# HG changeset patch
# User wenzelm
# Date 1377726525 -7200
# Node ID 082d0972096bb8b9c922b111fc889d739e9fbc15
# Parent  16235bb41881acea808e8234eff23daff634d4b7# Parent  220f306f5c4ead7792efcaeda423d1b2762a58d4
merged

diff -r 16235bb41881 -r 082d0972096b NEWS
--- a/NEWS	Wed Aug 28 18:44:50 2013 +0200
+++ b/NEWS	Wed Aug 28 23:48:45 2013 +0200
@@ -90,6 +90,11 @@
 according to Isabelle/Scala plugin option "jedit_font_reset_size"
 (cf. keyboard shortcut C+0).
 
+* More reactive and less intrusive completion.  Plain words need to be
+at least 3 characters long to be completed (was 2 before).  Symbols
+are only completed in backslash forms, e.g. \forall or \<forall> that
+both produce the Isabelle symbol \<forall> in its Unicode rendering.
+
 
 *** Pure ***
 
diff -r 16235bb41881 -r 082d0972096b etc/symbols
--- a/etc/symbols	Wed Aug 28 18:44:50 2013 +0200
+++ b/etc/symbols	Wed Aug 28 23:48:45 2013 +0200
@@ -240,10 +240,10 @@
 \<sqsupset>             code: 0x002290  group: relation
 \<sqsubseteq>           code: 0x002291  group: relation  abbrev: [=
 \<sqsupseteq>           code: 0x002292  group: relation  abbrev: =]
-\<inter>                code: 0x002229  group: operator  abbrev: Int
-\<Inter>                code: 0x0022c2  group: operator  abbrev: Inter
-\<union>                code: 0x00222a  group: operator  abbrev: Un
-\<Union>                code: 0x0022c3  group: operator  abbrev: Union
+\<inter>                code: 0x002229  group: operator
+\<Inter>                code: 0x0022c2  group: operator
+\<union>                code: 0x00222a  group: operator
+\<Union>                code: 0x0022c3  group: operator
 \<squnion>              code: 0x002294  group: operator
 \<Squnion>              code: 0x002a06  group: operator
 \<sqinter>              code: 0x002293  group: operator
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/Gar_Coll.thy
--- a/src/HOL/Hoare_Parallel/Gar_Coll.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/Gar_Coll.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -40,15 +40,15 @@
 edge but has not yet colored the new target.   *}
 
 definition Redirect_Edge :: "gar_coll_state ann_com" where
-  "Redirect_Edge \<equiv> .{\<acute>Mut_init \<and> \<acute>z}. \<langle>\<acute>E:=\<acute>E[R:=(fst(\<acute>E!R), T)],, \<acute>z:= (\<not>\<acute>z)\<rangle>"
+  "Redirect_Edge \<equiv> \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace> \<langle>\<acute>E:=\<acute>E[R:=(fst(\<acute>E!R), T)],, \<acute>z:= (\<not>\<acute>z)\<rangle>"
 
 definition Color_Target :: "gar_coll_state ann_com" where
-  "Color_Target \<equiv> .{\<acute>Mut_init \<and> \<not>\<acute>z}. \<langle>\<acute>M:=\<acute>M[T:=Black],, \<acute>z:= (\<not>\<acute>z)\<rangle>"
+  "Color_Target \<equiv> \<lbrace>\<acute>Mut_init \<and> \<not>\<acute>z\<rbrace> \<langle>\<acute>M:=\<acute>M[T:=Black],, \<acute>z:= (\<not>\<acute>z)\<rangle>"
 
 definition Mutator :: "gar_coll_state ann_com" where
   "Mutator \<equiv>
-  .{\<acute>Mut_init \<and> \<acute>z}. 
-  WHILE True INV .{\<acute>Mut_init \<and> \<acute>z}. 
+  \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace> 
+  WHILE True INV \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace> 
   DO  Redirect_Edge ;; Color_Target  OD"
 
 subsubsection {* Correctness of the mutator *}
@@ -64,14 +64,14 @@
 done
 
 lemma Color_Target:
-  "\<turnstile> Color_Target .{\<acute>Mut_init \<and> \<acute>z}."
+  "\<turnstile> Color_Target \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace>"
 apply (unfold mutator_defs)
 apply annhoare
 apply(simp_all)
 done
 
 lemma Mutator: 
- "\<turnstile> Mutator .{False}."
+ "\<turnstile> Mutator \<lbrace>False\<rbrace>"
 apply(unfold Mutator_def)
 apply annhoare
 apply(simp_all add:Redirect_Edge Color_Target)
@@ -101,21 +101,21 @@
 
 definition Blacken_Roots :: " gar_coll_state ann_com" where
   "Blacken_Roots \<equiv> 
-  .{\<acute>Proper}.
+  \<lbrace>\<acute>Proper\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Proper \<and> \<acute>ind=0}.
+  \<lbrace>\<acute>Proper \<and> \<acute>ind=0\<rbrace>
   WHILE \<acute>ind<length \<acute>M 
-   INV .{\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind\<le>length \<acute>M}.
-  DO .{\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
+   INV \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+  DO \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M\<rbrace>
    IF \<acute>ind\<in>Roots THEN 
-   .{\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<in>Roots}. 
+   \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<in>Roots\<rbrace> 
     \<acute>M:=\<acute>M[\<acute>ind:=Black] FI;;
-   .{\<acute>Proper \<and> (\<forall>i<\<acute>ind+1. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
+   \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind+1. i \<in> Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M\<rbrace>
     \<acute>ind:=\<acute>ind+1 
   OD"
 
 lemma Blacken_Roots: 
- "\<turnstile> Blacken_Roots .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M}."
+ "\<turnstile> Blacken_Roots \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M\<rbrace>"
 apply (unfold Blacken_Roots_def)
 apply annhoare
 apply(simp_all add:collector_defs Graph_defs)
@@ -135,28 +135,28 @@
 
 definition Propagate_Black_aux :: "gar_coll_state ann_com" where
   "Propagate_Black_aux \<equiv>
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M}.
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> \<acute>ind=0}. 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> \<acute>ind=0\<rbrace> 
   WHILE \<acute>ind<length \<acute>E 
-   INV .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-         \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>E}.
-  DO .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E}. 
+   INV \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+         \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>E\<rbrace>
+  DO \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E\<rbrace> 
    IF \<acute>M!(fst (\<acute>E!\<acute>ind)) = Black THEN 
-    .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E \<and> \<acute>M!fst(\<acute>E!\<acute>ind)=Black}.
+    \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E \<and> \<acute>M!fst(\<acute>E!\<acute>ind)=Black\<rbrace>
      \<acute>M:=\<acute>M[snd(\<acute>E!\<acute>ind):=Black];;
-    .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-       \<and> \<acute>PBInv (\<acute>ind + 1) \<and> \<acute>ind<length \<acute>E}.
+    \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+       \<and> \<acute>PBInv (\<acute>ind + 1) \<and> \<acute>ind<length \<acute>E\<rbrace>
      \<acute>ind:=\<acute>ind+1
    FI
   OD"
 
 lemma Propagate_Black_aux: 
   "\<turnstile>  Propagate_Black_aux
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-    \<and> ( \<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)}."
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+    \<and> ( \<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)\<rbrace>"
 apply (unfold Propagate_Black_aux_def  PBInv_def collector_defs)
 apply annhoare
 apply(simp_all add:Graph6 Graph7 Graph8 Graph12)
@@ -215,32 +215,32 @@
 
 definition Propagate_Black :: " gar_coll_state ann_com" where
   "Propagate_Black \<equiv>
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M}.
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> \<acute>ind=0}.
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> \<acute>ind=0\<rbrace>
   WHILE \<acute>ind<length \<acute>E 
-   INV .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-         \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>E}.
-  DO .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E}. 
+   INV \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+         \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>E\<rbrace>
+  DO \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E\<rbrace> 
    IF (\<acute>M!(fst (\<acute>E!\<acute>ind)))=Black THEN 
-    .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-      \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black}.
+    \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+      \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black\<rbrace>
      \<acute>k:=(snd(\<acute>E!\<acute>ind));;
-    .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+    \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
       \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black 
-      \<and> \<acute>Auxk}.
+      \<and> \<acute>Auxk\<rbrace>
      \<langle>\<acute>M:=\<acute>M[\<acute>k:=Black],, \<acute>ind:=\<acute>ind+1\<rangle>
-   ELSE .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-          \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E}. 
+   ELSE \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+          \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind<length \<acute>E\<rbrace> 
          \<langle>IF (\<acute>M!(fst (\<acute>E!\<acute>ind)))\<noteq>Black THEN \<acute>ind:=\<acute>ind+1 FI\<rangle> 
    FI
   OD"
 
 lemma Propagate_Black: 
   "\<turnstile>  Propagate_Black
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-    \<and> ( \<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)}."
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+    \<and> ( \<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)\<rbrace>"
 apply (unfold Propagate_Black_def  PBInv_def Auxk_def collector_defs)
 apply annhoare
 apply(simp_all add: Graph6 Graph7 Graph8 Graph12)
@@ -325,42 +325,42 @@
 
 definition Count :: " gar_coll_state ann_com" where
   "Count \<equiv>
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-    \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>bc={}}.
+    \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>bc={}\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
    \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>bc={} 
-   \<and> \<acute>ind=0}.
+   \<and> \<acute>ind=0\<rbrace>
    WHILE \<acute>ind<length \<acute>M 
-     INV .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+     INV \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
            \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
            \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>CountInv \<acute>ind
-           \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind\<le>length \<acute>M}.
-   DO .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+           \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+   DO \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
          \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
          \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>CountInv \<acute>ind 
-         \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M}. 
+         \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M\<rbrace> 
        IF \<acute>M!\<acute>ind=Black 
-          THEN .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+          THEN \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
                  \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
                  \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>CountInv \<acute>ind
-                 \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}.
+                 \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black\<rbrace>
           \<acute>bc:=insert \<acute>ind \<acute>bc
        FI;;
-      .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+      \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
         \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
         \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>CountInv (\<acute>ind+1)
-        \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M}.
+        \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind<length \<acute>M\<rbrace>
       \<acute>ind:=\<acute>ind+1
    OD"
 
 lemma Count: 
   "\<turnstile> Count 
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
    \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> length \<acute>Ma=length \<acute>M
-   \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)}."
+   \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>"
 apply(unfold Count_def)
 apply annhoare
 apply(simp_all add:CountInv_def Graph6 Graph7 Graph8 Graph12 Blacks_def collector_defs)
@@ -392,24 +392,24 @@
 
 definition Append :: "gar_coll_state ann_com" where
    "Append \<equiv>
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe}.
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind=0}.
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind=0\<rbrace>
     WHILE \<acute>ind<length \<acute>M 
-      INV .{\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>M}.
-    DO .{\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M}.
+      INV \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+    DO \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M\<rbrace>
        IF \<acute>M!\<acute>ind=Black THEN 
-          .{\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}. 
+          \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black\<rbrace> 
           \<acute>M:=\<acute>M[\<acute>ind:=White] 
-       ELSE .{\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<notin>Reach \<acute>E}.
+       ELSE \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<notin>Reach \<acute>E\<rbrace>
               \<acute>E:=Append_to_free(\<acute>ind,\<acute>E)
        FI;;
-     .{\<acute>Proper \<and> \<acute>AppendInv (\<acute>ind+1) \<and> \<acute>ind<length \<acute>M}. 
+     \<lbrace>\<acute>Proper \<and> \<acute>AppendInv (\<acute>ind+1) \<and> \<acute>ind<length \<acute>M\<rbrace> 
        \<acute>ind:=\<acute>ind+1
     OD"
 
 lemma Append: 
-  "\<turnstile> Append .{\<acute>Proper}."
+  "\<turnstile> Append \<lbrace>\<acute>Proper\<rbrace>"
 apply(unfold Append_def AppendInv_def)
 apply annhoare
 apply(simp_all add:collector_defs Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)
@@ -433,30 +433,30 @@
 
 definition Collector :: " gar_coll_state ann_com" where
   "Collector \<equiv>
-.{\<acute>Proper}.  
- WHILE True INV .{\<acute>Proper}. 
+\<lbrace>\<acute>Proper\<rbrace>  
+ WHILE True INV \<lbrace>\<acute>Proper\<rbrace> 
  DO  
   Blacken_Roots;; 
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M}.  
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M\<rbrace>  
    \<acute>obc:={};; 
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={}}. 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={}\<rbrace> 
    \<acute>bc:=Roots;; 
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots}. 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots\<rbrace> 
    \<acute>Ma:=M_init;;  
-  .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots \<and> \<acute>Ma=M_init}. 
+  \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots \<and> \<acute>Ma=M_init\<rbrace> 
    WHILE \<acute>obc\<noteq>\<acute>bc  
-     INV .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
+     INV \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M 
            \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-           \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)}. 
-   DO .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M}.
+           \<and> length \<acute>Ma=length \<acute>M \<and> (\<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace> 
+   DO \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M\<rbrace>
        \<acute>obc:=\<acute>bc;;
        Propagate_Black;; 
-      .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-        \<and> (\<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)}. 
+      \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+        \<and> (\<acute>obc < Blacks \<acute>M \<or> \<acute>Safe)\<rbrace> 
        \<acute>Ma:=\<acute>M;;
-      .{\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma 
+      \<lbrace>\<acute>Proper \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma 
         \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and> length \<acute>Ma=length \<acute>M 
-        \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)}.
+        \<and> ( \<acute>obc < Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>
        \<acute>bc:={};;
        Count 
    OD;; 
@@ -464,7 +464,7 @@
  OD"
 
 lemma Collector: 
-  "\<turnstile> Collector .{False}."
+  "\<turnstile> Collector \<lbrace>False\<rbrace>"
 apply(unfold Collector_def)
 apply annhoare
 apply(simp_all add: Blacken_Roots Propagate_Black Count Append)
@@ -806,15 +806,15 @@
 text {* In total there are 289 verification conditions.  *}
 
 lemma Gar_Coll: 
-  "\<parallel>- .{\<acute>Proper \<and> \<acute>Mut_init \<and> \<acute>z}.  
+  "\<parallel>- \<lbrace>\<acute>Proper \<and> \<acute>Mut_init \<and> \<acute>z\<rbrace>  
   COBEGIN  
    Collector
-  .{False}.
+  \<lbrace>False\<rbrace>
  \<parallel>  
    Mutator
-  .{False}. 
+  \<lbrace>False\<rbrace> 
  COEND 
-  .{False}."
+  \<lbrace>False\<rbrace>"
 apply oghoare
 apply(force simp add: Mutator_def Collector_def modules)
 apply(rule Collector)
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/Mul_Gar_Coll.thy
--- a/src/HOL/Hoare_Parallel/Mul_Gar_Coll.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/Mul_Gar_Coll.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -32,21 +32,21 @@
 
 definition Mul_Redirect_Edge  :: "nat \<Rightarrow> nat \<Rightarrow> mul_gar_coll_state ann_com" where
   "Mul_Redirect_Edge j n \<equiv>
-  .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.
+  \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)\<rbrace>
   \<langle>IF T(\<acute>Muts!j) \<in> Reach \<acute>E THEN  
   \<acute>E:= \<acute>E[R (\<acute>Muts!j):= (fst (\<acute>E!R(\<acute>Muts!j)), T (\<acute>Muts!j))] FI,, 
   \<acute>Muts:= \<acute>Muts[j:= (\<acute>Muts!j) \<lparr>Z:=False\<rparr>]\<rangle>"
 
 definition Mul_Color_Target :: "nat \<Rightarrow> nat \<Rightarrow> mul_gar_coll_state ann_com" where
   "Mul_Color_Target j n \<equiv>
-  .{\<acute>Mul_mut_init n \<and> \<not> Z (\<acute>Muts!j)}. 
+  \<lbrace>\<acute>Mul_mut_init n \<and> \<not> Z (\<acute>Muts!j)\<rbrace> 
   \<langle>\<acute>M:=\<acute>M[T (\<acute>Muts!j):=Black],, \<acute>Muts:=\<acute>Muts[j:= (\<acute>Muts!j) \<lparr>Z:=True\<rparr>]\<rangle>"
 
 definition Mul_Mutator :: "nat \<Rightarrow> nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Mutator j n \<equiv>
-  .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.  
+  \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)\<rbrace>  
   WHILE True  
-    INV .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.  
+    INV \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)\<rbrace>  
   DO Mul_Redirect_Edge j n ;; 
      Mul_Color_Target j n 
   OD"
@@ -67,7 +67,7 @@
 
 lemma Mul_Color_Target: "0\<le>j \<and> j<n \<Longrightarrow> 
   \<turnstile>  Mul_Color_Target j n  
-    .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}."
+    \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)\<rbrace>"
 apply (unfold mul_mutator_defs)
 apply annhoare
 apply(simp_all)
@@ -76,7 +76,7 @@
 done
 
 lemma Mul_Mutator: "0\<le>j \<and> j<n \<Longrightarrow>  
- \<turnstile> Mul_Mutator j n .{False}."
+ \<turnstile> Mul_Mutator j n \<lbrace>False\<rbrace>"
 apply(unfold Mul_Mutator_def)
 apply annhoare
 apply(simp_all add:Mul_Redirect_Edge Mul_Color_Target)
@@ -139,13 +139,13 @@
 subsubsection {* Modular Parameterized Mutators *}
 
 lemma Mul_Parameterized_Mutators: "0<n \<Longrightarrow>
- \<parallel>- .{\<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))}.
+ \<parallel>- \<lbrace>\<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))\<rbrace>
  COBEGIN
  SCHEME  [0\<le> j< n]
   Mul_Mutator j n
- .{False}.
+ \<lbrace>False\<rbrace>
  COEND
- .{False}."
+ \<lbrace>False\<rbrace>"
 apply oghoare
 apply(force simp add:Mul_Mutator_def mul_mutator_defs nth_list_update)
 apply(erule Mul_Mutator)
@@ -175,22 +175,22 @@
 
 definition Mul_Blacken_Roots :: "nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Blacken_Roots n \<equiv>
-  .{\<acute>Mul_Proper n}.
+  \<lbrace>\<acute>Mul_Proper n\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Mul_Proper n \<and> \<acute>ind=0}.
+  \<lbrace>\<acute>Mul_Proper n \<and> \<acute>ind=0\<rbrace>
   WHILE \<acute>ind<length \<acute>M 
-    INV .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind\<le>length \<acute>M}.
-  DO .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
+    INV \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+  DO \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M\<rbrace>
        IF \<acute>ind\<in>Roots THEN 
-     .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<in>Roots}. 
+     \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<in>Roots\<rbrace> 
        \<acute>M:=\<acute>M[\<acute>ind:=Black] FI;;
-     .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind+1. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
+     \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind+1. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M\<rbrace>
        \<acute>ind:=\<acute>ind+1 
   OD"
 
 lemma Mul_Blacken_Roots: 
   "\<turnstile> Mul_Blacken_Roots n  
-  .{\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M}."
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M\<rbrace>"
 apply (unfold Mul_Blacken_Roots_def)
 apply annhoare
 apply(simp_all add:mul_collector_defs Graph_defs)
@@ -213,40 +213,40 @@
 
 definition Mul_Propagate_Black :: "nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Propagate_Black n \<equiv>
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-  \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)}. 
+ \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+  \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)\<rbrace> 
  \<acute>ind:=0;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+ \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
    \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> Blacks \<acute>M\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-   \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) \<and> \<acute>ind=0}. 
+   \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) \<and> \<acute>ind=0\<rbrace> 
  WHILE \<acute>ind<length \<acute>E 
-  INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+  INV \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
         \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-        \<and> \<acute>Mul_PBInv \<and> \<acute>ind\<le>length \<acute>E}.
- DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+        \<and> \<acute>Mul_PBInv \<and> \<acute>ind\<le>length \<acute>E\<rbrace>
+ DO \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
      \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-     \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E}.
+     \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E\<rbrace>
    IF \<acute>M!(fst (\<acute>E!\<acute>ind))=Black THEN 
-   .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+   \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
      \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-     \<and> \<acute>Mul_PBInv \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black \<and> \<acute>ind<length \<acute>E}.
+     \<and> \<acute>Mul_PBInv \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black \<and> \<acute>ind<length \<acute>E\<rbrace>
     \<acute>k:=snd(\<acute>E!\<acute>ind);;
-   .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+   \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
      \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
      \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue \<or> (\<forall>i<\<acute>ind. \<not>BtoW(\<acute>E!i,\<acute>M)) 
         \<and> \<acute>l\<le>\<acute>Queue \<and> \<acute>Mul_Auxk ) \<and> \<acute>k<length \<acute>M \<and> \<acute>M!fst(\<acute>E!\<acute>ind)=Black 
-     \<and> \<acute>ind<length \<acute>E}.
+     \<and> \<acute>ind<length \<acute>E\<rbrace>
    \<langle>\<acute>M:=\<acute>M[\<acute>k:=Black],,\<acute>ind:=\<acute>ind+1\<rangle>
-   ELSE .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+   ELSE \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
          \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-         \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E}.
+         \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E\<rbrace>
          \<langle>IF \<acute>M!(fst (\<acute>E!\<acute>ind))\<noteq>Black THEN \<acute>ind:=\<acute>ind+1 FI\<rangle> FI
  OD"
 
 lemma Mul_Propagate_Black: 
   "\<turnstile> Mul_Propagate_Black n  
-   .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))}."
+   \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))\<rbrace>"
 apply(unfold Mul_Propagate_Black_def)
 apply annhoare
 apply(simp_all add:Mul_PBInv_def mul_collector_defs Mul_Auxk_def Graph6 Graph7 Graph8 Graph12 mul_collector_defs Queue_def)
@@ -295,51 +295,51 @@
 
 definition Mul_Count :: "nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Count n \<equiv> 
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
     \<and> length \<acute>Ma=length \<acute>M 
     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) ) 
-    \<and> \<acute>q<n+1 \<and> \<acute>bc={}}.
+    \<and> \<acute>q<n+1 \<and> \<acute>bc={}\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
     \<and> length \<acute>Ma=length \<acute>M 
     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) ) 
-    \<and> \<acute>q<n+1 \<and> \<acute>bc={} \<and> \<acute>ind=0}.
+    \<and> \<acute>q<n+1 \<and> \<acute>bc={} \<and> \<acute>ind=0\<rbrace>
   WHILE \<acute>ind<length \<acute>M 
-     INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+     INV \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
           \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M  
           \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind 
           \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
-          \<and> \<acute>q<n+1 \<and> \<acute>ind\<le>length \<acute>M}.
-  DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+          \<and> \<acute>q<n+1 \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+  DO \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
        \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
        \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind 
        \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
-       \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M}. 
+       \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M\<rbrace> 
      IF \<acute>M!\<acute>ind=Black 
-     THEN .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+     THEN \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
             \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M  
             \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind 
             \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
-            \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}.
+            \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black\<rbrace>
           \<acute>bc:=insert \<acute>ind \<acute>bc
      FI;;
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
     \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv (\<acute>ind+1) 
     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
-    \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M}.
+    \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M\<rbrace>
   \<acute>ind:=\<acute>ind+1
   OD"
  
 lemma Mul_Count: 
   "\<turnstile> Mul_Count n  
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
     \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
     \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc 
     \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) 
-    \<and> \<acute>q<n+1}."
+    \<and> \<acute>q<n+1\<rbrace>"
 apply (unfold Mul_Count_def)
 apply annhoare
 apply(simp_all add:Mul_CountInv_def mul_collector_defs Mul_Auxk_def Graph6 Graph7 Graph8 Graph12 mul_collector_defs Queue_def)
@@ -393,26 +393,26 @@
 
 definition Mul_Append :: "nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Append n \<equiv> 
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe}.
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe\<rbrace>
   \<acute>ind:=0;;
-  .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind=0}.
+  \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind=0\<rbrace>
   WHILE \<acute>ind<length \<acute>M 
-    INV .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>M}.
-  DO .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M}.
+    INV \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>M\<rbrace>
+  DO \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M\<rbrace>
       IF \<acute>M!\<acute>ind=Black THEN 
-     .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}. 
+     \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black\<rbrace> 
       \<acute>M:=\<acute>M[\<acute>ind:=White] 
       ELSE 
-     .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<notin>Reach \<acute>E}. 
+     \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<notin>Reach \<acute>E\<rbrace> 
       \<acute>E:=Append_to_free(\<acute>ind,\<acute>E)
       FI;;
-  .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv (\<acute>ind+1) \<and> \<acute>ind<length \<acute>M}. 
+  \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv (\<acute>ind+1) \<and> \<acute>ind<length \<acute>M\<rbrace> 
    \<acute>ind:=\<acute>ind+1
   OD"
 
 lemma Mul_Append: 
   "\<turnstile> Mul_Append n  
-     .{\<acute>Mul_Proper n}."
+     \<lbrace>\<acute>Mul_Proper n\<rbrace>"
 apply(unfold Mul_Append_def)
 apply annhoare
 apply(simp_all add: mul_collector_defs Mul_AppendInv_def 
@@ -431,52 +431,52 @@
 
 definition Mul_Collector :: "nat \<Rightarrow>  mul_gar_coll_state ann_com" where
   "Mul_Collector n \<equiv>
-.{\<acute>Mul_Proper n}.  
-WHILE True INV .{\<acute>Mul_Proper n}. 
+\<lbrace>\<acute>Mul_Proper n\<rbrace>  
+WHILE True INV \<lbrace>\<acute>Mul_Proper n\<rbrace> 
 DO  
 Mul_Blacken_Roots n ;; 
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M}.  
+\<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M\<rbrace>  
  \<acute>obc:={};; 
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={}}.  
+\<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={}\<rbrace>  
  \<acute>bc:=Roots;; 
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots}. 
+\<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots\<rbrace> 
  \<acute>l:=0;; 
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots \<and> \<acute>l=0}. 
+\<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots \<and> \<acute>l=0\<rbrace> 
  WHILE \<acute>l<n+1  
-   INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and>  
-         (\<acute>Safe \<or> (\<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M) \<and> \<acute>l<n+1)}. 
- DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
-      \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M)}.
+   INV \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and>  
+         (\<acute>Safe \<or> (\<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M) \<and> \<acute>l<n+1)\<rbrace> 
+ DO \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
+      \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M)\<rbrace>
     \<acute>obc:=\<acute>bc;;
     Mul_Propagate_Black n;; 
-    .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+    \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
       \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
       \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue 
-      \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))}. 
+      \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))\<rbrace> 
     \<acute>bc:={};;
-    .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+    \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
       \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
       \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue 
-      \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) \<and> \<acute>bc={}}. 
+      \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) \<and> \<acute>bc={}\<rbrace> 
        \<langle> \<acute>Ma:=\<acute>M,, \<acute>q:=\<acute>Queue \<rangle>;;
     Mul_Count n;; 
-    .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+    \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
       \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
       \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc 
       \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) 
-      \<and> \<acute>q<n+1}. 
+      \<and> \<acute>q<n+1\<rbrace> 
     IF \<acute>obc=\<acute>bc THEN
-    .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+    \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
       \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
       \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc 
       \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) 
-      \<and> \<acute>q<n+1 \<and> \<acute>obc=\<acute>bc}.  
+      \<and> \<acute>q<n+1 \<and> \<acute>obc=\<acute>bc\<rbrace>  
     \<acute>l:=\<acute>l+1  
-    ELSE .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
+    ELSE \<lbrace>\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M 
           \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M 
           \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc 
           \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) 
-          \<and> \<acute>q<n+1 \<and> \<acute>obc\<noteq>\<acute>bc}.  
+          \<and> \<acute>q<n+1 \<and> \<acute>obc\<noteq>\<acute>bc\<rbrace>  
         \<acute>l:=0 FI 
  OD;; 
  Mul_Append n  
@@ -488,7 +488,7 @@
 
 lemma Mul_Collector:
   "\<turnstile> Mul_Collector n 
-  .{False}."
+  \<lbrace>False\<rbrace>"
 apply(unfold Mul_Collector_def)
 apply annhoare
 apply(simp_all only:pre.simps Mul_Blacken_Roots 
@@ -1237,16 +1237,16 @@
 text {* The total number of verification conditions is 328 *}
 
 lemma Mul_Gar_Coll: 
- "\<parallel>- .{\<acute>Mul_Proper n \<and> \<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))}.  
+ "\<parallel>- \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))\<rbrace>  
  COBEGIN  
   Mul_Collector n
- .{False}.
+ \<lbrace>False\<rbrace>
  \<parallel>  
  SCHEME  [0\<le> j< n]
   Mul_Mutator j n
- .{False}.  
+ \<lbrace>False\<rbrace>  
  COEND  
- .{False}."
+ \<lbrace>False\<rbrace>"
 apply oghoare
 --{* Strengthening the precondition *}
 apply(rule Int_greatest)
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/OG_Examples.thy
--- a/src/HOL/Hoare_Parallel/OG_Examples.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/OG_Examples.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -16,30 +16,30 @@
  hold :: nat
 
