# HG changeset patch
# User huffman
# Date 1231986163 28800
# Node ID 247e4c816004b32e637943a9d8790eba7978e1f9
# Parent  7f4a32134447daa6706f8e819b569b9beab07f37
rename Dsum.thy to Sum_Cpo.thy

diff -r 7f4a32134447 -r 247e4c816004 src/HOLCF/Dsum.thy
--- a/src/HOLCF/Dsum.thy	Wed Jan 14 18:18:48 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,251 +0,0 @@
-(*  Title:      HOLCF/Dsum.thy
-    Author:     Brian Huffman
-*)
-
-header {* The cpo of disjoint sums *}
-
-theory Dsum
-imports Bifinite
-begin
-
-subsection {* Ordering on type @{typ "'a + 'b"} *}
-
-instantiation "+" :: (sq_ord, sq_ord) sq_ord
-begin
-
-definition
-  less_sum_def: "x \<sqsubseteq> y \<equiv> case x of
-         Inl a \<Rightarrow> (case y of Inl b \<Rightarrow> a \<sqsubseteq> b | Inr b \<Rightarrow> False) |
-         Inr a \<Rightarrow> (case y of Inl b \<Rightarrow> False | Inr b \<Rightarrow> a \<sqsubseteq> b)"
-
-instance ..
-end
-
-lemma Inl_less_iff [simp]: "Inl x \<sqsubseteq> Inl y = x \<sqsubseteq> y"
-unfolding less_sum_def by simp
-
-lemma Inr_less_iff [simp]: "Inr x \<sqsubseteq> Inr y = x \<sqsubseteq> y"
-unfolding less_sum_def by simp
-
-lemma Inl_less_Inr [simp]: "\<not> Inl x \<sqsubseteq> Inr y"
-unfolding less_sum_def by simp
-
-lemma Inr_less_Inl [simp]: "\<not> Inr x \<sqsubseteq> Inl y"
-unfolding less_sum_def by simp
-
-lemma Inl_mono: "x \<sqsubseteq> y \<Longrightarrow> Inl x \<sqsubseteq> Inl y"
-by simp
-
-lemma Inr_mono: "x \<sqsubseteq> y \<Longrightarrow> Inr x \<sqsubseteq> Inr y"
-by simp
-
-lemma Inl_lessE: "\<lbrakk>Inl a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
-by (cases x, simp_all)
-
-lemma Inr_lessE: "\<lbrakk>Inr a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
-by (cases x, simp_all)
-
-lemmas sum_less_elims = Inl_lessE Inr_lessE
-
-lemma sum_less_cases:
-  "\<lbrakk>x \<sqsubseteq> y;
-    \<And>a b. \<lbrakk>x = Inl a; y = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R;
-    \<And>a b. \<lbrakk>x = Inr a; y = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk>
-      \<Longrightarrow> R"
-by (cases x, safe elim!: sum_less_elims, auto)
-
-subsection {* Sum type is a complete partial order *}
-
-instance "+" :: (po, po) po
-proof
-  fix x :: "'a + 'b"
-  show "x \<sqsubseteq> x"
-    by (induct x, simp_all)
-next
-  fix x y :: "'a + 'b"
-  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> x" thus "x = y"
-    by (induct x, auto elim!: sum_less_elims intro: antisym_less)
-next
-  fix x y z :: "'a + 'b"
-  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
-    by (induct x, auto elim!: sum_less_elims intro: trans_less)
-qed
-
-lemma monofun_inv_Inl: "monofun (\<lambda>p. THE a. p = Inl a)"
-by (rule monofunI, erule sum_less_cases, simp_all)
-
-lemma monofun_inv_Inr: "monofun (\<lambda>p. THE b. p = Inr b)"
-by (rule monofunI, erule sum_less_cases, simp_all)
-
-lemma sum_chain_cases:
-  assumes Y: "chain Y"
-  assumes A: "\<And>A. \<lbrakk>chain A; Y = (\<lambda>i. Inl (A i))\<rbrakk> \<Longrightarrow> R"
