header {* The type of strict sums *}
theory Ssum
imports Cprod Tr
begin
defaultsort pcpo
subsection {* Definition of strict sum type *}
pcpodef (Ssum) ('a, 'b) "++" (infixr "++" 10) =
"{p :: tr × ('a × 'b).
(cfst·p \<sqsubseteq> TT <-> csnd·(csnd·p) = ⊥) ∧
(cfst·p \<sqsubseteq> FF <-> cfst·(csnd·p) = ⊥)}"
by simp_all
instance "++" :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po
by (rule typedef_finite_po [OF type_definition_Ssum])
instance "++" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
by (rule typedef_chfin [OF type_definition_Ssum less_Ssum_def])
syntax (xsymbols)
"++" :: "[type, type] => type" ("(_ ⊕/ _)" [21, 20] 20)
syntax (HTML output)
"++" :: "[type, type] => type" ("(_ ⊕/ _)" [21, 20] 20)
subsection {* Definitions of constructors *}
definition
sinl :: "'a -> ('a ++ 'b)" where
"sinl = (Λ a. Abs_Ssum <strictify·(Λ _. TT)·a, a, ⊥>)"
definition
sinr :: "'b -> ('a ++ 'b)" where
"sinr = (Λ b. Abs_Ssum <strictify·(Λ _. FF)·b, ⊥, b>)"
lemma sinl_Ssum: "<strictify·(Λ _. TT)·a, a, ⊥> ∈ Ssum"
by (simp add: Ssum_def strictify_conv_if)
lemma sinr_Ssum: "<strictify·(Λ _. FF)·b, ⊥, b> ∈ Ssum"
by (simp add: Ssum_def strictify_conv_if)
lemma sinl_Abs_Ssum: "sinl·a = Abs_Ssum <strictify·(Λ _. TT)·a, a, ⊥>"
by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum)
lemma sinr_Abs_Ssum: "sinr·b = Abs_Ssum <strictify·(Λ _. FF)·b, ⊥, b>"
by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum)
lemma Rep_Ssum_sinl: "Rep_Ssum (sinl·a) = <strictify·(Λ _. TT)·a, a, ⊥>"
by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum)
lemma Rep_Ssum_sinr: "Rep_Ssum (sinr·b) = <strictify·(Λ _. FF)·b, ⊥, b>"
by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum)
subsection {* Properties of @{term sinl} and @{term sinr} *}
text {* Ordering *}
lemma sinl_less [simp]: "(sinl·x \<sqsubseteq> sinl·y) = (x \<sqsubseteq> y)"
by (simp add: less_Ssum_def Rep_Ssum_sinl strictify_conv_if)
lemma sinr_less [simp]: "(sinr·x \<sqsubseteq> sinr·y) = (x \<sqsubseteq> y)"
by (simp add: less_Ssum_def Rep_Ssum_sinr strictify_conv_if)
lemma sinl_less_sinr [simp]: "(sinl·x \<sqsubseteq> sinr·y) = (x = ⊥)"
by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
lemma sinr_less_sinl [simp]: "(sinr·x \<sqsubseteq> sinl·y) = (x = ⊥)"
by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
text {* Equality *}
lemma sinl_eq [simp]: "(sinl·x = sinl·y) = (x = y)"
by (simp add: po_eq_conv)
lemma sinr_eq [simp]: "(sinr·x = sinr·y) = (x = y)"
by (simp add: po_eq_conv)
lemma sinl_eq_sinr [simp]: "(sinl·x = sinr·y) = (x = ⊥ ∧ y = ⊥)"
by (subst po_eq_conv, simp)
lemma sinr_eq_sinl [simp]: "(sinr·x = sinl·y) = (x = ⊥ ∧ y = ⊥)"
by (subst po_eq_conv, simp)
lemma sinl_inject: "sinl·x = sinl·y ==> x = y"
by (rule sinl_eq [THEN iffD1])
lemma sinr_inject: "sinr·x = sinr·y ==> x = y"
by (rule sinr_eq [THEN iffD1])
text {* Strictness *}
lemma sinl_strict [simp]: "sinl·⊥ = ⊥"
by (simp add: sinl_Abs_Ssum Abs_Ssum_strict)
lemma sinr_strict [simp]: "sinr·⊥ = ⊥"
by (simp add: sinr_Abs_Ssum Abs_Ssum_strict)
lemma sinl_defined_iff [simp]: "(sinl·x = ⊥) = (x = ⊥)"
by (cut_tac sinl_eq [of "x" "⊥"], simp)
lemma sinr_defined_iff [simp]: "(sinr·x = ⊥) = (x = ⊥)"
by (cut_tac sinr_eq [of "x" "⊥"], simp)
lemma sinl_defined [intro!]: "x ≠ ⊥ ==> sinl·x ≠ ⊥"
by simp
lemma sinr_defined [intro!]: "x ≠ ⊥ ==> sinr·x ≠ ⊥"
by simp
text {* Compactness *}
lemma compact_sinl: "compact x ==> compact (sinl·x)"
by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if)
lemma compact_sinr: "compact x ==> compact (sinr·x)"
by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if)
lemma compact_sinlD: "compact (sinl·x) ==> compact x"
unfolding compact_def
by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinl]], simp)
lemma compact_sinrD: "compact (sinr·x) ==> compact x"
unfolding compact_def