 lemma Petersons_mutex_1: 
-  "\<parallel>- .{\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2 }.  
-  COBEGIN .{\<acute>pr1=0 \<and> \<not>\<acute>in1}.  
-  WHILE True INV .{\<acute>pr1=0 \<and> \<not>\<acute>in1}.  
+  "\<parallel>- \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2 \<rbrace>  
+  COBEGIN \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>  
+  WHILE True INV \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>  
   DO  
-  .{\<acute>pr1=0 \<and> \<not>\<acute>in1}. \<langle> \<acute>in1:=True,,\<acute>pr1:=1 \<rangle>;;  
-  .{\<acute>pr1=1 \<and> \<acute>in1}.  \<langle> \<acute>hold:=1,,\<acute>pr1:=2 \<rangle>;;  
-  .{\<acute>pr1=2 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)}.  
+  \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace> \<langle> \<acute>in1:=True,,\<acute>pr1:=1 \<rangle>;;  
+  \<lbrace>\<acute>pr1=1 \<and> \<acute>in1\<rbrace>  \<langle> \<acute>hold:=1,,\<acute>pr1:=2 \<rangle>;;  
+  \<lbrace>\<acute>pr1=2 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)\<rbrace>  
   AWAIT (\<not>\<acute>in2 \<or> \<not>(\<acute>hold=1)) THEN \<acute>pr1:=3 END;;    
-  .{\<acute>pr1=3 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)}. 
+  \<lbrace>\<acute>pr1=3 \<and> \<acute>in1 \<and> (\<acute>hold=1 \<or> \<acute>hold=2 \<and> \<acute>pr2=2)\<rbrace> 
    \<langle>\<acute>in1:=False,,\<acute>pr1:=0\<rangle> 
-  OD .{\<acute>pr1=0 \<and> \<not>\<acute>in1}.  
+  OD \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1\<rbrace>  
   \<parallel>  
-  .{\<acute>pr2=0 \<and> \<not>\<acute>in2}.  
-  WHILE True INV .{\<acute>pr2=0 \<and> \<not>\<acute>in2}.  
+  \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>  
+  WHILE True INV \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>  
   DO  
-  .{\<acute>pr2=0 \<and> \<not>\<acute>in2}. \<langle> \<acute>in2:=True,,\<acute>pr2:=1 \<rangle>;;  
-  .{\<acute>pr2=1 \<and> \<acute>in2}. \<langle>  \<acute>hold:=2,,\<acute>pr2:=2 \<rangle>;;  
-  .{\<acute>pr2=2 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))}.  
+  \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace> \<langle> \<acute>in2:=True,,\<acute>pr2:=1 \<rangle>;;  
+  \<lbrace>\<acute>pr2=1 \<and> \<acute>in2\<rbrace> \<langle>  \<acute>hold:=2,,\<acute>pr2:=2 \<rangle>;;  
+  \<lbrace>\<acute>pr2=2 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))\<rbrace>  
   AWAIT (\<not>\<acute>in1 \<or> \<not>(\<acute>hold=2)) THEN \<acute>pr2:=3  END;;    
-  .{\<acute>pr2=3 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))}. 
+  \<lbrace>\<acute>pr2=3 \<and> \<acute>in2 \<and> (\<acute>hold=2 \<or> (\<acute>hold=1 \<and> \<acute>pr1=2))\<rbrace> 
     \<langle>\<acute>in2:=False,,\<acute>pr2:=0\<rangle> 
-  OD .{\<acute>pr2=0 \<and> \<not>\<acute>in2}.  
+  OD \<lbrace>\<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>  
   COEND  
-  .{\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2}."
+  \<lbrace>\<acute>pr1=0 \<and> \<not>\<acute>in1 \<and> \<acute>pr2=0 \<and> \<not>\<acute>in2\<rbrace>"
 apply oghoare
 --{* 104 verification conditions. *}
 apply auto
@@ -57,37 +57,37 @@
  after2 :: bool
 
 lemma Busy_wait_mutex: 
- "\<parallel>-  .{True}.  
+ "\<parallel>-  \<lbrace>True\<rbrace>  
   \<acute>flag1:=False,, \<acute>flag2:=False,,  
-  COBEGIN .{\<not>\<acute>flag1}.  
+  COBEGIN \<lbrace>\<not>\<acute>flag1\<rbrace>  
         WHILE True  
-        INV .{\<not>\<acute>flag1}.  
-        DO .{\<not>\<acute>flag1}. \<langle> \<acute>flag1:=True,,\<acute>after1:=False \<rangle>;;  
-           .{\<acute>flag1 \<and> \<not>\<acute>after1}. \<langle> \<acute>turn:=1,,\<acute>after1:=True \<rangle>;;  
-           .{\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}.  
+        INV \<lbrace>\<not>\<acute>flag1\<rbrace>  
+        DO \<lbrace>\<not>\<acute>flag1\<rbrace> \<langle> \<acute>flag1:=True,,\<acute>after1:=False \<rangle>;;  
+           \<lbrace>\<acute>flag1 \<and> \<not>\<acute>after1\<rbrace> \<langle> \<acute>turn:=1,,\<acute>after1:=True \<rangle>;;  
+           \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>  
             WHILE \<not>(\<acute>flag2 \<longrightarrow> \<acute>turn=2)  
-            INV .{\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}.  
-            DO .{\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}. SKIP OD;; 
-           .{\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>flag2 \<and> \<acute>after2 \<longrightarrow> \<acute>turn=2)}.
+            INV \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>  
+            DO \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace> SKIP OD;; 
+           \<lbrace>\<acute>flag1 \<and> \<acute>after1 \<and> (\<acute>flag2 \<and> \<acute>after2 \<longrightarrow> \<acute>turn=2)\<rbrace>
             \<acute>flag1:=False  
         OD  
-       .{False}.  
+       \<lbrace>False\<rbrace>  
   \<parallel>  
-     .{\<not>\<acute>flag2}.  
+     \<lbrace>\<not>\<acute>flag2\<rbrace>  
         WHILE True  
-        INV .{\<not>\<acute>flag2}.  
-        DO .{\<not>\<acute>flag2}. \<langle> \<acute>flag2:=True,,\<acute>after2:=False \<rangle>;;  
-           .{\<acute>flag2 \<and> \<not>\<acute>after2}. \<langle> \<acute>turn:=2,,\<acute>after2:=True \<rangle>;;  
-           .{\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}.  
+        INV \<lbrace>\<not>\<acute>flag2\<rbrace>  
+        DO \<lbrace>\<not>\<acute>flag2\<rbrace> \<langle> \<acute>flag2:=True,,\<acute>after2:=False \<rangle>;;  
+           \<lbrace>\<acute>flag2 \<and> \<not>\<acute>after2\<rbrace> \<langle> \<acute>turn:=2,,\<acute>after2:=True \<rangle>;;  
+           \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>  
             WHILE \<not>(\<acute>flag1 \<longrightarrow> \<acute>turn=1)  
-            INV .{\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}.  
-            DO .{\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)}. SKIP OD;;  
-           .{\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>flag1 \<and> \<acute>after1 \<longrightarrow> \<acute>turn=1)}. 
+            INV \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace>  
+            DO \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>turn=1 \<or> \<acute>turn=2)\<rbrace> SKIP OD;;  
+           \<lbrace>\<acute>flag2 \<and> \<acute>after2 \<and> (\<acute>flag1 \<and> \<acute>after1 \<longrightarrow> \<acute>turn=1)\<rbrace> 
             \<acute>flag2:=False  
         OD  
-       .{False}.  
+       \<lbrace>False\<rbrace>  
   COEND  
-  .{False}."
+  \<lbrace>False\<rbrace>"
 apply oghoare
 --{* 122 vc *}
 apply auto
@@ -100,21 +100,21 @@
  who :: nat
 
 lemma Semaphores_mutex: 
- "\<parallel>- .{i\<noteq>j}.  
+ "\<parallel>- \<lbrace>i\<noteq>j\<rbrace>  
   \<acute>out:=True ,,  
-  COBEGIN .{i\<noteq>j}.  
-       WHILE True INV .{i\<noteq>j}.  
-       DO .{i\<noteq>j}. AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;  
-          .{\<not>\<acute>out \<and> \<acute>who=i \<and> i\<noteq>j}. \<acute>out:=True OD  
-       .{False}.  
+  COBEGIN \<lbrace>i\<noteq>j\<rbrace>  
+       WHILE True INV \<lbrace>i\<noteq>j\<rbrace>  
+       DO \<lbrace>i\<noteq>j\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;  
+          \<lbrace>\<not>\<acute>out \<and> \<acute>who=i \<and> i\<noteq>j\<rbrace> \<acute>out:=True OD  
+       \<lbrace>False\<rbrace>  
   \<parallel>  
-       .{i\<noteq>j}.  
-       WHILE True INV .{i\<noteq>j}.  
-       DO .{i\<noteq>j}. AWAIT \<acute>out THEN  \<acute>out:=False,,\<acute>who:=j END;;  
-          .{\<not>\<acute>out \<and> \<acute>who=j \<and> i\<noteq>j}. \<acute>out:=True OD  
-       .{False}.  
+       \<lbrace>i\<noteq>j\<rbrace>  
+       WHILE True INV \<lbrace>i\<noteq>j\<rbrace>  
+       DO \<lbrace>i\<noteq>j\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,,\<acute>who:=j END;;  
+          \<lbrace>\<not>\<acute>out \<and> \<acute>who=j \<and> i\<noteq>j\<rbrace> \<acute>out:=True OD  
+       \<lbrace>False\<rbrace>  
   COEND  
-  .{False}."
+  \<lbrace>False\<rbrace>"
 apply oghoare
 --{* 38 vc *}
 apply auto
@@ -123,17 +123,17 @@
 subsubsection {* Peterson's Algorithm III: Parameterized version: *}
 
 lemma Semaphores_parameterized_mutex: 
- "0<n \<Longrightarrow> \<parallel>- .{True}.  
+ "0<n \<Longrightarrow> \<parallel>- \<lbrace>True\<rbrace>  
   \<acute>out:=True ,,  
  COBEGIN
   SCHEME [0\<le> i< n]
-    .{True}.  
-     WHILE True INV .{True}.  
-      DO .{True}. AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;  
-         .{\<not>\<acute>out \<and> \<acute>who=i}. \<acute>out:=True OD
-    .{False}. 
+    \<lbrace>True\<rbrace>  
+     WHILE True INV \<lbrace>True\<rbrace>  
+      DO \<lbrace>True\<rbrace> AWAIT \<acute>out THEN  \<acute>out:=False,, \<acute>who:=i END;;  
+         \<lbrace>\<not>\<acute>out \<and> \<acute>who=i\<rbrace> \<acute>out:=True OD
+    \<lbrace>False\<rbrace> 
  COEND
-  .{False}." 
+  \<lbrace>False\<rbrace>" 
 apply oghoare
 --{* 20 vc *}
 apply auto
@@ -150,22 +150,22 @@
 lemma Ticket_mutex: 
  "\<lbrakk> 0<n; I=\<guillemotleft>n=length \<acute>turn \<and> 0<\<acute>nextv \<and> (\<forall>k l. k<n \<and> l<n \<and> k\<noteq>l 
     \<longrightarrow> \<acute>turn!k < \<acute>num \<and> (\<acute>turn!k =0 \<or> \<acute>turn!k\<noteq>\<acute>turn!l))\<guillemotright> \<rbrakk>
-   \<Longrightarrow> \<parallel>- .{n=length \<acute>turn}.  
+   \<Longrightarrow> \<parallel>- \<lbrace>n=length \<acute>turn\<rbrace>  
    \<acute>index:= 0,,
-   WHILE \<acute>index < n INV .{n=length \<acute>turn \<and> (\<forall>i<\<acute>index. \<acute>turn!i=0)}. 
+   WHILE \<acute>index < n INV \<lbrace>n=length \<acute>turn \<and> (\<forall>i<\<acute>index. \<acute>turn!i=0)\<rbrace> 
     DO \<acute>turn:= \<acute>turn[\<acute>index:=0],, \<acute>index:=\<acute>index +1 OD,,
   \<acute>num:=1 ,, \<acute>nextv:=1 ,, 
  COBEGIN
   SCHEME [0\<le> i< n]
-    .{\<acute>I}.  
-     WHILE True INV .{\<acute>I}.  
-      DO .{\<acute>I}. \<langle> \<acute>turn :=\<acute>turn[i:=\<acute>num],, \<acute>num:=\<acute>num+1 \<rangle>;;  
-         .{\<acute>I}. WAIT \<acute>turn!i=\<acute>nextv END;;
-         .{\<acute>I \<and> \<acute>turn!i=\<acute>nextv}. \<acute>nextv:=\<acute>nextv+1
+    \<lbrace>\<acute>I\<rbrace>  
+     WHILE True INV \<lbrace>\<acute>I\<rbrace>  
+      DO \<lbrace>\<acute>I\<rbrace> \<langle> \<acute>turn :=\<acute>turn[i:=\<acute>num],, \<acute>num:=\<acute>num+1 \<rangle>;;  
+         \<lbrace>\<acute>I\<rbrace> WAIT \<acute>turn!i=\<acute>nextv END;;
+         \<lbrace>\<acute>I \<and> \<acute>turn!i=\<acute>nextv\<rbrace> \<acute>nextv:=\<acute>nextv+1
       OD
-    .{False}. 
+    \<lbrace>False\<rbrace> 
  COEND
-  .{False}." 
+  \<lbrace>False\<rbrace>" 
 apply oghoare
 --{* 35 vc *}
 apply simp_all
@@ -259,40 +259,40 @@
       \<and> (\<not>\<acute>found \<and> a<\<acute> x \<longrightarrow> f(\<acute>x)\<noteq>0) \<guillemotright> ;  
     I2= \<guillemotleft>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0)) 
       \<and> (\<not>\<acute>found \<and> \<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0) \<guillemotright> \<rbrakk> \<Longrightarrow>  
-  \<parallel>- .{\<exists> u. f(u)=0}.  
+  \<parallel>- \<lbrace>\<exists> u. f(u)=0\<rbrace>  
   \<acute>turn:=1,, \<acute>found:= False,,  
   \<acute>x:=a,, \<acute>y:=a+1 ,,  
-  COBEGIN .{\<acute>I1}.  
+  COBEGIN \<lbrace>\<acute>I1\<rbrace>  
        WHILE \<not>\<acute>found  
-       INV .{\<acute>I1}.  
-       DO .{a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)}.  
+       INV \<lbrace>\<acute>I1\<rbrace>  
+       DO \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>  
           WAIT \<acute>turn=1 END;;  
-          .{a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)}.  
+          \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>  
           \<acute>turn:=2;;  
-          .{a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)}.    
+          \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>    
           \<langle> \<acute>x:=\<acute>x+1,,  
             IF f(\<acute>x)=0 THEN \<acute>found:=True ELSE SKIP FI\<rangle>  
        OD;;  
-       .{\<acute>I1  \<and> \<acute>found}.  
+       \<lbrace>\<acute>I1  \<and> \<acute>found\<rbrace>  
        \<acute>turn:=2  
-       .{\<acute>I1 \<and> \<acute>found}.  
+       \<lbrace>\<acute>I1 \<and> \<acute>found\<rbrace>  
   \<parallel>  
-      .{\<acute>I2}.  
+      \<lbrace>\<acute>I2\<rbrace>  
        WHILE \<not>\<acute>found  
-       INV .{\<acute>I2}.  
-       DO .{\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)}.  
+       INV \<lbrace>\<acute>I2\<rbrace>  
+       DO \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>  
           WAIT \<acute>turn=2 END;;  
-          .{\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)}.  
+          \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>  
           \<acute>turn:=1;;  
-          .{\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)}.  
+          \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>  
           \<langle> \<acute>y:=(\<acute>y - 1),,  
             IF f(\<acute>y)=0 THEN \<acute>found:=True ELSE SKIP FI\<rangle>  
        OD;;  
-       .{\<acute>I2 \<and> \<acute>found}.  
+       \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>  
        \<acute>turn:=1  
-       .{\<acute>I2 \<and> \<acute>found}.  
+       \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>  
   COEND  
-  .{f(\<acute>x)=0 \<or> f(\<acute>y)=0}."
+  \<lbrace>f(\<acute>x)=0 \<or> f(\<acute>y)=0\<rbrace>"
 apply oghoare
 --{* 98 verification conditions *}
 apply auto 
@@ -306,26 +306,26 @@
     \<and> (\<not>\<acute>found \<and> a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<guillemotright>;  
  I2= \<guillemotleft>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> (a<\<acute>x \<and> f(\<acute>x)=0) \<or> (\<acute>y\<le>a \<and> f(\<acute>y)=0)) 
     \<and> (\<not>\<acute>found \<and> \<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<guillemotright>\<rbrakk> \<Longrightarrow>  
-  \<parallel>- .{\<exists>u. f(u)=0}.  
+  \<parallel>- \<lbrace>\<exists>u. f(u)=0\<rbrace>  
   \<acute>found:= False,,  
   \<acute>x:=a,, \<acute>y:=a+1,,  
-  COBEGIN .{\<acute>I1}.  
+  COBEGIN \<lbrace>\<acute>I1\<rbrace>  
        WHILE \<not>\<acute>found  
-       INV .{\<acute>I1}.  
-       DO .{a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)}.  
+       INV \<lbrace>\<acute>I1\<rbrace>  
+       DO \<lbrace>a\<le>\<acute>x \<and> (\<acute>found \<longrightarrow> \<acute>y\<le>a \<and> f(\<acute>y)=0) \<and> (a<\<acute>x \<longrightarrow> f(\<acute>x)\<noteq>0)\<rbrace>  
           \<langle> \<acute>x:=\<acute>x+1,,IF f(\<acute>x)=0 THEN  \<acute>found:=True ELSE  SKIP FI\<rangle>  
        OD  
-       .{\<acute>I1 \<and> \<acute>found}.  
+       \<lbrace>\<acute>I1 \<and> \<acute>found\<rbrace>  
   \<parallel>  
-      .{\<acute>I2}.  
+      \<lbrace>\<acute>I2\<rbrace>  
        WHILE \<not>\<acute>found  
-       INV .{\<acute>I2}.  
-       DO .{\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)}.  
+       INV \<lbrace>\<acute>I2\<rbrace>  
+       DO \<lbrace>\<acute>y\<le>a+1 \<and> (\<acute>found \<longrightarrow> a<\<acute>x \<and> f(\<acute>x)=0) \<and> (\<acute>y\<le>a \<longrightarrow> f(\<acute>y)\<noteq>0)\<rbrace>  
           \<langle> \<acute>y:=(\<acute>y - 1),,IF f(\<acute>y)=0 THEN  \<acute>found:=True ELSE  SKIP FI\<rangle>  
        OD  
-       .{\<acute>I2 \<and> \<acute>found}.  
+       \<lbrace>\<acute>I2 \<and> \<acute>found\<rbrace>  
   COEND  
-  .{f(\<acute>x)=0 \<or> f(\<acute>y)=0}."
+  \<lbrace>f(\<acute>x)=0 \<or> f(\<acute>y)=0\<rbrace>"
 apply oghoare
 --{* 20 vc *}
 apply auto
@@ -390,44 +390,44 @@
     p1= \<guillemotleft>\<acute>I1 \<and> \<acute>li=\<acute>ins\<guillemotright> ;  
     I2 = \<guillemotleft>\<acute>I \<and> (\<forall>k<\<acute>lj. (a ! k)=(\<acute>b ! k)) \<and> \<acute>lj\<le>length a\<guillemotright> ;
     p2 = \<guillemotleft>\<acute>I2 \<and> \<acute>lj=\<acute>outs\<guillemotright> \<rbrakk> \<Longrightarrow>   
-  \<parallel>- .{\<acute>INIT}.  
+  \<parallel>- \<lbrace>\<acute>INIT\<rbrace>  
  \<acute>ins:=0,, \<acute>outs:=0,, \<acute>li:=0,, \<acute>lj:=0,,
- COBEGIN .{\<acute>p1 \<and> \<acute>INIT}. 
+ COBEGIN \<lbrace>\<acute>p1 \<and> \<acute>INIT\<rbrace> 
    WHILE \<acute>li <length a 
-     INV .{\<acute>p1 \<and> \<acute>INIT}.   
-   DO .{\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a}.  
+     INV \<lbrace>\<acute>p1 \<and> \<acute>INIT\<rbrace>   
+   DO \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a\<rbrace>  
        \<acute>vx:= (a ! \<acute>li);;  
-      .{\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li)}. 
+      \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li)\<rbrace> 
         WAIT \<acute>ins-\<acute>outs < length \<acute>buffer END;; 
-      .{\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li) 
-         \<and> \<acute>ins-\<acute>outs < length \<acute>buffer}. 
+      \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a \<and> \<acute>vx=(a ! \<acute>li) 
+         \<and> \<acute>ins-\<acute>outs < length \<acute>buffer\<rbrace> 
        \<acute>buffer:=(list_update \<acute>buffer (\<acute>ins mod (length \<acute>buffer)) \<acute>vx);; 
-      .{\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a 
+      \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li<length a 
          \<and> (a ! \<acute>li)=(\<acute>buffer ! (\<acute>ins mod (length \<acute>buffer))) 
-         \<and> \<acute>ins-\<acute>outs <length \<acute>buffer}.  
+         \<and> \<acute>ins-\<acute>outs <length \<acute>buffer\<rbrace>  
        \<acute>ins:=\<acute>ins+1;; 
-      .{\<acute>I1 \<and> \<acute>INIT \<and> (\<acute>li+1)=\<acute>ins \<and> \<acute>li<length a}.  
+      \<lbrace>\<acute>I1 \<and> \<acute>INIT \<and> (\<acute>li+1)=\<acute>ins \<and> \<acute>li<length a\<rbrace>  
        \<acute>li:=\<acute>li+1  
    OD  
-  .{\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li=length a}.  
+  \<lbrace>\<acute>p1 \<and> \<acute>INIT \<and> \<acute>li=length a\<rbrace>  
   \<parallel>  
-  .{\<acute>p2 \<and> \<acute>INIT}.  
+  \<lbrace>\<acute>p2 \<and> \<acute>INIT\<rbrace>  
    WHILE \<acute>lj < length a  
-     INV .{\<acute>p2 \<and> \<acute>INIT}.  
-   DO .{\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>INIT}.  
+     INV \<lbrace>\<acute>p2 \<and> \<acute>INIT\<rbrace>  
+   DO \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>INIT\<rbrace>  
         WAIT \<acute>outs<\<acute>ins END;; 
-      .{\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>INIT}.  
+      \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>INIT\<rbrace>  
        \<acute>vy:=(\<acute>buffer ! (\<acute>outs mod (length \<acute>buffer)));; 
-      .{\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT}.  
+      \<lbrace>\<acute>p2 \<and> \<acute>lj<length a \<and> \<acute>outs<\<acute>ins \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT\<rbrace>  
        \<acute>outs:=\<acute>outs+1;;  
-      .{\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT}.  
+      \<lbrace>\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> \<acute>vy=(a ! \<acute>lj) \<and> \<acute>INIT\<rbrace>  
        \<acute>b:=(list_update \<acute>b \<acute>lj \<acute>vy);; 
-      .{\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> (a ! \<acute>lj)=(\<acute>b ! \<acute>lj) \<and> \<acute>INIT}.  
+      \<lbrace>\<acute>I2 \<and> (\<acute>lj+1)=\<acute>outs \<and> \<acute>lj<length a \<and> (a ! \<acute>lj)=(\<acute>b ! \<acute>lj) \<and> \<acute>INIT\<rbrace>  
        \<acute>lj:=\<acute>lj+1  
    OD  
-  .{\<acute>p2 \<and> \<acute>lj=length a \<and> \<acute>INIT}.  
+  \<lbrace>\<acute>p2 \<and> \<acute>lj=length a \<and> \<acute>INIT\<rbrace>  
  COEND  
- .{ \<forall>k<length a. (a ! k)=(\<acute>b ! k)}."
+ \<lbrace> \<forall>k<length a. (a ! k)=(\<acute>b ! k)\<rbrace>"
 apply oghoare
 --{* 138 vc  *}
 apply(tactic {* ALLGOALS (clarify_tac @{context}) *})
@@ -455,9 +455,9 @@
   a :: "nat \<Rightarrow> nat"
 
 lemma Example1: 
- "\<parallel>- .{True}.
-   COBEGIN SCHEME [0\<le>i<n] .{True}. \<acute>a:=\<acute>a (i:=0) .{\<acute>a i=0}. COEND 
-  .{\<forall>i < n. \<acute>a i = 0}."
+ "\<parallel>- \<lbrace>True\<rbrace>
+   COBEGIN SCHEME [0\<le>i<n] \<lbrace>True\<rbrace> \<acute>a:=\<acute>a (i:=0) \<lbrace>\<acute>a i=0\<rbrace> COEND 
+  \<lbrace>\<forall>i < n. \<acute>a i = 0\<rbrace>"
 apply oghoare
 apply simp_all
 done
@@ -466,11 +466,11 @@
 record Example1_list =
   A :: "nat list"
 lemma Example1_list: 
- "\<parallel>- .{n < length \<acute>A}. 
+ "\<parallel>- \<lbrace>n < length \<acute>A\<rbrace> 
    COBEGIN 
-     SCHEME [0\<le>i<n] .{n < length \<acute>A}. \<acute>A:=\<acute>A[i:=0] .{\<acute>A!i=0}. 
+     SCHEME [0\<le>i<n] \<lbrace>n < length \<acute>A\<rbrace> \<acute>A:=\<acute>A[i:=0] \<lbrace>\<acute>A!i=0\<rbrace> 
    COEND 
-    .{\<forall>i < n. \<acute>A!i = 0}."
+    \<lbrace>\<forall>i < n. \<acute>A!i = 0\<rbrace>"
 apply oghoare
 apply force+
 done
@@ -525,14 +525,14 @@
  x :: nat
 
 lemma Example_2: "0<n \<Longrightarrow> 
- \<parallel>- .{\<acute>x=0 \<and> (\<Sum>i=0..<n. \<acute>c i)=0}.  
+ \<parallel>- \<lbrace>\<acute>x=0 \<and> (\<Sum>i=0..<n. \<acute>c i)=0\<rbrace>  
  COBEGIN 
    SCHEME [0\<le>i<n] 
-  .{\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=0}. 
+  \<lbrace>\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=0\<rbrace> 
    \<langle> \<acute>x:=\<acute>x+(Suc 0),, \<acute>c:=\<acute>c (i:=(Suc 0)) \<rangle>
-  .{\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=(Suc 0)}.
+  \<lbrace>\<acute>x=(\<Sum>i=0..<n. \<acute>c i) \<and> \<acute>c i=(Suc 0)\<rbrace>
  COEND 
- .{\<acute>x=n}."
+ \<lbrace>\<acute>x=n\<rbrace>"
 apply oghoare
 apply (simp_all cong del: strong_setsum_cong)
 apply (tactic {* ALLGOALS (clarify_tac @{context}) *})
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/OG_Syntax.thy
--- a/src/HOL/Hoare_Parallel/OG_Syntax.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/OG_Syntax.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -44,15 +44,15 @@
                     ("(0WHILE _ //DO _ /OD)"  [0, 0] 61)
 
 translations
-  "IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST Cond .{b}. c1 c2"
+  "IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST Cond \<lbrace>b\<rbrace> c1 c2"
   "IF b THEN c FI" \<rightleftharpoons> "IF b THEN c ELSE SKIP FI"
-  "WHILE b INV i DO c OD" \<rightharpoonup> "CONST While .{b}. i c"
+  "WHILE b INV i DO c OD" \<rightharpoonup> "CONST While \<lbrace>b\<rbrace> i c"
   "WHILE b DO c OD" \<rightleftharpoons> "WHILE b INV CONST undefined DO c OD"
 