-  assumes B: "\<And>B. \<lbrakk>chain B; Y = (\<lambda>i. Inr (B i))\<rbrakk> \<Longrightarrow> R"
-  shows "R"
- apply (cases "Y 0")
-  apply (rule A)
-   apply (rule ch2ch_monofun [OF monofun_inv_Inl Y])
-  apply (rule ext)
-  apply (cut_tac j=i in chain_mono [OF Y le0], simp)
-  apply (erule Inl_lessE, simp)
- apply (rule B)
-  apply (rule ch2ch_monofun [OF monofun_inv_Inr Y])
- apply (rule ext)
- apply (cut_tac j=i in chain_mono [OF Y le0], simp)
- apply (erule Inr_lessE, simp)
-done
-
-lemma is_lub_Inl: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inl (S i)) <<| Inl x"
- apply (rule is_lubI)
-  apply (rule ub_rangeI)
-  apply (simp add: is_ub_lub)
- apply (frule ub_rangeD [where i=arbitrary])
- apply (erule Inl_lessE, simp)
- apply (erule is_lub_lub)
- apply (rule ub_rangeI)
- apply (drule ub_rangeD, simp)
-done
-
-lemma is_lub_Inr: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inr (S i)) <<| Inr x"
- apply (rule is_lubI)
-  apply (rule ub_rangeI)
-  apply (simp add: is_ub_lub)
- apply (frule ub_rangeD [where i=arbitrary])
- apply (erule Inr_lessE, simp)
- apply (erule is_lub_lub)
- apply (rule ub_rangeI)
- apply (drule ub_rangeD, simp)
-done
-
-instance "+" :: (cpo, cpo) cpo
- apply intro_classes
- apply (erule sum_chain_cases, safe)
-  apply (rule exI)
-  apply (rule is_lub_Inl)
-  apply (erule cpo_lubI)
- apply (rule exI)
- apply (rule is_lub_Inr)
- apply (erule cpo_lubI)
-done
-
-subsection {* Continuity of @{term Inl}, @{term Inr}, @{term sum_case} *}
-
-lemma cont2cont_Inl [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inl (f x))"
-by (fast intro: contI is_lub_Inl elim: contE)
-
-lemma cont2cont_Inr [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inr (f x))"
-by (fast intro: contI is_lub_Inr elim: contE)
-
-lemma cont_Inl: "cont Inl"
-by (rule cont2cont_Inl [OF cont_id])
-
-lemma cont_Inr: "cont Inr"
-by (rule cont2cont_Inr [OF cont_id])
-
-lemmas ch2ch_Inl [simp] = ch2ch_cont [OF cont_Inl]
-lemmas ch2ch_Inr [simp] = ch2ch_cont [OF cont_Inr]
-
-lemmas lub_Inl = cont2contlubE [OF cont_Inl, symmetric]
-lemmas lub_Inr = cont2contlubE [OF cont_Inr, symmetric]
-
-lemma cont_sum_case1:
-  assumes f: "\<And>a. cont (\<lambda>x. f x a)"
-  assumes g: "\<And>b. cont (\<lambda>x. g x b)"
-  shows "cont (\<lambda>x. case y of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
-by (induct y, simp add: f, simp add: g)
-
-lemma cont_sum_case2: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (sum_case f g)"
-apply (rule contI)
-apply (erule sum_chain_cases)
-apply (simp add: cont2contlubE [OF cont_Inl, symmetric] contE)
-apply (simp add: cont2contlubE [OF cont_Inr, symmetric] contE)
-done
-
-lemma cont2cont_sum_case [simp]:
-  assumes f1: "\<And>a. cont (\<lambda>x. f x a)" and f2: "\<And>x. cont (\<lambda>a. f x a)"
-  assumes g1: "\<And>b. cont (\<lambda>x. g x b)" and g2: "\<And>x. cont (\<lambda>b. g x b)"
-  assumes h: "cont (\<lambda>x. h x)"
-  shows "cont (\<lambda>x. case h x of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
