merged
authornoschinl
Fri, 06 Sep 2013 10:57:27 +0200
changeset 53431 d2a7b6fe953e
parent 53430 d92578436d47 (current diff)
parent 53425 f5b1f555b73b (diff)
child 53432 36ca6764027f
merged
src/Tools/jEdit/patches/jedit/annotation
src/Tools/jEdit/patches/jedit/completion
--- a/Admin/Release/CHECKLIST	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/Release/CHECKLIST	Fri Sep 06 10:57:27 2013 +0200
@@ -7,8 +7,6 @@
 
 - test polyml-5.4.1, polyml-5.4.0, polyml-5.3.0, smlnj;
 
-- test scala-2.9.2;
-
 - test Proof General 4.1, 3.7.1.1;
 
 - test 'display_drafts' command;
@@ -27,8 +25,7 @@
 
 - update https://isabelle.in.tum.de/repos/website;
 
-- maintain Docs:
-    doc/Contents
+- maintain doc/Contents;
 
 - maintain Logics:
     ROOTS
--- a/Admin/Windows/Installer/sfx.txt	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/Windows/Installer/sfx.txt	Fri Sep 06 10:57:27 2013 +0200
@@ -5,5 +5,5 @@
 ExtractPathText="Target directory"
 ExtractTitle="Unpacking {ISABELLE_NAME} ..."
 Shortcut="Du,{%%T\{ISABELLE_NAME}\{ISABELLE_NAME}.exe},{},{},{},{{ISABELLE_NAME}},{%%T\{ISABELLE_NAME}}"
-RunProgram="\"%%T\{ISABELLE_NAME}\{ISABELLE_NAME}.exe\" -i"
+RunProgram="\"%%T\{ISABELLE_NAME}\{ISABELLE_NAME}.exe\""
 ;!@InstallEnd@!
--- a/Admin/Windows/launch4j/isabelle.xml	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/Windows/launch4j/isabelle.xml	Fri Sep 06 10:57:27 2013 +0200
@@ -20,11 +20,11 @@
     <cp>%EXEDIR%\lib\classes\ext\scala-swing.jar</cp>
   </classPath>
   <jre>
-    <path>%EXEDIR%\contrib\jdk-7u21\x86-cygwin\jdk1.7.0_21</path>
+    <path>%EXEDIR%\contrib\jdk\x86-cygwin</path>
     <minVersion></minVersion>
     <maxVersion></maxVersion>
     <jdkPreference>jdkOnly</jdkPreference>
-    <opt>-Disabelle.home=&quot;%EXEDIR%&quot; -Dcygwin.root=&quot;%EXEDIR%\\contrib\\cygwin&quot;</opt>
+    <opt>-Disabelle.home=&quot;%EXEDIR%&quot;</opt>
   </jre>
   <splash>
     <file>isabelle.bmp</file>
--- a/Admin/components/bundled-windows	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/components/bundled-windows	Fri Sep 06 10:57:27 2013 +0200
@@ -1,3 +1,3 @@
 #additional components to be bundled for release
 cygwin-20130117
-windows_app-20130716
+windows_app-20130905
--- a/Admin/components/components.sha1	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/components/components.sha1	Fri Sep 06 10:57:27 2013 +0200
@@ -18,6 +18,7 @@
 38d2d2a91c66714c18430e136e7e5191af3996e6  jdk-7u11.tar.gz
 d765bc4ad2f34d494429b2a8c1563c49db224944  jdk-7u13.tar.gz
 13a265e4b706ece26fdfa6fc9f4a3dd1366016d2  jdk-7u21.tar.gz
+5080274f8721a18111a7f614793afe6c88726739  jdk-7u25.tar.gz
 ec740ee9ffd43551ddf1e5b91641405116af6291  jdk-7u6.tar.gz
 7d5b152ac70f720bb9e783fa45ecadcf95069584  jdk-7u9.tar.gz
 44775a22f42a9d665696bfb49e53c79371c394b0  jedit_build-20111217.tar.gz
@@ -31,6 +32,7 @@
 8fa0c67f59beba369ab836562eed4e56382f672a  jedit_build-20121201.tar.gz
 06e9be2627ebb95c45a9bcfa025d2eeef086b408  jedit_build-20130104.tar.gz
 c85c0829b8170f25aa65ec6852f505ce2a50639b  jedit_build-20130628.tar.gz
+5de3e399be2507f684b49dfd13da45228214bbe4  jedit_build-20130905.tar.gz
 8122526f1fc362ddae1a328bdbc2152853186fee  jfreechart-1.0.14.tar.gz
 6c737137cc597fc920943783382e928ea79e3feb  kodkodi-1.2.16.tar.gz
 5f95c96bb99927f3a026050f85bd056f37a9189e  kodkodi-1.5.2.tar.gz
@@ -52,6 +54,7 @@
 1f4a2053cc1f34fa36c4d9d2ac906ad4ebc863fd  sumatra_pdf-2.1.1.tar.gz
 869ea6d8ea35c8ba68d7fcb028f16b2b7064c5fd  vampire-1.0.tar.gz
 81d21dfd0ea5c58f375301f5166be9dbf8921a7a  windows_app-20130716.tar.gz
+fe15e1079cf5ad86f3cbab4553722a0d20002d11  windows_app-20130905.tar.gz
 2ae13aa17d0dc95ce254a52f1dba10929763a10d  xz-java-1.2.tar.gz
 4530a1aa6f4498ee3d78d6000fa71a3f63bd077f  yices-1.0.28.tar.gz
 12ae71acde43bd7bed1e005c43034b208c0cba4c  z3-3.2.tar.gz
--- a/Admin/components/main	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/components/main	Fri Sep 06 10:57:27 2013 +0200
@@ -3,8 +3,8 @@
 e-1.8
 exec_process-1.0.3
 Haskabelle-2013
-jdk-7u21
-jedit_build-20130628
+jdk-7u25
+jedit_build-20130905
 jfreechart-1.0.14
 kodkodi-1.5.2
 polyml-5.5.0-3
--- a/Admin/java/build	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/java/build	Fri Sep 06 10:57:27 2013 +0200
@@ -11,8 +11,8 @@
 
 ## parameters
 
-VERSION="7u21"
-FULL_VERSION="1.7.0_21"
+VERSION="7u25"
+FULL_VERSION="1.7.0_25"
 
 ARCHIVE_LINUX32="jdk-${VERSION}-linux-i586.tar.gz"
 ARCHIVE_LINUX64="jdk-${VERSION}-linux-x64.tar.gz"
@@ -37,8 +37,6 @@
 
 Linux, Mac OS X, Windows work uniformly, depending on certain
 platform-specific subdirectories.
-
-Note that Java 1.7 on Mac OS X requires 64bit hardware!
 EOF
 
 
@@ -53,10 +51,10 @@
     echo "### Java 1.7 unavailable on 32bit Macintosh!" >&2
     ;;
   x86_64-darwin)
-    ISABELLE_JDK_HOME="\$COMPONENT/\$ISABELLE_PLATFORM64/jdk${FULL_VERSION}.jdk/Contents/Home"
+    ISABELLE_JDK_HOME="\$COMPONENT/\$ISABELLE_PLATFORM64/Contents/Home"
     ;;
   *)
-    ISABELLE_JDK_HOME="\$COMPONENT/\${ISABELLE_PLATFORM64:-\$ISABELLE_PLATFORM32}/jdk${FULL_VERSION}"
+    ISABELLE_JDK_HOME="\$COMPONENT/\${ISABELLE_PLATFORM64:-\$ISABELLE_PLATFORM32}"
     ;;
 esac
 
@@ -82,6 +80,18 @@
 tar -C "$DIR/x86_64-darwin" -xf "$ARCHIVE_DARWIN"
 tar -C "$DIR/x86-cygwin" -xf "$ARCHIVE_WINDOWS"
 
+(
+  cd "$DIR"
+  for PLATFORM in x86-linux x86_64-linux x86-cygwin
+  do
+    mv "$PLATFORM/jdk${FULL_VERSION}"/* "$PLATFORM"/.
+    rmdir "$PLATFORM/jdk${FULL_VERSION}"
+  done
+  PLATFORM=x86_64-darwin
+  mv "$PLATFORM/jdk${FULL_VERSION}.jdk"/* "$PLATFORM"/.
+  rmdir "$PLATFORM/jdk${FULL_VERSION}.jdk"
+)
+
 chgrp -R isabelle "$DIR"
 chmod -R a+r "$DIR"
 chmod -R a+X "$DIR"
@@ -90,13 +100,13 @@
 
 echo "Sharing ..."
 (
-  cd "$DIR/x86-linux/jdk${FULL_VERSION}"
+  cd "$DIR/x86-linux"
   for FILE in $(find . -type f)
   do
     for OTHER in \
-      "../../x86_64-linux/jdk${FULL_VERSION}/$FILE" \
-      "../../x86_64-darwin/jdk${FULL_VERSION}.jdk/Contents/Home/$FILE" \
-      "../../x86-cygwin/jdk${FULL_VERSION}/$FILE"
+      "../../x86_64-linux/$FILE" \
+      "../../x86_64-darwin/Contents/Home/$FILE" \
+      "../../x86-cygwin/$FILE"
     do
       if cmp -s "$FILE" "$OTHER"
       then
--- a/Admin/lib/Tools/makedist_bundle	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/lib/Tools/makedist_bundle	Fri Sep 06 10:57:27 2013 +0200
@@ -48,7 +48,7 @@
 
 ISABELLE_TARGET="$TMP/$ISABELLE_NAME"
 
-tar -C "$TMP" -x -z -f "$ARCHIVE"
+tar -C "$TMP" -x -z -f "$ARCHIVE" || exit 2
 
 
 # bundled components
@@ -83,10 +83,18 @@
                   perl -e "exit((stat('${CONTRIB}'))[7] == 0 ? 0 : 1);" && exit 2
                 fi
 
-                tar -C "$ISABELLE_TARGET/contrib" -x -z -f "$CONTRIB"
+                tar -C "$ISABELLE_TARGET/contrib" -x -z -f "$CONTRIB" || exit 2
                 if [ -f "$COMPONENT_DIR/etc/settings" -o -f "$COMPONENT_DIR/etc/components" ]
                 then
-                  echo "contrib/$COMPONENT" >> "$ISABELLE_TARGET/etc/components"
+                  case "$COMPONENT" in
+                    jdk-*)
+                      mv "$ISABELLE_TARGET/contrib/$COMPONENT" "$ISABELLE_TARGET/contrib/jdk"
+                      echo "contrib/jdk" >> "$ISABELLE_TARGET/etc/components"
+                      ;;
+                    *)
+                      echo "contrib/$COMPONENT" >> "$ISABELLE_TARGET/etc/components"
+                      ;;
+                  esac
                 fi
                 ;;
             esac
@@ -154,6 +162,8 @@
 
       find . -type l -exec echo "{}" ";" -exec readlink "{}" ";" \
         > "contrib/cygwin/isabelle/symlinks"
+
+      touch "contrib/cygwin/isabelle/uninitialized"
     )
 
     perl -pi -e "s,/bin/rebaseall.*,/isabelle/rebaseall,g;" \
@@ -202,7 +212,7 @@
         cp -R "$APP_TEMPLATE/Resources/." "$APP/Contents/Resources/."
         cp "$APP_TEMPLATE/../isabelle.icns" "$APP/Contents/Resources/."
 
-        ln -sf "../Resources/${ISABELLE_NAME}/contrib/jdk-7u21/x86_64-darwin/jdk1.7.0_21.jdk" \
+        ln -sf "../Resources/${ISABELLE_NAME}/contrib/jdk/x86_64-darwin" \
           "$APP/Contents/PlugIns/jdk"
 
         cp macos_app/JavaAppLauncher "$APP/Contents/MacOS/." && \
--- a/Admin/lib/Tools/update_keywords	Fri Sep 06 10:56:40 2013 +0200
+++ b/Admin/lib/Tools/update_keywords	Fri Sep 06 10:57:27 2013 +0200
@@ -3,6 +3,7 @@
 # Author: Makarius
 #
 # DESCRIPTION: update standard keyword files for Emacs Proof General
+# (Proof General legacy)
 
 isabelle_admin_build jars || exit $?
 