-  "r IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST AnnCond1 r .{b}. c1 c2"
-  "r IF b THEN c FI" \<rightharpoonup> "CONST AnnCond2 r .{b}. c"
-  "r WHILE b INV i DO c OD" \<rightharpoonup> "CONST AnnWhile r .{b}. i c"
-  "r AWAIT b THEN c END" \<rightharpoonup> "CONST AnnAwait r .{b}. c"
+  "r IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST AnnCond1 r \<lbrace>b\<rbrace> c1 c2"
+  "r IF b THEN c FI" \<rightharpoonup> "CONST AnnCond2 r \<lbrace>b\<rbrace> c"
+  "r WHILE b INV i DO c OD" \<rightharpoonup> "CONST AnnWhile r \<lbrace>b\<rbrace> i c"
+  "r AWAIT b THEN c END" \<rightharpoonup> "CONST AnnAwait r \<lbrace>b\<rbrace> c"
   "r \<langle>c\<rangle>" \<rightleftharpoons> "r AWAIT CONST True THEN c END"
   "r WAIT b END" \<rightleftharpoons> "r AWAIT b THEN SKIP END"
  
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/Quote_Antiquote.thy
--- a/src/HOL/Hoare_Parallel/Quote_Antiquote.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/Quote_Antiquote.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -6,13 +6,10 @@
 syntax
   "_quote"     :: "'b \<Rightarrow> ('a \<Rightarrow> 'b)"                ("(\<guillemotleft>_\<guillemotright>)" [0] 1000)
   "_antiquote" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b"                ("\<acute>_" [1000] 1000)
-  "_Assert"    :: "'a \<Rightarrow> 'a set"                    ("(.{_}.)" [0] 1000)
-
-syntax (xsymbols)
-  "_Assert"    :: "'a \<Rightarrow> 'a set"            ("(\<lbrace>_\<rbrace>)" [0] 1000)
+  "_Assert"    :: "'a \<Rightarrow> 'a set"                    ("(\<lbrace>_\<rbrace>)" [0] 1000)
 
 translations
-  ".{b}." \<rightharpoonup> "CONST Collect \<guillemotleft>b\<guillemotright>"
+  "\<lbrace>b\<rbrace>" \<rightharpoonup> "CONST Collect \<guillemotleft>b\<guillemotright>"
 
 parse_translation {*
   let
diff -r 16235bb41881 -r 082d0972096b src/HOL/Hoare_Parallel/RG_Syntax.thy
--- a/src/HOL/Hoare_Parallel/RG_Syntax.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Hoare_Parallel/RG_Syntax.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -20,10 +20,10 @@
 
 translations
   "\<acute>x := a" \<rightharpoonup> "CONST Basic \<guillemotleft>\<acute>(_update_name x (\<lambda>_. a))\<guillemotright>"
-  "IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST Cond .{b}. c1 c2"
+  "IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST Cond \<lbrace>b\<rbrace> c1 c2"
   "IF b THEN c FI" \<rightleftharpoons> "IF b THEN c ELSE SKIP FI"
-  "WHILE b DO c OD" \<rightharpoonup> "CONST While .{b}. c"
-  "AWAIT b THEN c END" \<rightleftharpoons> "CONST Await .{b}. c"
+  "WHILE b DO c OD" \<rightharpoonup> "CONST While \<lbrace>b\<rbrace> c"
+  "AWAIT b THEN c END" \<rightleftharpoons> "CONST Await \<lbrace>b\<rbrace> c"
   "\<langle>c\<rangle>" \<rightleftharpoons> "AWAIT CONST True THEN c END"
   "WAIT b END" \<rightleftharpoons> "AWAIT b THEN SKIP END"
 
diff -r 16235bb41881 -r 082d0972096b src/HOL/Library/Infinite_Set.thy
--- a/src/HOL/Library/Infinite_Set.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Library/Infinite_Set.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -16,20 +16,18 @@
   lemmas may not work well with @{text "blast"}.
 *}
 
-abbreviation
-  infinite :: "'a set \<Rightarrow> bool" where
-  "infinite S == \<not> finite S"
+abbreviation infinite :: "'a set \<Rightarrow> bool"
+  where "infinite S \<equiv> \<not> finite S"
 
 text {*
   Infinite sets are non-empty, and if we remove some elements from an
   infinite set, the result is still infinite.
 *}
 
-lemma infinite_imp_nonempty: "infinite S ==> S \<noteq> {}"
+lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}"
   by auto
 
-lemma infinite_remove:
-  "infinite S \<Longrightarrow> infinite (S - {a})"
+lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})"
   by simp
 
 lemma Diff_infinite_finite:
@@ -72,7 +70,7 @@
 lemma finite_nat_bounded:
   assumes S: "finite (S::nat set)"
   shows "\<exists>k. S \<subseteq> {..<k}"  (is "\<exists>k. ?bounded S k")
-using S
+  using S
 proof induct
   have "?bounded {} 0" by simp
   then show "\<exists>k. ?bounded {} k" ..
@@ -92,7 +90,7 @@
 qed
 
 lemma finite_nat_iff_bounded:
-  "finite (S::nat set) = (\<exists>k. S \<subseteq> {..<k})"  (is "?lhs = ?rhs")
+  "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then show ?rhs by (rule finite_nat_bounded)
@@ -104,7 +102,7 @@
 qed
 
 lemma finite_nat_iff_bounded_le:
-  "finite (S::nat set) = (\<exists>k. S \<subseteq> {..k})"  (is "?lhs = ?rhs")
+  "finite (S::nat set) \<longleftrightarrow> (\<exists>k. S \<subseteq> {..k})"  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then obtain k where "S \<subseteq> {..<k}"
@@ -119,14 +117,14 @@
 qed
 
 lemma infinite_nat_iff_unbounded:
-  "infinite (S::nat set) = (\<forall>m. \<exists>n. m<n \<and> n\<in>S)"
-  (is "?lhs = ?rhs")
+  "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> n \<in> S)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   show ?rhs
   proof (rule ccontr)
     assume "\<not> ?rhs"
-    then obtain m where m: "\<forall>n. m<n \<longrightarrow> n\<notin>S" by blast
+    then obtain m where m: "\<forall>n. m < n \<longrightarrow> n \<notin> S" by blast
     then have "S \<subseteq> {..m}"
       by (auto simp add: sym [OF linorder_not_less])
     with `?lhs` show False
@@ -139,22 +137,22 @@
     assume "finite S"
     then obtain m where "S \<subseteq> {..m}"
       by (auto simp add: finite_nat_iff_bounded_le)
-    then have "\<forall>n. m<n \<longrightarrow> n\<notin>S" by auto
+    then have "\<forall>n. m < n \<longrightarrow> n \<notin> S" by auto
     with `?rhs` show False by blast
   qed
 qed
 
 lemma infinite_nat_iff_unbounded_le:
-  "infinite (S::nat set) = (\<forall>m. \<exists>n. m\<le>n \<and> n\<in>S)"
-  (is "?lhs = ?rhs")
+  "infinite (S::nat set) \<longleftrightarrow> (\<forall>m. \<exists>n. m \<le> n \<and> n \<in> S)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   show ?rhs
   proof
     fix m
-    from `?lhs` obtain n where "m<n \<and> n\<in>S"
+    from `?lhs` obtain n where "m < n \<and> n \<in> S"
       by (auto simp add: infinite_nat_iff_unbounded)
-    then have "m\<le>n \<and> n\<in>S" by simp
+    then have "m \<le> n \<and> n \<in> S" by simp
     then show "\<exists>n. m \<le> n \<and> n \<in> S" ..
   qed
 next
@@ -162,9 +160,9 @@
   show ?lhs
   proof (auto simp add: infinite_nat_iff_unbounded)
     fix m
-    from `?rhs` obtain n where "Suc m \<le> n \<and> n\<in>S"
+    from `?rhs` obtain n where "Suc m \<le> n \<and> n \<in> S"
       by blast
-    then have "m<n \<and> n\<in>S" by simp
+    then have "m < n \<and> n \<in> S" by simp
     then show "\<exists>n. m < n \<and> n \<in> S" ..
   qed
 qed
@@ -176,18 +174,18 @@
 *}
 
 lemma unbounded_k_infinite:
-  assumes k: "\<forall>m. k<m \<longrightarrow> (\<exists>n. m<n \<and> n\<in>S)"
+  assumes k: "\<forall>m. k < m \<longrightarrow> (\<exists>n. m < n \<and> n \<in> S)"
   shows "infinite (S::nat set)"
 proof -
   {
-    fix m have "\<exists>n. m<n \<and> n\<in>S"
-    proof (cases "k<m")
+    fix m have "\<exists>n. m < n \<and> n \<in> S"
+    proof (cases "k < m")
       case True
       with k show ?thesis by blast
     next
       case False
-      from k obtain n where "Suc k < n \<and> n\<in>S" by auto
-      with False have "m<n \<and> n\<in>S" by auto
+      from k obtain n where "Suc k < n \<and> n \<in> S" by auto
+      with False have "m < n \<and> n \<in> S" by auto
       then show ?thesis ..
     qed
   }
@@ -217,13 +215,14 @@
   then show False by simp
 qed
 
-lemma int_infinite [simp]:
-  shows "infinite (UNIV::int set)"
+lemma int_infinite [simp]: "infinite (UNIV::int set)"
 proof -
-  from inj_int have "infinite (range int)" by (rule range_inj_infinite)
+  from inj_int have "infinite (range int)"
+    by (rule range_inj_infinite)
   moreover 
   have "range int \<subseteq> (UNIV::int set)" by simp
-  ultimately show "infinite (UNIV::int set)" by (simp add: infinite_super)
+  ultimately show "infinite (UNIV::int set)"
+    by (simp add: infinite_super)
 qed
 
 text {*
@@ -235,7 +234,7 @@
 *}
 
 lemma linorder_injI:
-  assumes hyp: "!!x y. x < (y::'a::linorder) ==> f x \<noteq> f y"
+  assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
   shows "inj f"
 proof (rule inj_onI)
   fix x y
@@ -284,7 +283,8 @@
   proof -
     fix n
     show "pick n \<in> Sseq n"
-    proof (unfold pick_def, rule someI_ex)
+      unfolding pick_def
+    proof (rule someI_ex)
       from Sseq_inf have "infinite (Sseq n)" .
       then have "Sseq n \<noteq> {}" by auto
       then show "\<exists>x. x \<in> Sseq n" by auto
@@ -324,7 +324,7 @@
 qed
 
 lemma infinite_iff_countable_subset:
-    "infinite S = (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
+    "infinite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
   by (auto simp add: infinite_countable_subset range_inj_infinite infinite_super)
 
 text {*
@@ -359,13 +359,11 @@
   we introduce corresponding binders and their proof rules.
 *}
 
-definition
-  Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10) where
-  "Inf_many P = infinite {x. P x}"
+definition Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10)
+  where "Inf_many P \<longleftrightarrow> infinite {x. P x}"
 
-definition
-  Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10) where
-  "Alm_all P = (\<not> (INFM x. \<not> P x))"
+definition Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10)
+  where "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)"
 
 notation (xsymbols)
   Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
@@ -397,15 +395,21 @@
   unfolding Alm_all_def by simp
 
 lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)"
-  by (erule contrapos_pp, simp)
+  apply (erule contrapos_pp)
+  apply simp
+  done
 
 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x"
   by simp
 
-lemma INFM_E: assumes "INFM x. P x" obtains x where "P x"
+lemma INFM_E:
+  assumes "INFM x. P x"
+  obtains x where "P x"
   using INFM_EX [OF assms] by (rule exE)
 
-lemma MOST_I: assumes "\<And>x. P x" shows "MOST x. P x"
+lemma MOST_I:
+  assumes "\<And>x. P x"
+  shows "MOST x. P x"
   using assms by simp
 
 lemma INFM_mono:
@@ -476,15 +480,18 @@
 
 text {* Properties of quantifiers with injective functions. *}
 
-lemma INFM_inj:
-  "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
+lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
   unfolding INFM_iff_infinite
-  by (clarify, drule (1) finite_vimageI, simp)
+  apply clarify
+  apply (drule (1) finite_vimageI)
+  apply simp
+  done
 
-lemma MOST_inj:
-  "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
+lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
   unfolding MOST_iff_cofinite
-  by (drule (1) finite_vimageI, simp)
+  apply (drule (1) finite_vimageI)
+  apply simp
+  done
 
 text {* Properties of quantifiers with singletons. *}
 
@@ -515,16 +522,16 @@
 
 text {* Properties of quantifiers over the naturals. *}
 
-lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m<n \<and> P n)"
+lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n. m < n \<and> P n)"
   by (simp add: Inf_many_def infinite_nat_iff_unbounded)
 
-lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) = (\<forall>m. \<exists>n. m\<le>n \<and> P n)"
+lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<forall>m. \<exists>n. m \<le> n \<and> P n)"
   by (simp add: Inf_many_def infinite_nat_iff_unbounded_le)
 
-lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m<n \<longrightarrow> P n)"
+lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n. m < n \<longrightarrow> P n)"
   by (simp add: Alm_all_def INFM_nat)
 
-lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) = (\<exists>m. \<forall>n. m\<le>n \<longrightarrow> P n)"
+lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P (n::nat)) \<longleftrightarrow> (\<exists>m. \<forall>n. m \<le> n \<longrightarrow> P n)"
   by (simp add: Alm_all_def INFM_nat_le)
 
 
@@ -534,20 +541,20 @@
   The set's element type must be wellordered (e.g. the natural numbers).
 *}
 
-primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a" where
-    enumerate_0:   "enumerate S 0       = (LEAST n. n \<in> S)"
-  | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
+primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
+where
+  enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)"
+| enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
 
-lemma enumerate_Suc':
-    "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
+lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
   by simp
 
 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S"
-apply (induct n arbitrary: S)
- apply (fastforce intro: LeastI dest!: infinite_imp_nonempty)
-apply simp
-apply (metis DiffE infinite_remove)
-done
+  apply (induct n arbitrary: S)
+   apply (fastforce intro: LeastI dest!: infinite_imp_nonempty)
+  apply simp
+  apply (metis DiffE infinite_remove)
+  done
 
 declare enumerate_0 [simp del] enumerate_Suc [simp del]
 
@@ -573,31 +580,38 @@
   shows "n \<le> enumerate S n"
   using S 
 proof (induct n)
+  case 0
+  then show ?case by simp
+next
   case (Suc n)
   then have "n \<le> enumerate S n" by simp
   also note enumerate_mono[of n "Suc n", OF _ `infinite S`]
   finally show ?case by simp
-qed simp
+qed
 
 lemma enumerate_Suc'':
   fixes S :: "'a::wellorder set"
-  shows "infinite S  \<Longrightarrow> enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
+  assumes "infinite S"
+  shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
+  using assms
 proof (induct n arbitrary: S)
   case 0
-  then have "\<forall>s\<in>S. enumerate S 0 \<le> s"
+  then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
     by (auto simp: enumerate.simps intro: Least_le)
   then show ?case
     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
-    by (intro arg_cong[where f=Least] ext) auto
+    by (intro arg_cong[where f = Least] ext) auto
 next
   case (Suc n S)
   show ?case
     using enumerate_mono[OF zero_less_Suc `infinite S`, of n] `infinite S`
     apply (subst (1 2) enumerate_Suc')
     apply (subst Suc)
-    apply (insert `infinite S`, simp)
-    by (intro arg_cong[where f=Least] ext)
-       (auto simp: enumerate_Suc'[symmetric])
+    using `infinite S`
+    apply simp
+    apply (intro arg_cong[where f = Least] ext)
+    apply (auto simp: enumerate_Suc'[symmetric])
+    done
 qed
 
 lemma enumerate_Ex:
@@ -609,9 +623,12 @@
   proof cases
     let ?y = "Max {s'\<in>S. s' < s}"
     assume "\<exists>y\<in>S. y < s"
-    then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)" by (subst Max_less_iff) auto
-    then have y_in: "?y \<in> {s'\<in>S. s' < s}" by (intro Max_in) auto
-    with less.hyps[of ?y] obtain n where "enumerate S n = ?y" by auto
+    then have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
+      by (subst Max_less_iff) auto
+    then have y_in: "?y \<in> {s'\<in>S. s' < s}"
+      by (intro Max_in) auto
+    with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
+      by auto
     with S have "enumerate S (Suc n) = s"
       by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
     then show ?case by auto
@@ -632,7 +649,7 @@
     using enumerate_mono[OF _ `infinite S`] by (auto simp: neq_iff)
   then have "inj (enumerate S)"
     by (auto simp: inj_on_def)
-  moreover have "\<forall>s\<in>S. \<exists>i. enumerate S i = s"
+  moreover have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
     using enumerate_Ex[OF S] by auto
   moreover note `infinite S`
   ultimately show ?thesis
@@ -646,9 +663,8 @@
   These simplify the reasoning about deterministic automata.
 *}
 
-definition
-  atmost_one :: "'a set \<Rightarrow> bool" where
-  "atmost_one S = (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x=y)"
+definition atmost_one :: "'a set \<Rightarrow> bool"
+  where "atmost_one S \<longleftrightarrow> (\<forall>x y. x\<in>S \<and> y\<in>S \<longrightarrow> x = y)"
 
 lemma atmost_one_empty: "S = {} \<Longrightarrow> atmost_one S"
   by (simp add: atmost_one_def)
diff -r 16235bb41881 -r 082d0972096b src/HOL/Library/Lattice_Algebras.thy
--- a/src/HOL/Library/Lattice_Algebras.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Library/Lattice_Algebras.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -9,20 +9,19 @@
 class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
 begin
 
-lemma add_inf_distrib_left:
-  "a + inf b c = inf (a + b) (a + c)"
-apply (rule antisym)
-apply (simp_all add: le_infI)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc [symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
-done
+lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"
+  apply (rule antisym)
+  apply (simp_all add: le_infI)
+  apply (rule add_le_imp_le_left [of "uminus a"])
+  apply (simp only: add_assoc [symmetric], simp)
+  apply rule
+  apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
+  done
 
-lemma add_inf_distrib_right:
-  "inf a b + c = inf (a + c) (b + c)"
+lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
 proof -
-  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
+  have "c + inf a b = inf (c+a) (c+b)"
+    by (simp add: add_inf_distrib_left)
   thus ?thesis by (simp add: add_commute)
 qed
 
@@ -31,19 +30,17 @@
 class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
 begin
 
-lemma add_sup_distrib_left:
-  "a + sup b c = sup (a + b) (a + c)" 
-apply (rule antisym)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc[symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
-apply (rule le_supI)
-apply (simp_all)
-done
+lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
+  apply (rule antisym)
+  apply (rule add_le_imp_le_left [of "uminus a"])
+  apply (simp only: add_assoc[symmetric], simp)
+  apply rule
+  apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
+  apply (rule le_supI)
+  apply (simp_all)
+  done
 
-lemma add_sup_distrib_right:
-  "sup a b + c = sup (a+c) (b+c)"
+lemma add_sup_distrib_right: "sup a b + c = sup (a+c) (b+c)"
 proof -
   have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
   thus ?thesis by (simp add: add_commute)
@@ -57,69 +54,61 @@
 subclass semilattice_inf_ab_group_add ..
 subclass semilattice_sup_ab_group_add ..
 
-lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
+lemmas add_sup_inf_distribs =
+  add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
 
 lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
 proof (rule inf_unique)
-  fix a b :: 'a
+  fix a b c :: 'a
   show "- sup (-a) (-b) \<le> a"
     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
       (simp, simp add: add_sup_distrib_left)
-next
-  fix a b :: 'a
   show "- sup (-a) (-b) \<le> b"
     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
       (simp, simp add: add_sup_distrib_left)
-next
-  fix a b c :: 'a
   assume "a \<le> b" "a \<le> c"
-  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
-    (simp add: le_supI)
+  then show "a \<le> - sup (-b) (-c)"
+    by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
 qed
-  
+
 lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
 proof (rule sup_unique)
-  fix a b :: 'a
+  fix a b c :: 'a
   show "a \<le> - inf (-a) (-b)"
     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
       (simp, simp add: add_inf_distrib_left)
-next
-  fix a b :: 'a
   show "b \<le> - inf (-a) (-b)"
     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
       (simp, simp add: add_inf_distrib_left)
-next
-  fix a b c :: 'a
   assume "a \<le> c" "b \<le> c"
-  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
-    (simp add: le_infI)
+  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
 qed
 
 lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
-by (simp add: inf_eq_neg_sup)
+  by (simp add: inf_eq_neg_sup)
 
 lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
-by (simp add: sup_eq_neg_inf)
+  by (simp add: sup_eq_neg_inf)
 
 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
 proof -
-  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
-  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
+  have "0 = - inf 0 (a-b) + inf (a-b) 0"
+    by (simp add: inf_commute)
+  hence "0 = sup 0 (b-a) + inf (a-b) 0"
+    by (simp add: inf_eq_neg_sup)
   hence "0 = (-a + sup a b) + (inf a b + (-b))"
-    by (simp add: add_sup_distrib_left add_inf_distrib_right)
-       (simp add: algebra_simps)
+    by (simp add: add_sup_distrib_left add_inf_distrib_right) (simp add: algebra_simps)
   thus ?thesis by (simp add: algebra_simps)
 qed
 
+
 subsection {* Positive Part, Negative Part, Absolute Value *}
 
-definition
-  nprt :: "'a \<Rightarrow> 'a" where
-  "nprt x = inf x 0"
+definition nprt :: "'a \<Rightarrow> 'a"
+  where "nprt x = inf x 0"
 
-definition
-  pprt :: "'a \<Rightarrow> 'a" where
-  "pprt x = sup x 0"
+definition pprt :: "'a \<Rightarrow> 'a"
+  where "pprt x = sup x 0"
 
 lemma pprt_neg: "pprt (- x) = - nprt x"
 proof -
@@ -137,27 +126,29 @@
 qed
 
 lemma prts: "a = pprt a + nprt a"
-by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
+  by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
 
 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
-by (simp add: pprt_def)
+  by (simp add: pprt_def)
 
 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
-by (simp add: nprt_def)
+  by (simp add: nprt_def)
 
 lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
-proof -
-  have a: "?l \<longrightarrow> ?r"
-    apply (auto)
+proof
+  assume ?l
+  then show ?r
+    apply -
     apply (rule add_le_imp_le_right[of _ "uminus b" _])
     apply (simp add: add_assoc)
     done
-  have b: "?r \<longrightarrow> ?l"
-    apply (auto)
+next
+  assume ?r
+  then show ?l
+    apply -
     apply (rule add_le_imp_le_right[of _ "b" _])
-    apply (simp)
+    apply simp
     done
-  from a b show ?thesis by blast
 qed
 
 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
@@ -181,7 +172,7 @@
     fix a::'a
     assume hyp: "sup a (-a) = 0"
     hence "sup a (-a) + a = a" by (simp)
-    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
+    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
     hence "sup (a+a) 0 <= a" by (simp)
     hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
   }
@@ -192,16 +183,22 @@
 qed
 
 lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
-apply (simp add: inf_eq_neg_sup)
-apply (simp add: sup_commute)
-apply (erule sup_0_imp_0)
-done
+  apply (simp add: inf_eq_neg_sup)
+  apply (simp add: sup_commute)
+  apply (erule sup_0_imp_0)
+  done
 
 lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule inf_0_imp_0) simp
+  apply rule
+  apply (erule inf_0_imp_0)
+  apply simp
+  done
 
 lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule sup_0_imp_0) simp
+  apply rule
+  apply (erule sup_0_imp_0)
+  apply simp
+  done
 
 lemma zero_le_double_add_iff_zero_le_single_add [simp]:
   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
@@ -218,39 +215,48 @@
   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
 qed
 
-lemma double_zero [simp]:
-  "a + a = 0 \<longleftrightarrow> a = 0"
+lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
 proof
   assume assm: "a + a = 0"
   then have "a + a + - a = - a" by simp
   then have "a + (a + - a) = - a" by (simp only: add_assoc)
   then have a: "- a = a" by simp
-  show "a = 0" apply (rule antisym)
-  apply (unfold neg_le_iff_le [symmetric, of a])
-  unfolding a apply simp
-  unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
-  unfolding assm unfolding le_less apply simp_all done
+  show "a = 0"
+    apply (rule antisym)
+    apply (unfold neg_le_iff_le [symmetric, of a])
+    unfolding a
+    apply simp
+    unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
+    unfolding assm
+    unfolding le_less
+    apply simp_all
+    done
 next
-  assume "a = 0" then show "a + a = 0" by simp
+  assume "a = 0"
+  then show "a + a = 0" by simp
 qed
 
-lemma zero_less_double_add_iff_zero_less_single_add [simp]:
-  "0 < a + a \<longleftrightarrow> 0 < a"
+lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
 proof (cases "a = 0")
-  case True then show ?thesis by auto
+  case True
+  then show ?thesis by auto
 next
-  case False then show ?thesis (*FIXME tune proof*)
-  unfolding less_le apply simp apply rule
-  apply clarify
-  apply rule
-  apply assumption
-  apply (rule notI)
-  unfolding double_zero [symmetric, of a] apply simp
-  done
+  case False
+  then show ?thesis
+    unfolding less_le
+    apply simp
+    apply rule
+    apply clarify
+    apply rule
+    apply assumption
+    apply (rule notI)
+    unfolding double_zero [symmetric, of a]
+    apply simp
+    done
 qed
 
 lemma double_add_le_zero_iff_single_add_le_zero [simp]:
-  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
+  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
 proof -
   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
   moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by simp
@@ -270,7 +276,7 @@
 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
 proof -
   from add_le_cancel_left [of "uminus a" "plus a a" zero]
-  have "(a <= -a) = (a+a <= 0)" 
+  have "(a <= -a) = (a+a <= 0)"
     by (simp add: add_assoc[symmetric])
   thus ?thesis by simp
 qed
@@ -278,28 +284,28 @@
 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
 proof -
   from add_le_cancel_left [of "uminus a" zero "plus a a"]
-  have "(-a <= a) = (0 <= a+a)" 
+  have "(-a <= a) = (0 <= a+a)"
     by (simp add: add_assoc[symmetric])
   thus ?thesis by simp
 qed
 
 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
+  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
 
 lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
+  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
 
 lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
+  unfolding le_iff_sup by (simp add: pprt_def sup_commute)
 