-apply (rule cont2cont_app2 [OF cont2cont_lambda _ h])
-apply (rule cont_sum_case1 [OF f1 g1])
-apply (rule cont_sum_case2 [OF f2 g2])
-done
-
-subsection {* Compactness and chain-finiteness *}
-
-lemma compact_Inl: "compact a \<Longrightarrow> compact (Inl a)"
-apply (rule compactI2)
-apply (erule sum_chain_cases, safe)
-apply (simp add: lub_Inl)
-apply (erule (2) compactD2)
-apply (simp add: lub_Inr)
-done
-
-lemma compact_Inr: "compact a \<Longrightarrow> compact (Inr a)"
-apply (rule compactI2)
-apply (erule sum_chain_cases, safe)
-apply (simp add: lub_Inl)
-apply (simp add: lub_Inr)
-apply (erule (2) compactD2)
-done
-
-lemma compact_Inl_rev: "compact (Inl a) \<Longrightarrow> compact a"
-unfolding compact_def
-by (drule adm_subst [OF cont_Inl], simp)
-
-lemma compact_Inr_rev: "compact (Inr a) \<Longrightarrow> compact a"
-unfolding compact_def
-by (drule adm_subst [OF cont_Inr], simp)
-
-lemma compact_Inl_iff [simp]: "compact (Inl a) = compact a"
-by (safe elim!: compact_Inl compact_Inl_rev)
-
-lemma compact_Inr_iff [simp]: "compact (Inr a) = compact a"
-by (safe elim!: compact_Inr compact_Inr_rev)
-
-instance "+" :: (chfin, chfin) chfin
-apply intro_classes
-apply (erule compact_imp_max_in_chain)
-apply (case_tac "\<Squnion>i. Y i", simp_all)
-done
-
-instance "+" :: (finite_po, finite_po) finite_po ..
-
-instance "+" :: (discrete_cpo, discrete_cpo) discrete_cpo
-by intro_classes (simp add: less_sum_def split: sum.split)
-
-subsection {* Sum type is a bifinite domain *}
-
-instantiation "+" :: (profinite, profinite) profinite
-begin
-
-definition
-  approx_sum_def: "approx =
-    (\<lambda>n. \<Lambda> x. case x of Inl a \<Rightarrow> Inl (approx n\<cdot>a) | Inr b \<Rightarrow> Inr (approx n\<cdot>b))"
-
-lemma approx_Inl [simp]: "approx n\<cdot>(Inl x) = Inl (approx n\<cdot>x)"
-  unfolding approx_sum_def by simp
-
-lemma approx_Inr [simp]: "approx n\<cdot>(Inr x) = Inr (approx n\<cdot>x)"
-  unfolding approx_sum_def by simp
-
-instance proof
-  fix i :: nat and x :: "'a + 'b"
-  show "chain (approx :: nat \<Rightarrow> 'a + 'b \<rightarrow> 'a + 'b)"
-    unfolding approx_sum_def
-    by (rule ch2ch_LAM, case_tac x, simp_all)
-  show "(\<Squnion>i. approx i\<cdot>x) = x"
-    by (induct x, simp_all add: lub_Inl lub_Inr)
-  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
-    by (induct x, simp_all)
-  have "{x::'a + 'b. approx i\<cdot>x = x} \<subseteq>
-        {x::'a. approx i\<cdot>x = x} <+> {x::'b. approx i\<cdot>x = x}"
-    by (rule subsetI, case_tac x, simp_all add: InlI InrI)
-  thus "finite {x::'a + 'b. approx i\<cdot>x = x}"
-    by (rule finite_subset,
-        intro finite_Plus finite_fixes_approx)
-qed
-
-end
-
-end
diff -r 7f4a32134447 -r 247e4c816004 src/HOLCF/HOLCF.thy
--- a/src/HOLCF/HOLCF.thy	Wed Jan 14 18:18:48 2009 -0800
+++ b/src/HOLCF/HOLCF.thy	Wed Jan 14 18:22:43 2009 -0800
@@ -6,7 +6,7 @@
 
 theory HOLCF
 imports
-  Domain ConvexPD Algebraic Universal Dsum Main
+  Domain ConvexPD Algebraic Universal Sum_Cpo Main
