(*  Title:      HOL/MicroJava/DFA/Product.thy
    Author:     Tobias Nipkow
    Copyright   2000 TUM
*)

section \<open>Products as Semilattices\<close>

theory Product
imports Err
begin

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

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

definition esl :: "'a esl \<Rightarrow> 'b esl \<Rightarrow> ('a * 'b ) esl" where
"esl == %(A,rA,fA) (B,rB,fB). (A \<times> B, le rA rB, sup fA fB)"

abbreviation
  lesubprod_sntax :: "'a * 'b \<Rightarrow> 'a ord \<Rightarrow> 'b ord \<Rightarrow> 'a * 'b \<Rightarrow> 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"
  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 meson
done

lemma acc_le_prodI [intro!]:
  "\<lbrakk> acc r\<^sub>A; acc r\<^sub>B \<rbrakk> \<Longrightarrow> acc(Product.le r\<^sub>A r\<^sub>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 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 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:
  "\<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