 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
+  unfolding le_iff_inf by (simp add: nprt_def inf_commute)
 
 lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
-unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
+  unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
 
 lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
-unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
+  unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
 
 end
 
@@ -320,8 +326,7 @@
   then have "0 \<le> sup a (- a)" unfolding abs_lattice .
   then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
   then show ?thesis
-    by (simp add: add_sup_inf_distribs sup_aci
-      pprt_def nprt_def diff_minus abs_lattice)
+    by (simp add: add_sup_inf_distribs sup_aci pprt_def nprt_def diff_minus abs_lattice)
 qed
 
 subclass ordered_ab_group_add_abs
@@ -329,8 +334,10 @@
   have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
   proof -
     fix a b
-    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
-    show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
+    have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
+      by (auto simp add: abs_lattice)
+    show "0 \<le> \<bar>a\<bar>"
+      by (rule add_mono [OF a b, simplified])
   qed
   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
     by (simp add: abs_lattice le_supI)
@@ -340,18 +347,25 @@
     by (auto simp add: abs_lattice)
   show "\<bar>-a\<bar> = \<bar>a\<bar>"
     by (simp add: abs_lattice sup_commute)
-  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
+  {
+    assume "a \<le> b"
+    then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
+      by (rule abs_leI)
+  }
   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
   proof -
     have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
       by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus)
-    have a:"a+b <= sup ?m ?n" by (simp)
-    have b:"-a-b <= ?n" by (simp) 
-    have c:"?n <= sup ?m ?n" by (simp)
-    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
+    have a: "a + b <= sup ?m ?n" by simp
+    have b: "- a - b <= ?n" by simp
+    have c: "?n <= sup ?m ?n" by simp
+    from b c have d: "-a-b <= sup ?m ?n" by (rule order_trans)
     have e:"-a-b = -(a+b)" by (simp add: diff_minus)
-    from a d e have "abs(a+b) <= sup ?m ?n" 
-      by (drule_tac abs_leI, auto)
+    from a d e have "abs(a+b) <= sup ?m ?n"
+      apply -
+      apply (drule abs_leI)
+      apply auto
+      done
     with g[symmetric] show ?thesis by simp
   qed
 qed
@@ -370,10 +384,10 @@
 lemma abs_if_lattice:
   fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
-by auto
+  by auto
 
 lemma estimate_by_abs:
-  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b" 
+  "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
 proof -
   assume "a+b <= c"
   then have "a <= c+(-b)" by (simp add: algebra_simps)
@@ -390,7 +404,7 @@
 
 end
 
-lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))" 
+lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
 proof -
   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
@@ -398,11 +412,11 @@
     by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
   {
     fix u v :: 'a
-    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
-              u * v = pprt a * pprt b + pprt a * nprt b + 
+    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
+              u * v = pprt a * pprt b + pprt a * nprt b +
                       nprt a * pprt b + nprt a * nprt b"
       apply (subst prts[of u], subst prts[of v])
-      apply (simp add: algebra_simps) 
+      apply (simp add: algebra_simps)
       done
   }
   note b = this[OF refl[of a] refl[of b]]
@@ -432,7 +446,7 @@
   show "abs (a*b) = abs a * abs b"
   proof -
     have s: "(0 <= a*b) | (a*b <= 0)"
-      apply (auto)    
+      apply (auto)
       apply (rule_tac split_mult_pos_le)
       apply (rule_tac contrapos_np[of "a*b <= 0"])
       apply (simp)
@@ -448,8 +462,8 @@
       then show ?thesis
         apply (simp_all add: mulprts abs_prts)
         apply (insert a)
-        apply (auto simp add: 
-          algebra_simps 
+        apply (auto simp add:
+          algebra_simps
           iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
           iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
           apply(drule (1) mult_nonneg_nonpos[of a b], simp)
@@ -470,15 +484,14 @@
 qed
 
 lemma mult_le_prts:
-  assumes
-  "a1 <= (a::'a::lattice_ring)"
-  "a <= a2"
-  "b1 <= b"
-  "b <= b2"
-  shows
-  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
-proof - 
-  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
+  assumes "a1 <= (a::'a::lattice_ring)"
+    and "a <= a2"
+    and "b1 <= b"
+    and "b <= b2"
+  shows "a * b <=
+    pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
+proof -
+  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
     apply (subst prts[symmetric])+
     apply simp
     done
@@ -496,7 +509,7 @@
       by simp
   qed
   moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
-  proof - 
+  proof -
     have "nprt a * pprt b <= nprt a2 * pprt b"
       by (simp add: mult_right_mono assms)
     moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
@@ -514,29 +527,33 @@
       by simp
   qed
   ultimately show ?thesis
-    by - (rule add_mono | simp)+
+    apply -
+    apply (rule add_mono | simp)+
+    done
 qed
 
 lemma mult_ge_prts:
-  assumes
-  "a1 <= (a::'a::lattice_ring)"
-  "a <= a2"
-  "b1 <= b"
-  "b <= b2"
-  shows
-  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
-proof - 
-  from assms have a1:"- a2 <= -a" by auto
-  from assms have a2: "-a <= -a1" by auto
-  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg] 
-  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
+  assumes "a1 <= (a::'a::lattice_ring)"
+    and "a <= a2"
+    and "b1 <= b"
+    and "b <= b2"
+  shows "a * b >=
+    nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
+proof -
+  from assms have a1:"- a2 <= -a"
+    by auto
+  from assms have a2: "-a <= -a1"
+    by auto
+  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
+  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
+    by simp
   then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
     by (simp only: minus_le_iff)
   then show ?thesis by simp
 qed
 
 instance int :: lattice_ring
-proof  
+proof
   fix k :: int
   show "abs k = sup k (- k)"
     by (auto simp add: sup_int_def)
diff -r 16235bb41881 -r 082d0972096b src/HOL/Library/Permutation.thy
--- a/src/HOL/Library/Permutation.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Library/Permutation.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -8,13 +8,12 @@
 imports Multiset
 begin
 
-inductive
-  perm :: "'a list => 'a list => bool"  ("_ <~~> _"  [50, 50] 50)
-  where
-    Nil  [intro!]: "[] <~~> []"
-  | swap [intro!]: "y # x # l <~~> x # y # l"
-  | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
-  | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
+inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)
+where
+  Nil [intro!]: "[] <~~> []"
+| swap [intro!]: "y # x # l <~~> x # y # l"
+| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
+| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
 
 lemma perm_refl [iff]: "l <~~> l"
   by (induct l) auto
@@ -22,7 +21,7 @@
 
 subsection {* Some examples of rule induction on permutations *}
 
-lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
+lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
   by (induct xs == "[]::'a list" ys pred: perm) simp_all
 
 
@@ -30,13 +29,13 @@
   \medskip This more general theorem is easier to understand!
   *}
 
-lemma perm_length: "xs <~~> ys ==> length xs = length ys"
+lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
   by (induct pred: perm) simp_all
 
-lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
+lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
   by (drule perm_length) auto
 
-lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
+lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
   by (induct pred: perm) auto
 
 
@@ -64,10 +63,10 @@
   apply (blast intro!: perm_append_single intro: perm_sym)
   done
 
-lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
+lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
   by (induct l) auto
 
-lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
+lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
   by (blast intro!: perm_append_swap perm_append1)
 
 
@@ -81,7 +80,7 @@
   apply (erule perm_sym [THEN perm_empty_imp])
   done
 
-lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
+lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
   by (induct pred: perm) auto
 
 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
@@ -93,29 +92,26 @@
 
 subsection {* Removing elements *}
 
-lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"
+lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
   by (induct ys) auto
 
 
 text {* \medskip Congruence rule *}
 
-lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"
+lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
   by (induct pred: perm) auto
 
 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
   by auto
 
-lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
+lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
   by (drule_tac z = z in perm_remove_perm) auto
 
 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   by (blast intro: cons_perm_imp_perm)
 
-lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
-  apply (induct zs arbitrary: xs ys rule: rev_induct)
-   apply (simp_all (no_asm_use))
-  apply blast
-  done
+lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
+  by (induct zs arbitrary: xs ys rule: rev_induct) auto
 
 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   by (blast intro: append_perm_imp_perm perm_append1)
@@ -135,38 +131,38 @@
   apply (induct_tac xs, auto)
   apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
   apply (subgoal_tac "a \<in> set x")
-  apply (drule_tac z=a in perm.Cons)
+  apply (drule_tac z = a in perm.Cons)
   apply (erule perm.trans, rule perm_sym, erule perm_remove)
   apply (drule_tac f=set_of in arg_cong, simp)
   done
 
-lemma multiset_of_le_perm_append:
-    "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
+lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   apply (insert surj_multiset_of, drule surjD)
   apply (blast intro: sym)+
   done
 
-lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
+lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
   by (metis multiset_of_eq_perm multiset_of_eq_setD)
 
-lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
+lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
   apply (induct pred: perm)
      apply simp_all
    apply fastforce
   apply (metis perm_set_eq)
   done
 
-lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
+lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
   apply (induct xs arbitrary: ys rule: length_induct)
-  apply (case_tac "remdups xs", simp, simp)
-  apply (subgoal_tac "a : set (remdups ys)")
+  apply (case_tac "remdups xs")
+   apply simp_all
+  apply (subgoal_tac "a \<in> set (remdups ys)")
    prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
   apply (drule split_list) apply(elim exE conjE)
   apply (drule_tac x=list in spec) apply(erule impE) prefer 2
    apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
     apply simp
-    apply (subgoal_tac "a#list <~~> a#ysa@zs")
+    apply (subgoal_tac "a # list <~~> a # ysa @ zs")
      apply (metis Cons_eq_appendI perm_append_Cons trans)
     apply (metis Cons Cons_eq_appendI distinct.simps(2)
       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
@@ -180,21 +176,23 @@
    apply (rule length_remdups_leq)
   done
 
-lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
+lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
 
 lemma permutation_Ex_bij:
   assumes "xs <~~> ys"
   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
 using assms proof induct
-  case Nil then show ?case unfolding bij_betw_def by simp
+  case Nil
+  then show ?case unfolding bij_betw_def by simp
 next
   case (swap y x l)
   show ?case
   proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
     show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
       by (auto simp: bij_betw_def)
-    fix i assume "i < length(y#x#l)"
+    fix i
+    assume "i < length(y#x#l)"
     show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
       by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
   qed
@@ -202,19 +200,21 @@
   case (Cons xs ys z)
   then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
-  let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
+  let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
   show ?case
   proof (intro exI[of _ ?f] allI conjI impI)
     have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
             "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
       by (simp_all add: lessThan_Suc_eq_insert_0)
-    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *
+    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
+      unfolding *
     proof (rule bij_betw_combine)
       show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
         using bij unfolding bij_betw_def
         by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
     qed (auto simp: bij_betw_def)
-    fix i assume "i < length (z#xs)"
+    fix i
+    assume "i < length (z#xs)"
     then show "(z # xs) ! i = (z # ys) ! (?f i)"
       using perm by (cases i) auto
   qed
@@ -224,13 +224,13 @@
     bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
   show ?case
-  proof (intro exI[of _ "g\<circ>f"] conjI allI impI)
+  proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
     show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
       using bij by (rule bij_betw_trans)
     fix i assume "i < length xs"
     with bij have "f i < length ys" unfolding bij_betw_def by force
     with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
-      using trans(1,3)[THEN perm_length] perm by force
+      using trans(1,3)[THEN perm_length] perm by auto
   qed
 qed
 
diff -r 16235bb41881 -r 082d0972096b src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -29,7 +29,8 @@
   apply (cases "b=0")
   defer
   apply (rule divide_nonneg_pos)
-  using assms apply auto
+  using assms
+  apply auto
   done
 
 lemma brouwer_compactness_lemma:
@@ -95,12 +96,15 @@
   shows "setsum g C = setsum g A + setsum g B"
   using setsum_Un_disjoint[OF assms(1-3)] and assms(4) by auto
 
-lemma kuhn_counting_lemma: assumes "finite faces" "finite simplices"
-  "\<forall>f\<in>faces. bnd f  \<longrightarrow> (card {s \<in> simplices. face f s} = 1)"
-  "\<forall>f\<in>faces. \<not> bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 2)"
-  "\<forall>s\<in>simplices. compo s  \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 1)"
-  "\<forall>s\<in>simplices. \<not> compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 0) \<or>
-                             (card {f \<in> faces. face f s \<and> compo' f} = 2)"
+lemma kuhn_counting_lemma:
+  assumes
+    "finite faces"
+    "finite simplices"
+    "\<forall>f\<in>faces. bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 1)"
+    "\<forall>f\<in>faces. \<not> bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 2)"
+    "\<forall>s\<in>simplices. compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 1)"
+    "\<forall>s\<in>simplices. \<not> compo s \<longrightarrow>
+      (card {f \<in> faces. face f s \<and> compo' f} = 0) \<or> (card {f \<in> faces. face f s \<and> compo' f} = 2)"
     "odd(card {f \<in> faces. compo' f \<and> bnd f})"
   shows "odd(card {s \<in> simplices. compo s})"
 proof -
@@ -117,28 +121,39 @@
     using assms(1)
     apply (auto simp add: card_Un_Int, auto simp add:conj_commute)
     done
-  have lem2:"setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f j}) simplices =
-              1 * card {f \<in> faces. compo' f \<and> bnd f}"
-       "setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> \<not> bnd f}. face f j}) simplices =
-              2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}"
+  have lem2:
+    "setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f j}) simplices =
+      1 * card {f \<in> faces. compo' f \<and> bnd f}"
+    "setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> \<not> bnd f}. face f j}) simplices =
+      2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}"
     apply(rule_tac[!] setsum_multicount)
     using assms
     apply auto
     done
-  have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
-    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices.   compo s}+
-    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s}"
-    apply(rule setsum_Un_disjoint') using assms(2) by auto
-  have lem4:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s}
-    = setsum (\<lambda>s. 1) {s \<in> simplices. compo s}"
-    apply(rule setsum_cong2) using assms(5) by auto
+  have lem3:
+    "setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
+      setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices.   compo s}+
+      setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s}"
+    apply (rule setsum_Un_disjoint')
+    using assms(2)
+    apply auto
+    done
+  have lem4: "setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s} =
+    setsum (\<lambda>s. 1) {s \<in> simplices. compo s}"
+    apply (rule setsum_cong2)
+    using assms(5)
+    apply auto
+    done
   have lem5: "setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s} =
     setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
            {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 0)} +
     setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
            {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}"
-    apply(rule setsum_Un_disjoint') using assms(2,6) by auto
-  have *:"int (\<Sum>s\<in>{s \<in> simplices. compo s}. card {f \<in> faces. face f s \<and> compo' f}) =
+    apply (rule setsum_Un_disjoint')
+    using assms(2,6)
+    apply auto
+    done
+  have *: "int (\<Sum>s\<in>{s \<in> simplices. compo s}. card {f \<in> faces. face f s \<and> compo' f}) =
     int (card {f \<in> faces. compo' f \<and> bnd f} + 2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}) -
     int (card {s \<in> simplices. \<not> compo s \<and> card {f \<in> faces. face f s \<and> compo' f} = 2} * 2)"
     using lem1[unfolded lem3 lem2 lem5] by auto
@@ -175,7 +190,8 @@
     apply (rule as(2)[rule_format]) using as(1)
     apply auto
     done
-  show "card s = Suc 0" unfolding * using card_insert by auto
+  show "card s = Suc 0"
+    unfolding * using card_insert by auto
 qed auto
 
 lemma card_2_exists: "card s = 2 \<longleftrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. (z = x) \<or> (z = y)))"
@@ -227,7 +243,8 @@
     apply (rule set_eqI)
     unfolding singleton_iff
     apply (rule, rule inj[unfolded inj_on_def, rule_format])
-    unfolding a using a(2) and assms and inj[unfolded inj_on_def] apply auto
+    unfolding a using a(2) and assms and inj[unfolded inj_on_def]
+    apply auto
     done
   show ?thesis
     apply (rule image_lemma_0)
@@ -245,7 +262,8 @@
   then show ?thesis
     apply -
     apply (rule disjI1, rule image_lemma_0)
-    using assms(1) apply auto
+    using assms(1)
+    apply auto
     done
 next
   let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}"
@@ -317,7 +335,9 @@
     using assms(3) by (auto intro: card_ge_0_finite)
   show "finite {f. \<exists>s\<in>simplices. face f s}"
     unfolding assms(2)[rule_format] and *
-    apply (rule finite_UN_I[OF assms(1)]) using ** apply auto
+    apply (rule finite_UN_I[OF assms(1)])
+    using **
+    apply auto
     done
   have *: "\<And>P f s. s\<in>simplices \<Longrightarrow> (f \<in> {f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}}) \<and>
     (\<exists>a\<in>s. (f = s - {a})) \<and> P f \<longleftrightarrow> (\<exists>a\<in>s. (f = s - {a}) \<and> P f)" by auto
@@ -334,7 +354,8 @@
   show "rl ` s = {0..n+1} \<Longrightarrow> card ?S = 1" "rl ` s \<noteq> {0..n+1} \<Longrightarrow> card ?S = 0 \<or> card ?S = 2"
     unfolding S
     apply(rule_tac[!] image_lemma_1 image_lemma_2)
-    using ** assms(4) and s apply auto
+    using ** assms(4) and s
+    apply auto
     done
 qed
 
@@ -427,9 +448,9 @@
 
 lemma pointwise_antisym:
   fixes x :: "nat \<Rightarrow> nat"
-  shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> (x = y)"
+  shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> x = y"
   apply (rule, rule ext, erule conjE)
-  apply (erule_tac x=xa in allE)+
+  apply (erule_tac x = xa in allE)+
   apply auto
   done
 
@@ -489,11 +510,12 @@
     apply (rule pointwise_minimal_pointwise_maximal(1)[OF assms(1-2)])
     apply (rule, rule)
     apply (drule_tac assms(3)[rule_format], assumption)
-    using kle_imp_pointwise apply auto
+    using kle_imp_pointwise
+    apply auto
     done
   then guess a .. note a = this
   show ?thesis
-    apply (rule_tac x=a in bexI)
+    apply (rule_tac x = a in bexI)
   proof
     fix x
     assume "x \<in> s"
@@ -502,13 +524,14 @@
       apply -
     proof (erule disjE)
       assume "kle n x a"
-      hence "x = a"
+      then have "x = a"
         apply -
         unfolding pointwise_antisym[symmetric]
         apply (drule kle_imp_pointwise)
-        using a(2)[rule_format,OF `x\<in>s`] apply auto
+        using a(2)[rule_format,OF `x\<in>s`]
+        apply auto
         done
-      thus ?thesis using kle_refl by auto
+      then show ?thesis using kle_refl by auto
     qed
   qed (insert a, auto)
 qed
@@ -563,16 +586,18 @@
     "m1 \<le> card {k\<in>{1..n}. x k < y k}"
     "m2 \<le> card {k\<in>{1..n}. y k < z k}"
   shows "kle n x z \<and> m1 + m2 \<le> card {k\<in>{1..n}. x k < z k}"
-    apply (rule, rule kle_trans[OF assms(1-3)])
+  apply (rule, rule kle_trans[OF assms(1-3)])
 proof -
   have "\<And>j. x j < y j \<Longrightarrow> x j < z j"
     apply (rule less_le_trans)
-    using kle_imp_pointwise[OF assms(2)] apply auto
+    using kle_imp_pointwise[OF assms(2)]
+    apply auto
     done
   moreover
   have "\<And>j. y j < z j \<Longrightarrow> x j < z j"
     apply (rule le_less_trans)
-    using kle_imp_pointwise[OF assms(1)] apply auto
+    using kle_imp_pointwise[OF assms(1)]
+    apply auto
     done
   ultimately
   have *: "{k\<in>{1..n}. x k < y k} \<union> {k\<in>{1..n}. y k < z k} = {k\<in>{1..n}. x k < z k}"
@@ -653,12 +678,14 @@
       have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {1..n}. x k < b k}"
         apply (rule kle_strict_set)
         apply (rule a(2)[rule_format])
-        using a and xb apply auto
+        using a and xb
+        apply auto
         done
       thus ?thesis
         apply (rule_tac x=a in bexI, rule_tac x=b in bexI)
         using kle_range_combine[OF a(2)[rule_format] xb(3) Suc(5)[rule_format], of 1 "m"]
-        using a(1) xb(1-2) apply auto
+        using a(1) xb(1-2)
+        apply auto
         done
     next
       case True
@@ -691,7 +718,8 @@
   show ?thesis
     unfolding kle_def
     apply (rule_tac x="kk1 \<union> kk2" in exI)
-    apply (rule) defer
+    apply rule
+    defer
   proof
     fix i
     show "c i = a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)"
@@ -742,7 +770,7 @@
 
 lemma kle_adjacent:
   assumes "\<forall>j. b j = (if j = k then a(j) + 1 else a j)" "kle n a x" "kle n x b"
-  shows "(x = a) \<or> (x = b)"
+  shows "x = a \<or> x = b"
 proof (cases "x k = a k")
   case True
   show ?thesis
@@ -757,17 +785,22 @@
       using True
       apply auto
       done
-    thus "x j = a j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] by auto
+    then show "x j = a j"
+      using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] by auto
   qed
 next
   case False
   show ?thesis apply(rule disjI2,rule ext)
   proof -
     fix j
-    have "x j \<ge> b j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]]
-      unfolding assms(1)[rule_format] apply-apply(cases "j=k")
+    have "x j \<ge> b j"
+      using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]]
+      unfolding assms(1)[rule_format]
+      apply -
+      apply(cases "j = k")
       using False by auto
-    thus "x j = b j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]]
+    then show "x j = b j"
+      using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]]
     by auto
   qed
 qed
@@ -818,16 +851,18 @@
     using `s\<noteq>{}` assm using kle_minimal[of s n] by auto
   obtain b where b: "b \<in> s" "\<forall>x\<in>s. kle n x b"
     using `s\<noteq>{}` assm using kle_maximal[of s n] by auto
-  obtain c d where c_d: "c\<in>s" "d\<in>s" "kle n c d" "n \<le> card {k \<in> {1..n}. c k < d k}"
+  obtain c d where c_d: "c \<in> s" "d \<in> s" "kle n c d" "n \<le> card {k \<in> {1..n}. c k < d k}"
     using kle_range_induct[of s n n] using assm by auto
   have "kle n c b \<and> n \<le> card {k \<in> {1..n}. c k < b k}"
     apply (rule kle_range_combine_r[where y=d])
-    using c_d a b apply auto
+    using c_d a b
+    apply auto
     done
   hence "kle n a b \<and> n \<le> card {k\<in>{1..n}. a(k) < b(k)}"
     apply -
     apply (rule kle_range_combine_l[where y=c])
-    using a `c\<in>s` `b\<in>s` apply auto
+    using a `c \<in> s` `b \<in> s`
+    apply auto
     done
   moreover
   have "card {1..n} \<ge> card {k\<in>{1..n}. a(k) < b(k)}"
@@ -897,16 +932,18 @@
   case False
   then obtain b where b: "b\<in>s" "\<not> kle n b a" "\<forall>x\<in>{x \<in> s. \<not> kle n x a}. kle n b x"
     using kle_minimal[of "{x\<in>s. \<not> kle n x a}" n] and assm by auto
-  hence  **: "1 \<le> card {k\<in>{1..n}. a k < b k}"
+  then have **: "1 \<le> card {k\<in>{1..n}. a k < b k}"
     apply -
     apply (rule kle_strict_set)
-    using assm(6) and `a\<in>s` apply (auto simp add:kle_refl)
+    using assm(6) and `a\<in>s`
+    apply (auto simp add:kle_refl)
     done
 
   let ?kle1 = "{x \<in> s. \<not> kle n x a}"
   have "card ?kle1 > 0"
     apply (rule ccontr)
-    using assm(2) and False apply auto
+    using assm(2) and False
+    apply auto
     done
   hence sizekle1: "card ?kle1 = Suc (card ?kle1 - 1)"
     using assm(2) by auto
@@ -917,7 +954,8 @@
   let ?kle2 = "{x \<in> s. kle n x a}"
   have "card ?kle2 > 0"
     apply (rule ccontr)
-    using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2) apply auto
+    using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2)
+    apply auto
     done
   hence sizekle2: "card ?kle2 = Suc (card ?kle2 - 1)"
     using assm(2) by auto
@@ -940,11 +978,13 @@
       by auto
     have "kle n e a \<and> card {x \<in> s. kle n x a} - 1 \<le> card {k \<in> {1..n}. e k < a k}"
       apply (rule kle_range_combine_r[where y=f])
-      using e_f using `a\<in>s` assm(6) apply auto
+      using e_f using `a\<in>s` assm(6)
+      apply auto
       done
     moreover have "kle n b d \<and> card {x \<in> s. \<not> kle n x a} - 1 \<le> card {k \<in> {1..n}. b k < d k}"
       apply (rule kle_range_combine_l[where y=c])
-      using c_d using assm(6) and b apply auto
+      using c_d using assm(6) and b
+      apply auto
       done
     hence "kle n a d \<and> 2 + (card {x \<in> s. \<not> kle n x a} - 1) \<le> card {k \<in> {1..n}. a k < d k}"
       apply -
@@ -975,24 +1015,27 @@
     have kkk: "k \<in> kk"
       apply (rule ccontr)
       using k(1)
-      unfolding kk apply auto
+      unfolding kk
+      apply auto
       done
     show "b j = (if j = k then a j + 1 else a j)"
     proof (cases "j \<in> kk")
       case True
-      hence "j = k"
-      apply - apply (rule k(2)[rule_format])
-      using kk_raw kkk apply auto
-      done
-      thus ?thesis unfolding kk using kkk by auto
+      then have "j = k"
+        apply -
+        apply (rule k(2)[rule_format])
+        using kk_raw kkk
+        apply auto
+        done
+      then show ?thesis unfolding kk using kkk by auto
     next
       case False
-      hence "j \<noteq> k"
+      then have "j \<noteq> k"
         using k(2)[rule_format, of j k] and kk_raw kkk by auto
-      thus ?thesis unfolding kk using kkk and False
+      then show ?thesis unfolding kk using kkk and False
         by auto
     qed
-  qed(insert k(1) `b\<in>s`, auto)
+  qed (insert k(1) `b\<in>s`, auto)
 qed
 
 lemma ksimplex_predecessor:
@@ -1009,13 +1052,15 @@
   hence **: "1 \<le> card {k\<in>{1..n}. a k > b k}"
     apply -
     apply (rule kle_strict_set)
-    using assm(6) and `a\<in>s` apply (auto simp add: kle_refl)
+    using assm(6) and `a\<in>s`
+    apply (auto simp add: kle_refl)
     done
 
   let ?kle1 = "{x \<in> s. \<not> kle n a x}"
   have "card ?kle1 > 0"
     apply (rule ccontr)
-    using assm(2) and False apply auto
+    using assm(2) and False
+    apply auto
     done
   hence sizekle1: "card ?kle1 = Suc (card ?kle1 - 1)"
     using assm(2) by auto
@@ -1026,7 +1071,8 @@
   let ?kle2 = "{x \<in> s. kle n a x}"
   have "card ?kle2 > 0"
     apply (rule ccontr)
-    using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2) apply auto
+    using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2)
+    apply auto
     done
   hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)"
     using assm(2) by auto
@@ -1049,11 +1095,13 @@
       by auto
     have "kle n a e \<and> card {x \<in> s. kle n a x} - 1 \<le> card {k \<in> {1..n}. e k > a k}"
       apply (rule kle_range_combine_l[where y=f])
-      using e_f and `a\<in>s` assm(6) apply auto
+      using e_f and `a\<in>s` assm(6)
+      apply auto
       done
     moreover have "kle n d b \<and> card {x \<in> s. \<not> kle n a x} - 1 \<le> card {k \<in> {1..n}. b k > d k}"
       apply (rule kle_range_combine_r[where y=c])
-      using c_d and assm(6) and b apply auto
+      using c_d and assm(6) and b
+      apply auto
       done
     hence "kle n d a \<and> (card {x \<in> s. \<not> kle n a x} - 1) + 2 \<le> card {k \<in> {1..n}. a k > d k}"
       apply -
@@ -1063,7 +1111,8 @@
     ultimately have "kle n d e \<and> (card ?kle1 - 1 + 2) + (card ?kle2 - 1) \<le> card {k\<in>{1..n}. e k > d k}"
       apply -
       apply (rule kle_range_combine[where y=a])
-      using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply blast+
+      using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`]
+      apply blast+
       done
     moreover have "card {k \<in> {1..n}. e k > d k} \<le> card {1..n}"
       by (rule card_mono) auto
@@ -1092,7 +1141,8 @@
       hence "j = k"
         apply -
         apply (rule k(2)[rule_format])
-        using kk_raw kkk apply auto
+        using kk_raw kkk
+        apply auto
         done
       thus ?thesis unfolding kk using kkk by auto
     next
@@ -1115,14 +1165,14 @@
   using assms
   apply -
 proof (induct m arbitrary: s)
-  have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}"
+  case 0
+  have [simp]: "{f. \<forall>x. f x = d} = {\<lambda>x. d}"
     apply (rule set_eqI,rule)
     unfolding mem_Collect_eq
     apply (rule, rule ext)
     apply auto
     done
-  case 0
-  thus ?case by(auto simp add: *)
+  from 0 show ?case by auto
 next
   case (Suc m)
   guess a using card_eq_SucD[OF Suc(4)] ..
@@ -1246,9 +1296,8 @@
       then guess k unfolding kle_def .. note k = this
       hence *: "n + 1 \<notin> k" using xyp by auto
       have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})"
-        apply (rule ccontr)
-        unfolding not_not
-        apply(erule bexE)
+        apply (rule notI)
+        apply (erule bexE)
       proof -
         fix x
         assume as: "x \<in> k" "x \<notin> {1..n}"
@@ -1268,23 +1317,25 @@
       hence "kle (n + 1) y x"
         using ksimplexD(6)[OF sa(1),rule_format, of x y] and as by auto
       then guess k unfolding kle_def .. note k = this
-      hence *: "n + 1 \<notin> k" using xyp by auto
-      hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})"
+      then have *: "n + 1 \<notin> k" using xyp by auto
+      then have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})"
         apply -
-        apply (rule ccontr)
-        unfolding not_not
+        apply (rule notI)
         apply (erule bexE)
       proof -
         fix x
         assume as: "x \<in> k" "x \<notin> {1..n}"
         have "x \<noteq> n + 1" using as and * by auto
-        thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto
+        then show False using as and k[THEN conjunct1,unfolded subset_eq] by auto
       qed
-      thus ?thesis
+      then show ?thesis
         apply -
         apply (rule disjI2)
         unfolding kle_def
-        using k apply (rule_tac x = k in exI) by auto
+        using k
+        apply (rule_tac x = k in exI)
+        apply auto
+        done
     qed
   next
     fix x j
@@ -1298,13 +1349,14 @@
   qed (insert sa ksimplexD[OF sa(1)], auto)
 next
   assume ?rs note rs=ksimplexD[OF this]
-  guess a b apply (rule ksimplex_extrema[OF `?rs`]) . note ab = this
+  guess a b by (rule ksimplex_extrema[OF `?rs`]) note ab = this
   def c \<equiv> "\<lambda>i. if i = (n + 1) then p - 1 else a i"
   have "c \<notin> f"
-    apply (rule ccontr) unfolding not_not
+    apply (rule notI)
     apply (drule assms(2)[rule_format])
     unfolding c_def
-    using assms(1) apply auto
+    using assms(1)
+    apply auto
     done
   thus ?ls
     apply (rule_tac x = "insert c f" in exI, rule_tac x = c in exI)
@@ -1326,12 +1378,16 @@
       show ?thesis
         unfolding True c_def
         apply (cases "j=n+1")
-        using ab(1) and rs(4) apply auto
+        using ab(1) and rs(4)
+        apply auto
         done
     qed (insert x rs(4), auto simp add:c_def)
     show "j \<notin> {1..n + 1} \<longrightarrow> x j = p"
       apply (cases "x = c")
-      using x ab(1) rs(5) unfolding c_def by auto
+      using x ab(1) rs(5)
+      unfolding c_def
+      apply auto
+      done
     {
       fix z
       assume z: "z \<in> insert c f"
@@ -1359,9 +1415,9 @@
     assume y: "y \<in> insert c f"
     show "kle (n + 1) x y \<or> kle (n + 1) y x"
     proof (cases "x = c \<or> y = c")
-      case False hence **:"x\<in>f" "y\<in>f" using x y by auto
+      case False hence **: "x \<in> f" "y \<in> f" using x y by auto
       show ?thesis using rs(6)[rule_format,OF **]
-        by(auto dest: kle_Suc)
+        by (auto dest: kle_Suc)
     qed (insert * x y, auto)
   qed (insert rs, auto)
 qed
@@ -1377,7 +1433,9 @@
     apply (rule ccontr)
     using *[OF assms(3), of a0 a1]
     unfolding assms(6)[THEN spec[where x=j]]
-    using assms(1-2,4-5) by auto
+    using assms(1-2,4-5)
+    apply auto
+    done
 qed
 
 lemma ksimplex_fix_plane_0:
@@ -1399,11 +1457,11 @@
 proof (rule ccontr)
   note s = ksimplexD[OF assms(1),rule_format]
   assume as: "a \<noteq> a0"
-  hence *: "a0 \<in> s - {a}"
+  then have *: "a0 \<in> s - {a}"
     using assms(5) by auto
-  hence "a1 = a"
+  then have "a1 = a"
     using ksimplex_fix_plane[OF assms(2-)] by auto
-  thus False
+  then show False
     using as and assms(3,5) and assms(7)[rule_format,of j]
     unfolding assms(4)[rule_format,OF *]
     using s(4)[OF assms(6), of j]
@@ -1422,7 +1480,8 @@
     guess a0 a1 by (rule ksimplex_extrema_strong[OF assms(1,3)]) note exta = this[rule_format]
     have a:"a = a1"
       apply (rule ksimplex_fix_plane_0[OF assms(2,4-5)])
-      using exta(1-2,5) apply auto
+      using exta(1-2,5)
+      apply auto
       done
     moreover
     guess b0 b1 by (rule ksimplex_extrema_strong[OF goal1(1) assms(3)])
@@ -1430,7 +1489,9 @@
     have a': "a' = b1"
       apply (rule ksimplex_fix_plane_0[OF goal1(2) assms(4), of b0])
       unfolding goal1(3)
-      using assms extb goal1 apply auto done
+      using assms extb goal1
+      apply auto
+      done
     moreover
     have "b0 = a0"
       unfolding kle_antisym[symmetric, of b0 a0 n]
@@ -1446,7 +1507,8 @@
     show "s' = s"
       apply -
       apply (rule *[of _ a1 b1])
-      using exta(1-2) extb(1-2) goal1 apply auto
+      using exta(1-2) extb(1-2) goal1
+      apply auto
       done
   qed
   show ?thesis
@@ -1457,7 +1519,8 @@
     defer
     apply (erule conjE bexE)+
     apply (rule_tac a'=b in **)
-    using assms(1,2) apply auto
+    using assms(1,2)
+    apply auto
     done
 qed
 
