# Theory Sprod

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theory Sprod
imports Cprod
uses [Sprod.ML]
begin

    (*  Title:      HOLCF/Sprod.thy
        ID:         $Id: Sprod.thy,v 1.15 2005/07/26 16:27:16 huffman Exp $
        Author:     Franz Regensburger and Brian Huffman

    Strict product with typedef.
    *)

    header {* The type of strict products *}

    theory Sprod
    imports Cprod
    begin

    defaultsort pcpo

    subsection {* Definition of strict product type *}

    pcpodef (Sprod)  ('a, 'b) "**" (infixr 20) =
            "{p::'a × 'b. p = ⊥ ∨ (cfst·p ≠ ⊥ ∧ csnd·p ≠ ⊥)}"
    by simp

    syntax (xsymbols)
      "**"          :: "[type, type] => type"        ("(_ ⊗/ _)" [21,20] 20)
    syntax (HTML output)
      "**"          :: "[type, type] => type"        ("(_ ⊗/ _)" [21,20] 20)

    lemma spair_lemma:
      "<strictify·(Λ b. a)·b, strictify·(Λ a. b)·a> ∈ Sprod"
    by (simp add: Sprod_def strictify_conv_if cpair_strict)

    subsection {* Definitions of constants *}

    consts
      sfst :: "('a ** 'b) -> 'a"
      ssnd :: "('a ** 'b) -> 'b"
      spair :: "'a -> 'b -> ('a ** 'b)"
      ssplit :: "('a -> 'b -> 'c) -> ('a ** 'b) -> 'c"

    defs
      sfst_def: "sfst ≡ Λ p. cfst·(Rep_Sprod p)"
      ssnd_def: "ssnd ≡ Λ p. csnd·(Rep_Sprod p)"
      spair_def: "spair ≡ Λ a b. Abs_Sprod
                    <strictify·(Λ b. a)·b, strictify·(Λ a. b)·a>"
      ssplit_def: "ssplit ≡ Λ f. strictify·(Λ p. f·(sfst·p)·(ssnd·p))"

    syntax
      "@stuple"     :: "['a, args] => 'a ** 'b"     ("(1'(:_,/ _:'))")

    translations
            "(:x, y, z:)"   == "(:x, (:y, z:):)"
            "(:x, y:)"      == "spair$x$y"

    subsection {* Case analysis *}

    lemma spair_Abs_Sprod:
      "(:a, b:) = Abs_Sprod <strictify·(Λ b. a)·b, strictify·(Λ a. b)·a>"
    apply (unfold spair_def)
    apply (simp add: cont_Abs_Sprod spair_lemma)
    done

    lemma Exh_Sprod2:
      "z = ⊥ ∨ (∃a b. z = (:a, b:) ∧ a ≠ ⊥ ∧ b ≠ ⊥)"
    apply (rule_tac x=z in Abs_Sprod_cases)
    apply (simp add: Sprod_def)
    apply (erule disjE)
    apply (simp add: Abs_Sprod_strict)
    apply (rule disjI2)
    apply (rule_tac x="cfst·y" in exI)
    apply (rule_tac x="csnd·y" in exI)
    apply (simp add: spair_Abs_Sprod Abs_Sprod_inject spair_lemma)
    apply (simp add: surjective_pairing_Cprod2)
    done

    lemma sprodE:
      "[|p = ⊥ ==> Q; !!x y. [|p = (:x, y:); x ≠ ⊥; y ≠ ⊥|] ==> Q|] ==> Q"
    by (cut_tac z=p in Exh_Sprod2, auto)

    subsection {* Properties of @{term spair} *}

    lemma spair_strict1 [simp]: "(:⊥, y:) = ⊥"
    by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)

    lemma spair_strict2 [simp]: "(:x, ⊥:) = ⊥"
    by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)

    lemma spair_strict: "x = ⊥ ∨ y = ⊥ ==> (:x, y:) = ⊥"
    by auto

    lemma spair_strict_rev: "(:x, y:) ≠ ⊥ ==> x ≠ ⊥ ∧ y ≠ ⊥"
    by (erule contrapos_np, auto)

    lemma spair_defined [simp]:
      "[|x ≠ ⊥; y ≠ ⊥|] ==> (:x, y:) ≠ ⊥"
    by (simp add: spair_Abs_Sprod Abs_Sprod_defined cpair_defined_iff Sprod_def)

    lemma spair_defined_rev: "(:x, y:) = ⊥ ==> x = ⊥ ∨ y = ⊥"
    by (erule contrapos_pp, simp)

    lemma spair_eq:
      "[|x ≠ ⊥; y ≠ ⊥|] ==> ((:x, y:) = (:a, b:)) = (x = a ∧ y = b)"
    apply (simp add: spair_Abs_Sprod)
    apply (simp add: Abs_Sprod_inject [OF _ spair_lemma] Sprod_def)
    apply (simp add: strictify_conv_if)
    done

    lemma spair_inject:
      "[|x ≠ ⊥; y ≠ ⊥; (:x, y:) = (:a, b:)|] ==> x = a ∧ y = b"
    by (rule spair_eq [THEN iffD1])

    lemma inst_sprod_pcpo2: "UU = (:UU,UU:)"
    by simp

    subsection {* Properties of @{term sfst} and @{term ssnd} *}

    lemma sfst_strict [simp]: "sfst·⊥ = ⊥"
    by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict)

    lemma ssnd_strict [simp]: "ssnd·⊥ = ⊥"
    by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict)

    lemma Rep_Sprod_spair:
      "Rep_Sprod (:a, b:) = <strictify·(Λ b. a)·b, strictify·(Λ a. b)·a>"
    apply (unfold spair_def)
    apply (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)
    done

