Theory Sprod

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theory Sprod
imports Cprod
begin

(*  Title:      HOLCF/Sprod.thy
    ID:         $Id: Sprod.thy,v 1.25 2008/05/19 21:49:21 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

instance "**" :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po
by (rule typedef_finite_po [OF type_definition_Sprod])

instance "**" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
by (rule typedef_chfin [OF type_definition_Sprod less_Sprod_def])

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)

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 Rep_Sprod_spair:
  "Rep_Sprod (:a, b:) = <strictify·(Λ b. a)·b, strictify·(Λ a. b)·a>"
unfolding spair_def
by (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)

lemmas Rep_Sprod_simps =
  Rep_Sprod_inject [symmetric] less_Sprod_def
  Rep_Sprod_strict Rep_Sprod_spair

lemma Exh_Sprod2:
  "z = ⊥ ∨ (∃a b. z = (:a, b:) ∧ a ≠ ⊥ ∧ b ≠ ⊥)"
apply (insert Rep_Sprod [of z])
apply (simp add: Rep_Sprod_simps eq_cprod)
apply (simp add: Sprod_def)
apply (erule disjE, simp)
apply (simp add: strictify_conv_if)
apply fast
done

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

lemma sprod_induct [induct type: **]:
  "[|P ⊥; !!x y. [|x ≠ ⊥; y ≠ ⊥|] ==> P (:x, y:)|] ==> P x"
by (cases x, simp_all)

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

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

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

lemma spair_strict_iff [simp]: "((:x, y:) = ⊥) = (x = ⊥ ∨ y = ⊥)"
by (simp add: Rep_Sprod_simps strictify_conv_if)

lemma spair_less_iff:
  "((:a, b:) \<sqsubseteq> (:c, d:)) = (a = ⊥ ∨ b = ⊥ ∨ (a \<sqsubseteq> c ∧ b \<sqsubseteq> d))"
by (simp add: Rep_Sprod_simps strictify_conv_if)

lemma spair_eq_iff:
  "((:a, b:) = (:c, d:)) =
    (a = c ∧ b = d ∨ (a = ⊥ ∨ b = ⊥) ∧ (c = ⊥ ∨ d = ⊥))"
by (simp add: Rep_Sprod_simps strictify_conv_if)

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

lemma spair_strict_rev: "(:x, y:) ≠ ⊥ ==> x ≠ ⊥ ∧ y ≠ ⊥"
by simp

lemma spair_defined: "[|x ≠ ⊥; y ≠ ⊥|] ==> (:x, y:) ≠ ⊥"
by simp

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

lemma spair_eq:
  "[|x ≠ ⊥; y ≠ ⊥|] ==> ((:x, y:) = (:a, b:)) = (x = a ∧ y = b)"
by (simp add: spair_eq_iff)

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 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 (cases p, simp_all)

lemma ssnd_defined_iff [simp]: "(ssnd·p = ⊥) = (p = ⊥)"
by (cases p, 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 (cases p, 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 (cases "a = ⊥", simp)
apply (cases "b = ⊥", simp)
apply (simp add: less_sprod)
done

lemma sfst_less_iff: "sfst·x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd·x:)"
apply (cases "x = ⊥", simp, cases "y = ⊥", simp)
apply (simp add: less_sprod)
done

lemma ssnd_less_iff: "ssnd·x \<sqsubseteq> y = x \<sqsubseteq> (:sfst·x, y:)"
apply (cases "x = ⊥", simp, cases "y = ⊥", simp)
apply (simp add: less_sprod)
done

subsection {* Compactness *}

lemma compact_sfst: "compact x ==> compact (sfst·x)"
by (rule compactI, simp add: sfst_less_iff)

lemma compact_ssnd: "compact x ==> compact (ssnd·x)"
by (rule compactI, simp add: ssnd_less_iff)

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

lemma compact_spair_iff:
  "compact (:x, y:) = (x = ⊥ ∨ y = ⊥ ∨ (compact x ∧ compact y))"
apply (safe elim!: compact_spair)
apply (drule compact_sfst, simp)
apply (drule compact_ssnd, simp)
apply simp
apply simp
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 (cases z, simp_all)

subsection {* Strict product preserves flatness *}

instance "**" :: (flat, flat) flat
apply (intro_classes, clarify)
apply (rule_tac p=x in sprodE, simp)
apply (rule_tac p=y in sprodE, simp)
apply (simp add: flat_less_iff spair_less)
done

subsection {* Strict product is a bifinite domain *}

instantiation "**" :: (bifinite, bifinite) bifinite
begin

definition
  approx_sprod_def:
    "approx = (λn. Λ(:x, y:). (:approx n·x, approx n·y:))"

instance proof
  fix i :: nat and x :: "'a ⊗ 'b"
  show "chain (λi. approx i·x)"
    unfolding approx_sprod_def by simp
  show "(\<Squnion>i. approx i·x) = x"
    unfolding approx_sprod_def
    by (simp add: lub_distribs eta_cfun)
  show "approx i·(approx i·x) = approx i·x"
    unfolding approx_sprod_def
    by (simp add: ssplit_def strictify_conv_if)
  have "Rep_Sprod ` {x::'a ⊗ 'b. approx i·x = x} ⊆ {x. approx i·x = x}"
    unfolding approx_sprod_def
    apply (clarify, rule_tac p=x in sprodE)
     apply (simp add: Rep_Sprod_strict)
    apply (simp add: Rep_Sprod_spair spair_eq_iff)
    done
  hence "finite (Rep_Sprod ` {x::'a ⊗ 'b. approx i·x = x})"
    using finite_fixes_approx by (rule finite_subset)
  thus "finite {x::'a ⊗ 'b. approx i·x = x}"
    by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject)
qed

end

lemma approx_spair [simp]:
  "approx i·(:x, y:) = (:approx i·x, approx i·y:)"
unfolding approx_sprod_def
by (simp add: ssplit_def strictify_conv_if)

end

Definition of strict product type

lemma spair_lemma:

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

Definitions of constants

Case analysis

lemma Rep_Sprod_spair:

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

lemma Rep_Sprod_simps:

  (x = y) = (Rep_Sprod x = Rep_Sprod y)
  op << == λx y. Rep_Sprod x << Rep_Sprod y
  Rep_Sprod UU = UU
  Rep_Sprod (:a, b:) = <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

lemma sprod_induct:

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

Properties of @{term spair}

lemma spair_strict1:

  (:UU, y:) = UU

lemma spair_strict2:

  (:x, UU:) = UU

lemma spair_strict_iff:

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

lemma spair_less_iff:

  (:a, b:) << (:c, d:) = (a = UUb = UUa << cb << d)

lemma spair_eq_iff:

  ((:a, b:) = (:c, d:)) = (a = cb = d ∨ (a = UUb = UU) ∧ (c = UUd = 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:)

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)

lemma sfst_less_iff:

  sfst·x << y = x << (:y, ssnd·x:)

lemma ssnd_less_iff:

  ssnd·x << y = x << (:sfst·x, y:)

Compactness

lemma compact_sfst:

  compact x ==> compact (sfst·x)

lemma compact_ssnd:

  compact x ==> compact (ssnd·x)

lemma compact_spair:

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

lemma compact_spair_iff:

  compact (:x, y:) = (x = UUy = UUcompact xcompact y)

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

Strict product preserves flatness

Strict product is a bifinite domain

lemma approx_spair:

  approx i·(:x, y:) = (:approx i·x, approx i·y:)