Theory Sprod

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theory Sprod
imports Cprod
begin

(*  Title:      HOLCF/Sprod.thy
    ID:         $Id: Sprod.thy,v 1.20 2007/10/21 14:27:43 wenzelm 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 *}

definition
  sfst :: "('a ** 'b) -> 'a" where
  "sfst = (Λ p. cfst·(Rep_Sprod p))"

definition
  ssnd :: "('a ** 'b) -> 'b" where
  "ssnd = (Λ p. csnd·(Rep_Sprod p))"

definition
  spair :: "'a -> 'b -> ('a ** 'b)" where
  "spair = (Λ a b. Abs_Sprod
             <strictify·(Λ b. a)·b, strictify·(Λ a. b)·a>)"

definition
  ssplit :: "('a -> 'b -> 'c) -> ('a ** 'b) -> 'c" where
  "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:)"    == "CONST spair·x·y"

translations
  "Λ(CONST spair·x·y). t" == "CONST ssplit·(Λ x y. t)"


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 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

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 compact_spair: "[|compact x; compact y|] ==> compact (:x, y:)"
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)

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 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. ab, strictify·(LAM a. ba> ∈ Sprod

Definitions of constants

Case analysis

lemma spair_Abs_Sprod:

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

lemma Exh_Sprod2:

  z = UU ∨ (∃a b. z = (:a, b:) ∧ a  UUb  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 = UUy = UU ==> (:x, y:) = UU

lemma spair_strict_rev:

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

lemma spair_defined:

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

lemma spair_defined_rev:

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

lemma spair_eq:

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

lemma spair_inject:

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

lemma inst_sprod_pcpo2:

  UU = (:UU, UU:)

lemma Rep_Sprod_spair:

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

lemma compact_spair:

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

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

lemma sfst_strict:

  sfst·UU = UU

lemma ssnd_strict:

  ssnd·UU = UU

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·yssnd·x << ssnd·y)

lemma eq_sprod:

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

lemma spair_less:

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