
diff -r 3869b2400e22 -r 0d25e02759b7 src/HOL/HOLCF/Ssum.thy
--- a/src/HOL/HOLCF/Ssum.thy	Mon Jan 01 21:17:28 2018 +0100
+++ b/src/HOL/HOLCF/Ssum.thy	Mon Jan 01 23:07:24 2018 +0100
@@ -6,24 +6,24 @@
 section \<open>The type of strict sums\<close>
 
 theory Ssum
-imports Tr
+  imports Tr
 begin
 
 default_sort pcpo
 
+
 subsection \<open>Definition of strict sum type\<close>
 
-definition
-  "ssum =
-    {p :: tr \<times> ('a \<times> 'b). p = \<bottom> \<or>
-      (fst p = TT \<and> fst (snd p) \<noteq> \<bottom> \<and> snd (snd p) = \<bottom>) \<or>
-      (fst p = FF \<and> fst (snd p) = \<bottom> \<and> snd (snd p) \<noteq> \<bottom>)}"
+definition "ssum =
+  {p :: tr \<times> ('a \<times> 'b). p = \<bottom> \<or>
+    (fst p = TT \<and> fst (snd p) \<noteq> \<bottom> \<and> snd (snd p) = \<bottom>) \<or>
+    (fst p = FF \<and> fst (snd p) = \<bottom> \<and> snd (snd p) \<noteq> \<bottom>)}"
 
 pcpodef ('a, 'b) ssum  ("(_ \<oplus>/ _)" [21, 20] 20) = "ssum :: (tr \<times> 'a \<times> 'b) set"
-  unfolding ssum_def by simp_all
+  by (simp_all add: ssum_def)
 
 instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
-by (rule typedef_chfin [OF type_definition_ssum below_ssum_def])
+  by (rule typedef_chfin [OF type_definition_ssum below_ssum_def])
 
 type_notation (ASCII)
   ssum  (infixr "++" 10)
@@ -31,108 +31,108 @@
 
 subsection \<open>Definitions of constructors\<close>
 
-definition
-  sinl :: "'a \<rightarrow> ('a ++ 'b)" where
-  "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))"
+definition sinl :: "'a \<rightarrow> ('a ++ 'b)"
+  where "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))"
 
-definition
-  sinr :: "'b \<rightarrow> ('a ++ 'b)" where
-  "sinr = (\<Lambda> b. Abs_ssum (seq\<cdot>b\<cdot>FF, \<bottom>, b))"
+definition sinr :: "'b \<rightarrow> ('a ++ 'b)"
+  where "sinr = (\<Lambda> b. Abs_ssum (seq\<cdot>b\<cdot>FF, \<bottom>, b))"
 
 lemma sinl_ssum: "(seq\<cdot>a\<cdot>TT, a, \<bottom>) \<in> ssum"
-by (simp add: ssum_def seq_conv_if)
+  by (simp add: ssum_def seq_conv_if)
 
 lemma sinr_ssum: "(seq\<cdot>b\<cdot>FF, \<bottom>, b) \<in> ssum"
-by (simp add: ssum_def seq_conv_if)
+  by (simp add: ssum_def seq_conv_if)
 
 lemma Rep_ssum_sinl: "Rep_ssum (sinl\<cdot>a) = (seq\<cdot>a\<cdot>TT, a, \<bottom>)"
-by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)
+  by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)
 
 lemma Rep_ssum_sinr: "Rep_ssum (sinr\<cdot>b) = (seq\<cdot>b\<cdot>FF, \<bottom>, b)"
-by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)
+  by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)
 
 lemmas Rep_ssum_simps =
   Rep_ssum_inject [symmetric] below_ssum_def
   prod_eq_iff below_prod_def
   Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr
 
+
 subsection \<open>Properties of \emph{sinl} and \emph{sinr}\<close>
 
 text \<open>Ordering\<close>
 
-lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
-by (simp add: Rep_ssum_simps seq_conv_if)
+lemma sinl_below [simp]: "sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y"
+  by (simp add: Rep_ssum_simps seq_conv_if)
 
-lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
-by (simp add: Rep_ssum_simps seq_conv_if)
+lemma sinr_below [simp]: "sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y"
+  by (simp add: Rep_ssum_simps seq_conv_if)
 
-lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"
-by (simp add: Rep_ssum_simps seq_conv_if)
+lemma sinl_below_sinr [simp]: "sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y \<longleftrightarrow> x = \<bottom>"
+  by (simp add: Rep_ssum_simps seq_conv_if)
 
-lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"
-by (simp add: Rep_ssum_simps seq_conv_if)
+lemma sinr_below_sinl [simp]: "sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y \<longleftrightarrow> x = \<bottom>"
+  by (simp add: Rep_ssum_simps seq_conv_if)
 
 text \<open>Equality\<close>
 
-lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"
-by (simp add: po_eq_conv)
+lemma sinl_eq [simp]: "sinl\<cdot>x = sinl\<cdot>y \<longleftrightarrow> x = y"
+  by (simp add: po_eq_conv)
 
-lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"
-by (simp add: po_eq_conv)
+lemma sinr_eq [simp]: "sinr\<cdot>x = sinr\<cdot>y \<longleftrightarrow> x = y"
+  by (simp add: po_eq_conv)
 
-lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
-by (subst po_eq_conv, simp)
+lemma sinl_eq_sinr [simp]: "sinl\<cdot>x = sinr\<cdot>y \<longleftrightarrow> 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)
+lemma sinr_eq_sinl [simp]: "sinr\<cdot>x = sinl\<cdot>y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+  by (subst po_eq_conv) simp
 
 lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"
-by (rule sinl_eq [THEN iffD1])
+  by (rule sinl_eq [THEN iffD1])
 
 lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"
-by (rule sinr_eq [THEN iffD1])
+  by (rule sinr_eq [THEN iffD1])
 
 text \<open>Strictness\<close>
 
 lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"
-by (simp add: Rep_ssum_simps)
+  by (simp add: Rep_ssum_simps)
 
 lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"
-by (simp add: Rep_ssum_simps)
+  by (simp add: Rep_ssum_simps)
 
-lemma sinl_bottom_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"
-using sinl_eq [of "x" "\<bottom>"] by simp
+lemma sinl_bottom_iff [simp]: "sinl\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>"
+  using sinl_eq [of "x" "\<bottom>"] by simp
 
-lemma sinr_bottom_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"
-using sinr_eq [of "x" "\<bottom>"] by simp
+lemma sinr_bottom_iff [simp]: "sinr\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>"
+  using sinr_eq [of "x" "\<bottom>"] by simp
 
 lemma sinl_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"
-by simp
+  by simp
 
 lemma sinr_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"
-by simp
+  by simp
 
 text \<open>Compactness\<close>
 
 lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"
-by (rule compact_ssum, simp add: Rep_ssum_sinl)
+  by (rule compact_ssum) (simp add: Rep_ssum_sinl)
 
 lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"
-by (rule compact_ssum, simp add: Rep_ssum_sinr)
+  by (rule compact_ssum) (simp add: Rep_ssum_sinr)
 
 lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"
-unfolding compact_def
-by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinl]], simp)
+  unfolding compact_def
+  by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinl]], simp)
 
 lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x"
-unfolding compact_def
-by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinr]], simp)
+  unfolding compact_def
+  by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinr]], simp)
 
 lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x"
-by (safe elim!: compact_sinl compact_sinlD)
+  by (safe elim!: compact_sinl compact_sinlD)
 
 lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x"
-by (safe elim!: compact_sinr compact_sinrD)
+  by (safe elim!: compact_sinr compact_sinrD)
+
 
 subsection \<open>Case analysis\<close>
 
@@ -140,61 +140,61 @@
   obtains "p = \<bottom>"
   | x where "p = sinl\<cdot>x" and "x \<noteq> \<bottom>"
   | y where "p = sinr\<cdot>y" and "y \<noteq> \<bottom>"
-using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps)
+  using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps)
 
 lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]:
   "\<lbrakk>P \<bottom>;
    \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);
    \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"
-by (cases x, simp_all)
+  by (cases x) simp_all
 
 lemma ssumE2 [case_names sinl sinr]:
   "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-by (cases p, simp only: sinl_strict [symmetric], simp, simp)
+  by (cases p, simp only: sinl_strict [symmetric], simp, simp)
 
 lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"
-by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
+  by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
 
 lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"
-by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
+  by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
+
 
 subsection \<open>Case analysis combinator\<close>
 
-definition
-  sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where
-  "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s))"
+definition sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c"
+  where "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s))"
 
 translations
-  "case s of XCONST sinl\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
-  "case s of (XCONST sinl :: 'a)\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" => "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
+  "case s of XCONST sinl\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" \<rightleftharpoons> "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
+  "case s of (XCONST sinl :: 'a)\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" \<rightharpoonup> "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
 
 translations
-  "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
-  "\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"
+  "\<Lambda>(XCONST sinl\<cdot>x). t" \<rightleftharpoons> "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
+  "\<Lambda>(XCONST sinr\<cdot>y). t" \<rightleftharpoons> "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"
 
-lemma beta_sscase:
-  "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s)"
-unfolding sscase_def by (simp add: cont_Rep_ssum)
+lemma beta_sscase: "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s)"
+  by (simp add: sscase_def cont_Rep_ssum)
 
 lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
-unfolding beta_sscase by (simp add: Rep_ssum_strict)
+  by (simp add: beta_sscase Rep_ssum_strict)
 
 lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"
-unfolding beta_sscase by (simp add: Rep_ssum_sinl)
+  by (simp add: beta_sscase Rep_ssum_sinl)
 
 lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"
-unfolding beta_sscase by (simp add: Rep_ssum_sinr)
+  by (simp add: beta_sscase Rep_ssum_sinr)
 
 lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"
-by (cases z, simp_all)
+  by (cases z) simp_all
+
 
 subsection \<open>Strict sum preserves flatness\<close>
 
 instance ssum :: (flat, flat) flat
-apply (intro_classes, clarify)
-apply (case_tac x, simp)
-apply (case_tac y, simp_all add: flat_below_iff)
-apply (case_tac y, simp_all add: flat_below_iff)
-done
+  apply (intro_classes, clarify)
+  apply (case_tac x, simp)
+   apply (case_tac y, simp_all add: flat_below_iff)
+  apply (case_tac y, simp_all add: flat_below_iff)
+  done
 
 end