--- a/CONTRIBUTORS	Fri Sep 06 10:56:40 2013 +0200
+++ b/CONTRIBUTORS	Fri Sep 06 10:57:27 2013 +0200
@@ -6,7 +6,11 @@
 Contributions to this Isabelle version
 --------------------------------------
 
-* Spring and Summer 2013: Lorenz Panny, Dmitriy Traytel, and Jasmin Blanchette, TUM
+* September 2013: Nik Sultana, University of Cambridge
+  Improvements to HOL/TPTP parser and import facilities.
+
+* Spring and Summer 2013: Lorenz Panny, Dmitriy Traytel, and
+  Jasmin Blanchette, TUM
   Various improvements to BNF-based (co)datatype package, including a
   "primrec_new" command and a compatibility layer.
 
--- a/NEWS	Fri Sep 06 10:56:40 2013 +0200
+++ b/NEWS	Fri Sep 06 10:57:27 2013 +0200
@@ -101,7 +101,7 @@
 
   - Light-weight popup, which avoids explicit window (more reactive
     and more robust).  Interpreted key events include TAB, ESCAPE, UP,
-    DOWN, PAGE_UP, PAGE_DOWN.  Uninterpreted key events are passed to
+    DOWN, PAGE_UP, PAGE_DOWN.  All other key events are passed to
     the jEdit text area.
 
   - Explicit completion via standard jEdit shortcut C+b, which has
--- a/src/HOL/BNF/Tools/bnf_fp_rec_sugar.ML	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_fp_rec_sugar.ML	Fri Sep 06 10:57:27 2013 +0200
@@ -36,8 +36,7 @@
 
 fun permute_args n t = list_comb (t, map Bound (0 :: (n downto 1)))
   |> fold (K (fn u => Abs (Name.uu, dummyT, u))) (0 upto n);
-fun abs_tuple t = if t = undef_const then t else
-  strip_abs t |>> HOLogic.mk_tuple o map Free |-> HOLogic.tupled_lambda;
+val abs_tuple = HOLogic.tupled_lambda o HOLogic.mk_tuple;
 
 val simp_attrs = @{attributes [simp]};
 
@@ -107,7 +106,7 @@
      user_eqn = eqn'}
   end;
 
-fun rewrite_map_arg fun_name_ctr_pos_list rec_type res_type =
+fun rewrite_map_arg get_ctr_pos rec_type res_type =
   let
     val pT = HOLogic.mk_prodT (rec_type, res_type);
 
@@ -117,11 +116,9 @@
       | subst d t =
         let
           val (u, vs) = strip_comb t;
-          val maybe_fun_name_ctr_pos =
-            find_first (equal (free_name u) o SOME o fst) fun_name_ctr_pos_list;
-          val (fun_name, ctr_pos) = the_default ("", ~1) maybe_fun_name_ctr_pos;
+          val ctr_pos = try (get_ctr_pos o the) (free_name u) |> the_default ~1;
         in
-          if is_some maybe_fun_name_ctr_pos then
+          if ctr_pos >= 0 then
             if d = SOME ~1 andalso length vs = ctr_pos then
               list_comb (permute_args ctr_pos (snd_const pT), vs)
             else if length vs > ctr_pos andalso is_some d
@@ -138,7 +135,7 @@
     subst (SOME ~1)
   end;
 
-fun subst_rec_calls lthy fun_name_ctr_pos_list has_call ctr_args direct_calls indirect_calls t =
+fun subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls t =
   let
     fun subst bound_Ts (Abs (v, T, b)) = Abs (v, T, subst (T :: bound_Ts) b)
       | subst bound_Ts (t as g' $ y) =
@@ -146,19 +143,18 @@
           val maybe_direct_y' = AList.lookup (op =) direct_calls y;
           val maybe_indirect_y' = AList.lookup (op =) indirect_calls y;
           val (g, g_args) = strip_comb g';
-          val maybe_ctr_pos =
-            try (snd o the o find_first (equal (free_name g) o SOME o fst)) fun_name_ctr_pos_list;
-          val _ = is_none maybe_ctr_pos orelse length g_args >= the maybe_ctr_pos orelse
+          val ctr_pos = try (get_ctr_pos o the) (free_name g) |> the_default ~1;
+          val _ = ctr_pos < 0 orelse length g_args >= ctr_pos orelse
             primrec_error_eqn "too few arguments in recursive call" t;
         in
           if not (member (op =) ctr_args y) then
             pairself (subst bound_Ts) (g', y) |> (op $)
-          else if is_some maybe_ctr_pos then
+          else if ctr_pos >= 0 then
             list_comb (the maybe_direct_y', g_args)
           else if is_some maybe_indirect_y' then
             (if has_call g' then t else y)
             |> massage_indirect_rec_call lthy has_call
-              (rewrite_map_arg fun_name_ctr_pos_list) bound_Ts y (the maybe_indirect_y')
+              (rewrite_map_arg get_ctr_pos) bound_Ts y (the maybe_indirect_y')
             |> (if has_call g' then I else curry (op $) g')
           else
             t
@@ -211,16 +207,17 @@
             nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_indirect_type)))
           indirect_calls';
 
+      val fun_name_ctr_pos_list =
+        map (fn (x :: _) => (#fun_name x, length (#left_args x))) funs_data;
+      val get_ctr_pos = try (the o AList.lookup (op =) fun_name_ctr_pos_list) #> the_default ~1;
       val direct_calls = map (apfst (nth ctr_args) o apsnd (nth args)) direct_calls';
       val indirect_calls = map (apfst (nth ctr_args) o apsnd (nth args)) indirect_calls';
 
-      val abstractions = map dest_Free (args @ #left_args eqn_data @ #right_args eqn_data);
-      val fun_name_ctr_pos_list =
-        map (fn (x :: _) => (#fun_name x, length (#left_args x))) funs_data;
+      val abstractions = args @ #left_args eqn_data @ #right_args eqn_data;
     in
       t
-      |> subst_rec_calls lthy fun_name_ctr_pos_list has_call ctr_args direct_calls indirect_calls
-      |> fold_rev absfree abstractions
+      |> subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls
+      |> fold_rev lambda abstractions
     end;
 
 fun build_defs lthy bs mxs funs_data rec_specs has_call =
@@ -372,15 +369,16 @@
 
 type co_eqn_data_disc = {
   fun_name: string,
+  fun_args: term list,
   ctr_no: int, (*###*)
   cond: term,
   user_eqn: term
 };
 type co_eqn_data_sel = {
   fun_name: string,
+  fun_args: term list,
   ctr: term,
   sel: term,
-  fun_args: term list,
   rhs_term: term,
   user_eqn: term
 };
@@ -388,11 +386,10 @@
   Disc of co_eqn_data_disc |
   Sel of co_eqn_data_sel;
 
-fun co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps =
+fun co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds =
   let
     fun find_subterm p = let (* FIXME \<exists>? *)
-      fun f (t as u $ v) =
-        fold_rev (curry merge_options) [if p t then SOME t else NONE, f u, f v] NONE
+      fun f (t as u $ v) = if p t then SOME t else merge_options (f u, f v)
         | f t = if p t then SOME t else NONE
       in f end;
 
@@ -406,9 +403,8 @@
 
     val discs = #ctr_specs corec_spec |> map #disc;
     val ctrs = #ctr_specs corec_spec |> map #ctr;
-    val n_ctrs = length ctrs;
     val not_disc = head_of imp_rhs = @{term Not};
-    val _ = not_disc andalso n_ctrs <> 2 andalso
+    val _ = not_disc andalso length ctrs <> 2 andalso
       primrec_error_eqn "\<not>ed discriminator for a type with \<noteq> 2 constructors" imp_rhs;
     val disc = find_subterm (member (op =) discs o head_of) imp_rhs;
     val eq_ctr0 = imp_rhs |> perhaps (try (HOLogic.dest_not)) |> try (HOLogic.dest_eq #> snd)
@@ -428,32 +424,28 @@
     val mk_conjs = try (foldr1 HOLogic.mk_conj) #> the_default @{const True};
     val mk_disjs = try (foldr1 HOLogic.mk_disj) #> the_default @{const False};
     val catch_all = try (fst o dest_Free o the_single) imp_lhs' = SOME Name.uu_;
-    val matched_conds = filter (equal fun_name o fst) matched_conds_ps |> map snd;
-    val imp_lhs = mk_conjs imp_lhs';
+    val matched_cond = filter (equal fun_name o fst) matched_conds |> map snd |> mk_disjs;
+    val imp_lhs = mk_conjs imp_lhs'
+      |> incr_boundvars (length fun_args)
+      |> subst_atomic (fun_args ~~ map Bound (length fun_args - 1 downto 0))
     val cond =
       if catch_all then
-        if null matched_conds then fold_rev absfree (map dest_Free fun_args) @{const True} else
-          (strip_abs_vars (hd matched_conds),
-            mk_disjs (map strip_abs_body matched_conds) |> HOLogic.mk_not)
-          |-> fold_rev (fn (v, T) => fn u => Abs (v, T, u))
+        matched_cond |> HOLogic.mk_not
       else if sequential then
-        HOLogic.mk_conj (HOLogic.mk_not (mk_disjs (map strip_abs_body matched_conds)), imp_lhs)
-        |> fold_rev absfree (map dest_Free fun_args)
+        HOLogic.mk_conj (HOLogic.mk_not matched_cond, imp_lhs)
       else
-        imp_lhs |> fold_rev absfree (map dest_Free fun_args);
-    val matched_cond =
-      if sequential then fold_rev absfree (map dest_Free fun_args) imp_lhs else cond;
+        imp_lhs;
 
-    val matched_conds_ps' = if catch_all
-      then (fun_name, cond) :: filter (not_equal fun_name o fst) matched_conds_ps
-      else (fun_name, matched_cond) :: matched_conds_ps;
+    val matched_conds' =
+      (fun_name, if catch_all orelse not sequential then cond else imp_lhs) :: matched_conds;
   in
     (Disc {
       fun_name = fun_name,
+      fun_args = fun_args,
       ctr_no = ctr_no,
       cond = cond,
       user_eqn = eqn'
-    }, matched_conds_ps')
+    }, matched_conds')
   end;
 
 fun co_dissect_eqn_sel fun_name_corec_spec_list eqn' eqn =
@@ -473,15 +465,15 @@
   in
     Sel {
       fun_name = fun_name,
+      fun_args = fun_args,
       ctr = #ctr ctr_spec,
       sel = sel,
-      fun_args = fun_args,
       rhs_term = rhs,
       user_eqn = eqn'
     }
   end;
 
-fun co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps =
+fun co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds =
   let 
     val (lhs, rhs) = HOLogic.dest_eq imp_rhs;
     val fun_name = head_of lhs |> fst o dest_Free;
@@ -491,10 +483,10 @@
       handle Option.Option => primrec_error_eqn "not a constructor" ctr;
 
     val disc_imp_rhs = betapply (#disc ctr_spec, lhs);
-    val (maybe_eqn_data_disc, matched_conds_ps') = if length (#ctr_specs corec_spec) = 1
-      then (NONE, matched_conds_ps)
+    val (maybe_eqn_data_disc, matched_conds') = if length (#ctr_specs corec_spec) = 1
+      then (NONE, matched_conds)
       else apfst SOME (co_dissect_eqn_disc
-          sequential fun_name_corec_spec_list eqn' imp_lhs' disc_imp_rhs matched_conds_ps);
+          sequential fun_name_corec_spec_list eqn' imp_lhs' disc_imp_rhs matched_conds);
 
     val sel_imp_rhss = (#sels ctr_spec ~~ ctr_args)
       |> map (fn (sel, ctr_arg) => HOLogic.mk_eq (betapply (sel, lhs), ctr_arg));
@@ -506,10 +498,10 @@
     val eqns_data_sel =
       map (co_dissect_eqn_sel fun_name_corec_spec_list eqn') sel_imp_rhss;
   in
-    (map_filter I [maybe_eqn_data_disc] @ eqns_data_sel, matched_conds_ps')
+    (map_filter I [maybe_eqn_data_disc] @ eqns_data_sel, matched_conds')
   end;
 