@@ -2774,46 +2837,148 @@
 
 lemma kuhn_induction:
   assumes "0 < p" "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
-                  "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
-        "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
-  shows "odd (card {s. ksimplex p (n+1) s \<and>((reduced lab (n+1)) `  s = {0..n+1})})" proof-
-  have *:"\<And>s t. odd (card s) \<Longrightarrow> s = t \<Longrightarrow> odd (card t)" "\<And>s f. (\<And>x. f x \<le> n +1 ) \<Longrightarrow> f ` s \<subseteq> {0..n+1}" by auto
-  show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling)
-    apply(rule *(1)[OF assms(4)]) apply(rule set_eqI) unfolding mem_Collect_eq apply(rule,erule conjE) defer apply(rule) proof-(*(rule,rule)*)
-    fix f assume as:"ksimplex p n f" "reduced lab n ` f = {0..n}"
-    have *:"\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = 0 \<longrightarrow> lab x j = 0" "\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = p \<longrightarrow> lab x j = 1"
+    "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
+    "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
+  shows "odd (card {s. ksimplex p (n+1) s \<and>((reduced lab (n+1)) `  s = {0..n+1})})"
+proof -
+  have *: "\<And>s t. odd (card s) \<Longrightarrow> s = t \<Longrightarrow> odd (card t)"
+    "\<And>s f. (\<And>x. f x \<le> n +1 ) \<Longrightarrow> f ` s \<subseteq> {0..n+1}" by auto
+  show ?thesis
+    apply (rule kuhn_simplex_lemma[unfolded mem_Collect_eq])
+    apply (rule, rule, rule *, rule reduced_labelling)
+    apply (rule *(1)[OF assms(4)])
+    apply (rule set_eqI)
+    unfolding mem_Collect_eq
+    apply (rule, erule conjE)
+    defer
+    apply rule
+  proof -
+    fix f
+    assume as: "ksimplex p n f" "reduced lab n ` f = {0..n}"
+    have *: "\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = 0 \<longrightarrow> lab x j = 0"
+      "\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = p \<longrightarrow> lab x j = 1"
       using assms(2-3) using as(1)[unfolded ksimplex_def] by auto
-    have allp:"\<forall>x\<in>f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto
-    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_nonzero)
-        defer using assms(3) using as(1)[unfolded ksimplex_def] by auto
-      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto }
-    hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[symmetric] apply- apply(rule set_eqI) unfolding image_iff by auto
-    moreover guess s using as(1)[unfolded simplex_top_face[OF assms(1) allp,symmetric]] .. then guess a ..
+    have allp: "\<forall>x\<in>f. x (n + 1) = p"
+      using assms(2) using as(1)[unfolded ksimplex_def] by auto
+    {
+      fix x
+      assume "x \<in> f"
+      hence "reduced lab (n + 1) x < n + 1"
+        apply -
+        apply (rule reduced_labelling_nonzero)
+        defer using assms(3) using as(1)[unfolded ksimplex_def]
+        apply auto
+        done
+      hence "reduced lab (n + 1) x = reduced lab n x"
+        apply -
+        apply (rule reduced_labelling_Suc)
+        using reduced_labelling(1)
+        apply auto
+        done
+    }
+    hence "reduced lab (n + 1) ` f = {0..n}"
+      unfolding as(2)[symmetric]
+      apply -
+      apply (rule set_eqI)
+      unfolding image_iff
+      apply auto
+      done
+    moreover guess s using as(1)[unfolded simplex_top_face[OF assms(1) allp,symmetric]] ..
+    then guess a ..
     ultimately show "\<exists>s a. ksimplex p (n + 1) s \<and>
       a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex)
-      apply(rule_tac x=s in exI,rule_tac x=a in exI) unfolding complete_face_top[OF *] using allp as(1) by auto
-  next fix f assume as:"\<exists>s a. ksimplex p (n + 1) s \<and>
+      apply (rule_tac x = s in exI)
+      apply (rule_tac x = a in exI)
+      unfolding complete_face_top[OF *]
+      using allp as(1)
+      apply auto
+      done
+  next
+    fix f
+    assume as: "\<exists>s a. ksimplex p (n + 1) s \<and>
       a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex)
-    then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
-    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x \<in> reduced lab (n + 1) ` f" by auto
-      hence "reduced lab (n + 1) x < n + 1" using sa(4) by auto
-      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc)
-        using reduced_labelling(1) by auto }
-    thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[symmetric] apply-apply(rule set_eqI) unfolding image_iff by auto
-    have *:"\<forall>x\<in>f. x (n + 1) = p" proof(cases "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0")
-      case True then guess j .. hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab (n + 1) x \<noteq> j - 1" apply-apply(rule reduced_labelling_zero) apply assumption
-        apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover
+    then guess s ..
+    then guess a
+      apply -
+      apply (erule exE,(erule conjE)+)
+      done
+    note sa = this
+    {
+      fix x
+      assume "x \<in> f"
+      hence "reduced lab (n + 1) x \<in> reduced lab (n + 1) ` f"
+        by auto
+      hence "reduced lab (n + 1) x < n + 1"
+        using sa(4) by auto
+      hence "reduced lab (n + 1) x = reduced lab n x"
+        apply -
+        apply (rule reduced_labelling_Suc)
+        using reduced_labelling(1)
+        apply auto
+        done
+    }
+    thus part1: "reduced lab n ` f = {0..n}"
+      unfolding sa(4)[symmetric]
+      apply -
+      apply (rule set_eqI)
+      unfolding image_iff
+      apply auto
+      done
+    have *: "\<forall>x\<in>f. x (n + 1) = p"
+    proof (cases "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0")
+      case True
+      then guess j ..
+      hence "\<And>x. x \<in> f \<Longrightarrow> reduced lab (n + 1) x \<noteq> j - 1"
+        apply -
+        apply (rule reduced_labelling_zero)
+        apply assumption
+        apply (rule assms(2)[rule_format])
+        using sa(1)[unfolded ksimplex_def]
+        unfolding sa
+        apply auto
+        done
+      moreover
       have "j - 1 \<in> {0..n}" using `j\<in>{1..n+1}` by auto
-      ultimately have False unfolding sa(4)[symmetric] unfolding image_iff by fastforce thus ?thesis by auto next
-      case False hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p" using sa(5) by fastforce then guess j .. note j=this
-      thus ?thesis proof(cases "j = n+1")
-        case False hence *:"j\<in>{1..n}" using j by auto
-        hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab n x < j" apply(rule reduced_labelling_nonzero) proof- fix x assume "x\<in>f"
-          hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)])
-            using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j \<noteq> 0" by auto qed
-        moreover have "j\<in>{0..n}" using * by auto
-        ultimately have False unfolding part1[symmetric] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed
-    thus "ksimplex p n f" using as unfolding simplex_top_face[OF assms(1) *,symmetric] by auto qed qed
+      ultimately have False
+        unfolding sa(4)[symmetric]
+        unfolding image_iff
+        by fastforce
+      thus ?thesis by auto
+    next
+      case False
+      hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p"
+        using sa(5) by fastforce then guess j .. note j=this
+      thus ?thesis
+      proof (cases "j = n + 1")
+        case False hence *: "j \<in> {1..n}"
+          using j by auto
+        hence "\<And>x. x \<in> f \<Longrightarrow> reduced lab n x < j"
+          apply (rule reduced_labelling_nonzero)
+        proof -
+          fix x
+          assume "x \<in> f"
+          hence "lab x j = 1"
+            apply -
+            apply (rule assms(3)[rule_format,OF j(1)])
+            using sa(1)[unfolded ksimplex_def]
+            using j
+            unfolding sa
+            apply auto
+            done
+          thus "lab x j \<noteq> 0" by auto
+        qed
+        moreover have "j \<in> {0..n}" using * by auto
+        ultimately have False
+          unfolding part1[symmetric]
+          using * unfolding image_iff
+          by auto
+        thus ?thesis by auto
+      qed auto
+    qed
+    thus "ksimplex p n f"
+      using as unfolding simplex_top_face[OF assms(1) *,symmetric] by auto
+  qed
+qed
 
 lemma kuhn_induction_Suc:
   assumes "0 < p" "\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
@@ -2822,51 +2987,135 @@
   shows "odd (card {s. ksimplex p (Suc n) s \<and>((reduced lab (Suc n)) `  s = {0..Suc n})})"
   using assms unfolding Suc_eq_plus1 by(rule kuhn_induction)
 
+
 subsection {* And so we get the final combinatorial result. *}
 
-lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}" (is "?l = ?r") proof
-  assume l:?l guess a using ksimplexD(3)[OF l, unfolded add_0] unfolding card_1_exists .. note a=this
-  have "a = (\<lambda>x. p)" using ksimplexD(5)[OF l, rule_format, OF a(1)] by(rule,auto) thus ?r using a by auto next
-  assume r:?r show ?l unfolding r ksimplex_eq by auto qed
+lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}" (is "?l = ?r")
+proof
+  assume l: ?l
+  guess a using ksimplexD(3)[OF l, unfolded add_0] unfolding card_1_exists .. note a = this
+  have "a = (\<lambda>x. p)"
+    using ksimplexD(5)[OF l, rule_format, OF a(1)] by rule auto
+  thus ?r using a by auto
+next
+  assume r: ?r
+  show ?l unfolding r ksimplex_eq by auto
+qed
 
-lemma reduce_labelling_zero[simp]: "reduced lab 0 x = 0" apply(rule reduced_labelling_unique) by auto
+lemma reduce_labelling_zero[simp]: "reduced lab 0 x = 0"
+  by (rule reduced_labelling_unique) auto
 
 lemma kuhn_combinatorial:
   assumes "0 < p" "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
-  "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n  \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
-  shows " odd (card {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})})" using assms proof(induct n)
+    "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n  \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
+  shows " odd (card {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})})"
+  using assms
+proof (induct n)
   let ?M = "\<lambda>n. {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})}"
-  { case 0 have *:"?M 0 = {{(\<lambda>x. p)}}" unfolding ksimplex_0 by auto show ?case unfolding * by auto }
-  case (Suc n) have "odd (card (?M n))" apply(rule Suc(1)[OF Suc(2)]) using Suc(3-) by auto
-  thus ?case apply-apply(rule kuhn_induction_Suc) using Suc(2-) by auto qed
+  {
+    case 0
+    have *: "?M 0 = {{(\<lambda>x. p)}}"
+      unfolding ksimplex_0 by auto
+    show ?case unfolding * by auto
+  next
+    case (Suc n)
+    have "odd (card (?M n))"
+      apply (rule Suc(1)[OF Suc(2)])
+      using Suc(3-)
+      apply auto
+      done
+    thus ?case
+      apply -
+      apply (rule kuhn_induction_Suc)
+      using Suc(2-)
+      apply auto
+      done
+  }
+qed
 
-lemma kuhn_lemma: assumes "0 < (p::nat)" "0 < (n::nat)"
-  "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (label x i = (0::nat)) \<or> (label x i = 1))"
-  "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = 0) \<longrightarrow> (label x i = 0))"
-  "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = p) \<longrightarrow> (label x i = 1))"
+lemma kuhn_lemma:
+  assumes "0 < (p::nat)" "0 < (n::nat)"
+    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (label x i = (0::nat)) \<or> (label x i = 1))"
+    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = 0) \<longrightarrow> (label x i = 0))"
+    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = p) \<longrightarrow> (label x i = 1))"
   obtains q where "\<forall>i\<in>{1..n}. q i < p"
-  "\<forall>i\<in>{1..n}. \<exists>r s. (\<forall>j\<in>{1..n}. q(j) \<le> r(j) \<and> r(j) \<le> q(j) + 1) \<and>
-                               (\<forall>j\<in>{1..n}. q(j) \<le> s(j) \<and> s(j) \<le> q(j) + 1) \<and>
-                               ~(label r i = label s i)" proof-
-  let ?A = "{s. ksimplex p n s \<and> reduced label n ` s = {0..n}}" have "n\<noteq>0" using assms by auto
-  have conjD:"\<And>P Q. P \<and> Q \<Longrightarrow> P" "\<And>P Q. P \<and> Q \<Longrightarrow> Q" by auto
-  have "odd (card ?A)" apply(rule kuhn_combinatorial[of p n label]) using assms by auto
-  hence "card ?A \<noteq> 0" apply-apply(rule ccontr) by auto hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto
-  then obtain s where "s\<in>?A" by auto note s=conjD[OF this[unfolded mem_Collect_eq]]
-  guess a b apply(rule ksimplex_extrema_strong[OF s(1) `n\<noteq>0`]) . note ab=this
-  show ?thesis apply(rule that[of a]) proof(rule_tac[!] ballI) fix i assume "i\<in>{1..n}"
-    hence "a i + 1 \<le> p" apply-apply(rule order_trans[of _ "b i"]) apply(subst ab(5)[THEN spec[where x=i]])
-      using s(1)[unfolded ksimplex_def] defer apply- apply(erule conjE)+ apply(drule_tac bspec[OF _ ab(2)])+ by auto
+    "\<forall>i\<in>{1..n}. \<exists>r s. (\<forall>j\<in>{1..n}. q(j) \<le> r(j) \<and> r(j) \<le> q(j) + 1) \<and>
+                                 (\<forall>j\<in>{1..n}. q(j) \<le> s(j) \<and> s(j) \<le> q(j) + 1) \<and>
+                                 ~(label r i = label s i)"
+proof -
+  let ?A = "{s. ksimplex p n s \<and> reduced label n ` s = {0..n}}"
+  have "n \<noteq> 0" using assms by auto
+  have conjD:"\<And>P Q. P \<and> Q \<Longrightarrow> P" "\<And>P Q. P \<and> Q \<Longrightarrow> Q"
+    by auto
+  have "odd (card ?A)"
+    apply (rule kuhn_combinatorial[of p n label])
+    using assms
+    apply auto
+    done
+  hence "card ?A \<noteq> 0"
+    apply -
+    apply (rule ccontr)
+    apply auto
+    done
+  hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto
+  then obtain s where "s \<in> ?A"
+    by auto note s=conjD[OF this[unfolded mem_Collect_eq]]
+  guess a b by (rule ksimplex_extrema_strong[OF s(1) `n\<noteq>0`]) note ab = this
+  show ?thesis
+    apply (rule that[of a])
+    apply (rule_tac[!] ballI)
+  proof -
+    fix i
+    assume "i\<in>{1..n}"
+    hence "a i + 1 \<le> p"
+      apply -
+      apply (rule order_trans[of _ "b i"])
+      apply (subst ab(5)[THEN spec[where x=i]])
+      using s(1)[unfolded ksimplex_def]
+      defer
+      apply -
+      apply (erule conjE)+
+      apply (drule_tac bspec[OF _ ab(2)])+
+      apply auto
+      done
     thus "a i < p" by auto
-    case goal2 hence "i \<in> reduced label n ` s" using s by auto then guess u unfolding image_iff .. note u=this
-    from goal2 have "i - 1 \<in> reduced label n ` s" using s by auto then guess v unfolding image_iff .. note v=this
-    show ?case apply(rule_tac x=u in exI, rule_tac x=v in exI) apply(rule conjI) defer apply(rule conjI) defer 2 proof(rule_tac[1-2] ballI)
-      show "label u i \<noteq> label v i" using reduced_labelling[of label n u] reduced_labelling[of label n v]
-        unfolding u(2)[symmetric] v(2)[symmetric] using goal2 by auto
-      fix j assume j:"j\<in>{1..n}" show "a j \<le> u j \<and> u j \<le> a j + 1" "a j \<le> v j \<and> v j \<le> a j + 1"
-        using conjD[OF ab(4)[rule_format, OF u(1)]] and conjD[OF ab(4)[rule_format, OF v(1)]] apply-
-        apply(drule_tac[!] kle_imp_pointwise)+ apply(erule_tac[!] x=j in allE)+ unfolding ab(5)[rule_format] using j
-        by auto qed qed qed
+  next
+    case goal2
+    hence "i \<in> reduced label n ` s" using s by auto
+    then guess u unfolding image_iff .. note u = this
+    from goal2 have "i - 1 \<in> reduced label n ` s"
+      using s by auto
+    then guess v unfolding image_iff .. note v = this
+    show ?case
+      apply (rule_tac x = u in exI)
+      apply (rule_tac x = v in exI)
+      apply (rule conjI)
+      defer
+      apply (rule conjI)
+      defer 2
+      apply (rule_tac[1-2] ballI)
+    proof -
+      show "label u i \<noteq> label v i"
+        using reduced_labelling [of label n u] reduced_labelling [of label n v]
+        unfolding u(2)[symmetric] v(2)[symmetric]
+        using goal2
+        apply auto
+        done
+      fix j
+      assume j: "j \<in> {1..n}"
+      show "a j \<le> u j \<and> u j \<le> a j + 1" "a j \<le> v j \<and> v j \<le> a j + 1"
+        using conjD[OF ab(4)[rule_format, OF u(1)]]
+          and conjD[OF ab(4)[rule_format, OF v(1)]]
+        apply -
+        apply (drule_tac[!] kle_imp_pointwise)+
+        apply (erule_tac[!] x=j in allE)+
+        unfolding ab(5)[rule_format]
+        using j
+        apply auto
+        done
+    qed
+  qed
+qed
 
 
 subsection {* The main result for the unit cube. *}
@@ -2946,11 +3195,13 @@
         using i label(1)[of i y] by auto
       show "x \<bullet> i \<le> f x \<bullet> i"
         apply (rule label(4)[rule_format])
-        using xy lx i(2) apply auto
+        using xy lx i(2)
+        apply auto
         done
       show "f y \<bullet> i \<le> y \<bullet> i"
         apply (rule label(5)[rule_format])
-        using xy ly i(2) apply auto
+        using xy ly i(2)
+        apply auto
         done
     next
       assume "label x i \<noteq> 0"
@@ -2958,11 +3209,13 @@
         using i label(1)[of i x] label(1)[of i y] by auto
       show "f x \<bullet> i \<le> x \<bullet> i"
         apply (rule label(5)[rule_format])
-        using xy l i(2) apply auto
+        using xy l i(2)
+        apply auto
         done
       show "y \<bullet> i \<le> f y \<bullet> i"
         apply (rule label(4)[rule_format])
-        using xy l i(2) apply auto
+        using xy l i(2)
+        apply auto
         done
     qed
     also have "\<dots> \<le> norm (f y - f x) + norm (y - x)"
@@ -2986,7 +3239,8 @@
     note e=this[rule_format,unfolded dist_norm]
     show ?thesis
       apply (rule_tac x="min (e/2) (d/real n/8)" in exI)
-    proof safe
+      apply safe
+    proof -
       show "0 < min (e / 2) (d / real n / 8)"
         using d' e by auto
       fix x y z i
@@ -3011,21 +3265,25 @@
           unfolding norm_minus_commute by auto
         also have "\<dots> < e / 2 + e / 2"
           apply (rule add_strict_mono)
-          using as(4,5) apply auto
+          using as(4,5)
+          apply auto
           done
         finally show "norm (f y - f x) < d / real n / 8"
           apply -
           apply (rule e(2))
-          using as apply auto
+          using as
+          apply auto
           done
         have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8"
           apply (rule add_strict_mono)
-          using as apply auto
+          using as
+          apply auto
           done
         thus "norm (y - x) < 2 * (d / real n / 8)" using tria by auto
         show "norm (f x - f z) < d / real n / 8"
           apply (rule e(2))
-          using as e(1) apply auto
+          using as e(1)
+          apply auto
           done
       qed (insert as, auto)
     qed
@@ -3036,9 +3294,10 @@
     apply (rule add_pos_pos)
     defer
     apply (rule divide_pos_pos)
-    using e(1) n apply auto
+    using e(1) n
+    apply auto
     done
-  hence "p > 0" using p by auto
+  then have "p > 0" using p by auto
 
   obtain b :: "nat \<Rightarrow> 'a" where b: "bij_betw b {1..n} Basis"
     by atomize_elim (auto simp: n_def intro!: finite_same_card_bij)
@@ -3121,7 +3380,8 @@
   have "\<And>i. i \<in> Basis \<Longrightarrow> r (b' i) \<le> p"
     apply (rule order_trans)
     apply (rule rs(1)[OF b'_im,THEN conjunct2])
-    using q(1)[rule_format,OF b'_im] apply (auto simp add: Suc_le_eq)
+    using q(1)[rule_format,OF b'_im]
+    apply (auto simp add: Suc_le_eq)
     done
   hence "r' \<in> {0..\<Sum>Basis}"
     unfolding r'_def mem_interval using b'_Basis
@@ -3130,7 +3390,8 @@
   have "\<And>i. i\<in>Basis \<Longrightarrow> s (b' i) \<le> p"
     apply (rule order_trans)
     apply (rule rs(2)[OF b'_im, THEN conjunct2])
-    using q(1)[rule_format,OF b'_im] apply (auto simp add: Suc_le_eq)
+    using q(1)[rule_format,OF b'_im]
+    apply (auto simp add: Suc_le_eq)
     done
   hence "s' \<in> {0..\<Sum>Basis}"
     unfolding s'_def mem_interval using b'_Basis
@@ -3141,7 +3402,8 @@
   have *: "\<And>x. 1 + real x = real (Suc x)" by auto
   { have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       apply (rule setsum_mono)
-      using rs(1)[OF b'_im] apply (auto simp add:* field_simps)
+      using rs(1)[OF b'_im]
+      apply (auto simp add:* field_simps)
       done
     also have "\<dots> < e * real p" using p `e>0` `p>0`
       by (auto simp add: field_simps n_def real_of_nat_def)
@@ -3150,7 +3412,8 @@
   moreover
   { have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       apply (rule setsum_mono)
-      using rs(2)[OF b'_im] apply (auto simp add:* field_simps)
+      using rs(2)[OF b'_im]
+      apply (auto simp add:* field_simps)
       done
     also have "\<dots> < e * real p" using p `e > 0` `p > 0`
       by (auto simp add: field_simps n_def real_of_nat_def)
@@ -3216,7 +3479,7 @@
     and t :: "('b::euclidean_space) set"
   assumes "s homeomorphic t"
   shows "(\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)) \<longleftrightarrow>
-         (\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))"
+    (\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))"
 proof -
   guess r using assms[unfolded homeomorphic_def homeomorphism_def] ..
   then guess i ..
@@ -3244,8 +3507,9 @@
     apply -
     apply (rule invertible_fixpoint_property[OF continuous_on_id _ _ _ _ assms(2), of t h g])
     prefer 7
-    apply (rule_tac y=y in that)
-    using assms apply auto
+    apply (rule_tac y = y in that)
+    using assms
+    apply auto
     done
 qed
 
@@ -3253,7 +3517,7 @@
 subsection {*So the Brouwer theorem for any set with nonempty interior. *}
 
 lemma brouwer_weak:
-  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
   assumes "compact s" "convex s" "interior s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
   obtains x where "x \<in> s" "f x = x"
 proof -
@@ -3271,7 +3535,8 @@
     defer
     apply (erule bexE)
     apply (rule_tac x=y in that)
-    using assms apply auto
+    using assms
+    apply auto
     done
 qed
 
@@ -3290,7 +3555,8 @@
   rather involved @{text "HOMEOMORPHIC_CONVEX_COMPACT"} theorem, just using
   a scaling and translation to put the set inside the unit cube. *}
 
-lemma brouwer: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
+lemma brouwer:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
   assumes "compact s" "convex s" "s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
   obtains x where "x \<in> s" "f x = x"
 proof -
@@ -3353,7 +3619,8 @@
     apply (simp add: * norm_minus_commute)
     done
   note x = this
-  hence "scaleR 2 a = scaleR 1 x + scaleR 1 x" by (auto simp add:algebra_simps)
+  hence "scaleR 2 a = scaleR 1 x + scaleR 1 x"
+    by (auto simp add: algebra_simps)
   hence "a = x" unfolding scaleR_left_distrib[symmetric] by auto
   thus False using x assms by auto
 qed
@@ -3361,14 +3628,15 @@
 
 subsection {*Bijections between intervals. *}
 
-definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::ordered_euclidean_space" where
-  "interval_bij \<equiv> \<lambda>(a, b) (u, v) x. (\<Sum>i\<in>Basis. (u\<bullet>i + (x\<bullet>i - a\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (v\<bullet>i - u\<bullet>i)) *\<^sub>R i)"
+definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::ordered_euclidean_space"
+  where "interval_bij =
+    (\<lambda>(a, b) (u, v) x. (\<Sum>i\<in>Basis. (u\<bullet>i + (x\<bullet>i - a\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (v\<bullet>i - u\<bullet>i)) *\<^sub>R i))"
 
 lemma interval_bij_affine:
   "interval_bij (a,b) (u,v) = (\<lambda>x. (\<Sum>i\<in>Basis. ((v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (x\<bullet>i)) *\<^sub>R i) +
     (\<Sum>i\<in>Basis. (u\<bullet>i - (v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (a\<bullet>i)) *\<^sub>R i))"
   by (auto simp: setsum_addf[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff
-                 field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum_cong)
+    field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum_cong)
 
 lemma continuous_interval_bij:
   "continuous (at x) (interval_bij (a,b::'a::ordered_euclidean_space) (u,v::'a))"
@@ -3384,7 +3652,8 @@
   assumes "x \<in> {a..b}" "{u..v} \<noteq> {}"
   shows "interval_bij (a,b) (u,v) x \<in> {u..v}"
   apply (simp only: interval_bij_def split_conv mem_interval inner_setsum_left_Basis cong: ball_cong)
-proof safe
+  apply safe
+proof -
   fix i :: 'a
   assume i: "i \<in> Basis"
   have "{a..b} \<noteq> {}" using assms by auto
@@ -3399,7 +3668,8 @@
   have "((x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (v \<bullet> i - u \<bullet> i) \<le> 1 * (v \<bullet> i - u \<bullet> i)"
     apply (rule mult_right_mono)
     unfolding divide_le_eq_1
-    using * x apply auto
+    using * x
+    apply auto
     done
   thus "u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i) \<le> v \<bullet> i"
     using * by auto
diff -r 16235bb41881 -r 082d0972096b src/HOL/Multivariate_Analysis/Determinants.thy
--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -11,14 +11,18 @@
 begin
 
 subsection{* First some facts about products*}
-lemma setprod_insert_eq: "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
-apply clarsimp
-by(subgoal_tac "insert a A = A", auto)
+
+lemma setprod_insert_eq:
+  "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
+  apply clarsimp
+  apply (subgoal_tac "insert a A = A")
+  apply auto
+  done
 
 lemma setprod_add_split:
   assumes mn: "(m::nat) <= n + 1"
   shows "setprod f {m.. n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
-proof-
+proof -
   let ?A = "{m .. n+p}"
   let ?B = "{m .. n}"
   let ?C = "{n+1..n+p}"
@@ -30,47 +34,56 @@
 
 
 lemma setprod_offset: "setprod f {(m::nat) + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
-apply (rule setprod_reindex_cong[where f="op + p"])
-apply (auto simp add: image_iff Bex_def inj_on_def)
-apply arith
-apply (rule ext)
-apply (simp add: add_commute)
-done
+  apply (rule setprod_reindex_cong[where f="op + p"])
+  apply (auto simp add: image_iff Bex_def inj_on_def)
+  apply presburger
+  apply (rule ext)
+  apply (simp add: add_commute)
+  done
 
-lemma setprod_singleton: "setprod f {x} = f x" by simp
-
-lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
+lemma setprod_singleton: "setprod f {x} = f x"
+  by simp
 
-lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
-  "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
-                             else setprod f {m..n})"
+lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)"
+  by simp
+
+lemma setprod_numseg:
+  "setprod f {m..0} = (if m = 0 then f 0 else 1)"
+  "setprod f {m .. Suc n} =
+    (if m \<le> Suc n then f (Suc n) * setprod f {m..n} else setprod f {m..n})"
   by (auto simp add: atLeastAtMostSuc_conv)
 
-lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::linordered_idom)"
+lemma setprod_le:
+  assumes fS: "finite S"
+    and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::linordered_idom)"
   shows "setprod f S \<le> setprod g S"
-using fS fg
-apply(induct S)
-apply simp
-apply auto
-apply (rule mult_mono)
-apply (auto intro: setprod_nonneg)
-done
+  using fS fg
+  apply (induct S)
+  apply simp
+  apply auto
+  apply (rule mult_mono)
+  apply (auto intro: setprod_nonneg)
+  done
 
   (* FIXME: In Finite_Set there is a useless further assumption *)
-lemma setprod_inversef: "finite A ==> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: field_inverse_zero)"
+lemma setprod_inversef:
+  "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: field_inverse_zero)"
   apply (erule finite_induct)
   apply (simp)
   apply simp
   done
 
-lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::linordered_idom)"
+lemma setprod_le_1:
+  assumes fS: "finite S"
+    and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::linordered_idom)"
   shows "setprod f S \<le> 1"
-using setprod_le[OF fS f] unfolding setprod_1 .
+  using setprod_le[OF fS f] unfolding setprod_1 .
 