by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinr]], simp)
lemma compact_sinl_iff [simp]: "compact (sinl·x) = compact x"
by (safe elim!: compact_sinl compact_sinlD)
lemma compact_sinr_iff [simp]: "compact (sinr·x) = compact x"
by (safe elim!: compact_sinr compact_sinrD)
subsection {* Case analysis *}
lemma Exh_Ssum:
"z = ⊥ ∨ (∃a. z = sinl·a ∧ a ≠ ⊥) ∨ (∃b. z = sinr·b ∧ b ≠ ⊥)"
apply (rule_tac x=z in Abs_Ssum_induct)
apply (rule_tac p=y in cprodE, rename_tac t x)
apply (rule_tac p=x in cprodE, rename_tac a b)
apply (rule_tac p=t in trE)
apply (rule disjI1)
apply (simp add: Ssum_def cpair_strict Abs_Ssum_strict)
apply (rule disjI2, rule disjI1, rule_tac x=a in exI)
apply (simp add: sinl_Abs_Ssum Ssum_def)
apply (rule disjI2, rule disjI2, rule_tac x=b in exI)
apply (simp add: sinr_Abs_Ssum Ssum_def)
done
lemma ssumE [cases type: ++]:
"[|p = ⊥ ==> Q;
!!x. [|p = sinl·x; x ≠ ⊥|] ==> Q;
!!y. [|p = sinr·y; y ≠ ⊥|] ==> Q|] ==> Q"
by (cut_tac z=p in Exh_Ssum, auto)
lemma ssum_induct [induct type: ++]:
"[|P ⊥;
!!x. x ≠ ⊥ ==> P (sinl·x);
!!y. y ≠ ⊥ ==> P (sinr·y)|] ==> P x"
by (cases x, simp_all)
lemma ssumE2:
"[|!!x. p = sinl·x ==> Q; !!y. p = sinr·y ==> Q|] ==> Q"
by (cases p, simp only: sinl_strict [symmetric], simp, simp)
lemma less_sinlD: "p \<sqsubseteq> sinl·x ==> ∃y. p = sinl·y ∧ y \<sqsubseteq> x"
by (cases p, rule_tac x="⊥" in exI, simp_all)
lemma less_sinrD: "p \<sqsubseteq> sinr·x ==> ∃y. p = sinr·y ∧ y \<sqsubseteq> x"
by (cases p, rule_tac x="⊥" in exI, simp_all)
subsection {* Case analysis combinator *}
definition
sscase :: "('a -> 'c) -> ('b -> 'c) -> ('a ++ 'b) -> 'c" where
"sscase = (Λ f g s. (Λ<t, x, y>. If t then f·x else g·y fi)·(Rep_Ssum s))"
translations
"case s of XCONST sinl·x => t1 | XCONST sinr·y => t2" == "CONST sscase·(Λ x. t1)·(Λ y. t2)·s"
translations
"Λ(XCONST sinl·x). t" == "CONST sscase·(Λ x. t)·⊥"
"Λ(XCONST sinr·y). t" == "CONST sscase·⊥·(Λ y. t)"
lemma beta_sscase:
"sscase·f·g·s = (Λ<t, x, y>. If t then f·x else g·y fi)·(Rep_Ssum s)"
unfolding sscase_def by (simp add: cont_Rep_Ssum cont2cont_LAM)
lemma sscase1 [simp]: "sscase·f·g·⊥ = ⊥"
unfolding beta_sscase by (simp add: Rep_Ssum_strict)
lemma sscase2 [simp]: "x ≠ ⊥ ==> sscase·f·g·(sinl·x) = f·x"
unfolding beta_sscase by (simp add: Rep_Ssum_sinl)
lemma sscase3 [simp]: "y ≠ ⊥ ==> sscase·f·g·(sinr·y) = g·y"
unfolding beta_sscase by (simp add: Rep_Ssum_sinr)
lemma sscase4 [simp]: "sscase·sinl·sinr·z = z"
by (cases z, simp_all)
subsection {* Strict sum preserves flatness *}
instance "++" :: (flat, flat) flat
apply (intro_classes, clarify)
apply (rule_tac p=x in ssumE, simp)
apply (rule_tac p=y in ssumE, simp_all add: flat_less_iff)
apply (rule_tac p=y in ssumE, simp_all add: flat_less_iff)
done
subsection {* Strict sum is a bifinite domain *}
instantiation "++" :: (bifinite, bifinite) bifinite
begin
definition
approx_ssum_def:
"approx = (λn. sscase·(Λ x. sinl·(approx n·x))·(Λ y. sinr·(approx n·y)))"
lemma approx_sinl [simp]: "approx i·(sinl·x) = sinl·(approx i·x)"
unfolding approx_ssum_def by (cases "x = ⊥") simp_all
lemma approx_sinr [simp]: "approx i·(sinr·x) = sinr·(approx i·x)"
unfolding approx_ssum_def by (cases "x = ⊥") simp_all
instance proof
fix i :: nat and x :: "'a ⊕ 'b"
show "chain (approx :: nat => 'a ⊕ 'b -> 'a ⊕ 'b)"
unfolding approx_ssum_def by simp
show "(\<Squnion>i. approx i·x) = x"
unfolding approx_ssum_def
by (simp add: lub_distribs eta_cfun)
show "approx i·(approx i·x) = approx i·x"
by (cases x, simp add: approx_ssum_def, simp, simp)
have "{x::'a ⊕ 'b. approx i·x = x} ⊆
(λx. sinl·x) ` {x. approx i·x = x} ∪
(λx. sinr·x) ` {x. approx i·x = x}"
by (rule subsetI, case_tac x rule: ssumE2, simp, simp)
thus "finite {x::'a ⊕ 'b. approx i·x = x}"
by (rule finite_subset,
intro finite_UnI finite_imageI finite_fixes_approx)
qed
end
end