(* Title: HOLCF/Ssum.thy
ID: $Id$
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 \<times> 'b. cfst\<cdot>p = \<bottom> \<or> csnd\<cdot>p = \<bottom>}"
by simp
syntax (xsymbols)
"++" :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20)
syntax (HTML output)
"++" :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20)
subsection {* Definitions of constructors *}
definition
sinl :: "'a \<rightarrow> ('a ++ 'b)" where
"sinl = (\<Lambda> a. Abs_Ssum <a, \<bottom>>)"
definition
sinr :: "'b \<rightarrow> ('a ++ 'b)" where
"sinr = (\<Lambda> b. Abs_Ssum <\<bottom>, b>)"
subsection {* Properties of @{term sinl} and @{term sinr} *}
lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum <a, \<bottom>>"
by (unfold sinl_def, simp add: cont_Abs_Ssum Ssum_def)
lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum <\<bottom>, b>"
by (unfold sinr_def, simp add: cont_Abs_Ssum Ssum_def)
lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = <a, \<bottom>>"
by (unfold sinl_def, simp add: cont_Abs_Ssum Abs_Ssum_inverse Ssum_def)
lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = <\<bottom>, b>"
by (unfold sinr_def, simp add: cont_Abs_Ssum Abs_Ssum_inverse Ssum_def)
lemma compact_sinl [simp]: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"
by (rule compact_Ssum, simp add: Rep_Ssum_sinl)
lemma compact_sinr [simp]: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"
by (rule compact_Ssum, simp add: Rep_Ssum_sinr)
lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"
by (simp add: sinl_Abs_Ssum Abs_Ssum_strict cpair_strict)
lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"
by (simp add: sinr_Abs_Ssum Abs_Ssum_strict cpair_strict)
lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"
by (simp add: sinl_Abs_Ssum Abs_Ssum_inject Ssum_def)
lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"
by (simp add: sinr_Abs_Ssum Abs_Ssum_inject Ssum_def)
lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"
by (rule sinl_eq [THEN iffD1])
lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"
by (rule sinr_eq [THEN iffD1])
lemma sinl_defined_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"
by (cut_tac sinl_eq [of "x" "\<bottom>"], simp)
lemma sinr_defined_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"
by (cut_tac sinr_eq [of "x" "\<bottom>"], simp)
lemma sinl_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"
by simp
lemma sinr_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"
by simp
subsection {* Case analysis *}
lemma Exh_Ssum:
"z = \<bottom> \<or> (\<exists>a. z = sinl\<cdot>a \<and> a \<noteq> \<bottom>) \<or> (\<exists>b. z = sinr\<cdot>b \<and> b \<noteq> \<bottom>)"
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:
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q;
\<And>x. \<lbrakk>p = sinl\<cdot>x; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q;
\<And>y. \<lbrakk>p = sinr\<cdot>y; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
by (cut_tac z=p in Exh_Ssum, auto)
lemma ssumE2:
"\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> 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\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
by (simp add: less_Ssum_def Rep_Ssum_sinl)
lemma sinr_less [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
by (simp add: less_Ssum_def Rep_Ssum_sinr)
lemma sinl_less_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"
by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr)
lemma sinr_less_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"
by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr)
lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
by (subst po_eq_conv, simp)
lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
by (subst po_eq_conv, simp)
subsection {* Chains of strict sums *}
lemma less_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"
apply (rule_tac p=p in ssumE)
apply (rule_tac x="\<bottom>" in exI, simp)
apply simp
apply simp
done
lemma less_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"
apply (rule_tac p=p in ssumE)
apply (rule_tac x="\<bottom>" in exI, simp)
apply simp
apply simp
done
lemma ssum_chain_lemma:
"chain Y \<Longrightarrow> (\<exists>A. chain A \<and> Y = (\<lambda>i. sinl\<cdot>(A i))) \<or>
(\<exists>B. chain B \<and> Y = (\<lambda>i. sinr\<cdot>(B i)))"
apply (rule_tac p="lub (range Y)" in ssumE2)
apply (rule disjI1)
apply (rule_tac x="\<lambda>i. cfst\<cdot>(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="\<lambda>i. csnd\<cdot>(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 \<rightarrow> 'c, 'b \<rightarrow> 'c, 'a ++ 'b] \<Rightarrow> 'c" where
"Iwhen = (\<lambda>f g s.
if cfst\<cdot>(Rep_Ssum s) \<noteq> \<bottom> then f\<cdot>(cfst\<cdot>(Rep_Ssum s)) else
if csnd\<cdot>(Rep_Ssum s) \<noteq> \<bottom> then g\<cdot>(csnd\<cdot>(Rep_Ssum s)) else \<bottom>)"
text {* rewrites for @{term Iwhen} *}
lemma Iwhen1 [simp]: "Iwhen f g \<bottom> = \<bottom>"
by (simp add: Iwhen_def Rep_Ssum_strict)
lemma Iwhen2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> Iwhen f g (sinl\<cdot>x) = f\<cdot>x"
by (simp add: Iwhen_def Rep_Ssum_sinl)
lemma Iwhen3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> Iwhen f g (sinr\<cdot>y) = g\<cdot>y"
by (simp add: Iwhen_def Rep_Ssum_sinr)
lemma Iwhen4: "Iwhen f g (sinl\<cdot>x) = strictify\<cdot>f\<cdot>x"
by (simp add: strictify_conv_if)
lemma Iwhen5: "Iwhen f g (sinr\<cdot>y) = strictify\<cdot>g\<cdot>y"
by (simp add: strictify_conv_if)
subsection {* Continuity of @{term Iwhen} *}
text {* @{term Iwhen} is continuous in all arguments *}
lemma cont_Iwhen1: "cont (\<lambda>f. Iwhen f g s)"
by (rule_tac p=s in ssumE, simp_all)
lemma cont_Iwhen2: "cont (\<lambda>g. Iwhen f g s)"
by (rule_tac p=s in ssumE, simp_all)
lemma cont_Iwhen3: "cont (\<lambda>s. 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 \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where
"sscase = (\<Lambda> f g s. Iwhen f g s)"
translations
"case s of CONST sinl\<cdot>x \<Rightarrow> t1 | CONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
translations
"\<Lambda>(CONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
"\<Lambda>(CONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"
text {* continuous versions of lemmas for @{term sscase} *}
lemma beta_sscase: "sscase\<cdot>f\<cdot>g\<cdot>s = Iwhen f g s"
by (simp add: sscase_def cont_Iwhen1 cont_Iwhen2 cont_Iwhen3)
lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
by (simp add: beta_sscase)
lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"
by (simp add: beta_sscase)
lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"
by (simp add: beta_sscase)
lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"
by (rule_tac p=z in ssumE, simp_all)
end