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_all
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_Sprod:
"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_Sprod, 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
proof
fix x y :: "'a ⊗ 'b"
assume "x \<sqsubseteq> y" thus "x = ⊥ ∨ x = y"
apply (induct x, simp)
apply (induct y, simp)
apply (simp add: spair_less_iff flat_less_iff)
done
qed
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 (approx :: nat => 'a ⊗ 'b -> 'a ⊗ 'b)"
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, case_tac x)
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