-subsection{* Trace *}
+
+subsection {* Trace *}
 
-definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
-  "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
+definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
+  where "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
 
 lemma trace_0: "trace(mat 0) = 0"
   by (simp add: trace_def mat_def)
@@ -87,14 +100,17 @@
 lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
   apply (simp add: trace_def matrix_matrix_mult_def)
   apply (subst setsum_commute)
-  by (simp add: mult_commute)
+  apply (simp add: mult_commute)
+  done
 
 (* ------------------------------------------------------------------------- *)
 (* Definition of determinant.                                                *)
 (* ------------------------------------------------------------------------- *)
 
 definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
-  "det A = setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)) {p. p permutes (UNIV :: 'n set)}"
+  "det A =
+    setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))
+      {p. p permutes (UNIV :: 'n set)}"
 
 (* ------------------------------------------------------------------------- *)
 (* A few general lemmas we need below.                                       *)
@@ -105,7 +121,8 @@
   shows "setprod f S = setprod (f o p) S"
   using assms by (fact setprod.permute)
 
-lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
+lemma setproduct_permute_nat_interval:
+  "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
   by (blast intro!: setprod_permute)
 
 (* ------------------------------------------------------------------------- *)
@@ -113,52 +130,71 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma det_transpose: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
-proof-
+proof -
   let ?di = "\<lambda>A i j. A$i$j"
   let ?U = "(UNIV :: 'n set)"
   have fU: "finite ?U" by simp
-  {fix p assume p: "p \<in> {p. p permutes ?U}"
+  {
+    fix p
+    assume p: "p \<in> {p. p permutes ?U}"
     from p have pU: "p permutes ?U" by blast
     have sth: "sign (inv p) = sign p"
       by (metis sign_inverse fU p mem_Collect_eq permutation_permutes)
     from permutes_inj[OF pU]
     have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
     from permutes_image[OF pU]
-    have "setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U = setprod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)" by simp
+    have "setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U =
+      setprod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)" by simp
     also have "\<dots> = setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) o p) ?U"
       unfolding setprod_reindex[OF pi] ..
     also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
-    proof-
-      {fix i assume i: "i \<in> ?U"
+    proof -
+      {
+        fix i
+        assume i: "i \<in> ?U"
         from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
         have "((\<lambda>i. ?di (transpose A) i (inv p i)) o p) i = ?di A i (p i)"
-          unfolding transpose_def by (simp add: fun_eq_iff)}
-      then show "setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
+          unfolding transpose_def by (simp add: fun_eq_iff)
+      }
+      then show "setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) o p) ?U =
+        setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
     qed
-    finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
-      by simp}
-  then show ?thesis unfolding det_def apply (subst setsum_permutations_inverse)
-  apply (rule setsum_cong2) by blast
+    finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U) =
+      of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth by simp
+  }
+  then show ?thesis
+    unfolding det_def
+    apply (subst setsum_permutations_inverse)
+    apply (rule setsum_cong2)
+    apply blast
+    done
 qed
 
 lemma det_lowerdiagonal:
   fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
   assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
-proof-
+proof -
   let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
   let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
   have fU: "finite ?U" by simp
   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
-  {fix p assume p: "p \<in> ?PU -{id}"
-    from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
-    from permutes_natset_le[OF pU] pid obtain i where
-      i: "p i > i" by (metis not_le)
-    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
-    from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
-  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
+  {
+    fix p
+    assume p: "p \<in> ?PU -{id}"
+    from p have pU: "p permutes ?U" and pid: "p \<noteq> id"
+      by blast+
+    from permutes_natset_le[OF pU] pid obtain i where i: "p i > i"
+      by (metis not_le)
+    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0"
+      by blast
+    from setprod_zero[OF fU ex] have "?pp p = 0"
+      by simp
+  }
+  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
+    by blast
   from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
@@ -167,21 +203,26 @@
   fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
   assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
-proof-
+proof -
   let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
   let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
   have fU: "finite ?U" by simp
   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
-  {fix p assume p: "p \<in> ?PU -{id}"
-    from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
-    from permutes_natset_ge[OF pU] pid obtain i where
-      i: "p i < i" by (metis not_le)
+  {
+    fix p
+    assume p: "p \<in> ?PU -{id}"
+    from p have pU: "p permutes ?U" and pid: "p \<noteq> id"
+      by blast+
+    from permutes_natset_ge[OF pU] pid obtain i where i: "p i < i"
+      by (metis not_le)
     from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
-    from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
-  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
-  from   setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
+    from setprod_zero[OF fU ex] have "?pp p = 0" by simp
+  }
+  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
+    by blast
+  from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
 
@@ -189,14 +230,16 @@
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV::'n set)"
-proof-
+proof -
   let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
   let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
   have fU: "finite ?U" by simp
   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
-  {fix p assume p: "p \<in> ?PU - {id}"
+  {
+    fix p
+    assume p: "p \<in> ?PU - {id}"
     then have "p \<noteq> id" by simp
     then obtain i where i: "p i \<noteq> i" unfolding fun_eq_iff by auto
     from ld [OF i [symmetric]] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
@@ -207,16 +250,22 @@
 qed
 
 lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
-proof-
+proof -
   let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
   let ?U = "UNIV :: 'n set"
   let ?f = "\<lambda>i j. ?A$i$j"
-  {fix i assume i: "i \<in> ?U"
-    have "?f i i = 1" using i by (vector mat_def)}
-  hence th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
+  {
+    fix i
+    assume i: "i \<in> ?U"
+    have "?f i i = 1" using i by (vector mat_def)
+  }
+  then have th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
     by (auto intro: setprod_cong)
-  {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
-    have "?f i j = 0" using i j ij by (vector mat_def) }
+  {
+    fix i j
+    assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
+    have "?f i j = 0" using i j ij by (vector mat_def)
+  }
   then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_diagonal
     by blast
   also have "\<dots> = 1" unfolding th setprod_1 ..
@@ -232,23 +281,27 @@
   shows "det(\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
   apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric])
   apply (subst sum_permutations_compose_right[OF p])
-proof(rule setsum_cong2)
+proof (rule setsum_cong2)
   let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
-  fix q assume qPU: "q \<in> ?PU"
+  fix q
+  assume qPU: "q \<in> ?PU"
   have fU: "finite ?U" by simp
-  from qPU have q: "q permutes ?U" by blast
+  from qPU have q: "q permutes ?U"
+    by blast
   from p q have pp: "permutation p" and qp: "permutation q"
     by (metis fU permutation_permutes)+
   from permutes_inv[OF p] have ip: "inv p permutes ?U" .
-    have "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
-      by (simp only: setprod_permute[OF ip, symmetric])
-    also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
-      by (simp only: o_def)
-    also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
-    finally   have thp: "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
-      by blast
-  show "of_int (sign (q o p)) * setprod (\<lambda>i. A$ p i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
+  have "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
+    by (simp only: setprod_permute[OF ip, symmetric])
+  also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
+    by (simp only: o_def)
+  also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U"
+    by (simp only: o_def permutes_inverses[OF p])
+  finally have thp: "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
+    by blast
+  show "of_int (sign (q o p)) * setprod (\<lambda>i. A$ p i$ (q o p) i) ?U =
+    of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
     by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
 qed
 
@@ -256,7 +309,7 @@
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes p: "p permutes (UNIV :: 'n set)"
   shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
-proof-
+proof -
   let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
   let ?At = "transpose A"
   have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A $ p i))"
@@ -270,16 +323,16 @@
 lemma det_identical_rows:
   fixes A :: "'a::linordered_idom^'n^'n"
   assumes ij: "i \<noteq> j"
-  and r: "row i A = row j A"
+    and r: "row i A = row j A"
   shows "det A = 0"
 proof-
-  have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
+  have tha: "\<And>(a::'a) b. a = b \<Longrightarrow> b = - a \<Longrightarrow> a = 0"
     by simp
   have th1: "of_int (-1) = - 1" by simp
   let ?p = "Fun.swap i j id"
   let ?A = "\<chi> i. A $ ?p i"
   from r have "A = ?A" by (simp add: vec_eq_iff row_def swap_def)
-  hence "det A = det ?A" by simp
+  then have "det A = det ?A" by simp
   moreover have "det A = - det ?A"
     by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
   ultimately show "det A = 0" by (metis tha)
@@ -288,21 +341,22 @@
 lemma det_identical_columns:
   fixes A :: "'a::linordered_idom^'n^'n"
   assumes ij: "i \<noteq> j"
-  and r: "column i A = column j A"
+    and r: "column i A = column j A"
   shows "det A = 0"
-apply (subst det_transpose[symmetric])
-apply (rule det_identical_rows[OF ij])
-by (metis row_transpose r)
+  apply (subst det_transpose[symmetric])
+  apply (rule det_identical_rows[OF ij])
+  apply (metis row_transpose r)
+  done
 
 lemma det_zero_row:
   fixes A :: "'a::{idom, ring_char_0}^'n^'n"
   assumes r: "row i A = 0"
   shows "det A = 0"
-using r
-apply (simp add: row_def det_def vec_eq_iff)
-apply (rule setsum_0')
-apply (auto simp: sign_nz)
-done
+  using r
+  apply (simp add: row_def det_def vec_eq_iff)
+  apply (rule setsum_0')
+  apply (auto simp: sign_nz)
+  done
 
 lemma det_zero_column:
   fixes A :: "'a::{idom,ring_char_0}^'n^'n"
@@ -310,27 +364,32 @@
   shows "det A = 0"
   apply (subst det_transpose[symmetric])
   apply (rule det_zero_row [of i])
-  by (metis row_transpose r)
+  apply (metis row_transpose r)
+  done
 
 lemma det_row_add:
   fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
-             det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
-             det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
-unfolding det_def vec_lambda_beta setsum_addf[symmetric]
+    det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
+    det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
+    unfolding det_def vec_lambda_beta setsum_addf[symmetric]
 proof (rule setsum_cong2)
   let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
   let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
-  fix p assume p: "p \<in> ?pU"
+  fix p
+  assume p: "p \<in> ?pU"
   let ?Uk = "?U - {k}"
   from p have pU: "p permutes ?U" by blast
   have kU: "?U = insert k ?Uk" by blast
-  {fix j assume j: "j \<in> ?Uk"
+  {
+    fix j
+    assume j: "j \<in> ?Uk"
     from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
-      by simp_all}
+      by simp_all
+  }
   then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
     and th2: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?h i $ p i) ?Uk"
     apply -
@@ -342,36 +401,45 @@
   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
     apply (rule setprod_insert)
     apply simp
-    by blast
-  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)" by (simp add: field_simps)
-  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?h i $ p i) ?Uk)" by (metis th1 th2)
+    apply blast
+    done
+  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)"
+    by (simp add: field_simps)
+  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?h i $ p i) ?Uk)"
+    by (metis th1 th2)
   also have "\<dots> = setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i $ p i) (insert k ?Uk)"
     unfolding  setprod_insert[OF th3] by simp
-  finally have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?g i $ p i) ?U + setprod (\<lambda>i. ?h i $ p i) ?U" unfolding kU[symmetric] .
-  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U = of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i $ p i) ?U"
+  finally have "setprod (\<lambda>i. ?f i $ p i) ?U =
+    setprod (\<lambda>i. ?g i $ p i) ?U + setprod (\<lambda>i. ?h i $ p i) ?U" unfolding kU[symmetric] .
+  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U =
+    of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i $ p i) ?U"
     by (simp add: field_simps)
 qed
 
 lemma det_row_mul:
   fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
-             c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
-
-unfolding det_def vec_lambda_beta setsum_right_distrib
+    c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
+  unfolding det_def vec_lambda_beta setsum_right_distrib
 proof (rule setsum_cong2)
   let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
   let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
-  fix p assume p: "p \<in> ?pU"
+  fix p
+  assume p: "p \<in> ?pU"
   let ?Uk = "?U - {k}"
   from p have pU: "p permutes ?U" by blast
   have kU: "?U = insert k ?Uk" by blast
-  {fix j assume j: "j \<in> ?Uk"
-    from j have "?f j $ p j = ?g j $ p j" by simp}
+  {
+    fix j
+    assume j: "j \<in> ?Uk"
+    from j have "?f j $ p j = ?g j $ p j" by simp
+  }
   then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
     apply -
-    apply (rule setprod_cong, simp_all)
+    apply (rule setprod_cong)
+    apply simp_all
     done
   have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
   have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
@@ -379,29 +447,34 @@
   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
     apply (rule setprod_insert)
     apply simp
-    by blast
-  also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk" by (simp add: field_simps)
+    apply blast
+    done
+  also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk"
+    by (simp add: field_simps)
   also have "\<dots> = c* (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk)"
     unfolding th1 by (simp add: mult_ac)
   also have "\<dots> = c* (setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk))"
-    unfolding  setprod_insert[OF th3] by simp
-  finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)" unfolding kU[symmetric] .
-  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U = c * (of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U)"
+    unfolding setprod_insert[OF th3] by simp
+  finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)"
+    unfolding kU[symmetric] .
+  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U =
+    c * (of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U)"
     by (simp add: field_simps)
 qed
 
 lemma det_row_0:
   fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
-using det_row_mul[of k 0 "\<lambda>i. 1" b]
-apply (simp)
-  unfolding vector_smult_lzero .
+  using det_row_mul[of k 0 "\<lambda>i. 1" b]
+  apply simp
+  apply (simp only: vector_smult_lzero)
+  done
 
 lemma det_row_operation:
   fixes A :: "'a::linordered_idom^'n^'n"
   assumes ij: "i \<noteq> j"
   shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
-proof-
+proof -
   let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
   have th: "row i ?Z = row j ?Z" by (vector row_def)
   have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
@@ -415,30 +488,38 @@
   fixes A :: "real^'n^'n"
   assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
   shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
-proof-
+proof -
   let ?U = "UNIV :: 'n set"
   let ?S = "{row j A |j. j \<noteq> i}"
   let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
   let ?P = "\<lambda>x. ?d (row i A + x) = det A"
-  {fix k
-
-    have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
+  {
+    fix k
+    have "(if k = i then row i A + 0 else row k A) = row k A" by simp
+  }
   then have P0: "?P 0"
     apply -
     apply (rule cong[of det, OF refl])
-    by (vector row_def)
+    apply (vector row_def)
+    done
   moreover
-  {fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
+  {
+    fix c z y
+    assume zS: "z \<in> ?S" and Py: "?P y"
     from zS obtain j where j: "z = row j A" "i \<noteq> j" by blast
     let ?w = "row i A + y"
     have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector
     have thz: "?d z = 0"
       apply (rule det_identical_rows[OF j(2)])
-      using j by (vector row_def)
-    have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
-    then have "?P (c*s z + y)" unfolding thz Py det_row_mul[of i] det_row_add[of i]
-      by simp }
-
+      using j
+      apply (vector row_def)
+      done
+    have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)"
+      unfolding th0 ..
+    then have "?P (c*s z + y)"
+      unfolding thz Py det_row_mul[of i] det_row_add[of i]
+      by simp
+  }
   ultimately show ?thesis
     apply -
     apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR])
@@ -456,53 +537,68 @@
   fixes A:: "real^'n^'n"
   assumes d: "dependent (rows A)"
   shows "det A = 0"
-proof-
+proof -
   let ?U = "UNIV :: 'n set"
   from d obtain i where i: "row i A \<in> span (rows A - {row i A})"
     unfolding dependent_def rows_def by blast
-  {fix j k assume jk: "j \<noteq> k"
-    and c: "row j A = row k A"
-    from det_identical_rows[OF jk c] have ?thesis .}
+  {
+    fix j k
+    assume jk: "j \<noteq> k" and c: "row j A = row k A"
+    from det_identical_rows[OF jk c] have ?thesis .
+  }
   moreover
-  {assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
+  {
+    assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
     have th0: "- row i A \<in> span {row j A|j. j \<noteq> i}"
       apply (rule span_neg)
       apply (rule set_rev_mp)
       apply (rule i)
       apply (rule span_mono)
-      using H i by (auto simp add: rows_def)
+      using H i
+      apply (auto simp add: rows_def)
+      done
     from det_row_span[OF th0]
     have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
       unfolding right_minus vector_smult_lzero ..
     with det_row_mul[of i "0::real" "\<lambda>i. 1"]
-    have "det A = 0" by simp}
+    have "det A = 0" by simp
+  }
   ultimately show ?thesis by blast
 qed
 
-lemma det_dependent_columns: assumes d: "dependent(columns (A::real^'n^'n))" shows "det A = 0"
-by (metis d det_dependent_rows rows_transpose det_transpose)
+lemma det_dependent_columns:
+  assumes d: "dependent (columns (A::real^'n^'n))"
+  shows "det A = 0"
+  by (metis d det_dependent_rows rows_transpose det_transpose)
 
 (* ------------------------------------------------------------------------- *)
 (* Multilinearity and the multiplication formula.                            *)
 (* ------------------------------------------------------------------------- *)
 
 lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
-  apply (rule iffD1[OF vec_lambda_unique]) by vector
+  by (rule iffD1[OF vec_lambda_unique]) vector
 
 lemma det_linear_row_setsum:
   assumes fS: "finite S"
-  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case apply simp  unfolding setsum_empty det_row_0[of k] ..
+  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
+    setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
+proof (induct rule: finite_induct[OF fS])
+  case 1
+  then show ?case
+    apply simp
+    unfolding setsum_empty det_row_0[of k]
+    apply rule
+    done
 next
   case (2 x F)
-  then  show ?case by (simp add: det_row_add cong del: if_weak_cong)
+  then show ?case
+    by (simp add: det_row_add cong del: if_weak_cong)
 qed
 
 lemma finite_bounded_functions:
   assumes fS: "finite S"
   shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
-proof(induct k)
+proof (induct k)
   case 0
   have th: "{f. \<forall>i. f i = i} = {id}" by auto
   show ?case by (auto simp add: th)
@@ -526,13 +622,14 @@
 lemma det_linear_rows_setsum_lemma:
   assumes fS: "finite S" and fT: "finite T"
   shows "det((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
-             setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
-                 {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
-using fT
-proof(induct T arbitrary: a c set: finite)
+    setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
+      {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
+  using fT
+proof (induct T arbitrary: a c set: finite)
   case empty
-  have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)" by vector
-  from "empty.prems"  show ?case unfolding th0 by simp
+  have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
+    by vector
+  from empty.prems show ?case unfolding th0 by simp
 next
   case (insert z T a c)
   let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
@@ -540,42 +637,48 @@
   let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
   let ?s = "\<lambda> k a c f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n)"
   let ?c = "\<lambda>i. if i = z then a i j else c i"
-  have thif: "\<And>a b c d. (if a \<or> b then c else d) = (if a then c else if b then c else d)" by simp
+  have thif: "\<And>a b c d. (if a \<or> b then c else d) = (if a then c else if b then c else d)"
+    by simp
   have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
-     (if c then (if a then b else d) else (if a then b else e))" by simp
-  from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False" by auto
+     (if c then (if a then b else d) else (if a then b else e))"
+    by simp
+  from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False"
+    by auto
   have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
-        det (\<chi> i. if i = z then setsum (a i) S
-                 else if i \<in> T then setsum (a i) S else c i)"
+    det (\<chi> i. if i = z then setsum (a i) S else if i \<in> T then setsum (a i) S else c i)"
     unfolding insert_iff thif ..
-  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S
-                    else if i = z then a i j else c i))"
+  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S else if i = z then a i j else c i))"
     unfolding det_linear_row_setsum[OF fS]
     apply (subst thif2)
-    using nz by (simp cong del: if_weak_cong cong add: if_cong)
+    using nz
+    apply (simp cong del: if_weak_cong cong add: if_cong)
+    done
   finally have tha:
     "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
      (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
                                 else if i = z then a i j
                                 else c i))"
-    unfolding  insert.hyps unfolding setsum_cartesian_product by blast
+    unfolding insert.hyps unfolding setsum_cartesian_product by blast
   show ?case unfolding tha
-    apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
+    apply (rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
       blast intro: finite_cartesian_product fS finite,
       blast intro: finite_cartesian_product fS finite)
     using `z \<notin> T`
     apply auto
     apply (rule cong[OF refl[of det]])
-    by vector
+    apply vector
+    done
 qed
 
 lemma det_linear_rows_setsum:
   assumes fS: "finite (S::'n::finite set)"
-  shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
-proof-
-  have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
-
-  from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp
+  shows "det (\<chi> i. setsum (a i) S) =
+    setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
+proof -
+  have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)"
+    by vector
+  from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
+  show ?thesis by simp
 qed
 
 lemma matrix_mul_setsum_alt:
@@ -585,75 +688,93 @@
 
 lemma det_rows_mul:
   "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
-  setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
+    setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
 proof (simp add: det_def setsum_right_distrib cong add: setprod_cong, rule setsum_cong2)
   let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
-  fix p assume pU: "p \<in> ?PU"
+  fix p
+  assume pU: "p \<in> ?PU"
   let ?s = "of_int (sign p)"
-  from pU have p: "p permutes ?U" by blast
+  from pU have p: "p permutes ?U"
+    by blast
   have "setprod (\<lambda>i. c i * a i $ p i) ?U = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
     unfolding setprod_timesf ..
   then show "?s * (\<Prod>xa\<in>?U. c xa * a xa $ p xa) =
-        setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))" by (simp add: field_simps)
+    setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))" by (simp add: field_simps)
 qed
 
 lemma det_mul:
   fixes A B :: "'a::linordered_idom^'n^'n"
   shows "det (A ** B) = det A * det B"
-proof-
+proof -
   let ?U = "UNIV :: 'n set"
   let ?F = "{f. (\<forall>i\<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
   let ?PU = "{p. p permutes ?U}"
   have fU: "finite ?U" by simp
   have fF: "finite ?F" by (rule finite)
-  {fix p assume p: "p permutes ?U"
-
+  {
+    fix p
+    assume p: "p permutes ?U"
     have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
-      using p[unfolded permutes_def] by simp}
+      using p[unfolded permutes_def] by simp
+  }
   then have PUF: "?PU \<subseteq> ?F"  by blast
-  {fix f assume fPU: "f \<in> ?F - ?PU"
+  {
+    fix f
+    assume fPU: "f \<in> ?F - ?PU"
     have fUU: "f ` ?U \<subseteq> ?U" using fPU by auto
-    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U"
-      "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
-      by auto
+    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U" "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)"
+      unfolding permutes_def by auto
 
     let ?A = "(\<chi> i. A$i$f i *s B$f i) :: 'a^'n^'n"
     let ?B = "(\<chi> i. B$f i) :: 'a^'n^'n"
-    {assume fni: "\<not> inj_on f ?U"
+    {
+      assume fni: "\<not> inj_on f ?U"
       then obtain i j where ij: "f i = f j" "i \<noteq> j"
         unfolding inj_on_def by blast
       from ij
       have rth: "row i ?B = row j ?B" by (vector row_def)
       from det_identical_rows[OF ij(2) rth]
       have "det (\<chi> i. A$i$f i *s B$f i) = 0"
-        unfolding det_rows_mul by simp}
+        unfolding det_rows_mul by simp
+    }
     moreover
-    {assume fi: "inj_on f ?U"
+    {
+      assume fi: "inj_on f ?U"
       from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
         unfolding inj_on_def by metis
       note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
 
