(* Title: HOLCF/Ssum.thy ID: $Id: Ssum.thy,v 1.20 2007/10/21 12:21:49 wenzelm Exp $ Author: Franz Regensburger and Brian Huffman Strict sum with typedef. *) header {* The type of strict sums *} theory Ssum imports Cprod begin defaultsort pcpo subsection {* Definition of strict sum type *} pcpodef (Ssum) ('a, 'b) "++" (infixr "++" 10) = "{p::'a × 'b. cfst·p = ⊥ ∨ csnd·p = ⊥}" by simp 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 <a, ⊥>)" definition sinr :: "'b -> ('a ++ 'b)" where "sinr = (Λ b. Abs_Ssum <⊥, b>)" subsection {* Properties of @{term sinl} and @{term sinr} *} lemma sinl_Abs_Ssum: "sinl·a = Abs_Ssum <a, ⊥>" by (unfold sinl_def, simp add: cont_Abs_Ssum Ssum_def) lemma sinr_Abs_Ssum: "sinr·b = Abs_Ssum <⊥, b>" by (unfold sinr_def, simp add: cont_Abs_Ssum Ssum_def) lemma Rep_Ssum_sinl: "Rep_Ssum (sinl·a) = <a, ⊥>" by (unfold sinl_def, simp add: cont_Abs_Ssum Abs_Ssum_inverse Ssum_def) lemma Rep_Ssum_sinr: "Rep_Ssum (sinr·b) = <⊥, b>" by (unfold sinr_def, simp add: cont_Abs_Ssum Abs_Ssum_inverse Ssum_def) lemma compact_sinl [simp]: "compact x ==> compact (sinl·x)" by (rule compact_Ssum, simp add: Rep_Ssum_sinl) lemma compact_sinr [simp]: "compact x ==> compact (sinr·x)" by (rule compact_Ssum, simp add: Rep_Ssum_sinr) lemma sinl_strict [simp]: "sinl·⊥ = ⊥" by (simp add: sinl_Abs_Ssum Abs_Ssum_strict cpair_strict) lemma sinr_strict [simp]: "sinr·⊥ = ⊥" by (simp add: sinr_Abs_Ssum Abs_Ssum_strict cpair_strict) lemma sinl_eq [simp]: "(sinl·x = sinl·y) = (x = y)" by (simp add: sinl_Abs_Ssum Abs_Ssum_inject Ssum_def) lemma sinr_eq [simp]: "(sinr·x = sinr·y) = (x = y)" by (simp add: sinr_Abs_Ssum Abs_Ssum_inject Ssum_def) 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]) 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 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) apply (simp add: sinl_Abs_Ssum sinr_Abs_Ssum) apply (simp add: Abs_Ssum_inject Ssum_def) apply (auto simp add: cpair_strict Abs_Ssum_strict) done lemma ssumE: "[|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 ssumE2: "[|!!x. p = sinl·x ==> Q; !!y. p = sinr·y ==> Q|] ==> Q" apply (rule_tac p=p in ssumE) apply (simp only: sinl_strict [symmetric]) apply simp apply simp done subsection {* Ordering properties of @{term sinl} and @{term sinr} *} lemma sinl_less [simp]: "(sinl·x \<sqsubseteq> sinl·y) = (x \<sqsubseteq> y)" by (simp add: less_Ssum_def Rep_Ssum_sinl) lemma sinr_less [simp]: "(sinr·x \<sqsubseteq> sinr·y) = (x \<sqsubseteq> y)" by (simp add: less_Ssum_def Rep_Ssum_sinr) lemma sinl_less_sinr [simp]: "(sinl·x \<sqsubseteq> sinr·y) = (x = ⊥)" by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr) lemma sinr_less_sinl [simp]: "(sinr·x \<sqsubseteq> sinl·y) = (x = ⊥)" by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr) 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) subsection {* Chains of strict sums *} lemma less_sinlD: "p \<sqsubseteq> sinl·x ==> ∃y. p = sinl·y ∧ y \<sqsubseteq> x" apply (rule_tac p=p in ssumE) apply (rule_tac x="⊥" in exI, simp) apply simp apply simp done lemma less_sinrD: "p \<sqsubseteq> sinr·x ==> ∃y. p = sinr·y ∧ y \<sqsubseteq> x" apply (rule_tac p=p in ssumE) apply (rule_tac x="⊥" in exI, simp) apply simp apply simp done lemma ssum_chain_lemma: "chain Y ==> (∃A. chain A ∧ Y = (λi. sinl·(A i))) ∨ (∃B. chain B ∧ Y = (λi. sinr·(B i)))" apply (rule_tac p="lub (range Y)" in ssumE2) apply (rule disjI1) apply (rule_tac x="λi. cfst·(Rep_Ssum (Y i))" in exI) apply (rule conjI) apply (rule chain_monofun) apply (erule cont_Rep_Ssum [THEN ch2ch_cont]) apply (rule ext, drule_tac x=i in is_ub_thelub, simp) apply (drule less_sinlD, clarify) apply (simp add: Rep_Ssum_sinl) apply (rule disjI2) apply (rule_tac x="λi. csnd·(Rep_Ssum (Y i))" in exI) apply (rule conjI) apply (rule chain_monofun) apply (erule cont_Rep_Ssum [THEN ch2ch_cont]) apply (rule ext, drule_tac x=i in is_ub_thelub, simp) apply (drule less_sinrD, clarify) apply (simp add: Rep_Ssum_sinr) done subsection {* Definitions of constants *} definition Iwhen :: "['a -> 'c, 'b -> 'c, 'a ++ 'b] => 'c" where "Iwhen = (λf g s. if cfst·(Rep_Ssum s) ≠ ⊥ then f·(cfst·(Rep_Ssum s)) else if csnd·(Rep_Ssum s) ≠ ⊥ then g·(csnd·(Rep_Ssum s)) else ⊥)" text {* rewrites for @{term Iwhen} *} lemma Iwhen1 [simp]: "Iwhen f g ⊥ = ⊥" by (simp add: Iwhen_def Rep_Ssum_strict) lemma Iwhen2 [simp]: "x ≠ ⊥ ==> Iwhen f g (sinl·x) = f·x" by (simp add: Iwhen_def Rep_Ssum_sinl) lemma Iwhen3 [simp]: "y ≠ ⊥ ==> Iwhen f g (sinr·y) = g·y" by (simp add: Iwhen_def Rep_Ssum_sinr) lemma Iwhen4: "Iwhen f g (sinl·x) = strictify·f·x" by (simp add: strictify_conv_if) lemma Iwhen5: "Iwhen f g (sinr·y) = strictify·g·y" by (simp add: strictify_conv_if) subsection {* Continuity of @{term Iwhen} *} text {* @{term Iwhen} is continuous in all arguments *} lemma cont_Iwhen1: "cont (λf. Iwhen f g s)" by (rule_tac p=s in ssumE, simp_all) lemma cont_Iwhen2: "cont (λg. Iwhen f g s)" by (rule_tac p=s in ssumE, simp_all) lemma cont_Iwhen3: "cont (λs. Iwhen f g s)" apply (rule contI) apply (drule ssum_chain_lemma, safe) apply (simp add: contlub_cfun_arg [symmetric]) apply (simp add: Iwhen4 cont_cfun_arg) apply (simp add: contlub_cfun_arg [symmetric]) apply (simp add: Iwhen5 cont_cfun_arg) done subsection {* Continuous versions of constants *} definition sscase :: "('a -> 'c) -> ('b -> 'c) -> ('a ++ 'b) -> 'c" where "sscase = (Λ f g s. Iwhen f g s)" translations "case s of CONST sinl·x => t1 | CONST sinr·y => t2" == "CONST