 uses
   "holcf_logic.ML"
   "Tools/cont_consts.ML"
diff -r 7f4a32134447 -r 247e4c816004 src/HOLCF/IsaMakefile
--- a/src/HOLCF/IsaMakefile	Wed Jan 14 18:18:48 2009 -0800
+++ b/src/HOLCF/IsaMakefile	Wed Jan 14 18:22:43 2009 -0800
@@ -39,7 +39,6 @@
   Discrete.thy \
   Deflation.thy \
   Domain.thy \
-  Dsum.thy \
   Eventual.thy \
   Ffun.thy \
   Fixrec.thy \
@@ -55,6 +54,7 @@
   Product_Cpo.thy \
   Sprod.thy \
   Ssum.thy \
+  Sum_Cpo.thy \
   Tr.thy \
   Universal.thy \
   UpperPD.thy \
diff -r 7f4a32134447 -r 247e4c816004 src/HOLCF/Sum_Cpo.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Sum_Cpo.thy	Wed Jan 14 18:22:43 2009 -0800
@@ -0,0 +1,251 @@
+(*  Title:      HOLCF/Sum_Cpo.thy
+    Author:     Brian Huffman
+*)
+
+header {* The cpo of disjoint sums *}
+
+theory Sum_Cpo
+imports Bifinite
+begin
+
+subsection {* Ordering on type @{typ "'a + 'b"} *}
+
+instantiation "+" :: (sq_ord, sq_ord) sq_ord
+begin
+
+definition
+  less_sum_def: "x \<sqsubseteq> y \<equiv> case x of
+         Inl a \<Rightarrow> (case y of Inl b \<Rightarrow> a \<sqsubseteq> b | Inr b \<Rightarrow> False) |
+         Inr a \<Rightarrow> (case y of Inl b \<Rightarrow> False | Inr b \<Rightarrow> a \<sqsubseteq> b)"
+
+instance ..
+end
+
+lemma Inl_less_iff [simp]: "Inl x \<sqsubseteq> Inl y = x \<sqsubseteq> y"
+unfolding less_sum_def by simp
+
+lemma Inr_less_iff [simp]: "Inr x \<sqsubseteq> Inr y = x \<sqsubseteq> y"
+unfolding less_sum_def by simp
+
+lemma Inl_less_Inr [simp]: "\<not> Inl x \<sqsubseteq> Inr y"
+unfolding less_sum_def by simp
+
+lemma Inr_less_Inl [simp]: "\<not> Inr x \<sqsubseteq> Inl y"
+unfolding less_sum_def by simp
+
+lemma Inl_mono: "x \<sqsubseteq> y \<Longrightarrow> Inl x \<sqsubseteq> Inl y"
+by simp
+
+lemma Inr_mono: "x \<sqsubseteq> y \<Longrightarrow> Inr x \<sqsubseteq> Inr y"
+by simp
+
+lemma Inl_lessE: "\<lbrakk>Inl a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by (cases x, simp_all)
+
+lemma Inr_lessE: "\<lbrakk>Inr a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by (cases x, simp_all)
+
+lemmas sum_less_elims = Inl_lessE Inr_lessE
+
+lemma sum_less_cases:
+  "\<lbrakk>x \<sqsubseteq> y;
+    \<And>a b. \<lbrakk>x = Inl a; y = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R;
+    \<And>a b. \<lbrakk>x = Inr a; y = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk>
+      \<Longrightarrow> R"
+by (cases x, safe elim!: sum_less_elims, auto)
+
+subsection {* Sum type is a complete partial order *}
+
+instance "+" :: (po, po) po
+proof
+  fix x :: "'a + 'b"
+  show "x \<sqsubseteq> x"
+    by (induct x, simp_all)
+next
+  fix x y :: "'a + 'b"
+  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> x" thus "x = y"
+    by (induct x, auto elim!: sum_less_elims intro: antisym_less)
+next
+  fix x y z :: "'a + 'b"
+  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
+    by (induct x, auto elim!: sum_less_elims intro: trans_less)
+qed
+
+lemma monofun_inv_Inl: "monofun (\<lambda>p. THE a. p = Inl a)"