-      {fix y
+      {
+        fix y
         from fs f have "\<exists>x. f x = y" by blast
         then obtain x where x: "f x = y" by blast
-        {fix z assume z: "f z = y" from fith x z have "z = x" by metis}
-        with x have "\<exists>!x. f x = y" by blast}
-      with f(3) have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
-    ultimately have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
-  hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A$i$f i *s B$f i) = 0" by simp
-  {fix p assume pU: "p \<in> ?PU"
+        {
+          fix z
+          assume z: "f z = y"
+          from fith x z have "z = x" by metis
+        }
+        with x have "\<exists>!x. f x = y" by blast
+      }
+      with f(3) have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast
+    }
+    ultimately have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast
+  }
+  hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A$i$f i *s B$f i) = 0"
+    by simp
+  {
+    fix p
+    assume pU: "p \<in> ?PU"
     from pU have p: "p permutes ?U" by blast
     let ?s = "\<lambda>p. of_int (sign p)"
-    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) *
-               (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))"
+    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))"
     have "(setsum (\<lambda>q. ?s q *
-            (\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
-        (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) *
-               (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
+        (\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
+      (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
       unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
     proof(rule setsum_cong2)
-      fix q assume qU: "q \<in> ?PU"
+      fix q
+      assume qU: "q \<in> ?PU"
       hence q: "q permutes ?U" by blast
       from p q have pp: "permutation p" and pq: "permutation q"
         unfolding permutation_permutes by auto
@@ -666,11 +787,15 @@
         by (simp add:  th00 mult_ac sign_idempotent sign_compose)
       have th001: "setprod (\<lambda>i. B$i$ q (inv p i)) ?U = setprod ((\<lambda>i. B$i$ q (inv p i)) o p) ?U"
         by (rule setprod_permute[OF p])
-      have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
+      have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U =
+        setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
         unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
         apply (rule setprod_cong[OF refl])
-        using permutes_in_image[OF q] by vector
-      show "?s q * setprod (\<lambda>i. (((\<chi> i. A$i$p i *s B$p i) :: 'a^'n^'n)$i$q i)) ?U = ?s p * (setprod (\<lambda>i. A$i$p i) ?U) * (?s (q o inv p) * setprod (\<lambda>i. B$i$(q o inv p) i) ?U)"
+        using permutes_in_image[OF q]
+        apply vector
+        done
+      show "?s q * setprod (\<lambda>i. (((\<chi> i. A$i$p i *s B$p i) :: 'a^'n^'n)$i$q i)) ?U =
+        ?s p * (setprod (\<lambda>i. A$i$p i) ?U) * (?s (q o inv p) * setprod (\<lambda>i. B$i$(q o inv p) i) ?U)"
         using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
         by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
     qed
@@ -703,22 +828,24 @@
 lemma invertible_det_nz:
   fixes A::"real ^'n^'n"
   shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
-proof-
-  {assume "invertible A"
+proof -
+  {
+    assume "invertible A"
     then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
       unfolding invertible_righ_inverse by blast
     hence "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp
-    hence "det A \<noteq> 0"
-      apply (simp add: det_mul det_I) by algebra }
+    hence "det A \<noteq> 0" by (simp add: det_mul det_I) algebra
+  }
   moreover
-  {assume H: "\<not> invertible A"
+  {
+    assume H: "\<not> invertible A"
     let ?U = "UNIV :: 'n set"
     have fU: "finite ?U" by simp
     from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
       and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
       unfolding invertible_righ_inverse
       unfolding matrix_right_invertible_independent_rows by blast
-    have stupid: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
+    have *: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
       apply (drule_tac f="op + (- a)" in cong[OF refl])
       apply (simp only: ab_left_minus add_assoc[symmetric])
       apply simp
@@ -729,7 +856,7 @@
       apply -
       apply (rule vector_mul_lcancel_imp[OF ci])
       apply (auto simp add: field_simps)
-      unfolding stupid ..
+      unfolding * ..
     have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
       unfolding thr0
       apply (rule span_setsum)
@@ -743,7 +870,8 @@
     have thrb: "row i ?B = 0" using iU by (vector row_def)
     have "det A = 0"
       unfolding det_row_span[OF thr, symmetric] right_minus
-      unfolding  det_zero_row[OF thrb]  ..}
+      unfolding det_zero_row[OF thrb] ..
+  }
   ultimately show ?thesis by blast
 qed
 
@@ -756,7 +884,7 @@
   shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
                            else row i A)::real^'n^'n) = x$k * det A"
   (is "?lhs = ?rhs")
-proof-
+proof -
   let ?U = "UNIV :: 'n set"
   let ?Uk = "?U - {k}"
   have U: "?U = insert k ?Uk" by blast
@@ -766,7 +894,8 @@
     by (vector field_simps)
   have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by auto
   have "(\<chi> i. row i A) = A" by (vector row_def)
-  then have thd1: "det (\<chi> i. row i A) = det A"  by simp
+  then have thd1: "det (\<chi> i. row i A) = det A"
+    by simp
   have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
     apply (rule det_row_span)
     apply (rule span_setsum[OF fUk])
@@ -784,37 +913,44 @@
     unfolding thd0
     unfolding det_row_mul
     unfolding th001[of k "\<lambda>i. row i A"]
-    unfolding thd1  by (simp add: field_simps)
+    unfolding thd1
+    apply (simp add: field_simps)
+    done
 qed
 
 lemma cramer_lemma:
   fixes A :: "real^'n^'n"
   shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: real^'n^'n) = x$k * det A"
-proof-
+proof -
   let ?U = "UNIV :: 'n set"
-  have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
+  have *: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
     by (auto simp add: row_transpose intro: setsum_cong2)
   show ?thesis  unfolding matrix_mult_vsum
-  unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
-  unfolding stupid[of "\<lambda>i. x$i"]
-  apply (subst det_transpose[symmetric])
-  apply (rule cong[OF refl[of det]]) by (vector transpose_def column_def row_def)
+    unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
+    unfolding *[of "\<lambda>i. x$i"]
+    apply (subst det_transpose[symmetric])
+    apply (rule cong[OF refl[of det]])
+    apply (vector transpose_def column_def row_def)
+    done
 qed
 
 lemma cramer:
   fixes A ::"real^'n^'n"
   assumes d0: "det A \<noteq> 0"
   shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
-proof-
+proof -
   from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
     unfolding invertible_det_nz[symmetric] invertible_def by blast
   have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
-  hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
+  then have "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
   then have xe: "\<exists>x. A*v x = b" by blast
-  {fix x assume x: "A *v x = b"
-  have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
-    unfolding x[symmetric]
-    using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)}
+  {
+    fix x
+    assume x: "A *v x = b"
+    have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
+      unfolding x[symmetric]
+      using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)
+  }
   with xe show ?thesis by auto
 qed
 
@@ -824,16 +960,19 @@
 
 definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
 
-lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
+lemma orthogonal_transformation:
+  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
   unfolding orthogonal_transformation_def
   apply auto
   apply (erule_tac x=v in allE)+
   apply (simp add: norm_eq_sqrt_inner)
-  by (simp add: dot_norm  linear_add[symmetric])
+  apply (simp add: dot_norm  linear_add[symmetric])
+  done
 
-definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
+definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
+  transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
 
-lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n)  \<longleftrightarrow> transpose Q ** Q = mat 1"
+lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
   by (metis matrix_left_right_inverse orthogonal_matrix_def)
 
 lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
@@ -842,28 +981,31 @@
 lemma orthogonal_matrix_mul:
   fixes A :: "real ^'n^'n"
   assumes oA : "orthogonal_matrix A"
-  and oB: "orthogonal_matrix B"
+    and oB: "orthogonal_matrix B"
   shows "orthogonal_matrix(A ** B)"
   using oA oB
   unfolding orthogonal_matrix matrix_transpose_mul
   apply (subst matrix_mul_assoc)
   apply (subst matrix_mul_assoc[symmetric])
-  by (simp add: matrix_mul_rid)
+  apply (simp add: matrix_mul_rid)
+  done
 
 lemma orthogonal_transformation_matrix:
   fixes f:: "real^'n \<Rightarrow> real^'n"
   shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
   (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
+proof -
   let ?mf = "matrix f"
   let ?ot = "orthogonal_transformation f"
   let ?U = "UNIV :: 'n set"
   have fU: "finite ?U" by simp
   let ?m1 = "mat 1 :: real ^'n^'n"
-  {assume ot: ?ot
+  {
+    assume ot: ?ot
     from ot have lf: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
       unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
-    {fix i j
+    {
+      fix i j
       let ?A = "transpose ?mf ** ?mf"
       have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
         "\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
@@ -871,16 +1013,22 @@
       from fd[rule_format, of "axis i 1" "axis j 1", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
       have "?A$i$j = ?m1 $ i $ j"
         by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
-            th0 setsum_delta[OF fU] mat_def axis_def) }
-    hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
-    with lf have ?rhs by blast}
+            th0 setsum_delta[OF fU] mat_def axis_def)
+    }
+    then have "orthogonal_matrix ?mf" unfolding orthogonal_matrix
+      by vector
+    with lf have ?rhs by blast
+  }
   moreover
-  {assume lf: "linear f" and om: "orthogonal_matrix ?mf"
+  {
+    assume lf: "linear f" and om: "orthogonal_matrix ?mf"
     from lf om have ?lhs
       unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
       unfolding matrix_works[OF lf, symmetric]
       apply (subst dot_matrix_vector_mul)
-      by (simp add: dot_matrix_product matrix_mul_lid)}
+      apply (simp add: dot_matrix_product matrix_mul_lid)
+      done
+  }
   ultimately show ?thesis by blast
 qed
 
@@ -888,21 +1036,26 @@
   fixes Q:: "'a::linordered_idom^'n^'n"
   assumes oQ: "orthogonal_matrix Q"
   shows "det Q = 1 \<or> det Q = - 1"
-proof-
-
+proof -
   have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
-  proof-
+  proof -
     fix x:: 'a
-    have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: field_simps)
+    have th0: "x*x - 1 = (x - 1)*(x + 1)"
+      by (simp add: field_simps)
     have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
-      apply (subst eq_iff_diff_eq_0) by simp
-    have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
+      apply (subst eq_iff_diff_eq_0)
+      apply simp
+      done
+    have "x * x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
     also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
     finally show "?ths x" ..
   qed
-  from oQ have "Q ** transpose Q = mat 1" by (metis orthogonal_matrix_def)
-  hence "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)" by simp
-  hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transpose)
+  from oQ have "Q ** transpose Q = mat 1"
+    by (metis orthogonal_matrix_def)
+  then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
+    by simp
+  then have "det Q * det Q = 1"
+    by (simp add: det_mul det_I det_transpose)
   then show ?thesis unfolding th .
 qed
 
@@ -911,25 +1064,29 @@
 (* ------------------------------------------------------------------------- *)
 lemma scaling_linear:
   fixes f :: "real ^'n \<Rightarrow> real ^'n"
-  assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
+  assumes f0: "f 0 = 0"
+    and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
   shows "linear f"
-proof-
-  {fix v w
-    {fix x note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right] }
+proof -
+  {
+    fix v w
+    {
+      fix x
+      note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right]
+    }
     note th0 = this
     have "f v \<bullet> f w = c\<^sup>2 * (v \<bullet> w)"
       unfolding dot_norm_neg dist_norm[symmetric]
       unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
   note fc = this
   show ?thesis
-    unfolding linear_def vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR 
+    unfolding linear_def vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR
     by (simp add: inner_add fc field_simps)
 qed
 
 lemma isometry_linear:
-  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
-        \<Longrightarrow> linear f"
-by (rule scaling_linear[where c=1]) simp_all
+  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
+  by (rule scaling_linear[where c=1]) simp_all
 
 (* ------------------------------------------------------------------------- *)
 (* Hence another formulation of orthogonal transformation.                   *)
@@ -948,7 +1105,8 @@
   apply clarify
   apply (erule_tac x=v in allE)
   apply (erule_tac x=0 in allE)
-  by (simp add: dist_norm)
+  apply (simp add: dist_norm)
+  done
 
 (* ------------------------------------------------------------------------- *)
 (* Can extend an isometry from unit sphere.                                  *)
@@ -957,15 +1115,19 @@
 lemma isometry_sphere_extend:
   fixes f:: "real ^'n \<Rightarrow> real ^'n"
   assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
-  and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
+    and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
   shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
-proof-
-  {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
-    assume H: "x = norm x *\<^sub>R x0" "y = norm y *\<^sub>R y0"
-    "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'"
-    "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
-    "norm(x0' - y0') = norm(x0 - y0)"
-    hence *:"x0 \<bullet> y0 = x0' \<bullet> y0' + y0' \<bullet> x0' - y0 \<bullet> x0 " by(simp add: norm_eq norm_eq_1 inner_add inner_diff)
+proof -
+  {
+    fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
+    assume H:
+      "x = norm x *\<^sub>R x0"
+      "y = norm y *\<^sub>R y0"
+      "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'"
+      "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
+      "norm(x0' - y0') = norm(x0 - y0)"
+    hence *: "x0 \<bullet> y0 = x0' \<bullet> y0' + y0' \<bullet> x0' - y0 \<bullet> x0 "
+      by (simp add: norm_eq norm_eq_1 inner_add inner_diff)
     have "norm(x' - y') = norm(x - y)"
       apply (subst H(1))
       apply (subst H(2))
@@ -974,48 +1136,71 @@
       using H(5-9)
       apply (simp add: norm_eq norm_eq_1)
       apply (simp add: inner_diff scalar_mult_eq_scaleR) unfolding *
-      by (simp add: field_simps) }
+      apply (simp add: field_simps)
+      done
+  }
   note th0 = this
   let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
-  {fix x:: "real ^'n" assume nx: "norm x = 1"
-    have "?g x = f x" using nx by auto}
-  hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
+  {
+    fix x:: "real ^'n"
+    assume nx: "norm x = 1"
+    have "?g x = f x" using nx by auto
+  }
+  then have thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x"
+    by blast
   have g0: "?g 0 = 0" by simp
-  {fix x y :: "real ^'n"
-    {assume "x = 0" "y = 0"
-      then have "dist (?g x) (?g y) = dist x y" by simp }
+  {
+    fix x y :: "real ^'n"
+    {
+      assume "x = 0" "y = 0"
+      then have "dist (?g x) (?g y) = dist x y" by simp
+    }
     moreover
-    {assume "x = 0" "y \<noteq> 0"
+    {
+      assume "x = 0" "y \<noteq> 0"
       then have "dist (?g x) (?g y) = dist x y"
         apply (simp add: dist_norm)
         apply (rule f1[rule_format])
-        by(simp add: field_simps)}
+        apply (simp add: field_simps)
+        done
+    }
     moreover
-    {assume "x \<noteq> 0" "y = 0"
+    {
+      assume "x \<noteq> 0" "y = 0"
       then have "dist (?g x) (?g y) = dist x y"
         apply (simp add: dist_norm)
         apply (rule f1[rule_format])
-        by(simp add: field_simps)}
+        apply (simp add: field_simps)
+        done
+    }
     moreover
-    {assume z: "x \<noteq> 0" "y \<noteq> 0"
-      have th00: "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)" "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)" "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
+    {
+      assume z: "x \<noteq> 0" "y \<noteq> 0"
+      have th00:
+        "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)"
+        "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)"
+        "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
         "norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y) = norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y)"
         "norm (inverse (norm x) *\<^sub>R x) = 1"
         "norm (f (inverse (norm x) *\<^sub>R x)) = 1"
         "norm (inverse (norm y) *\<^sub>R y) = 1"
         "norm (f (inverse (norm y) *\<^sub>R y)) = 1"
         "norm (f (inverse (norm x) *\<^sub>R x) - f (inverse (norm y) *\<^sub>R y)) =
-        norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)"
+          norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)"
         using z
         by (auto simp add: field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
       from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
-        by (simp add: dist_norm)}
-    ultimately have "dist (?g x) (?g y) = dist x y" by blast}
+        by (simp add: dist_norm)
+    }
+    ultimately have "dist (?g x) (?g y) = dist x y" by blast
+  }
   note thd = this
     show ?thesis
     apply (rule exI[where x= ?g])
     unfolding orthogonal_transformation_isometry
-      using  g0 thfg thd by metis
+    using g0 thfg thd
+    apply metis
+    done
 qed
 
 (* ------------------------------------------------------------------------- *)
@@ -1029,14 +1214,17 @@
   fixes Q :: "'a::linordered_idom^'n^'n"
   shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
   by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
+
 (* ------------------------------------------------------------------------- *)
 (* Explicit formulas for low dimensions.                                     *)
 (* ------------------------------------------------------------------------- *)
 
-lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
+lemma setprod_1: "setprod f {(1::nat)..1} = f 1"
+  by simp
 
 lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
   by (simp add: eval_nat_numeral setprod_numseg mult_commute)
+
 lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
   by (simp add: eval_nat_numeral setprod_numseg mult_commute)
 
@@ -1044,33 +1232,33 @@
   by (simp add: det_def sign_id)
 
 lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
-proof-
+proof -
   have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
   show ?thesis
-  unfolding det_def UNIV_2
-  unfolding setsum_over_permutations_insert[OF f12]
-  unfolding permutes_sing
-  by (simp add: sign_swap_id sign_id swap_id_eq)
+    unfolding det_def UNIV_2
+    unfolding setsum_over_permutations_insert[OF f12]
+    unfolding permutes_sing
+    by (simp add: sign_swap_id sign_id swap_id_eq)
 qed
 
-lemma det_3: "det (A::'a::comm_ring_1^3^3) =
-  A$1$1 * A$2$2 * A$3$3 +
-  A$1$2 * A$2$3 * A$3$1 +
-  A$1$3 * A$2$1 * A$3$2 -
-  A$1$1 * A$2$3 * A$3$2 -
-  A$1$2 * A$2$1 * A$3$3 -
-  A$1$3 * A$2$2 * A$3$1"
-proof-
+lemma det_3:
+  "det (A::'a::comm_ring_1^3^3) =
+    A$1$1 * A$2$2 * A$3$3 +
+    A$1$2 * A$2$3 * A$3$1 +
+    A$1$3 * A$2$1 * A$3$2 -
+    A$1$1 * A$2$3 * A$3$2 -
+    A$1$2 * A$2$1 * A$3$3 -
+    A$1$3 * A$2$2 * A$3$1"
+proof -
   have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}" by auto
   have f23: "finite {3::3}" "2 \<notin> {3::3}" by auto
 
   show ?thesis
-  unfolding det_def UNIV_3
-  unfolding setsum_over_permutations_insert[OF f123]
-  unfolding setsum_over_permutations_insert[OF f23]
-
-  unfolding permutes_sing
-  by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
+    unfolding det_def UNIV_3
+    unfolding setsum_over_permutations_insert[OF f123]
+    unfolding setsum_over_permutations_insert[OF f23]
+    unfolding permutes_sing
+    by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
 qed
 
 end
diff -r 16235bb41881 -r 082d0972096b src/HOL/Multivariate_Analysis/Operator_Norm.thy
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 28 23:48:45 2013 +0200
@@ -11,72 +11,83 @@
 definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
 
 lemma norm_bound_generalize:
-  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes lf: "linear f"
-  shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
-  {assume H: ?rhs
-    {fix x :: "'a" assume x: "norm x = 1"
-      from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
-    then have ?lhs by blast }
+  shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume H: ?rhs
+  {
+    fix x :: "'a"
+    assume x: "norm x = 1"
+    from H[rule_format, of x] x have "norm (f x) \<le> b" by simp
+  }
+  then show ?lhs by blast
+next
+  assume H: ?lhs
+  have bp: "b \<ge> 0"
+    apply -
+    apply (rule order_trans [OF norm_ge_zero])
+    apply (rule H[rule_format, of "SOME x::'a. x \<in> Basis"])
+    apply (auto intro: SOME_Basis norm_Basis)
+    done
+  {
+    fix x :: "'a"
+    {
+      assume "x = 0"
+      then have "norm (f x) \<le> b * norm x"
+        by (simp add: linear_0[OF lf] bp)
+    }
+    moreover
+    {
+      assume x0: "x \<noteq> 0"
+      then have n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
+      let ?c = "1/ norm x"
+      have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0)
+      with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast
+      then have "?c * norm (f x) \<le> b"
+        by (simp add: linear_cmul[OF lf])
+      then have "norm (f x) \<le> b * norm x"
+        using n0 norm_ge_zero[of x] by (auto simp add: field_simps)
+    }
+    ultimately have "norm (f x) \<le> b * norm x" by blast
+  }
+  then show ?rhs by blast
+qed
 
-  moreover
-  {assume H: ?lhs
-    have bp: "b \<ge> 0"
-      apply -
-      apply(rule order_trans [OF norm_ge_zero])
-      apply(rule H[rule_format, of "SOME x::'a. x \<in> Basis"])
-      by (auto intro: SOME_Basis norm_Basis)
-    {fix x :: "'a"
-      {assume "x = 0"
-        then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
-      moreover
-      {assume x0: "x \<noteq> 0"
-        hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
-        let ?c = "1/ norm x"
-        have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0)
-        with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast
-        hence "?c * norm (f x) \<le> b"
-          by (simp add: linear_cmul[OF lf])
-        hence "norm (f x) \<le> b * norm x"
-          using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
-      ultimately have "norm (f x) \<le> b * norm x" by blast}
-    then have ?rhs by blast}
-  ultimately show ?thesis by blast
-qed
- 
 lemma onorm:
   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes lf: "linear f"
-  shows "norm (f x) <= onorm f * norm x"
-  and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
-proof-
-  {
-    let ?S = "{norm (f x) |x. norm x = 1}"
-    have "norm (f (SOME i. i \<in> Basis)) \<in> ?S"
-      by (auto intro!: exI[of _ "SOME i. i \<in> Basis"] norm_Basis SOME_Basis)
-    hence Se: "?S \<noteq> {}" by auto
-    from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
-      unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
-    { from isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
-      show "norm (f x) <= onorm f * norm x"
-        apply -
-        apply (rule spec[where x = x])
-        unfolding norm_bound_generalize[OF lf, symmetric]
-        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
-    {
-      show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
-        using isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
-        unfolding norm_bound_generalize[OF lf, symmetric]
-        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
-  }
+  shows "norm (f x) \<le> onorm f * norm x"
+    and "\<forall>x. norm (f x) \<le> b * norm x \<Longrightarrow> onorm f \<le> b"
+proof -
+  let ?S = "{norm (f x) |x. norm x = 1}"
+  have "norm (f (SOME i. i \<in> Basis)) \<in> ?S"
+    by (auto intro!: exI[of _ "SOME i. i \<in> Basis"] norm_Basis SOME_Basis)
+  then have Se: "?S \<noteq> {}" by auto
+  from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
+    unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
+  from isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
+  show "norm (f x) <= onorm f * norm x"
+    apply -
+    apply (rule spec[where x = x])
+    unfolding norm_bound_generalize[OF lf, symmetric]
+    apply (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)
+    done
+  show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
+    using isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
+    unfolding norm_bound_generalize[OF lf, symmetric]
+    by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)
 qed
 
-lemma onorm_pos_le: assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)" shows "0 <= onorm f"
-  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "SOME i. i \<in> Basis"]] 
+lemma onorm_pos_le:
+  assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
+  shows "0 \<le> onorm f"
+  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "SOME i. i \<in> Basis"]]
   by (simp add: SOME_Basis)
 
-lemma onorm_eq_0: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_eq_0:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
   using onorm[OF lf]
   apply (auto simp add: onorm_pos_le)
@@ -87,47 +98,53 @@
   done
 
 lemma onorm_const: "onorm(\<lambda>x::'a::euclidean_space. (y::'b::euclidean_space)) = norm y"
-proof-
+proof -
   let ?f = "\<lambda>x::'a. (y::'b)"
   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
     by (auto simp: SOME_Basis intro!: exI[of _ "SOME i. i \<in> Basis"])
   show ?thesis
     unfolding onorm_def th
-    apply (rule cSup_unique) by (simp_all  add: setle_def)
+    apply (rule cSup_unique)
+    apply (simp_all  add: setle_def)
+    done
 qed
 
-lemma onorm_pos_lt: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_pos_lt:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
   unfolding onorm_eq_0[OF lf, symmetric]
   using onorm_pos_le[OF lf] by arith
 
 lemma onorm_compose:
   assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
-  and lg: "linear (g::'k::euclidean_space \<Rightarrow> 'n::euclidean_space)"
-  shows "onorm (f o g) <= onorm f * onorm g"
-  apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
-  unfolding o_def
-  apply (subst mult_assoc)
-  apply (rule order_trans)
-  apply (rule onorm(1)[OF lf])
-  apply (rule mult_left_mono)
-  apply (rule onorm(1)[OF lg])
-  apply (rule onorm_pos_le[OF lf])
-  done
+    and lg: "linear (g::'k::euclidean_space \<Rightarrow> 'n::euclidean_space)"
+  shows "onorm (f o g) \<le> onorm f * onorm g"
+    apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
+    unfolding o_def
+    apply (subst mult_assoc)
+    apply (rule order_trans)
+    apply (rule onorm(1)[OF lf])
+    apply (rule mult_left_mono)
+    apply (rule onorm(1)[OF lg])
+    apply (rule onorm_pos_le[OF lf])
+    done
 
-lemma onorm_neg_lemma: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_neg_lemma:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
   unfolding norm_minus_cancel by metis
 
-lemma onorm_neg: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_neg:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
   shows "onorm (\<lambda>x. - f x) = onorm f"
   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
   by simp
 
 lemma onorm_triangle:
-  assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)" and lg: "linear g"
-  shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
+  assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
+    and lg: "linear g"
+  shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
   apply (rule order_trans)
   apply (rule norm_triangle_ineq)
@@ -137,17 +154,20 @@
   apply (rule onorm(1)[OF lg])
   done
 
-lemma onorm_triangle_le: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
-  \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
+lemma onorm_triangle_le:
+  "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow>
+    linear g \<Longrightarrow> onorm f + onorm g \<le> e \<Longrightarrow> onorm (\<lambda>x. f x + g x) \<le> e"
   apply (rule order_trans)
   apply (rule onorm_triangle)
   apply assumption+
   done
 
-lemma onorm_triangle_lt: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
-  ==> onorm(\<lambda>x. f x + g x) < e"
+lemma onorm_triangle_lt:
+  "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow>
+    onorm f + onorm g < e \<Longrightarrow> onorm(\<lambda>x. f x + g x) < e"
   apply (rule order_le_less_trans)
   apply (rule onorm_triangle)
-  by assumption+
+  apply assumption+
+  done
 
 end
diff -r 16235bb41881 -r 082d0972096b src/Pure/Thy/completion.scala
--- a/src/Pure/Thy/completion.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Pure/Thy/completion.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -27,12 +27,13 @@
 
     def reverse_symbol: Parser[String] = """>[A-Za-z0-9_']+\^?<\\""".r
     def reverse_symb: Parser[String] = """[A-Za-z0-9_']{2,}\^?<\\""".r
-    def word: Parser[String] = "[a-zA-Z0-9_']{2,}".r
+    def escape: Parser[String] = """[a-zA-Z0-9_']+\\""".r
+    def word: Parser[String] = """[a-zA-Z0-9_']{3,}""".r
 
     def read(in: CharSequence): Option[String] =
     {
       val reverse_in = new Library.Reverse(in)
-      parse((reverse_symbol | reverse_symb | word) ^^ (_.reverse), reverse_in) match {
+      parse((reverse_symbol | reverse_symb | escape | word) ^^ (_.reverse), reverse_in) match {
         case Success(result, _) => Some(result)
         case _ => None
       }
@@ -61,7 +62,7 @@
   {
     val words =
       (for ((x, _) <- Symbol.names) yield (x, x)).toList :::
-      (for ((x, y) <- Symbol.names) yield (y, x)).toList :::
+      (for ((x, y) <- Symbol.names) yield ("\\" + y, x)).toList :::
       (for ((x, y) <- Symbol.abbrevs if Completion.is_word(y)) yield (y, x)).toList
 
     val abbrs =
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/etc/options
--- a/src/Tools/jEdit/etc/options	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/etc/options	Wed Aug 28 23:48:45 2013 +0200
@@ -9,18 +9,18 @@
 public option jedit_font_scale : real = 1.0
   -- "scale factor of add-on panels wrt. main text area"
 
+public option jedit_popup_font_scale : real = 0.85
+  -- "scale factor of popups wrt. main text area"
+
+public option jedit_popup_bounds : real = 0.5
+  -- "relative bounds of popup window wrt. logical screen size"
+
 public option jedit_tooltip_delay : real = 0.75
   -- "open/close delay for document tooltips"
 
-public option jedit_tooltip_font_scale : real = 0.85
-  -- "scale factor of tooltips wrt. main text area"
-
 public option jedit_tooltip_margin : int = 60
   -- "margin for tooltip pretty-printing"
 
-public option jedit_tooltip_bounds : real = 0.5
-  -- "relative bounds of tooltip window wrt. logical screen size"
-
 public option jedit_text_overview_limit : int = 100000
   -- "maximum amount of text to visualize in overview column"
 
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/lib/Tools/jedit
--- a/src/Tools/jEdit/lib/Tools/jedit	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/lib/Tools/jedit	Wed Aug 28 23:48:45 2013 +0200
@@ -33,6 +33,7 @@
   "src/osx_adapter.scala"
   "src/output_dockable.scala"
   "src/plugin.scala"
+  "src/popup.scala"
   "src/pretty_text_area.scala"
   "src/pretty_tooltip.scala"
   "src/process_indicator.scala"
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/completion_popup.scala
--- a/src/Tools/jEdit/src/completion_popup.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/completion_popup.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -1,7 +1,7 @@
 /*  Title:      Tools/jEdit/src/completion_popup.scala
     Author:     Makarius
 