-fun co_dissect_eqn sequential fun_name_corec_spec_list eqn' matched_conds_ps =
+fun co_dissect_eqn sequential fun_name_corec_spec_list eqn' matched_conds =
   let
     val eqn = subst_bounds (strip_qnt_vars @{const_name all} eqn' |> map Free |> rev,
         strip_qnt_body @{const_name all} eqn')
@@ -531,79 +523,112 @@
     if member (op =) discs head orelse
       is_some maybe_rhs andalso
         member (op =) (filter (null o binder_types o fastype_of) ctrs) (the maybe_rhs) then
-      co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps
+      co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds
       |>> single
     else if member (op =) sels head then
-      ([co_dissect_eqn_sel fun_name_corec_spec_list eqn' imp_rhs], matched_conds_ps)
+      ([co_dissect_eqn_sel fun_name_corec_spec_list eqn' imp_rhs], matched_conds)
     else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) then
-      co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps
+      co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds
     else
       primrec_error_eqn "malformed function equation" eqn
   end;
 
 fun build_corec_args_discs disc_eqns ctr_specs =
-  let
-    val conds = map #cond disc_eqns;
-    val args' =
-      if length ctr_specs = 1 then []
-      else if length disc_eqns = length ctr_specs then
-        fst (split_last conds)
-      else if length disc_eqns = length ctr_specs - 1 then
-        let val n = 0 upto length ctr_specs - 1
-            |> the o find_first (fn idx => not (exists (equal idx o #ctr_no) disc_eqns)) (*###*) in
-          if n = length ctr_specs - 1 then
-            conds
-          else
-            split_last conds
-            ||> (fn t => fold_rev absfree (strip_abs_vars t) (strip_abs_body t |> HOLogic.mk_not))
-            |>> chop n
-            |> (fn ((l, r), x) => l @ (x :: r))
-        end
-      else
-        0 upto length ctr_specs - 1
-        |> map (fn idx => find_first (equal idx o #ctr_no) disc_eqns
-          |> Option.map #cond
-          |> the_default undef_const)
-        |> fst o split_last;
-  in
-    (* FIXME: deal with #preds above *)
-    fold2 (fn idx => nth_map idx o K o abs_tuple) (map_filter #pred ctr_specs) args'
-  end;
+  if null disc_eqns then I else
+    let
+(*val _ = tracing ("d/p:\<cdot> " ^ space_implode "\n    \<cdot> " (map ((op ^) o
+ apfst (Syntax.string_of_term @{context}) o apsnd (curry (op ^) " / " o @{make_string}))
+  (ctr_specs |> map_filter (fn {disc, pred = SOME pred, ...} => SOME (disc, pred) | _ => NONE))));*)
+      val conds = map #cond disc_eqns;
+      val fun_args = #fun_args (hd disc_eqns);
+      val args =
+        if length ctr_specs = 1 then []
+        else if length disc_eqns = length ctr_specs then
+          fst (split_last conds)
+        else if length disc_eqns = length ctr_specs - 1 then
+          let val n = 0 upto length ctr_specs - 1
+              |> the(*###*) o find_first (fn idx => not (exists (equal idx o #ctr_no) disc_eqns)) in
+            if n = length ctr_specs - 1 then
+              conds
+            else
+              split_last conds
+              ||> HOLogic.mk_not
+              |> `(uncurry (fold (curry HOLogic.mk_conj o HOLogic.mk_not)))
+              ||> chop n o fst
+              |> (fn (x, (l, r)) => l @ (x :: r))
+          end
+        else
+          0 upto length ctr_specs - 1
+          |> map (fn idx => find_first (equal idx o #ctr_no) disc_eqns
+            |> Option.map #cond
+            |> the_default undef_const)
+          |> fst o split_last;
+    in
+      (* FIXME deal with #preds above *)
+      (map_filter #pred ctr_specs, args)
+      |-> fold2 (fn idx => fn t => nth_map idx
+        (K (subst_bounds (List.rev fun_args, t)
+          |> HOLogic.tupled_lambda (HOLogic.mk_tuple fun_args))))
+    end;
 
 fun build_corec_arg_no_call sel_eqns sel = find_first (equal sel o #sel) sel_eqns
-  |> try (fn SOME sel_eqn => (#fun_args sel_eqn |> map dest_Free, #rhs_term sel_eqn))
+  |> try (fn SOME sel_eqn => (#fun_args sel_eqn, #rhs_term sel_eqn))
   |> the_default ([], undef_const)
-  |-> abs_tuple oo fold_rev absfree;
+  |-> abs_tuple
+  |> K;
 
 fun build_corec_arg_direct_call lthy has_call sel_eqns sel =
   let
-    val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns
-
-    fun rewrite U T t =
+    val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;
+    fun rewrite is_end U T t =
       if U = @{typ bool} then @{term True} |> has_call t ? K @{term False} (* stop? *)
-      else if T = U = has_call t then undef_const
-      else if T = U then t (* end *)
+      else if is_end = has_call t then undef_const
+      else if is_end then t (* end *)
       else HOLogic.mk_tuple (snd (strip_comb t)); (* continue *)
-    fun massage rhs_term t =
-      massage_direct_corec_call lthy has_call rewrite [] (body_type (fastype_of t)) rhs_term;
-    val abstract = abs_tuple oo fold_rev absfree o map dest_Free;
+    fun massage rhs_term is_end t = massage_direct_corec_call
+      lthy has_call (rewrite is_end) [] (range_type (fastype_of t)) rhs_term;
+  in
+    if is_none maybe_sel_eqn then K I else
+      abs_tuple (#fun_args (the maybe_sel_eqn)) oo massage (#rhs_term (the maybe_sel_eqn))
+  end;
+
+fun build_corec_arg_indirect_call lthy has_call sel_eqns sel =
+  let
+    val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;
+    fun rewrite _ _ =
+      let
+        fun subst (Abs (v, T, b)) = Abs (v, T, subst b)
+          | subst (t as _ $ _) =
+            let val (u, vs) = strip_comb t in
+              if is_Free u andalso has_call u then
+                Const (@{const_name Inr}, dummyT) $ (*HOLogic.mk_tuple vs*)
+                  (try (foldr1 (fn (x, y) => Const (@{const_name Pair}, dummyT) $ x $ y)) vs
+                    |> the_default (hd vs))
+              else if try (fst o dest_Const) u = SOME @{const_name prod_case} then
+                list_comb (u |> map_types (K dummyT), map subst vs)
+              else
+                list_comb (subst u, map subst vs)
+            end
+          | subst t = t;
+      in
+        subst
+      end;
+    fun massage rhs_term t = massage_indirect_corec_call
+      lthy has_call rewrite [] (fastype_of t |> range_type) rhs_term;
   in
     if is_none maybe_sel_eqn then I else
-      massage (#rhs_term (the maybe_sel_eqn)) #> abstract (#fun_args (the maybe_sel_eqn))
+      abs_tuple (#fun_args (the maybe_sel_eqn)) o massage (#rhs_term (the maybe_sel_eqn))
   end;
 
-fun build_corec_arg_indirect_call sel_eqns sel =
-  primrec_error "indirect corecursion not implemented yet";
-
 fun build_corec_args_sel lthy has_call all_sel_eqns ctr_spec =
   let val sel_eqns = filter (equal (#ctr ctr_spec) o #ctr) all_sel_eqns in
     if null sel_eqns then I else
       let
         val sel_call_list = #sels ctr_spec ~~ #calls ctr_spec;
 
-val _ = tracing ("sels / calls:\n    \<cdot> " ^ space_implode "\n    \<cdot> " (map ((op ^) o
- apfst (Syntax.string_of_term @{context}) o apsnd (curry (op ^) " / " o @{make_string}))
-  (sel_call_list)));
+(*val _ = tracing ("s/c:\<cdot> " ^ space_implode "\n    \<cdot> " (map ((op ^) o
+ apfst (Syntax.string_of_term lthy) o apsnd (curry (op ^) " / " o @{make_string}))
+  sel_call_list));*)
 
         val no_calls' = map_filter (try (apsnd (fn No_Corec n => n))) sel_call_list;
         val direct_calls' = map_filter (try (apsnd (fn Direct_Corec n => n))) sel_call_list;
@@ -611,12 +636,12 @@
       in
         I
         #> fold (fn (sel, n) => nth_map n
-          (build_corec_arg_no_call sel_eqns sel |> K)) no_calls'
+          (build_corec_arg_no_call sel_eqns sel)) no_calls'
         #> fold (fn (sel, (q, g, h)) =>
           let val f = build_corec_arg_direct_call lthy has_call sel_eqns sel in
-            nth_map h f o nth_map g f o nth_map q f end) direct_calls'
+            nth_map h (f false) o nth_map g (f true) o nth_map q (f true) end) direct_calls'
         #> fold (fn (sel, n) => nth_map n
-          (build_corec_arg_indirect_call sel_eqns sel |> K)) indirect_calls'
+          (build_corec_arg_indirect_call lthy has_call sel_eqns sel)) indirect_calls'
       end
   end;
 
@@ -651,24 +676,26 @@
       |> fold2 build_corec_args_discs disc_eqnss ctr_specss
       |> fold2 (fold o build_corec_args_sel lthy has_call) sel_eqnss ctr_specss;
 
+    fun currys Ts t = if length Ts <= 1 then t else
+      t $ foldr1 (fn (u, v) => HOLogic.pair_const dummyT dummyT $ u $ v)
+        (length Ts - 1 downto 0 |> map Bound)
+      |> fold_rev (fn T => fn u => Abs (Name.uu, T, u)) Ts;
+
 val _ = tracing ("corecursor arguments:\n    \<cdot> " ^
- space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) corec_args));
+ space_implode "\n    \<cdot> " (map (Syntax.string_of_term lthy) corec_args));
 
     fun uneq_pairs_rev xs = xs (* FIXME \<exists>? *)
       |> these o try (split_last #> (fn (ys, y) => uneq_pairs_rev ys @ map (pair y) ys));
     val proof_obligations = if sequential then [] else
-      maps (uneq_pairs_rev o map #cond) disc_eqnss
-      |> map (fn (x, y) => ((strip_abs_body x, strip_abs_body y), strip_abs_vars x))
-      |> map (apfst (apsnd HOLogic.mk_not #> pairself HOLogic.mk_Trueprop
-        #> apfst (curry (op $) @{const ==>}) #> (op $)))
-      |> map (fn (t, abs_vars) => fold_rev (fn (v, T) => fn u =>
-          Const (@{const_name all}, (T --> @{typ prop}) --> @{typ prop}) $
-            Abs (v, T, u)) abs_vars t);
+      disc_eqnss
+      |> maps (uneq_pairs_rev o map (fn {fun_args, cond, ...} => (fun_args, cond)))
+      |> map (fn ((fun_args, x), (_, y)) => [x, HOLogic.mk_not y]
+        |> map (HOLogic.mk_Trueprop o curry subst_bounds (List.rev fun_args))
+        |> curry list_comb @{const ==>});
 