    lemma sfst_spair [simp]: "y ≠ ⊥ ==> sfst·(:x, y:) = x"
    by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair)

    lemma ssnd_spair [simp]: "x ≠ ⊥ ==> ssnd·(:x, y:) = y"
    by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair)

    lemma sfst_defined_iff [simp]: "(sfst·p = ⊥) = (p = ⊥)"
    by (rule_tac p=p in sprodE, simp_all)

    lemma ssnd_defined_iff [simp]: "(ssnd·p = ⊥) = (p = ⊥)"
    by (rule_tac p=p in sprodE, simp_all)

    lemma sfst_defined: "p ≠ ⊥ ==> sfst·p ≠ ⊥"
    by simp

    lemma ssnd_defined: "p ≠ ⊥ ==> ssnd·p ≠ ⊥"
    by simp

    lemma surjective_pairing_Sprod2: "(:sfst·p, ssnd·p:) = p"
    by (rule_tac p=p in sprodE, simp_all)

    lemma less_sprod: "x \<sqsubseteq> y = (sfst·x \<sqsubseteq> sfst·y ∧ ssnd·x \<sqsubseteq> ssnd·y)"
    apply (simp add: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)
    apply (rule less_cprod)
    done

    lemma eq_sprod: "(x = y) = (sfst·x = sfst·y ∧ ssnd·x = ssnd·y)"
    by (auto simp add: po_eq_conv less_sprod)

    lemma spair_less:
      "[|x ≠ ⊥; y ≠ ⊥|] ==> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a ∧ y \<sqsubseteq> b)"
    apply (case_tac "a = ⊥")
    apply (simp add: eq_UU_iff [symmetric])
    apply (case_tac "b = ⊥")
    apply (simp add: eq_UU_iff [symmetric])
    apply (simp add: less_sprod)
    done

    subsection {* Properties of @{term ssplit} *}

    lemma ssplit1 [simp]: "ssplit·f·⊥ = ⊥"
    by (simp add: ssplit_def)

    lemma ssplit2 [simp]: "[|x ≠ ⊥; y ≠ ⊥|] ==> ssplit·f·(:x, y:) = f·x·y"
    by (simp add: ssplit_def)

    lemma ssplit3 [simp]: "ssplit·spair·z = z"
    by (rule_tac p=z in sprodE, simp_all)

    end

### Definition of strict product type

lemma spair_lemma:

      <strictify·(LAM b. a)·b, strictify·(LAM a. b)·a> ∈ Sprod

### Definitions of constants

### Case analysis

lemma spair_Abs_Sprod:

      (:a, b:) = Abs_Sprod <strictify·(LAM b. a)·b, strictify·(LAM a. b)·a>

lemma Exh_Sprod2:

      z = UU ∨ (∃a b. z = (:a, b:) ∧ a ≠ UU ∧ b ≠ UU)

lemma sprodE:

      [| p = UU ==> Q; !!x y. [| p = (:x, y:); x ≠ UU; y ≠ UU |] ==> Q |] ==> Q

### Properties of @{term spair}

lemma spair_strict1:

      (:UU, y:) = UU

lemma spair_strict2:

      (:x, UU:) = UU

lemma spair_strict:

      x = UU ∨ y = UU ==> (:x, y:) = UU

lemma spair_strict_rev:

      (:x, y:) ≠ UU ==> x ≠ UU ∧ y ≠ UU

lemma spair_defined:

      [| x ≠ UU; y ≠ UU |] ==> (:x, y:) ≠ UU

lemma spair_defined_rev:

      (:x, y:) = UU ==> x = UU ∨ y = UU

lemma spair_eq:

      [| x ≠ UU; y ≠ UU |] ==> ((:x, y:) = (:a, b:)) = (x = a ∧ y = b)

lemma spair_inject:

      [| x ≠ UU; y ≠ UU; (:x, y:) = (:a, b:) |] ==> x = a ∧ y = b

lemma inst_sprod_pcpo2:

      UU = (:UU, UU:)

### Properties of @{term sfst} and @{term ssnd}

lemma sfst_strict:

      sfst·UU = UU

lemma ssnd_strict:

      ssnd·UU = UU

lemma Rep_Sprod_spair:

      Rep_Sprod (:a, b:) = <strictify·(LAM b. a)·b, strictify·(LAM a. b)·a>

lemma sfst_spair:

      y ≠ UU ==> sfst·(:x, y:) = x

lemma ssnd_spair:

      x ≠ UU ==> ssnd·(:x, y:) = y

lemma sfst_defined_iff:

      (sfst·p = UU) = (p = UU)

lemma ssnd_defined_iff:

      (ssnd·p = UU) = (p = UU)

lemma sfst_defined:

      p ≠ UU ==> sfst·p ≠ UU

lemma ssnd_defined:

      p ≠ UU ==> ssnd·p ≠ UU

lemma surjective_pairing_Sprod2:

      (:sfst·p, ssnd·p:) = p

lemma less_sprod:

      x << y = (sfst·x << sfst·y ∧ ssnd·x << ssnd·y)

lemma eq_sprod:

      (x = y) = (sfst·x = sfst·y ∧ ssnd·x = ssnd·y)

lemma spair_less:

      [| x ≠ UU; y ≠ UU |] ==> (:x, y:) << (:a, b:) = (x << a ∧ y << b)

### Properties of @{term ssplit}

lemma ssplit1:

      ssplit·f·UU = UU

lemma ssplit2:

      [| x ≠ UU; y ≠ UU |] ==> ssplit·f·(:x, y:) = f·x·y

lemma ssplit3:

      ssplit·spair·z = z