-Completion popup based on javax.swing.PopupFactory.
+Completion popup.
 */
 
 package isabelle.jedit
@@ -9,56 +9,162 @@
 
 import isabelle._
 
-import java.awt.{Point, BorderLayout, Dimension}
-import java.awt.event.{KeyEvent, KeyAdapter, MouseEvent, MouseAdapter, FocusAdapter, FocusEvent}
-import javax.swing.{JPanel, JComponent, PopupFactory}
+import java.awt.{Font, Point, BorderLayout, Dimension}
+import java.awt.event.{KeyEvent, MouseEvent, MouseAdapter, FocusAdapter, FocusEvent}
+import javax.swing.{JPanel, JComponent, JLayeredPane, SwingUtilities}
 
 import scala.swing.{ListView, ScrollPane}
 import scala.swing.event.MouseClicked
 
 import org.gjt.sp.jedit.View
+import org.gjt.sp.jedit.textarea.JEditTextArea
 
 
 object Completion_Popup
 {
-  sealed case class Item(name: String, text: String)
-  { override def toString: String = text }
+  /* items */
+
+  private sealed case class Item(original: String, replacement: String, description: String)
+  { override def toString: String = description }
+
+
+  /* maintain single instance */
 
-  def apply(
-    opt_view: Option[View],
-    parent: JComponent,
-    screen_point: Point,
-    items: List[Item],
-    complete: Item => Unit): Completion_Popup =
+  def dismissed(layered: JLayeredPane): Boolean =
+  {
+    Swing_Thread.require()
+
+    layered.getClientProperty(Completion_Popup) match {
+      case old_completion: Completion_Popup =>
+        old_completion.hide_popup()
+        true
+      case _ =>
+        false
+    }
+  }
+
+  private def register(layered: JLayeredPane, completion: Completion_Popup)
   {
     Swing_Thread.require()
 
-    require(!items.isEmpty)
+    dismissed(layered)
+    layered.putClientProperty(Completion_Popup, completion)
+  }
+
+
+  /* jEdit text area operations */
+
+  object Text_Area
+  {
+    private def insert(text_area: JEditTextArea, item: Item)
+    {
+      Swing_Thread.require()
+
+      val buffer = text_area.getBuffer
+      val len = item.original.length
+      if (buffer.isEditable && len > 0) {
+        JEdit_Lib.buffer_edit(buffer) {
+          val caret = text_area.getCaretPosition
+          JEdit_Lib.try_get_text(buffer, Text.Range(caret - len, caret)) match {
+            case Some(text) if text == item.original =>
+              buffer.remove(caret - len, len)
+              buffer.insert(caret - len, item.replacement)
+            case _ =>
+          }
+        }
+      }
+    }
+
+    def input(text_area: JEditTextArea, get_syntax: => Option[Outer_Syntax], evt: KeyEvent)
+    {
+      Swing_Thread.require()
+
+      val view = text_area.getView
+      val layered = view.getLayeredPane
+      val buffer = text_area.getBuffer
+      val painter = text_area.getPainter
+
+      if (buffer.isEditable && !evt.isConsumed) {
+        dismissed(layered)
 
-    val completion = new Completion_Popup(opt_view, parent, screen_point, items, complete)
-    completion.show_popup()
-    completion
+        get_syntax match {
+          case Some(syntax) =>
+            val caret = text_area.getCaretPosition
+            val result =
+            {
+              val line = buffer.getLineOfOffset(caret)
+              val start = buffer.getLineStartOffset(line)
+              val text = buffer.getSegment(start, caret - start)
+
+              syntax.completion.complete(text) match {
+                case Some((word, cs)) =>
+                  val ds =
+                    (if (Isabelle_Encoding.is_active(buffer))
+                      cs.map(Symbol.decode(_)).sorted
+                     else cs).filter(_ != word)
+                  if (ds.isEmpty) None
+                  else Some((word, ds.map(s => Item(word, s, s))))
+                case None => None
+              }
+            }
+            result match {
+              case Some((original, items)) =>
+                val popup_font =
+                  painter.getFont.deriveFont(
+                    (painter.getFont.getSize2D * PIDE.options.real("jedit_popup_font_scale")).toFloat
+                      max 10.0f)
+
+                val loc1 = text_area.offsetToXY(caret - original.length)
+                if (loc1 != null) {
+                  val loc2 =
+                    SwingUtilities.convertPoint(painter,
+                      loc1.x, loc1.y + painter.getFontMetrics.getHeight, layered)
+                  val completion =
+                    new Completion_Popup(layered, loc2, popup_font, items) {
+                      override def complete(item: Item) { insert(text_area, item) }
+                      override def propagate(e: KeyEvent) {
+                        JEdit_Lib.propagate_key(view, e)
+                        input(text_area, get_syntax, e)
+                      }
+                      override def refocus() { text_area.requestFocus }
+                    }
+                  register(layered, completion)
+                  completion.show_popup()
+                }
+              case None =>
+            }
+          case None =>
+        }
+      }
+    }
   }
 }
 
 
 class Completion_Popup private(
-  opt_view: Option[View],
-  parent: JComponent,
-  screen_point: Point,
-  items: List[Completion_Popup.Item],
-  complete: Completion_Popup.Item => Unit) extends JPanel(new BorderLayout)
+  layered: JLayeredPane,
+  location: Point,
+  popup_font: Font,
+  items: List[Completion_Popup.Item]) extends JPanel(new BorderLayout)
 {
   completion =>
 
   Swing_Thread.require()
+  require(!items.isEmpty)
+
+
+  /* actions */
+
+  def complete(item: Completion_Popup.Item) { }
+  def propagate(evt: KeyEvent) { }
+  def refocus() { }
 
 
   /* list view */
 
   private val list_view = new ListView(items)
   {
-    font = parent.getFont
+    font = popup_font
   }
   list_view.selection.intervalMode = ListView.IntervalMode.Single
   list_view.peer.setFocusTraversalKeysEnabled(false)
@@ -92,49 +198,33 @@
 
   /* event handling */
 
-  private val key_handler = new KeyAdapter {
-    override def keyPressed(e: KeyEvent)
-    {
-      if (!e.isConsumed) {
-        e.getKeyCode match {
-          case KeyEvent.VK_TAB =>
-            if (complete_selected()) e.consume
-            hide_popup()
-          case KeyEvent.VK_ESCAPE =>
-            hide_popup()
-            e.consume
-          case KeyEvent.VK_UP => move_items(-1); e.consume
-          case KeyEvent.VK_DOWN => move_items(1); e.consume
-          case KeyEvent.VK_PAGE_UP => move_pages(-1); e.consume
-          case KeyEvent.VK_PAGE_DOWN => move_pages(1); e.consume
-          case _ =>
-            if (e.isActionKey || e.isAltDown || e.isMetaDown || e.isControlDown)
-              hide_popup()
-        }
-      }
-      if (!e.isConsumed) pass_to_view(e)
-    }
+  private val key_listener =
+    JEdit_Lib.key_listener(
+      key_pressed = (e: KeyEvent) =>
+        {
+          if (!e.isConsumed) {
+            e.getKeyCode match {
+              case KeyEvent.VK_TAB =>
+                if (complete_selected()) e.consume
+                hide_popup()
+              case KeyEvent.VK_ESCAPE =>
+                hide_popup()
+                e.consume
+              case KeyEvent.VK_UP => move_items(-1); e.consume
+              case KeyEvent.VK_DOWN => move_items(1); e.consume
+              case KeyEvent.VK_PAGE_UP => move_pages(-1); e.consume
+              case KeyEvent.VK_PAGE_DOWN => move_pages(1); e.consume
+              case _ =>
+                if (e.isActionKey || e.isAltDown || e.isMetaDown || e.isControlDown)
+                  hide_popup()
+            }
+          }
+          propagate(e)
+        },
+      key_typed = propagate _
+    )
 
-    override def keyTyped(e: KeyEvent)
-    {
-      if (!e.isConsumed) pass_to_view(e)
-    }
-  }
-
-  private def pass_to_view(e: KeyEvent)
-  {
-    opt_view match {
-      case Some(view) if view.getKeyEventInterceptor == key_handler =>
-        try {
-          view.setKeyEventInterceptor(null)
-          view.getInputHandler().processKeyEvent(e, View.ACTION_BAR, false)
-        }
-        finally { view.setKeyEventInterceptor(key_handler) }
-      case _ =>
-    }
-  }
-
-  list_view.peer.addKeyListener(key_handler)
+  list_view.peer.addKeyListener(key_listener)
 
   list_view.peer.addMouseListener(new MouseAdapter {
     override def mouseClicked(e: MouseEvent) {
@@ -159,37 +249,29 @@
 
   private val popup =
   {
-    val screen_bounds = JEdit_Lib.screen_bounds(screen_point)
-
-    val geometry = JEdit_Lib.window_geometry(completion, completion)
-
-    val w = geometry.width min (screen_bounds.width / 2)
-    val h = geometry.height min (screen_bounds.height / 2)
-
-    completion.setSize(new Dimension(w, h))
-    completion.setPreferredSize(new Dimension(w, h))
-
-    val x = screen_point.x min (screen_bounds.x + screen_bounds.width - w)
-    val y = screen_point.y min (screen_bounds.y + screen_bounds.height - h)
-    PopupFactory.getSharedInstance.getPopup(parent, completion, x, y)
+    val screen = JEdit_Lib.screen_location(layered, location)
+    val size =
+    {
+      val geometry = JEdit_Lib.window_geometry(completion, completion)
+      val bounds = Rendering.popup_bounds
+      val w = geometry.width min (screen.bounds.width * bounds).toInt min layered.getWidth
+      val h = geometry.height min (screen.bounds.height * bounds).toInt min layered.getHeight
+      new Dimension(w, h)
+    }
+    new Popup(layered, completion, screen.relative(layered, size), size)
   }
 
-  def show_popup()
+  private def show_popup()
   {
-    opt_view.foreach(view => view.setKeyEventInterceptor(key_handler))
     popup.show
     list_view.requestFocus
   }
 
-  def hide_popup()
+  private def hide_popup()
   {
-    opt_view match {
-      case Some(view) if view.getKeyEventInterceptor == key_handler =>
-        view.setKeyEventInterceptor(null)
-      case _ =>
-    }
+    val had_focus = list_view.peer.isFocusOwner
     popup.hide
-    parent.requestFocus
+    if (had_focus) refocus()
   }
 }
 
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/document_view.scala
--- a/src/Tools/jEdit/src/document_view.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/document_view.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -17,7 +17,7 @@
 import java.lang.System
 import java.text.BreakIterator
 import java.awt.{Color, Graphics2D, Point}
-import java.awt.event.{KeyEvent, KeyAdapter}
+import java.awt.event.KeyEvent
 import javax.swing.event.{CaretListener, CaretEvent}
 
 import org.gjt.sp.jedit.{jEdit, Debug}
@@ -149,13 +149,15 @@
 
   /* key listener */
 
-  private val key_listener = new KeyAdapter {
-    override def keyTyped(evt: KeyEvent)
-    {
-      if (evt.getKeyChar == 27 && Pretty_Tooltip.dismissed_all())
-        evt.consume
-    }
-  }
+  private val key_listener =
+    JEdit_Lib.key_listener(
+      key_pressed = (evt: KeyEvent) =>
+        {
+          if (evt.getKeyCode == KeyEvent.VK_ESCAPE && PIDE.dismissed_popups(text_area.getView))
+            evt.consume
+        },
+      key_typed = Completion_Popup.Text_Area.input(text_area, PIDE.get_recent_syntax, _)
+    )
 
 
   /* caret handling */
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/graphview_dockable.scala
--- a/src/Tools/jEdit/src/graphview_dockable.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/graphview_dockable.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -9,7 +9,7 @@
 
 import isabelle._
 
-import javax.swing.{SwingUtilities, JComponent}
+import javax.swing.JComponent
 import java.awt.Point
 import java.awt.event.{WindowFocusListener, WindowEvent}
 
@@ -66,10 +66,8 @@
             Pretty_Tooltip.invoke(() =>
               {
                 val rendering = Rendering(snapshot, PIDE.options.value)
-                val screen_point = new Point(x, y)
-                SwingUtilities.convertPointToScreen(screen_point, parent)
                 val info = Text.Info(Text.Range(~1), body)
-                Pretty_Tooltip(view, parent, rendering, screen_point, Command.Results.empty, info)
+                Pretty_Tooltip(view, parent, new Point(x, y), rendering, Command.Results.empty, info)
               })
             null
           }
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/isabelle_sidekick.scala
--- a/src/Tools/jEdit/src/isabelle_sidekick.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/isabelle_sidekick.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -188,6 +188,9 @@
 
 class Isabelle_Sidekick_Default extends Isabelle_Sidekick_Structure(
   "isabelle", PIDE.get_recent_syntax, PIDE.thy_load.buffer_node_name)
+{
+  override def supportsCompletion = false
+}
 
 
 class Isabelle_Sidekick_Options extends Isabelle_Sidekick_Structure(
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/jedit_lib.scala
--- a/src/Tools/jEdit/src/jedit_lib.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/jedit_lib.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -9,28 +9,56 @@
 
 import isabelle._
 
-import java.awt.{Component, Container, Window, GraphicsEnvironment, Point, Rectangle}
-import javax.swing.{Icon, ImageIcon, JWindow}
+import java.awt.{Component, Container, Window, GraphicsEnvironment, Point, Rectangle, Dimension}
+import java.awt.event.{KeyEvent, KeyListener}
+import javax.swing.{Icon, ImageIcon, JWindow, SwingUtilities}
 
 import scala.annotation.tailrec
 
 import org.gjt.sp.jedit.{jEdit, Buffer, View, GUIUtilities}
+import org.gjt.sp.jedit.gui.KeyEventWorkaround
 import org.gjt.sp.jedit.buffer.JEditBuffer
 import org.gjt.sp.jedit.textarea.{JEditTextArea, TextArea, TextAreaPainter}
 
 
 object JEdit_Lib
 {
-  /* bounds within multi-screen environment */
+  /* location within multi-screen environment */
+
+  final case class Screen_Location(point: Point, bounds: Rectangle)
+  {
+    def relative(parent: Component, size: Dimension): Point =
+    {
+      val w = size.width
+      val h = size.height
 
-  def screen_bounds(screen_point: Point): Rectangle =
+      val x0 = parent.getLocationOnScreen.x
+      val y0 = parent.getLocationOnScreen.y
+      val x1 = x0 + parent.getWidth - w
+      val y1 = y0 + parent.getHeight - h
+      val x2 = point.x min (bounds.x + bounds.width - w)
+      val y2 = point.y min (bounds.y + bounds.height - h)
+
+      val location = new Point((x2 min x1) max x0, (y2 min y1) max y0)
+      SwingUtilities.convertPointFromScreen(location, parent)
+      location
+    }
+  }
+
+  def screen_location(component: Component, point: Point): Screen_Location =
   {
+    val screen_point = new Point(point.x, point.y)
+    SwingUtilities.convertPointToScreen(screen_point, component)
+
     val ge = GraphicsEnvironment.getLocalGraphicsEnvironment
-    (for {
-      device <- ge.getScreenDevices.iterator
-      config <- device.getConfigurations.iterator
-      bounds = config.getBounds
-    } yield bounds).find(_.contains(screen_point)) getOrElse ge.getMaximumWindowBounds
+    val screen_bounds =
+      (for {
+        device <- ge.getScreenDevices.iterator
+        config <- device.getConfigurations.iterator
+        bounds = config.getBounds
+      } yield bounds).find(_.contains(screen_point)) getOrElse ge.getMaximumWindowBounds
+
+    Screen_Location(screen_point, screen_bounds)
   }
 
 
@@ -106,21 +134,6 @@
   def buffer_name(buffer: Buffer): String = buffer.getSymlinkPath
 
 
-  /* focus of main window */
-
-  def request_focus_view: Unit =
-  {
-    jEdit.getActiveView() match {
-      case null =>
-      case view =>
-        view.getTextArea match {
-          case null =>
-          case text_area => text_area.requestFocus
-        }
-    }
-  }
-
-
   /* main jEdit components */
 
   def jedit_buffers(): Iterator[Buffer] = jEdit.getBuffers().iterator
@@ -300,5 +313,42 @@
       case icon: ImageIcon => icon
       case _ => error("Bad image icon: " + name)
     }
+
+
+  /* key event handling */
+
+  def request_focus_view
+  {
+    val view = jEdit.getActiveView()
+    if (view != null) {
+      val text_area = view.getTextArea
+      if (text_area != null) text_area.requestFocus
+    }
+  }
+
+  def propagate_key(view: View, evt: KeyEvent)
+  {
+    if (view != null && !evt.isConsumed)
+      view.getInputHandler().processKeyEvent(evt, View.ACTION_BAR, false)
+  }
+
+  def key_listener(
+    key_typed: KeyEvent => Unit = _ => (),
+    key_pressed: KeyEvent => Unit = _ => (),
+    key_released: KeyEvent => Unit = _ => ()): KeyListener =
+  {
+    def process_key_event(evt0: KeyEvent, handle: KeyEvent => Unit)
+    {
+      val evt = KeyEventWorkaround.processKeyEvent(evt0)
+      if (evt != null) handle(evt)
+    }
+
+    new KeyListener
+    {
+      def keyTyped(evt: KeyEvent) { process_key_event(evt, key_typed) }
+      def keyPressed(evt: KeyEvent) { process_key_event(evt, key_pressed) }
+      def keyReleased(evt: KeyEvent) { process_key_event(evt, key_released) }
+    }
+  }
 }
 
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/plugin.scala
--- a/src/Tools/jEdit/src/plugin.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/plugin.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -53,6 +53,16 @@
   lazy val editor = new JEdit_Editor
 
 
+  /* popups */
+
+  def dismissed_popups(view: View): Boolean =
+  {
+    val b1 = Completion_Popup.dismissed(view.getLayeredPane)
+    val b2 = Pretty_Tooltip.dismissed_all()
+    b1 || b2
+  }
+
+
   /* document model and view */
 
   def document_model(buffer: Buffer): Option[Document_Model] = Document_Model(buffer)
@@ -265,7 +275,7 @@
                 PIDE.init_view(buffer, text_area)
             }
             else {
-              Pretty_Tooltip.dismissed_all()
+              PIDE.dismissed_popups(text_area.getView)
               PIDE.exit_view(buffer, text_area)
             }
           }
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/popup.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/src/popup.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -0,0 +1,40 @@
+/*  Title:      Tools/jEdit/src/popup.scala
+    Author:     Makarius
+
+Popup within layered pane.
+*/
+
+package isabelle.jedit
+
+
+import isabelle._
+
+import java.awt.{Point, Dimension}
+import javax.swing.{JLayeredPane, JComponent}
+
+
+class Popup(
+  layered: JLayeredPane,
+  component: JComponent,
+  location: Point,
+  size: Dimension)
+{
+  def show
+  {
+    component.setLocation(location)
+    component.setSize(size)
+    component.setPreferredSize(size)
+    component.setOpaque(true)
+    layered.add(component, JLayeredPane.POPUP_LAYER)
+    layered.moveToFront(component)
+    layered.repaint(component.getBounds())
+  }
+
+  def hide
+  {
+    val bounds = component.getBounds()
+    layered.remove(component)
+    layered.repaint(bounds)
+  }
+}
+
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/pretty_text_area.scala
--- a/src/Tools/jEdit/src/pretty_text_area.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/pretty_text_area.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -11,7 +11,7 @@
 import isabelle._
 
 import java.awt.{Color, Font, FontMetrics, Toolkit, Window}
-import java.awt.event.{KeyEvent, KeyAdapter}
+import java.awt.event.KeyEvent
 
 import org.gjt.sp.jedit.{jEdit, View, Registers}
 import org.gjt.sp.jedit.textarea.{AntiAlias, JEditEmbeddedTextArea}
@@ -162,33 +162,33 @@
 
   /* key handling */
 
-  addKeyListener(new KeyAdapter {
-    override def keyPressed(evt: KeyEvent)
-    {
-      evt.getKeyCode match {
-        case KeyEvent.VK_C
-        if (evt.getModifiers & Toolkit.getDefaultToolkit.getMenuShortcutKeyMask) != 0 =>
-          Registers.copy(text_area, '$')
-          evt.consume
-        case KeyEvent.VK_A
-        if (evt.getModifiers & Toolkit.getDefaultToolkit.getMenuShortcutKeyMask) != 0 =>
-          text_area.selectAll
-          evt.consume
-        case _ =>
+  addKeyListener(JEdit_Lib.key_listener(
+    key_pressed = (evt: KeyEvent) =>
+      {
+        evt.getKeyCode match {
+          case KeyEvent.VK_C
+          if (evt.getModifiers & Toolkit.getDefaultToolkit.getMenuShortcutKeyMask) != 0 =>
+            Registers.copy(text_area, '$')
+            evt.consume
+
+          case KeyEvent.VK_A
+          if (evt.getModifiers & Toolkit.getDefaultToolkit.getMenuShortcutKeyMask) != 0 =>
+            text_area.selectAll
+            evt.consume
+
+          case KeyEvent.VK_ESCAPE =>
+            if (PIDE.dismissed_popups(view)) evt.consume
+
+          case _ =>
+        }
+        if (propagate_keys) JEdit_Lib.propagate_key(view, evt)
+      },
+    key_typed = (evt: KeyEvent) =>
+      {
+        if (propagate_keys) JEdit_Lib.propagate_key(view, evt)
       }
-      if (propagate_keys && !evt.isConsumed)
-        view.getInputHandler.processKeyEvent(evt, View.ACTION_BAR, false)
-    }
-
-    override def keyTyped(evt: KeyEvent)
-    {
-      if (evt.getKeyChar == 27 && Pretty_Tooltip.dismissed_all())
-        evt.consume
-
-      if (propagate_keys && !evt.isConsumed)
-        view.getInputHandler.processKeyEvent(evt, View.ACTION_BAR, false)
-    }
-  })
+    )
+  )
 
 
   /* init */
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/pretty_tooltip.scala
--- a/src/Tools/jEdit/src/pretty_tooltip.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/pretty_tooltip.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -1,7 +1,7 @@
 /*  Title:      Tools/jEdit/src/pretty_tooltip.scala
     Author:     Makarius
 
-Enhanced tooltip window based on Pretty_Text_Area.
+Tooltip based on Pretty_Text_Area.
 */
 
 package isabelle.jedit
@@ -11,7 +11,7 @@
 
 import java.awt.{Color, Point, BorderLayout, Dimension}
 import java.awt.event.{FocusAdapter, FocusEvent}
-import javax.swing.{JPanel, JComponent, PopupFactory}
+import javax.swing.{JPanel, JComponent, SwingUtilities}
 import javax.swing.border.LineBorder
 
 import scala.swing.{FlowPanel, Label}
@@ -39,8 +39,8 @@
   def apply(
     view: View,
     parent: JComponent,
+    location: Point,
     rendering: Rendering,
-    screen_point: Point,
     results: Command.Results,
     info: Text.Info[XML.Body]): Pretty_Tooltip =
   {
@@ -56,7 +56,8 @@
           }
         old.foreach(_.hide_popup)
 
-        val tip = new Pretty_Tooltip(view, rendering, screen_point, results, info)
+        val loc = SwingUtilities.convertPoint(parent, location, view.getLayeredPane)
+        val tip = new Pretty_Tooltip(view, loc, rendering, results, info)
         stack = tip :: rest
         tip.show_popup
         tip
@@ -138,8 +139,8 @@
 
 class Pretty_Tooltip private(
   view: View,
+  location: Point,
   rendering: Rendering,
-  screen_point: Point,
   private val results: Command.Results,
   private val info: Text.Info[XML.Body]) extends JPanel(new BorderLayout)
 {
@@ -195,7 +196,7 @@
   })
 
   pretty_text_area.resize(Rendering.font_family(),
-    Rendering.font_size("jedit_tooltip_font_scale").round)
+    Rendering.font_size("jedit_popup_font_scale").round)
 
 
   /* main content */
@@ -217,15 +218,9 @@
 
   private val popup =
   {
-    val screen_bounds = JEdit_Lib.screen_bounds(screen_point)
-
-    val root = view.getRootPane
-    val x0 = root.getLocationOnScreen.x
-    val y0 = root.getLocationOnScreen.y
-    val w0 = root.getWidth
-    val h0 = root.getHeight
-
-    val (w, h) =
+    val layered = view.getLayeredPane
+    val screen = JEdit_Lib.screen_location(layered, location)
+    val size =
     {
       val painter = pretty_text_area.getPainter
       val metric = JEdit_Lib.pretty_metric(painter)
@@ -237,10 +232,11 @@
           (n: Int, s: String) => n + s.iterator.filter(_ == '\n').length)
 
       val geometry = JEdit_Lib.window_geometry(tip, tip.pretty_text_area.getPainter)
-      val bounds = rendering.tooltip_bounds
+      val bounds = Rendering.popup_bounds
 
+      val h0 = layered.getHeight
       val h1 = painter.getFontMetrics.getHeight * (lines + 1) + geometry.deco_height
-      val h2 = h0 min (screen_bounds.height * bounds).toInt
+      val h2 = h0 min (screen.bounds.height * bounds).toInt
       val h = h1 min h2
 
       val margin1 =
@@ -248,25 +244,14 @@
           (0.0 /: split_lines(XML.content(formatted)))({ case (m, line) => m max metric(line) })
         else margin
 
+      val w0 = layered.getWidth
       val w1 = (metric.unit * (margin1 + metric.average)).round.toInt + geometry.deco_width
-      val w2 = w0 min (screen_bounds.width * bounds).toInt
+      val w2 = w0 min (screen.bounds.width * bounds).toInt
       val w = w1 min w2
 
-      (w, h)
+      new Dimension(w, h)
     }
-
-    val (x, y) =
-    {
-      val x1 = x0 + w0 - w
-      val y1 = y0 + h0 - h
-      val x2 = screen_point.x min (screen_bounds.x + screen_bounds.width - w)
-      val y2 = screen_point.y min (screen_bounds.y + screen_bounds.height - h)
-      ((x2 min x1) max x0, (y2 min y1) max y0)
-    }
-
-    tip.setSize(new Dimension(w, h))
-    tip.setPreferredSize(new Dimension(w, h))
-    PopupFactory.getSharedInstance.getPopup(root, tip, x, y)
+    new Popup(layered, tip, screen.relative(layered, size), size)
   }
 
   private def show_popup
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/rendering.scala
--- a/src/Tools/jEdit/src/rendering.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/rendering.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -49,6 +49,11 @@
     (jEdit.getIntegerProperty("view.fontsize", 16) * PIDE.options.real(scale)).toFloat
 
 
+  /* popup window bounds */
+
+  def popup_bounds: Double = (PIDE.options.real("jedit_popup_bounds") max 0.2) min 0.8
+
+
   /* token markup -- text styles */
 
   private val command_style: Map[String, Byte] =
@@ -372,7 +377,6 @@
   }
 
   val tooltip_margin: Int = options.int("jedit_tooltip_margin")
-  val tooltip_bounds: Double = (options.real("jedit_tooltip_bounds") max 0.2) min 0.8
 
   lazy val tooltip_close_icon = JEdit_Lib.load_icon(options.string("tooltip_close_icon"))
   lazy val tooltip_detach_icon = JEdit_Lib.load_icon(options.string("tooltip_detach_icon"))
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/rich_text_area.scala
--- a/src/Tools/jEdit/src/rich_text_area.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/rich_text_area.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -208,10 +208,9 @@
                       case None =>
                       case Some(tip) =>
                         val painter = text_area.getPainter
-                        val screen_point = evt.getLocationOnScreen
-                        screen_point.translate(0, painter.getFontMetrics.getHeight / 2)
+                        val loc = new Point(x, y + painter.getFontMetrics.getHeight / 2)
                         val results = rendering.command_results(range)
-                        Pretty_Tooltip(view, painter, rendering, screen_point, results, tip)
+                        Pretty_Tooltip(view, painter, loc, rendering, results, tip)
                     }
                 }
               }
diff -r 16235bb41881 -r 082d0972096b src/Tools/jEdit/src/token_markup.scala
--- a/src/Tools/jEdit/src/token_markup.scala	Wed Aug 28 18:44:50 2013 +0200
+++ b/src/Tools/jEdit/src/token_markup.scala	Wed Aug 28 23:48:45 2013 +0200
@@ -285,7 +285,11 @@
   def refresh_buffer(buffer: JEditBuffer)
   {
     buffer.setTokenMarker(jEdit.getMode("text").getTokenMarker)
-    markers.get(buffer.getMode.getName).map(buffer.setTokenMarker(_))
+    // FIXME potential NPE ahead!?!
+    val mode = buffer.getMode
+    val name = mode.getName
+    val marker = markers.get(name)
+    marker.map(buffer.setTokenMarker(_))
   }
 }