-    fun currys Ts t = if length Ts <= 1 then t else
-      t $ foldr1 (fn (u, v) => HOLogic.pair_const dummyT dummyT $ u $ v)
-        (length Ts - 1 downto 0 |> map Bound)
-      |> fold_rev (fn T => fn u => Abs (Name.uu, T, u)) Ts;
+val _ = tracing ("proof obligations:\n    \<cdot> " ^
+ space_implode "\n    \<cdot> " (map (Syntax.string_of_term lthy) proof_obligations));
+
   in
     map (list_comb o rpair corec_args) corecs
     |> map2 (fn Ts => fn t => if length Ts = 0 then t $ HOLogic.unit else t) arg_Tss
--- a/src/HOL/BNF/Tools/bnf_fp_rec_sugar_util.ML	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_fp_rec_sugar_util.ML	Fri Sep 06 10:57:27 2013 +0200
@@ -198,11 +198,11 @@
 fun massage_indirect_corec_call ctxt has_call massage_direct_call bound_Ts res_U t =
   let
     val typof = curry fastype_of1 bound_Ts;
-    val build_map_Inl = build_map ctxt (uncurry Inl_const o dest_sumT o fst);
+    val build_map_Inl = build_map ctxt (uncurry Inl_const o dest_sumT o snd)
 
     fun check_and_massage_direct_call U T t =
       if has_call t then factor_out_types ctxt massage_direct_call dest_sumT U T t
-      else build_map_Inl (U, T) $ t;
+      else build_map_Inl (T, U) $ t;
 
     fun check_and_massage_unapplied_direct_call U T t =
       let val var = Var ((Name.uu, Term.maxidx_of_term t + 1), domain_type (typof t)) in
@@ -241,11 +241,11 @@
               | NONE =>
                 (case t of
                   t1 $ t2 =>
-                  if has_call t2 then
+                  (if has_call t2 then
                     check_and_massage_direct_call U T t
                   else
                     massage_map U T t1 $ t2
-                    handle AINT_NO_MAP _ => check_and_massage_direct_call U T t
+                    handle AINT_NO_MAP _ => check_and_massage_direct_call U T t)
                 | _ => check_and_massage_direct_call U T t))
             | _ => ill_formed_corec_call ctxt t))
         U T
--- a/src/HOL/List.thy	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/List.thy	Fri Sep 06 10:57:27 2013 +0200
@@ -548,9 +548,9 @@
         fun check (i, case_t) s =
           (case strip_abs_body case_t of
             (Const (@{const_name List.Nil}, _)) => s
-          | _ => (case s of NONE => SOME i | SOME _ => NONE))
+          | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE))
       in
-        fold_index check cases NONE
+        fold_index check cases (SOME NONE) |> the_default NONE
       end
     (* returns (case_expr type index chosen_case) option  *)
     fun dest_case case_term =
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Fri Sep 06 10:57:27 2013 +0200
@@ -34,8 +34,8 @@
   using assms convex_def[of S] by auto
 
 lemma mem_convex_alt:
-  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
-  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
+  assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
+  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
   apply (subst mem_convex_2)
   using assms
   apply (auto simp add: algebra_simps zero_le_divide_iff)
@@ -74,20 +74,20 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
     and fi: "inj_on f (span S)"
-  shows "dim (f ` S) = dim (S:: 'n::euclidean_space set)"
-proof -
-  obtain B where B_def: "B \<subseteq> S \<and> independent B \<and> S \<subseteq> span B \<and> card B = dim S"
+  shows "dim (f ` S) = dim (S::'n::euclidean_space set)"
+proof -
+  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
     using basis_exists[of S] by auto
   then have "span S = span B"
     using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
   then have "independent (f ` B)"
-    using independent_injective_on_span_image[of B f] B_def assms by auto
+    using independent_injective_on_span_image[of B f] B assms by auto
   moreover have "card (f ` B) = card B"
-    using assms card_image[of f B] subset_inj_on[of f "span S" B] B_def span_inc by auto
+    using assms card_image[of f B] subset_inj_on[of f "span S" B] B span_inc by auto
   moreover have "(f ` B) \<subseteq> (f ` S)"
-    using B_def by auto
+    using B by auto
   ultimately have "dim (f ` S) \<ge> dim S"
-    using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
+    using independent_card_le_dim[of "f ` B" "f ` S"] B by auto
   then show ?thesis
     using dim_image_le[of f S] assms by auto
 qed
@@ -220,8 +220,6 @@
     and C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T"
   shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
 proof -
-(* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
-*)
   from B independent_bound have fB: "finite B"
     by blast
   from C independent_bound have fC: "finite C"
@@ -293,9 +291,6 @@
   then show ?thesis using closure_linear_image[of f S] assms by auto
 qed
 
-lemma closure_direct_sum: "closure (S \<times> T) = closure S \<times> closure T"
-  by (rule closure_Times)
-
 lemma closure_scaleR:
   fixes S :: "'a::real_normed_vector set"
   shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
@@ -367,7 +362,7 @@
     by (auto simp add:norm_minus_commute)
 qed
 
-lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
+lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
 
 lemma Min_grI:
   assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
@@ -8668,7 +8663,7 @@
   have "(closure S) + (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
     by (simp add: set_plus_image)
   also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
-    using closure_direct_sum by auto
+    using closure_Times by auto
   also have "... \<subseteq> closure (S + T)"
     using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp: set_plus_image)
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Fri Sep 06 10:57:27 2013 +0200
@@ -1,6 +1,8 @@
+(*  Author:     John Harrison
+    Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light)
+*)
+
 header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
-(*  Author:                     John Harrison
-    Translation from HOL light: Robert Himmelmann, TU Muenchen *)
 
 theory Integration
 imports
@@ -11,62 +13,76 @@
 lemma cSup_abs_le: (* TODO: is this really needed? *)
   fixes S :: "real set"
   shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
-by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2) 
+  by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2)
 
 lemma cInf_abs_ge: (* TODO: is this really needed? *)
   fixes S :: "real set"
   shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
-by (simp add: Inf_real_def) (rule cSup_abs_le, auto) 
+  by (simp add: Inf_real_def) (rule cSup_abs_le, auto)
 
 lemma cSup_asclose: (* TODO: is this really needed? *)
   fixes S :: "real set"
-  assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Sup S - l\<bar> \<le> e"
-proof-
-  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
-  thus ?thesis using S b cSup_bounds[of S "l - e" "l+e"] unfolding th
-    by  (auto simp add: setge_def setle_def)
+  assumes S: "S \<noteq> {}"
+    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
+  shows "\<bar>Sup S - l\<bar> \<le> e"
+proof -
+  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
+    by arith
+  then show ?thesis
+    using S b cSup_bounds[of S "l - e" "l+e"]
+    unfolding th
+    by (auto simp add: setge_def setle_def)
 qed
 
 lemma cInf_asclose: (* TODO: is this really needed? *)
   fixes S :: "real set"
-  assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Inf S - l\<bar> \<le> e"
+  assumes S: "S \<noteq> {}"
+    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
+  shows "\<bar>Inf S - l\<bar> \<le> e"
 proof -
   have "\<bar>- Sup (uminus ` S) - l\<bar> =  \<bar>Sup (uminus ` S) - (-l)\<bar>"
     by auto
-  also have "... \<le> e" 
-    apply (rule cSup_asclose) 
+  also have "\<dots> \<le> e"
+    apply (rule cSup_asclose)
     apply (auto simp add: S)
     apply (metis abs_minus_add_cancel b add_commute diff_minus)
     done
   finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
-  thus ?thesis
+  then show ?thesis
     by (simp add: Inf_real_def)
 qed
 
-lemma cSup_finite_ge_iff: 
-  fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
+lemma cSup_finite_ge_iff:
+  fixes S :: "real set"
+  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
   by (metis cSup_eq_Max Max_ge_iff)
 
-lemma cSup_finite_le_iff: 
-  fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
+lemma cSup_finite_le_iff:
+  fixes S :: "real set"
+  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
   by (metis cSup_eq_Max Max_le_iff)
 
-lemma cInf_finite_ge_iff: 
-  fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
+lemma cInf_finite_ge_iff:
+  fixes S :: "real set"
+  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
   by (metis cInf_eq_Min Min_ge_iff)
 
-lemma cInf_finite_le_iff: 
-  fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
+lemma cInf_finite_le_iff:
+  fixes S :: "real set"
+  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
   by (metis cInf_eq_Min Min_le_iff)
 
 lemma Inf: (* rename *)
   fixes S :: "real set"
-  shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
-by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def intro: cInf_lower cInf_greatest) 
- 
+  shows "S \<noteq> {} \<Longrightarrow> (\<exists>b. b <=* S) \<Longrightarrow> isGlb UNIV S (Inf S)"
+  by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def
+    intro: cInf_lower cInf_greatest)
+
 lemma real_le_inf_subset:
-  assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s"
-  shows "Inf s <= Inf (t::real set)"
+  assumes "t \<noteq> {}"
+    and "t \<subseteq> s"
+    and "\<exists>b. b <=* s"
+  shows "Inf s \<le> Inf (t::real set)"
   apply (rule isGlb_le_isLb)
   apply (rule Inf[OF assms(1)])
   apply (insert assms)
@@ -76,8 +92,11 @@
   done
 
 lemma real_ge_sup_subset:
-  assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b"
-  shows "Sup s >= Sup (t::real set)"
+  fixes t :: "real set"
+  assumes "t \<noteq> {}"
+    and "t \<subseteq> s"
+    and "\<exists>b. s *<= b"
+  shows "Sup s \<ge> Sup t"
   apply (rule isLub_le_isUb)
   apply (rule isLub_cSup[OF assms(1)])
   apply (insert assms)
@@ -104,9 +123,10 @@
 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
 
-declare norm_triangle_ineq4[intro] 
-
-lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast
+declare norm_triangle_ineq4[intro]
+
+lemma simple_image: "{f x |x . x \<in> s} = f ` s"
+  by blast
 
 lemma linear_simps:
   assumes "bounded_linear f"
@@ -123,24 +143,30 @@
 
 lemma bounded_linearI:
   assumes "\<And>x y. f (x + y) = f x + f y"
-    and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K"
+    and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
+    and "\<And>x. norm (f x) \<le> norm x * K"
   shows "bounded_linear f"
-  unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto
+  unfolding bounded_linear_def additive_def bounded_linear_axioms_def
+  using assms by auto
 
 lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \<bullet> k)"
   by (rule bounded_linear_inner_left)
 
 lemma transitive_stepwise_lt_eq:
   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
-  shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r")
-proof (safe)
+  shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))"
+  (is "?l = ?r")
+proof safe
   assume ?r
   fix n m :: nat
   assume "m < n"
   then show "R m n"
   proof (induct n arbitrary: m)
+    case 0
+    then show ?case by auto
+  next
     case (Suc n)
-    show ?case 
+    show ?case
     proof (cases "m < n")
       case True
       show ?thesis
@@ -150,14 +176,16 @@
         done
     next
       case False
-      then have "m = n" using Suc(2) by auto
-      then show ?thesis using `?r` by auto
+      then have "m = n"
+        using Suc(2) by auto
+      then show ?thesis
+        using `?r` by auto
     qed
-  qed auto
+  qed
 qed auto
 
 lemma transitive_stepwise_gt:
-  assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
+  assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n)"
   shows "\<forall>n>m. R m n"
 proof -
   have "\<forall>m. \<forall>n>m. R m n"
@@ -172,12 +200,13 @@
 
 lemma transitive_stepwise_le_eq:
   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
-  shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r")
+  shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))"
+  (is "?l = ?r")
 proof safe
   assume ?r
   fix m n :: nat
   assume "m \<le> n"
-  thus "R m n"
+  then show "R m n"
   proof (induct n arbitrary: m)
     case 0
     with assms show ?case by auto
@@ -193,21 +222,25 @@
         done
     next
       case False
-      hence "m = Suc n" using Suc(2) by auto
-      thus ?thesis using assms(1) by auto
+      then have "m = Suc n"
+        using Suc(2) by auto
+      then show ?thesis
+        using assms(1) by auto
     qed
   qed
 qed auto
 
 lemma transitive_stepwise_le:
-  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
+  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+    and "\<And>n. R n (Suc n)"
   shows "\<forall>n\<ge>m. R m n"
 proof -
   have "\<forall>m. \<forall>n\<ge>m. R m n"
     apply (subst transitive_stepwise_le_eq)
     apply (rule assms)
     apply (rule assms,assumption,assumption)
-    using assms(3) apply auto
+    using assms(3)
+    apply auto
     done
   then show ?thesis by auto
 qed
@@ -215,14 +248,18 @@
 
 subsection {* Some useful lemmas about intervals. *}
 
-abbreviation One where "One \<equiv> ((\<Sum>Basis)::_::euclidean_space)"
+abbreviation One :: "'a::euclidean_space"
+  where "One \<equiv> \<Sum>Basis"
 
 lemma empty_as_interval: "{} = {One..(0::'a::ordered_euclidean_space)}"
   by (auto simp: set_eq_iff eucl_le[where 'a='a] intro!: bexI[OF _ SOME_Basis])
 