+by (rule monofunI, erule sum_less_cases, simp_all)
+
+lemma monofun_inv_Inr: "monofun (\<lambda>p. THE b. p = Inr b)"
+by (rule monofunI, erule sum_less_cases, simp_all)
+
+lemma sum_chain_cases:
+  assumes Y: "chain Y"
+  assumes A: "\<And>A. \<lbrakk>chain A; Y = (\<lambda>i. Inl (A i))\<rbrakk> \<Longrightarrow> R"
+  assumes B: "\<And>B. \<lbrakk>chain B; Y = (\<lambda>i. Inr (B i))\<rbrakk> \<Longrightarrow> R"
+  shows "R"
+ apply (cases "Y 0")
+  apply (rule A)
+   apply (rule ch2ch_monofun [OF monofun_inv_Inl Y])
+  apply (rule ext)
+  apply (cut_tac j=i in chain_mono [OF Y le0], simp)
+  apply (erule Inl_lessE, simp)
+ apply (rule B)
+  apply (rule ch2ch_monofun [OF monofun_inv_Inr Y])
+ apply (rule ext)
+ apply (cut_tac j=i in chain_mono [OF Y le0], simp)
+ apply (erule Inr_lessE, simp)
+done
+
+lemma is_lub_Inl: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inl (S i)) <<| Inl x"
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (simp add: is_ub_lub)
+ apply (frule ub_rangeD [where i=arbitrary])
+ apply (erule Inl_lessE, simp)
+ apply (erule is_lub_lub)
+ apply (rule ub_rangeI)
+ apply (drule ub_rangeD, simp)
+done
+
+lemma is_lub_Inr: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inr (S i)) <<| Inr x"
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (simp add: is_ub_lub)
+ apply (frule ub_rangeD [where i=arbitrary])
+ apply (erule Inr_lessE, simp)
+ apply (erule is_lub_lub)
+ apply (rule ub_rangeI)
+ apply (drule ub_rangeD, simp)
+done
+
+instance "+" :: (cpo, cpo) cpo
+ apply intro_classes
+ apply (erule sum_chain_cases, safe)
+  apply (rule exI)
+  apply (rule is_lub_Inl)
+  apply (erule cpo_lubI)
+ apply (rule exI)
+ apply (rule is_lub_Inr)
+ apply (erule cpo_lubI)
+done
+
+subsection {* Continuity of @{term Inl}, @{term Inr}, @{term sum_case} *}
+
+lemma cont2cont_Inl [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inl (f x))"
+by (fast intro: contI is_lub_Inl elim: contE)
+
+lemma cont2cont_Inr [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inr (f x))"
+by (fast intro: contI is_lub_Inr elim: contE)
+
+lemma cont_Inl: "cont Inl"
+by (rule cont2cont_Inl [OF cont_id])
+
+lemma cont_Inr: "cont Inr"
+by (rule cont2cont_Inr [OF cont_id])
+
+lemmas ch2ch_Inl [simp] = ch2ch_cont [OF cont_Inl]
+lemmas ch2ch_Inr [simp] = ch2ch_cont [OF cont_Inr]
+
+lemmas lub_Inl = cont2contlubE [OF cont_Inl, symmetric]
+lemmas lub_Inr = cont2contlubE [OF cont_Inr, symmetric]
+
+lemma cont_sum_case1:
+  assumes f: "\<And>a. cont (\<lambda>x. f x a)"
+  assumes g: "\<And>b. cont (\<lambda>x. g x b)"
+  shows "cont (\<lambda>x. case y of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
+by (induct y, simp add: f, simp add: g)
+
+lemma cont_sum_case2: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (sum_case f g)"
+apply (rule contI)
+apply (erule sum_chain_cases)
+apply (simp add: cont2contlubE [OF cont_Inl, symmetric] contE)
+apply (simp add: cont2contlubE [OF cont_Inr, symmetric] contE)
+done
+
+lemma cont2cont_sum_case [simp]:
