) q"

lemma unfold_lesub_prod:
"p <=(rA,rB) q == le rA rB p q"

lemma le_prod_Pair_conv [iff]:
"((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)"
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 meson
done

lemma acc_le_prodI [intro!]:
"⟦ acc r⇩A; acc r⇩B ⟧ ⟹ acc(Product.le r⇩A r⇩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:
"⟦ closed (err A) (lift2 f); closed (err B) (lift2 g) ⟧ ⟹
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"

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:
"⋀r f z. ⟦ z ∈ err A; semilat (err A, r, f); OK x ∈ err A; OK y ∈ err A;
OK x +_f OK y <=_r z⟧ ⟹ OK x <=_r z ∧ OK y <=_r z"
by (rule Semilat.plus_le_conv [OF Semilat.intro, THEN iffD1])
from assms 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:
"⋀L1 L2. ⟦ err_semilat L1; err_semilat L2 ⟧ ⟹ 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
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
```