-lemma interior_subset_union_intervals: 
-  assumes "i = {a..b::'a::ordered_euclidean_space}" "j = {c..d}"
-    "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
+lemma interior_subset_union_intervals:
+  assumes "i = {a..b::'a::ordered_euclidean_space}"
+    and "j = {c..d}"
+    and "interior j \<noteq> {}"
+    and "i \<subseteq> j \<union> s"
+    and "interior i \<inter> interior j = {}"
   shows "interior i \<subseteq> interior s"
 proof -
   have "{a<..<b} \<inter> {c..d} = {}"
@@ -247,9 +284,12 @@
 
 lemma inter_interior_unions_intervals:
   fixes f::"('a::ordered_euclidean_space) set set"
-  assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
-  shows "s \<inter> interior(\<Union>f) = {}"
-proof (rule ccontr, unfold ex_in_conv[THEN sym])
+  assumes "finite f"
+    and "open s"
+    and "\<forall>t\<in>f. \<exists>a b. t = {a..b}"
+    and "\<forall>t\<in>f. s \<inter> (interior t) = {}"
+  shows "s \<inter> interior (\<Union>f) = {}"
+proof (rule ccontr, unfold ex_in_conv[symmetric])
   case goal1
   have lem1: "\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U"
     apply rule
@@ -260,42 +300,53 @@
     apply auto
     done
   have lem2: "\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
-  have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow>
-    (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)"
+  have "\<And>f. finite f \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = {a..b} \<Longrightarrow>
+    \<exists>x. x \<in> s \<inter> interior (\<Union>f) \<Longrightarrow> \<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t"
   proof -
     case goal1
     then show ?case
     proof (induct rule: finite_induct)
-      case empty from this(2) guess x ..
-      hence False unfolding Union_empty interior_empty by auto
-      thus ?case by auto
+      case empty
+      obtain x where "x \<in> s \<inter> interior (\<Union>{})"
+        using empty(2) ..
+      then have False
+        unfolding Union_empty interior_empty by auto
+      then show ?case by auto
     next
-      case (insert i f) guess x using insert(5) .. note x = this
-      then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
-      guess a using insert(4)[rule_format,OF insertI1] ..
-      then guess b .. note ab = this
+      case (insert i f)
+      obtain x where x: "x \<in> s \<inter> interior (\<Union>insert i f)"
+        using insert(5) ..
+      then obtain e where e: "0 < e \<and> ball x e \<subseteq> s \<inter> interior (\<Union>insert i f)"
+        unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
+      obtain a where "\<exists>b. i = {a..b}"
+        using insert(4)[rule_format,OF insertI1] ..
+      then obtain b where ab: "i = {a..b}" ..
       show ?case
-      proof (cases "x\<in>i")
+      proof (cases "x \<in> i")
         case False
-        hence "x \<in> UNIV - {a..b}" unfolding ab by auto
-        then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
-        hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" unfolding ab ball_min_Int by auto
-        hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
+        then have "x \<in> UNIV - {a..b}"
+          unfolding ab by auto
+        then obtain d where "0 < d \<and> ball x d \<subseteq> UNIV - {a..b}"
+          unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
+        then have "0 < d" "ball x (min d e) \<subseteq> UNIV - i"
+          unfolding ab ball_min_Int by auto
+        then have "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
           using e unfolding lem1 unfolding  ball_min_Int by auto
-        hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
-        hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t"
+        then have "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
+        then have "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t"
           apply -
           apply (rule insert(3))
           using insert(4)
           apply auto
           done
-        thus ?thesis by auto
+        then show ?thesis by auto
       next
         case True show ?thesis
         proof (cases "x\<in>{a<..<b}")
           case True
-          then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
-          thus ?thesis
+          then obtain d where "0 < d \<and> ball x d \<subseteq> {a<..<b}"
+            unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
+          then show ?thesis
             apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
             unfolding ab
             using interval_open_subset_closed[of a b] and e
@@ -303,38 +354,40 @@
             done
         next
           case False
-          then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k:"k\<in>Basis"
+          then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k: "k \<in> Basis"
             unfolding mem_interval by (auto simp add: not_less)
-          hence "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
+          then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
             using True unfolding ab and mem_interval
               apply (erule_tac x = k in ballE)
               apply auto
               done
-          hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
-          proof (erule_tac disjE)
+          then have "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
+          proof (rule disjE)
             let ?z = "x - (e/2) *\<^sub>R k"
             assume as: "x\<bullet>k = a\<bullet>k"
             have "ball ?z (e / 2) \<inter> i = {}"
               apply (rule ccontr)
-              unfolding ex_in_conv[THEN sym]
-            proof (erule exE)
+              unfolding ex_in_conv[symmetric]
+              apply (erule exE)
+            proof -
               fix y
               assume "y \<in> ball ?z (e / 2) \<inter> i"
-              hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
-              hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+              then have "dist ?z y < e/2" and yi:"y\<in>i" by auto
+              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
                 using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
-              hence "y\<bullet>k < a\<bullet>k"
-                using e[THEN conjunct1] k by (auto simp add: field_simps as inner_Basis inner_simps)
-              hence "y \<notin> i"
+              then have "y\<bullet>k < a\<bullet>k"
+                using e[THEN conjunct1] k
+                by (auto simp add: field_simps as inner_Basis inner_simps)
+              then have "y \<notin> i"
                 unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
-              thus False using yi by auto
+              then show False using yi by auto
             qed
             moreover
             have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
-              apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
+              apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
             proof
               fix y
-              assume as: "y\<in> ball ?z (e/2)"
+              assume as: "y \<in> ball ?z (e/2)"
               have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
                 apply -
                 apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
@@ -348,7 +401,7 @@
                 using e
                 apply (auto simp add: field_simps)
                 done
-              finally show "y\<in>ball x e"
+              finally show "y \<in> ball x e"
                 unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
             qed
             ultimately show ?thesis
@@ -361,18 +414,22 @@
             assume as: "x\<bullet>k = b\<bullet>k"
             have "ball ?z (e / 2) \<inter> i = {}"
               apply (rule ccontr)
-              unfolding ex_in_conv[THEN sym]
-            proof(erule exE)
+              unfolding ex_in_conv[symmetric]
+              apply (erule exE)
+            proof -
               fix y
               assume "y \<in> ball ?z (e / 2) \<inter> i"
-              hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
-              hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
-                using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
-              hence "y\<bullet>k > b\<bullet>k"
-                using e[THEN conjunct1] k by(auto simp add:field_simps inner_simps inner_Basis as)
-              hence "y \<notin> i"
+              then have "dist ?z y < e/2" and yi: "y \<in> i"
+                by auto
+              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+                using Basis_le_norm[OF k, of "?z - y"]
+                unfolding dist_norm by auto
+              then have "y\<bullet>k > b\<bullet>k"
+                using e[THEN conjunct1] k
+                by (auto simp add:field_simps inner_simps inner_Basis as)
+              then have "y \<notin> i"
                 unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
-              thus False using yi by auto
+              then show False using yi by auto
             qed
             moreover
             have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
@@ -382,7 +439,7 @@
               assume as: "y\<in> ball ?z (e/2)"
               have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
                 apply -
-                apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
+                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
                 unfolding norm_scaleR
                 apply (auto simp: k)
                 done
@@ -391,79 +448,81 @@
                 using as unfolding mem_ball dist_norm
                 using e apply (auto simp add: field_simps)
                 done
-              finally show "y\<in>ball x e"
-                unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
+              finally show "y \<in> ball x e"
+                unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
             qed
             ultimately show ?thesis
               apply (rule_tac x="?z" in exI)
               unfolding Union_insert
               apply auto
               done
-          qed 
-          then guess x ..
-          hence "x \<in> s \<inter> interior (\<Union>f)"
-            unfolding lem1[where U="\<Union>f",THEN sym]
+          qed
+          then obtain x where "ball x (e / 2) \<subseteq> s \<inter> \<Union>f" ..
+          then have "x \<in> s \<inter> interior (\<Union>f)"
+            unfolding lem1[where U="\<Union>f", symmetric]
             using centre_in_ball e[THEN conjunct1] by auto
-          thus ?thesis
+          then show ?thesis
             apply -
             apply (rule lem2, rule insert(3))
-            using insert(4) apply auto
+            using insert(4)
+            apply auto
             done
         qed
       qed
     qed
   qed
-  note * = this
-  guess t using *[OF assms(1,3) goal1] ..
-  from this(2) guess x ..
-  then guess e ..
-  hence "x \<in> s" "x\<in>interior t"
-    defer
-    using open_subset_interior[OF open_ball, of x e t] apply auto
-    done
-  thus False using `t\<in>f` assms(4) by auto
+  from this[OF assms(1,3) goal1]
+  obtain t x e where "t \<in> f" "0 < e" "ball x e \<subseteq> s \<inter> t"
+    by blast
+  then have "x \<in> s" "x \<in> interior t"
+    using open_subset_interior[OF open_ball, of x e t]
+    by auto
+  then show False
+    using `t \<in> f` assms(4) by auto
 qed
 
 
 subsection {* Bounds on intervals where they exist. *}
 
-definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
-  "interval_upperbound s = (\<Sum>i\<in>Basis. Sup {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
-
-definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
-  "interval_lowerbound s = (\<Sum>i\<in>Basis. Inf {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
+definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
+  where "interval_upperbound s = (\<Sum>i\<in>Basis. Sup {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
+
+definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
+  where "interval_lowerbound s = (\<Sum>i\<in>Basis. Inf {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
 
 lemma interval_upperbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_upperbound {a..b} = (b::'a::ordered_euclidean_space)"
   unfolding interval_upperbound_def euclidean_representation_setsum
   by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setle_def
-           intro!: cSup_unique)
+      intro!: cSup_unique)
 
 lemma interval_lowerbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_lowerbound {a..b} = (a::'a::ordered_euclidean_space)"
   unfolding interval_lowerbound_def euclidean_representation_setsum
   by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setge_def
-           intro!: cInf_unique)
+      intro!: cInf_unique)
 
 lemmas interval_bounds = interval_upperbound interval_lowerbound
 
 lemma interval_bounds'[simp]:
-  assumes "{a..b}\<noteq>{}"
-  shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
+  assumes "{a..b} \<noteq> {}"
+  shows "interval_upperbound {a..b} = b"
+    and "interval_lowerbound {a..b} = a"
   using assms unfolding interval_ne_empty by auto
 
+
 subsection {* Content (length, area, volume...) of an interval. *}
 
 definition "content (s::('a::ordered_euclidean_space) set) =
   (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
 
-lemma interval_not_empty:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
+lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
   unfolding interval_eq_empty unfolding not_ex not_less by auto
 
 lemma content_closed_interval:
-  fixes a::"'a::ordered_euclidean_space"
+  fixes a :: "'a::ordered_euclidean_space"
   assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
   shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   using interval_not_empty[OF assms]
@@ -471,8 +530,8 @@
   by auto
 
 lemma content_closed_interval':
-  fixes a::"'a::ordered_euclidean_space"
-  assumes "{a..b}\<noteq>{}"
+  fixes a :: "'a::ordered_euclidean_space"
+  assumes "{a..b} \<noteq> {}"
   shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   apply (rule content_closed_interval)
   using assms
@@ -480,13 +539,8 @@
   apply assumption
   done
 