sscase·(Λ x. t1)·(Λ y. t2)·s" translations "Λ(CONST sinl·x). t" == "CONST sscase·(Λ x. t)·⊥" "Λ(CONST sinr·y). t" == "CONST sscase·⊥·(Λ y. t)" text {* continuous versions of lemmas for @{term sscase} *} lemma beta_sscase: "sscase·f·g·s = Iwhen f g s" by (simp add: sscase_def cont_Iwhen1 cont_Iwhen2 cont_Iwhen3) lemma sscase1 [simp]: "sscase·f·g·⊥ = ⊥" by (simp add: beta_sscase) lemma sscase2 [simp]: "x ≠ ⊥ ==> sscase·f·g·(sinl·x) = f·x" by (simp add: beta_sscase) lemma sscase3 [simp]: "y ≠ ⊥ ==> sscase·f·g·(sinr·y) = g·y" by (simp add: beta_sscase) lemma sscase4 [simp]: "sscase·sinl·sinr·z = z" by (rule_tac p=z in ssumE, simp_all) end
lemma sinl_Abs_Ssum:
sinl·a = Abs_Ssum <a, UU>
lemma sinr_Abs_Ssum:
sinr·b = Abs_Ssum <UU, b>
lemma Rep_Ssum_sinl:
Rep_Ssum (sinl·a) = <a, UU>
lemma Rep_Ssum_sinr:
Rep_Ssum (sinr·b) = <UU, b>
lemma compact_sinl:
compact x ==> compact (sinl·x)
lemma compact_sinr:
compact x ==> compact (sinr·x)
lemma sinl_strict:
sinl·UU = UU
lemma sinr_strict:
sinr·UU = UU
lemma sinl_eq:
(sinl·x = sinl·y) = (x = y)
lemma sinr_eq:
(sinr·x = sinr·y) = (x = y)
lemma sinl_inject:
sinl·x = sinl·y ==> x = y
lemma sinr_inject:
sinr·x = sinr·y ==> x = y
lemma sinl_defined_iff:
(sinl·x = UU) = (x = UU)
lemma sinr_defined_iff:
(sinr·x = UU) = (x = UU)
lemma sinl_defined:
x ≠ UU ==> sinl·x ≠ UU
lemma sinr_defined:
x ≠ UU ==> sinr·x ≠ UU
lemma Exh_Ssum:
z = UU ∨ (∃a. z = sinl·a ∧ a ≠ UU) ∨ (∃b. z = sinr·b ∧ b ≠ UU)
lemma ssumE:
[| p = UU ==> Q; !!x. [| p = sinl·x; x ≠ UU |] ==> Q;
!!y. [| p = sinr·y; y ≠ UU |] ==> Q |]
==> Q
lemma ssumE2:
[| !!x. p = sinl·x ==> Q; !!y. p = sinr·y ==> Q |] ==> Q
lemma sinl_less:
sinl·x << sinl·y = x << y
lemma sinr_less:
sinr·x << sinr·y = x << y
lemma sinl_less_sinr:
sinl·x << sinr·y = (x = UU)
lemma sinr_less_sinl:
sinr·x << sinl·y = (x = UU)
lemma sinl_eq_sinr:
(sinl·x = sinr·y) = (x = UU ∧ y = UU)
lemma sinr_eq_sinl:
(sinr·x = sinl·y) = (x = UU ∧ y = UU)
lemma less_sinlD:
p << sinl·x ==> ∃y. p = sinl·y ∧ y << x
lemma less_sinrD:
p << sinr·x ==> ∃y. p = sinr·y ∧ y << x
lemma ssum_chain_lemma:
chain Y
==> (∃A. chain A ∧ Y = (λi. sinl·(A i))) ∨ (∃B. chain B ∧ Y = (λi. sinr·(B i)))
lemma Iwhen1:
Iwhen f g UU = UU
lemma Iwhen2:
x ≠ UU ==> Iwhen f g (sinl·x) = f·x
lemma Iwhen3:
y ≠ UU ==> Iwhen f g (sinr·y) = g·y
lemma Iwhen4:
Iwhen f g (sinl·x) = strictify·f·x
lemma Iwhen5:
Iwhen f g (sinr·y) = strictify·g·y
lemma cont_Iwhen1:
cont (λf. Iwhen f g s)
lemma cont_Iwhen2:
cont (λg. Iwhen f g s)
lemma cont_Iwhen3:
cont (Iwhen f g)
lemma beta_sscase:
sscase·f·g·s = Iwhen f g s
lemma sscase1:
sscase·f·g·UU = UU
lemma sscase2:
x ≠ UU ==> sscase·f·g·(sinl·x) = f·x
lemma sscase3:
y ≠ UU ==> sscase·f·g·(sinr·y) = g·y
lemma sscase4:
sscase·sinl·sinr·z = z