+  assumes f1: "\<And>a. cont (\<lambda>x. f x a)" and f2: "\<And>x. cont (\<lambda>a. f x a)"
+  assumes g1: "\<And>b. cont (\<lambda>x. g x b)" and g2: "\<And>x. cont (\<lambda>b. g x b)"
+  assumes h: "cont (\<lambda>x. h x)"
+  shows "cont (\<lambda>x. case h x of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
+apply (rule cont2cont_app2 [OF cont2cont_lambda _ h])
+apply (rule cont_sum_case1 [OF f1 g1])
+apply (rule cont_sum_case2 [OF f2 g2])
+done
+
+subsection {* Compactness and chain-finiteness *}
+
+lemma compact_Inl: "compact a \<Longrightarrow> compact (Inl a)"
+apply (rule compactI2)
+apply (erule sum_chain_cases, safe)
+apply (simp add: lub_Inl)
+apply (erule (2) compactD2)
+apply (simp add: lub_Inr)
+done
+
+lemma compact_Inr: "compact a \<Longrightarrow> compact (Inr a)"
+apply (rule compactI2)
+apply (erule sum_chain_cases, safe)
+apply (simp add: lub_Inl)
+apply (simp add: lub_Inr)
+apply (erule (2) compactD2)
+done
+
+lemma compact_Inl_rev: "compact (Inl a) \<Longrightarrow> compact a"
+unfolding compact_def
+by (drule adm_subst [OF cont_Inl], simp)
+
+lemma compact_Inr_rev: "compact (Inr a) \<Longrightarrow> compact a"
+unfolding compact_def
+by (drule adm_subst [OF cont_Inr], simp)
+
+lemma compact_Inl_iff [simp]: "compact (Inl a) = compact a"
+by (safe elim!: compact_Inl compact_Inl_rev)
+
+lemma compact_Inr_iff [simp]: "compact (Inr a) = compact a"
+by (safe elim!: compact_Inr compact_Inr_rev)
+
+instance "+" :: (chfin, chfin) chfin
+apply intro_classes
+apply (erule compact_imp_max_in_chain)
+apply (case_tac "\<Squnion>i. Y i", simp_all)
+done
+
+instance "+" :: (finite_po, finite_po) finite_po ..
+
+instance "+" :: (discrete_cpo, discrete_cpo) discrete_cpo
+by intro_classes (simp add: less_sum_def split: sum.split)
+
+subsection {* Sum type is a bifinite domain *}
+
+instantiation "+" :: (profinite, profinite) profinite
+begin
+
+definition
+  approx_sum_def: "approx =
+    (\<lambda>n. \<Lambda> x. case x of Inl a \<Rightarrow> Inl (approx n\<cdot>a) | Inr b \<Rightarrow> Inr (approx n\<cdot>b))"
+
+lemma approx_Inl [simp]: "approx n\<cdot>(Inl x) = Inl (approx n\<cdot>x)"
+  unfolding approx_sum_def by simp
+
+lemma approx_Inr [simp]: "approx n\<cdot>(Inr x) = Inr (approx n\<cdot>x)"
+  unfolding approx_sum_def by simp
+
+instance proof
+  fix i :: nat and x :: "'a + 'b"
+  show "chain (approx :: nat \<Rightarrow> 'a + 'b \<rightarrow> 'a + 'b)"
+    unfolding approx_sum_def
+    by (rule ch2ch_LAM, case_tac x, simp_all)
+  show "(\<Squnion>i. approx i\<cdot>x) = x"
+    by (induct x, simp_all add: lub_Inl lub_Inr)
+  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
+    by (induct x, simp_all)
+  have "{x::'a + 'b. approx i\<cdot>x = x} \<subseteq>
+        {x::'a. approx i\<cdot>x = x} <+> {x::'b. approx i\<cdot>x = x}"
+    by (rule subsetI, case_tac x, simp_all add: InlI InrI)
+  thus "finite {x::'a + 'b. approx i\<cdot>x = x}"
+    by (rule finite_subset,
+        intro finite_Plus finite_fixes_approx)
+qed
+
+end
+
+end