-lemma content_real:
-  assumes "a\<le>b"
-  shows "content {a..b} = b-a"
-proof -
-  have *: "{..<Suc 0} = {0}" by auto
-  show ?thesis unfolding content_def using assms by (auto simp: *)
-qed
+lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
+  unfolding content_def by auto
 
 lemma content_singleton[simp]: "content {a} = 0"
 proof -
@@ -497,9 +551,12 @@
 
 lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1"
 proof -
-  have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i" by auto
-  have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
-  thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto
+  have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
+    by auto
+  have "0 \<in> {0..One::'a}"
+    unfolding mem_interval by auto
+  then show ?thesis
+    unfolding content_def interval_bounds[OF *] using setprod_1 by auto
 qed
 
 lemma content_pos_le[intro]:
@@ -507,7 +564,8 @@
   shows "0 \<le> content {a..b}"
 proof (cases "{a..b} = {}")
   case False
-  hence *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty .
+  then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
+    unfolding interval_ne_empty .
   have "(\<Prod>i\<in>Basis. interval_upperbound {a..b} \<bullet> i - interval_lowerbound {a..b} \<bullet> i) \<ge> 0"
     apply (rule setprod_nonneg)
     unfolding interval_bounds[OF *]
@@ -515,29 +573,38 @@
     apply (erule_tac x=x in ballE)
     apply auto
     done
-  thus ?thesis unfolding content_def by (auto simp del:interval_bounds')
-qed (unfold content_def, auto)
+  then show ?thesis
+    unfolding content_def by (auto simp del:interval_bounds')
+next
+  case True
+  then show ?thesis
+    unfolding content_def by auto
+qed
 
 lemma content_pos_lt:
-  fixes a::"'a::ordered_euclidean_space"
+  fixes a :: "'a::ordered_euclidean_space"
   assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
   shows "0 < content {a..b}"
 proof -
   have help_lemma1: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> \<forall>i\<in>Basis. a\<bullet>i \<le> ((b\<bullet>i)::real)"
-    apply (rule, erule_tac x=i in ballE)
+    apply rule
+    apply (erule_tac x=i in ballE)
     apply auto
     done
-  show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]]
-    apply(rule setprod_pos)
-    using assms apply (erule_tac x=x in ballE)
+  show ?thesis
+    unfolding content_closed_interval[OF help_lemma1[OF assms]]
+    apply (rule setprod_pos)
+    using assms
+    apply (erule_tac x=x in ballE)
     apply auto
     done
 qed
 
-lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
+lemma content_eq_0:
+  "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
 proof (cases "{a..b} = {}")
   case True
-  thus ?thesis
+  then show ?thesis
     unfolding content_def if_P[OF True]
     unfolding interval_eq_empty
     apply -
@@ -547,15 +614,16 @@
     done
 next
   case False
-  from this[unfolded interval_eq_empty not_ex not_less]
-  have as: "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i" by fastforce
-  show ?thesis
+  then have "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i"
+    unfolding interval_eq_empty not_ex not_less
+    by fastforce
+  then show ?thesis
     unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_Basis]
-    using as
     by (auto intro!: bexI)
 qed
 
-lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
+lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
+  by auto
 
 lemma content_closed_interval_cases:
   "content {a..b::'a::ordered_euclidean_space} =
@@ -563,42 +631,51 @@
   by (auto simp: not_le content_eq_0 intro: less_imp_le content_closed_interval)
 
 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
-  unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
-
-lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
+  unfolding content_eq_0 interior_closed_interval interval_eq_empty
+  by auto
+
+lemma content_pos_lt_eq:
+  "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
   apply rule
   defer
   apply (rule content_pos_lt, assumption)
 proof -
   assume "0 < content {a..b}"
-  hence "content {a..b} \<noteq> 0" by auto
-  thus "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+  then have "content {a..b} \<noteq> 0" by auto
+  then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
     unfolding content_eq_0 not_ex not_le by fastforce
 qed
 
-lemma content_empty [simp]: "content {} = 0" unfolding content_def by auto
+lemma content_empty [simp]: "content {} = 0"
+  unfolding content_def by auto
 
 lemma content_subset:
   assumes "{a..b} \<subseteq> {c..d}"
   shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}"
 proof (cases "{a..b} = {}")
   case True
-  thus ?thesis using content_pos_le[of c d] by auto
+  then show ?thesis
+    using content_pos_le[of c d] by auto
 next
   case False
-  hence ab_ne:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty by auto
-  hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
+  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
+    unfolding interval_ne_empty by auto
+  then have ab_ab: "a\<in>{a..b}" "b\<in>{a..b}"
+    unfolding mem_interval by auto
   have "{c..d} \<noteq> {}" using assms False by auto
-  hence cd_ne:"\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i" using assms unfolding interval_ne_empty by auto
+  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
+    using assms unfolding interval_ne_empty by auto
   show ?thesis
     unfolding content_def
     unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`]
-    apply (rule setprod_mono, rule)
+    apply (rule setprod_mono)
+    apply rule
   proof
     fix i :: 'a
-    assume i: "i\<in>Basis"
-    show "0 \<le> b \<bullet> i - a \<bullet> i" using ab_ne[THEN bspec, OF i] i by auto
+    assume i: "i \<in> Basis"
+    show "0 \<le> b \<bullet> i - a \<bullet> i"
+      using ab_ne[THEN bspec, OF i] i by auto
     show "b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i]
@@ -612,58 +689,78 @@
 
 subsection {* The notion of a gauge --- simply an open set containing the point. *}
 
-definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
-
-lemma gaugeI: assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
+definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
+
+lemma gaugeI:
+  assumes "\<And>x. x \<in> g x"
+    and "\<And>x. open (g x)"
+  shows "gauge g"
   using assms unfolding gauge_def by auto
 
-lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)"
+lemma gaugeD[dest]:
+  assumes "gauge d"
+  shows "x \<in> d x"
+    and "open (d x)"
   using assms unfolding gauge_def by auto
 
 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
-  unfolding gauge_def by auto 
-
-lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
+  unfolding gauge_def by auto
+
+lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
+  unfolding gauge_def by auto
 
 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)"
   by (rule gauge_ball) auto
 
-lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
-  unfolding gauge_def by auto 
+lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
+  unfolding gauge_def by auto
 
 lemma gauge_inters:
-  assumes "finite s" "\<forall>d\<in>s. gauge (f d)"
-  shows "gauge(\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
+  assumes "finite s"
+    and "\<forall>d\<in>s. gauge (f d)"
+  shows "gauge (\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
 proof -
-  have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto
+  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
+    by auto
   show ?thesis
-    unfolding gauge_def unfolding * 
+    unfolding gauge_def unfolding *
     using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
 qed
 
-lemma gauge_existence_lemma: "(\<forall>x. \<exists>d::real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
-  by(meson zero_less_one)
+lemma gauge_existence_lemma:
+  "(\<forall>x. \<exists>d :: real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
+  by (metis zero_less_one)
 
 
 subsection {* Divisions. *}
 
-definition division_of (infixl "division'_of" 40) where
-  "s division_of i \<equiv>
-        finite s \<and>
-        (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
-        (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
-        (\<Union>s = i)"
+definition division_of (infixl "division'_of" 40)
+where
+  "s division_of i \<longleftrightarrow>
+    finite s \<and>
+    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
+    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
+    (\<Union>s = i)"
 
 lemma division_ofD[dest]:
   assumes "s division_of i"
-  shows "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
-    "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
+  shows "finite s"
+    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
+    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
+    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
+    and "\<Union>s = i"
   using assms unfolding division_of_def by auto
 
 lemma division_ofI:
-  assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
-    "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
-  shows "s division_of i" using assms unfolding division_of_def by auto
+  assumes "finite s"
+    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
+    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
+    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+    and "\<Union>s = i"
+  shows "s division_of i"
+  using assms unfolding division_of_def by auto
 
 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   unfolding division_of_def by auto
@@ -671,28 +768,38 @@
 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   unfolding division_of_def by auto
 
-lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
+lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
+  unfolding division_of_def by auto
 
 lemma division_of_sing[simp]:
-  "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r")
+  "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}"
+  (is "?l = ?r")
 proof
   assume ?r
-  moreover {
+  moreover
+  {
     assume "s = {{a}}"
-    moreover fix k assume "k\<in>s" 
+    moreover fix k assume "k\<in>s"
     ultimately have"\<exists>x y. k = {x..y}"
       apply (rule_tac x=a in exI)+
       unfolding interval_sing
       apply auto
       done
   }
-  ultimately show ?l unfolding division_of_def interval_sing by auto
+  ultimately show ?l
+    unfolding division_of_def interval_sing by auto
 next
   assume ?l
-  note as=conjunctD4[OF this[unfolded division_of_def interval_sing]]
-  { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
-  moreover have "s \<noteq> {}" using as(4) by auto
-  ultimately show ?r unfolding interval_sing by auto
+  note * = conjunctD4[OF this[unfolded division_of_def interval_sing]]
+  {
+    fix x
+    assume x: "x \<in> s" have "x = {a}"
+      using *(2)[rule_format,OF x] by auto
+  }
+  moreover have "s \<noteq> {}"
+    using *(4) by auto
+  ultimately show ?r
+    unfolding interval_sing by auto
 qed
 
 lemma elementary_empty: obtains p where "p division_of {}"
@@ -705,27 +812,38 @@
   unfolding division_of_def by auto
 
 lemma forall_in_division:
- "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
+  "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b})"
   unfolding division_of_def by fastforce
 
-lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
-  apply (rule division_ofI)
-proof -
-  note as=division_ofD[OF assms(1)]
+lemma division_of_subset:
+  assumes "p division_of (\<Union>p)"
+    and "q \<subseteq> p"
+  shows "q division_of (\<Union>q)"
+proof (rule division_ofI)
+  note * = division_ofD[OF assms(1)]
   show "finite q"
     apply (rule finite_subset)
-    using as(1) assms(2) apply auto
+    using *(1) assms(2)
+    apply auto
     done
-  { fix k
+  {
+    fix k
     assume "k \<in> q"
-    hence kp:"k\<in>p" using assms(2) by auto
-    show "k\<subseteq>\<Union>q" using `k \<in> q` by auto
-    show "\<exists>a b. k = {a..b}" using as(4)[OF kp]
-      by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
+    then have kp: "k \<in> p"
+      using assms(2) by auto
+    show "k \<subseteq> \<Union>q"
+      using `k \<in> q` by auto
+    show "\<exists>a b. k = {a..b}"
+      using *(4)[OF kp] by auto
+    show "k \<noteq> {}"
+      using *(3)[OF kp] by auto
+  }
   fix k1 k2
   assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
-  hence *: "k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto
-  show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto
+  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
+    using assms(2) by auto
+  show "interior k1 \<inter> interior k2 = {}"
+    using *(5)[OF **] by auto
 qed auto
 
