# Theory Product

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theory Product
imports Err
begin

    (*  Title:      HOL/MicroJava/BV/Product.thy
        ID:         $Id: Product.thy,v 1.9 2005/06/17 14:13:08 haftmann Exp $
        Author:     Tobias Nipkow
        Copyright   2000 TUM

    Products as semilattices
    *)

    header {* \isaheader{Products as Semilattices} *}

    theory Product imports Err begin

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

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

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

    syntax "@lesubprod" :: "'a*'b => 'a ord => 'b ord => 'b => 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!]:
      "[| acc rA; acc rB |] ==> acc(Product.le rA rB)"
    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"
      by (simp add: plussub_def)

    lemma plus_eq_Err_conv [simp]:
      "[| x:A; y:A; semilat(err A, Err.le r, lift2 f) |]
      ==> (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 [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:
      "!!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 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:

      ((a1.0, b1.0) <=(rA,rB) (a2.0, b2.0)) = (a1.0 <=_rA a2.0 ∧ b1.0 <=_rB b2.0)

lemma less_prod_Pair_conv:

      ((a1.0, b1.0) <_(Product.le rA rB) (a2.0, b2.0)) =
      (a1.0 <_rA a2.0 ∧ b1.0 <=_rB b2.0 ∨ a1.0 <=_rA a2.0 ∧ b1.0 <_rB b2.0)

lemma order_le_prod:

      order (Product.le rA rB) = (order rA ∧ order rB)

lemma acc_le_prodI:

      [| acc rA; acc rB |] ==> acc (Product.le rA rB)

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:

      e1.0 +_(lift2 f) e2.0 == lift2 f e1.0 e2.0

lemma plus_eq_Err_conv:

      [| x ∈ A; y ∈ A; semilat (err A, Err.le r, lift2 f) |]
      ==> (x +_f y = Err) = (¬ (∃z∈A. x <=_r z ∧ y <=_r z))

lemma err_semilat_Product_esl:

      [| semilat (sl L1.0); semilat (sl L2.0) |]
      ==> semilat (sl (Product.esl L1.0 L2.0))