 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
@@ -740,52 +858,65 @@
   apply (drule content_subset) unfolding assms(1)
 proof -
   case goal1
-  thus ?case using content_pos_le[of a b] by auto
+  then show ?case using content_pos_le[of a b] by auto
 qed
 
 lemma division_inter:
-  assumes "p1 division_of s1" "p2 division_of (s2::('a::ordered_euclidean_space) set)"
+  fixes s1 s2 :: "'a::ordered_euclidean_space set"
+  assumes "p1 division_of s1"
+    and "p2 division_of s2"
   shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
   (is "?A' division_of _")
 proof -
   let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
-  have *:"?A' = ?A" by auto
-  show ?thesis unfolding *
+  have *: "?A' = ?A" by auto
+  show ?thesis
+    unfolding *
   proof (rule division_ofI)
-    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
-    moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto
+    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
+      by auto
+    moreover have "finite (p1 \<times> p2)"
+      using assms unfolding division_of_def by auto
     ultimately show "finite ?A" by auto
-    have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto
+    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
+      by auto
     show "\<Union>?A = s1 \<inter> s2"
       apply (rule set_eqI)
       unfolding * and Union_image_eq UN_iff
       using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
       apply auto
       done
-    { fix k
-      assume "k\<in>?A"
-      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto
-      thus "k \<noteq> {}" by auto
+    {
+      fix k
+      assume "k \<in> ?A"
+      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
+        by auto
+      then show "k \<noteq> {}"
+        by auto
       show "k \<subseteq> s1 \<inter> s2"
         using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
         unfolding k by auto
-      guess a1 using division_ofD(4)[OF assms(1) k(2)] ..
-      then guess b1 .. note ab1=this
-      guess a2 using division_ofD(4)[OF assms(2) k(3)] ..
-      then guess b2 .. note ab2=this
+      obtain a1 b1 where k1: "k1 = {a1..b1}"
+        using division_ofD(4)[OF assms(1) k(2)] by blast
+      obtain a2 b2 where k2: "k2 = {a2..b2}"
+        using division_ofD(4)[OF assms(2) k(3)] by blast
       show "\<exists>a b. k = {a..b}"
-        unfolding k ab1 ab2 unfolding inter_interval by auto }
+        unfolding k k1 k2 unfolding inter_interval by auto
+    }
     fix k1 k2
-    assume "k1\<in>?A"
-    then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
-    assume "k2\<in>?A"
-    then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
+    assume "k1 \<in> ?A"
+    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
+      by auto
+    assume "k2 \<in> ?A"
+    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
+      by auto
     assume "k1 \<noteq> k2"
-    hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
-    have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
-      interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
-      interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
-      \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
+    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
+      unfolding k1 k2 by auto
+    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
+      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
+      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
+      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
     show "interior k1 \<inter> interior k2 = {}"
       unfolding k1 k2
       apply (rule *)
@@ -793,31 +924,41 @@
       apply (rule_tac[1-4] interior_mono)
       using division_ofD(5)[OF assms(1) k1(2) k2(2)]
       using division_ofD(5)[OF assms(2) k1(3) k2(3)]
-      using th apply auto done
+      using th
+      apply auto
+      done
   qed
 qed
 
 lemma division_inter_1:
-  assumes "d division_of i" "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
-  shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}"
+  assumes "d division_of i"
+    and "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
+  shows "{{a..b} \<inter> k | k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {}} division_of {a..b}"
 proof (cases "{a..b} = {}")
   case True
-  show ?thesis unfolding True and division_of_trivial by auto
+  show ?thesis
+    unfolding True and division_of_trivial by auto
 next
   case False
   have *: "{a..b} \<inter> i = {a..b}" using assms(2) by auto
-  show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto
+  show ?thesis
+    using division_inter[OF division_of_self[OF False] assms(1)]
+    unfolding * by auto
 qed
 
 lemma elementary_inter:
-  assumes "p1 division_of s" "p2 division_of (t::('a::ordered_euclidean_space) set)"
+  fixes s t :: "'a::ordered_euclidean_space set"
+  assumes "p1 division_of s"
+    and "p2 division_of t"
   shows "\<exists>p. p division_of (s \<inter> t)"
   apply rule
   apply (rule division_inter[OF assms])
   done
 
 lemma elementary_inters:
-  assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
+  assumes "finite f"
+    and "f \<noteq> {}"
+    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
   shows "\<exists>p. p division_of (\<Inter> f)"
   using assms
 proof (induct f rule: finite_induct)
@@ -825,14 +966,18 @@
   show ?case
   proof (cases "f = {}")
     case True
-    thus ?thesis unfolding True using insert by auto
+    then show ?thesis
+      unfolding True using insert by auto
   next
     case False
-    guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
-    moreover guess px using insert(5)[rule_format,OF insertI1] ..
+    obtain p where "p division_of \<Inter>f"
+      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
+    moreover obtain px where "px division_of x"
+      using insert(5)[rule_format,OF insertI1] ..
     ultimately show ?thesis
+      apply -
       unfolding Inter_insert
-      apply (rule_tac elementary_inter)
+      apply (rule elementary_inter)
       apply assumption
       apply assumption
       done
@@ -840,12 +985,17 @@
 qed auto
 
 lemma division_disjoint_union:
-  assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
+  assumes "p1 division_of s1"
+    and "p2 division_of s2"
+    and "interior s1 \<inter> interior s2 = {}"
   shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
 proof (rule division_ofI)
-  note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
-  show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
-  show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
+  note d1 = division_ofD[OF assms(1)]
+  note d2 = division_ofD[OF assms(2)]
+  show "finite (p1 \<union> p2)"
+    using d1(1) d2(1) by auto
+  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
+    using d1(6) d2(6) by auto
   {
     fix k1 k2
     assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
@@ -864,28 +1014,33 @@
         using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
         using assms(3) by blast
     }
-    ultimately show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
+    ultimately show ?g
+      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
   }
   fix k
   assume k: "k \<in> p1 \<union> p2"
-  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
-  show "k \<noteq> {}" using k d1(3) d2(3) by auto
-  show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto
+  show "k \<subseteq> s1 \<union> s2"
+    using k d1(2) d2(2) by auto
+  show "k \<noteq> {}"
+    using k d1(3) d2(3) by auto
+  show "\<exists>a b. k = {a..b}"
+    using k d1(4) d2(4) by auto
 qed
 
 lemma partial_division_extend_1:
-  assumes incl: "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}"
+  fixes a b c d :: "'a::ordered_euclidean_space"
+  assumes incl: "{c..d} \<subseteq> {a..b}"
     and nonempty: "{c..d} \<noteq> {}"
   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
 proof
-  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a. {(\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)}"
+  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
+    {(\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)}"
   def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
 
   show "{c .. d} \<in> p"
     unfolding p_def
     by (auto simp add: interval_eq_empty eucl_le[where 'a='a]
-             intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
-
+        intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
   {
     fix i :: 'a
     assume "i \<in> Basis"
@@ -896,13 +1051,15 @@
 
   show "p division_of {a..b}"
   proof (rule division_ofI)
-    show "finite p" unfolding p_def by (auto intro!: finite_PiE)
+    show "finite p"
+      unfolding p_def by (auto intro!: finite_PiE)
     {
       fix k
       assume "k \<in> p"
       then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
         by (auto simp: p_def)
-      then show "\<exists>a b. k = {a..b}" by auto
+      then show "\<exists>a b. k = {a..b}"
+        by auto
       have "k \<subseteq> {a..b} \<and> k \<noteq> {}"
       proof (simp add: k interval_eq_empty subset_interval not_less, safe)
         fix i :: 'a
@@ -913,50 +1070,55 @@
         show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
           by (auto simp: subset_iff eucl_le[where 'a='a])
       qed
-      then show "k \<noteq> {}" "k \<subseteq> {a .. b}" by auto
+      then show "k \<noteq> {}" "k \<subseteq> {a .. b}"
+        by auto
       {
-        fix l assume "l \<in> p"
+        fix l
+        assume "l \<in> p"
         then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
           by (auto simp: p_def)
         assume "l \<noteq> k"
         have "\<exists>i\<in>Basis. f i \<noteq> g i"
         proof (rule ccontr)
-          assume "\<not> (\<exists>i\<in>Basis. f i \<noteq> g i)"
+          assume "\<not> ?thesis"
           with f g have "f = g"
             by (auto simp: PiE_iff extensional_def intro!: ext)
           with `l \<noteq> k` show False
             by (simp add: l k)
         qed
-        then guess i .. note * = this
-        moreover from * have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
+        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
+        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
             "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
           using f g by (auto simp: PiE_iff)
-        moreover note ord[of i]
-        ultimately show "interior l \<inter> interior k = {}"
+        with * ord[of i] show "interior l \<inter> interior k = {}"
           by (auto simp add: l k interior_closed_interval disjoint_interval intro!: bexI[of _ i])
       }
-      note `k \<subseteq> { a.. b}`
+      note `k \<subseteq> {a.. b}`
     }
     moreover
     {
       fix x assume x: "x \<in> {a .. b}"
       have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
       proof
-        fix i :: 'a assume "i \<in> Basis"
-        with x ord[of i] 
+        fix i :: 'a
+        assume "i \<in> Basis"
+        with x ord[of i]
         have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
             (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
           by (auto simp: eucl_le[where 'a='a])
         then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
           by auto
       qed
-      then guess f unfolding bchoice_iff .. note f = this
+      then obtain f where
+        f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}"
+        unfolding bchoice_iff ..
       moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
         by auto
       moreover from f have "x \<in> ?B (restrict f Basis)"
         by (auto simp: mem_interval eucl_le[where 'a='a])
       ultimately have "\<exists>k\<in>p. x \<in> k"
-        unfolding p_def by blast }
+        unfolding p_def by blast
+    }
     ultimately show "\<Union>p = {a..b}"
       by auto
   qed
@@ -967,8 +1129,9 @@
   obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}"
 proof (cases "p = {}")
   case True
-  guess q apply (rule elementary_interval[of a b]) .
-  thus ?thesis
+  obtain q where "q division_of {a..b}"
+    by (rule elementary_interval)
+  then show ?thesis
     apply -
     apply (rule that[of q])
     unfolding True
@@ -977,31 +1140,36 @@
 next
   case False
   note p = division_ofD[OF assms(1)]
-  have *: "\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q"
+  have *: "\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k \<in> q"
   proof
     case goal1
-    guess c using p(4)[OF goal1] ..
-    then guess d .. note "cd" = this
+    obtain c d where k: "k = {c..d}"
+      using p(4)[OF goal1] by blast
     have *: "{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}"
-      using p(2,3)[OF goal1, unfolded "cd"] using assms(2) by auto
-    guess q apply(rule partial_division_extend_1[OF *]) .
-    thus ?case unfolding "cd" by auto
+      using p(2,3)[OF goal1, unfolded k] using assms(2) by auto
+    obtain q where "q division_of {a..b}" "{c..d} \<in> q"
+      by (rule partial_division_extend_1[OF *])
+    then show ?case
+      unfolding k by auto
   qed
-  guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
-  have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
-    apply (rule, rule_tac p="q x" in division_of_subset)
+  obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of {a..b}" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
+    using bchoice[OF *] by blast
+  have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
+    apply rule
+    apply (rule_tac p="q x" in division_of_subset)
   proof -
     fix x
-    assume x: "x\<in>p"
+    assume x: "x \<in> p"
     show "q x division_of \<Union>q x"
       apply -
       apply (rule division_ofI)
       using division_ofD[OF q(1)[OF x]]
       apply auto
       done
-    show "q x - {x} \<subseteq> q x" by auto
+    show "q x - {x} \<subseteq> q x"
+      by auto
   qed
-  hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
+  then have "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
     apply -
     apply (rule elementary_inters)
     apply (rule finite_imageI[OF p(1)])
@@ -1009,16 +1177,16 @@
     apply (rule False)
     apply auto
     done
-  then guess d .. note d = this
+  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
   show ?thesis
     apply (rule that[of "d \<union> p"])
   proof -
-    have *: "\<And>s f t. s \<noteq> {} \<Longrightarrow> (\<forall>i\<in>s. f i \<union> i = t) \<Longrightarrow> t = \<Inter> (f ` s) \<union> (\<Union>s)" by auto
-    have *: "{a..b} = \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
+    have *: "\<And>s f t. s \<noteq> {} \<Longrightarrow> \<forall>i\<in>s. f i \<union> i = t \<Longrightarrow> t = \<Inter>(f ` s) \<union> \<Union>s" by auto
+    have *: "{a..b} = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
       apply (rule *[OF False])
     proof
       fix i
-      assume i: "i\<in>p"
+      assume i: "i \<in> p"
       show "\<Union>(q i - {i}) \<union> i = {a..b}"
         using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
     qed
@@ -1027,10 +1195,12 @@
       apply (rule division_disjoint_union[OF d assms(1)])
       apply (rule inter_interior_unions_intervals)
       apply (rule p open_interior ballI)+
-    proof (assumption, rule)
+      apply assumption
+    proof
       fix k
-      assume k: "k\<in>p"
-      have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto
+      assume k: "k \<in> p"
+      have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}"
+        by auto
       show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<inter> interior k = {}"
         apply (rule *[of _ "interior (\<Union>(q k - {k}))"])
         defer
@@ -1038,27 +1208,34 @@
         apply (rule inter_interior_unions_intervals)
       proof -
         note qk=division_ofD[OF q(1)[OF k]]
-        show "finite (q k - {k})" "open (interior k)"
-          "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
+        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}"
+          using qk by auto
         show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
           using qk(5) using q(2)[OF k] by auto
-        have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto
+        have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x"
+          by auto
         show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<subseteq> interior (\<Union>(q k - {k}))"
           apply (rule interior_mono *)+
-          using k by auto
+          using k
+          apply auto
+          done
       qed
     qed
   qed auto
 qed
 
-lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::('a::ordered_euclidean_space) set)"
-  unfolding division_of_def by(metis bounded_Union bounded_interval) 
-
-lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
+lemma elementary_bounded[dest]:
+  fixes s :: "'a::ordered_euclidean_space set"
+  shows "p division_of s \<Longrightarrow> bounded s"
+  unfolding division_of_def by (metis bounded_Union bounded_interval)
+
+lemma elementary_subset_interval:
+  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
   by (meson elementary_bounded bounded_subset_closed_interval)
 
 lemma division_union_intervals_exists:
-  assumes "{a..b::'a::ordered_euclidean_space} \<noteq> {}"
+  fixes a b :: "'a::ordered_euclidean_space"
+  assumes "{a..b} \<noteq> {}"
   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})"
 proof (cases "{c..d} = {}")
   case True
@@ -1070,16 +1247,15 @@
     done
 next
   case False
-  note false=this
   show ?thesis
   proof (cases "{a..b} \<inter> {c..d} = {}")
     case True
-    have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
+    have *: "\<And>a b. {a, b} = {a} \<union> {b}" by auto
     show ?thesis
       apply (rule that[of "{{c..d}}"])
       unfolding *
       apply (rule division_disjoint_union)
-      using false True assms
+      using `{c..d} \<noteq> {}` True assms
       using interior_subset
       apply auto
       done
@@ -1088,10 +1264,11 @@
     obtain u v where uv: "{a..b} \<inter> {c..d} = {u..v}"
       unfolding inter_interval by auto
     have *: "{u..v} \<subseteq> {c..d}" using uv by auto
-    guess p apply (rule partial_division_extend_1[OF * False[unfolded uv]]) .
-    note p=this division_ofD[OF this(1)]
+    obtain p where "p division_of {c..d}" "{u..v} \<in> p"
+      by (rule partial_division_extend_1[OF * False[unfolded uv]])
+    note p = this division_ofD[OF this(1)]
     have *: "{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s"
-      using p(8) unfolding uv[THEN sym] by auto
+      using p(8) unfolding uv[symmetric] by auto
     show ?thesis
       apply (rule that[of "p - {{u..v}}"])
       unfolding *(1)
@@ -1101,10 +1278,10 @@
       apply (rule division_of_subset[of p])
       apply (rule division_of_union_self[OF p(1)])
       defer
-      unfolding interior_inter[THEN sym]
+      unfolding interior_inter[symmetric]
     proof -
       have *: "\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
-      have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
+      have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
         apply (rule arg_cong[of _ _ interior])
         apply (rule *[OF _ uv])
         using p(8)
@@ -1121,270 +1298,611 @@
   qed
 qed
 
-lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
-  "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
-  shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
-  apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
-  using division_ofD[OF assms(2)] by auto
-  
-lemma elementary_union_interval: assumes "p division_of \<Union>p"
-  obtains q where "q division_of ({a..b::'a::ordered_euclidean_space} \<union> \<Union>p)" proof-
-  note assm=division_ofD[OF assms]
-  have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>((\<lambda>x.\<Union>(f x)) ` s)" by auto
-  have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
-{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
-    "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
-  thus thesis by auto
-next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
-  thus thesis apply(rule_tac that[of p]) unfolding as by auto 
-next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
-next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
-  show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
-    unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
-    using assm(2-4) as apply- by(fastforce dest: assm(5))+
-next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
-  have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
-    from assm(4)[OF this] guess c .. then guess d ..
-    thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
-  qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
-  let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
-  show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
-    have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
-    show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
-    show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
-      using q(6) by auto
-    fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
-    show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
-    fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
-    obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
-    obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
-    show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
-      case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
-    next case False 
-      { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
-        "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
-        thus ?thesis by auto }
-      { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
-      { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
-      assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
-      guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
-      have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
-      hence "interior k \<subseteq> interior x" apply-
-        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
-      guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
-      have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
-      hence "interior k' \<subseteq> interior x'" apply-
-        apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
-      ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
-    qed qed } qed
+lemma division_of_unions:
+  assumes "finite f"
+    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
+    and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+  shows "\<Union>f division_of \<Union>\<Union>f"
+  apply (rule division_ofI)
+  prefer 5
+  apply (rule assms(3)|assumption)+
+  apply (rule finite_Union assms(1))+
+  prefer 3
+  apply (erule UnionE)
+  apply (rule_tac s=X in division_ofD(3)[OF assms(2)])
+  using division_ofD[OF assms(2)]
+  apply auto
+  done
+
+lemma elementary_union_interval:
+  fixes a b :: "'a::ordered_euclidean_space"
+  assumes "p division_of \<Union>p"
+  obtains q where "q division_of ({a..b} \<union> \<Union>p)"
+proof -
+  note assm = division_ofD[OF assms]
+  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
+    by auto
+  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
+    by auto
+  {
+    presume "p = {} \<Longrightarrow> thesis"
+      "{a..b} = {} \<Longrightarrow> thesis"
+      "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
+      "p \<noteq> {} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
+    then show thesis by auto
+  next
+    assume as: "p = {}"
+    obtain p where "p division_of {a..b}"
+      by (rule elementary_interval)
+    then show thesis
+      apply -
+      apply (rule that[of p])
+      unfolding as
+      apply auto
+      done
+  next
+    assume as: "{a..b} = {}"
+    show thesis
+      apply (rule that)
+      unfolding as
+      using assms
+      apply auto
+      done
+  next
+    assume as: "interior {a..b} = {}" "{a..b} \<noteq> {}"
+    show thesis
+      apply (rule that[of "insert {a..b} p"],rule division_ofI)
+      unfolding finite_insert
+      apply (rule assm(1)) unfolding Union_insert
+      using assm(2-4) as
+      apply -
+      apply (fastforce dest: assm(5))+
+      done
+  next
+    assume as: "p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b} \<noteq> {}"
+    have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)"
+    proof
+      case goal1
+      from assm(4)[OF this] obtain c d where "k = {c..d}" by blast
+      then show ?case
+        apply -
+        apply (rule division_union_intervals_exists[OF as(3), of c d])
+        apply auto
+        done
+    qed
+    from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert {a..b} (q x) division_of {a..b} \<union> x" ..
+    note q = division_ofD[OF this[rule_format]]
+    let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
+    show thesis
+      apply (rule that[of "?D"])
+      apply (rule division_ofI)
+    proof -
+      have *: "{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p"
+        by auto
+      show "finite ?D"
+        apply (rule finite_Union)
+        unfolding *
+        apply (rule finite_imageI)
+        using assm(1) q(1)
+        apply auto
+        done
+      show "\<Union>?D = {a..b} \<union> \<Union>p"
+        unfolding * lem1
+        unfolding lem2[OF as(1), of "{a..b}", symmetric]
+        using q(6)
+        by auto
+      fix k
+      assume k: "k \<in> ?D"
+      then show "k \<subseteq> {a..b} \<union> \<Union>p"
+        using q(2) by auto
+      show "k \<noteq> {}"
+        using q(3) k by auto
+      show "\<exists>a b. k = {a..b}"
+        using q(4) k by auto
+      fix k'
+      assume k': "k' \<in> ?D" "k \<noteq> k'"
+      obtain x where x: "k \<in> insert {a..b} (q x)" "x\<in>p"
+        using k by auto
+      obtain x' where x': "k'\<in>insert {a..b} (q x')" "x'\<in>p"
+        using k' by auto
+      show "interior k \<inter> interior k' = {}"
+      proof (cases "x = x'")
+        case True
+        show ?thesis
+          apply(rule q(5))
+          using x x' k'
+          unfolding True
+          apply auto
+          done
+      next
+        case False
+        {
+          presume "k = {a..b} \<Longrightarrow> ?thesis"
+            and "k' = {a..b} \<Longrightarrow> ?thesis"
+            and "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
+          then show ?thesis by auto
+        next
+          assume as': "k  = {a..b}"
+          show ?thesis
+            apply (rule q(5))
+            using x' k'(2)
+            unfolding as'
+            apply auto
+            done
+        next
+          assume as': "k' = {a..b}"
+          show ?thesis
+            apply (rule q(5))
+            using x  k'(2)
+            unfolding as'
+            apply auto
+            done
+        }
+        assume as': "k \<noteq> {a..b}" "k' \<noteq> {a..b}"
+        obtain c d where k: "k = {c..d}"
+          using q(4)[OF x(2,1)] by blast
+        have "interior k \<inter> interior {a..b} = {}"
+          apply (rule q(5))
+          using x k'(2)
+          using as'
+          apply auto
+          done
+        then have "interior k \<subseteq> interior x"
+          apply -
+          apply (rule interior_subset_union_intervals[OF k _ as(2) q(2)[OF x(2,1)]])
+          apply auto
+          done
+        moreover
+        obtain c d where c_d: "k' = {c..d}"
+          using q(4)[OF x'(2,1)] by blast
+        have "interior k' \<inter> interior {a..b} = {}"
+          apply (rule q(5))
+          using x' k'(2)
+          using as'
+          apply auto
+          done
+        then have "interior k' \<subseteq> interior x'"
+          apply -
+          apply (rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]])
+          apply auto
+          done
+        ultimately show ?thesis
+          using assm(5)[OF x(2) x'(2) False] by auto
+      qed
+    qed
+  }
+qed
 
 lemma elementary_unions_intervals:
-  assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
-  obtains p where "p division_of (\<Union>f)" proof-
-  have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
+  assumes fin: "finite f"
+    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
+  obtains p where "p division_of (\<Union>f)"
+proof -
+  have "\<exists>p. p division_of (\<Union>f)"
+  proof (induct_tac f rule:finite_subset_induct)
     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
-    fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
-    from this(3) guess p .. note p=this
-    from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
-    have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
-    show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
-      unfolding Union_insert ab * by auto
-  qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
-
-lemma elementary_union: assumes "ps division_of s" "pt division_of (t::('a::ordered_euclidean_space) set)"
+  next
+    fix x F
+    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
+    from this(3) obtain p where p: "p division_of \<Union>F" ..
+    from assms(2)[OF as(4)] obtain a b where x: "x = {a..b}" by blast
+    have *: "\<Union>F = \<Union>p"
+      using division_ofD[OF p] by auto
+    show "\<exists>p. p division_of \<Union>insert x F"
+      using elementary_union_interval[OF p[unfolded *], of a b]
+      unfolding Union_insert x * by auto
+  qed (insert assms, auto)
+  then show ?thesis
+    apply -
+    apply (erule exE)
+    apply (rule that)
+    apply auto
+    done
+qed
+
+lemma elementary_union:
+  fixes s t :: "'a::ordered_euclidean_space set"
+  assumes "ps division_of s"
+    and "pt division_of t"
   obtains p where "p division_of (s \<union> t)"
-proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
-  hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
-  show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
-    unfolding * prefer 3 apply(rule_tac p=p in that)
-    using assms[unfolded division_of_def] by auto qed
-
-lemma partial_division_extend: fixes t::"('a::ordered_euclidean_space) set"
-  assumes "p division_of s" "q division_of t" "s \<subseteq> t"
-  obtains r where "p \<subseteq> r" "r division_of t" proof-
+proof -
+  have "s \<union> t = \<Union>ps \<union> \<Union>pt"
+    using assms unfolding division_of_def by auto
+  then have *: "\<Union>(ps \<union> pt) = s \<union> t" by auto
+  show ?thesis
+    apply -
+    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
+    unfolding *
+    prefer 3
+    apply (rule_tac p=p in that)
+    using assms[unfolded division_of_def]
+    apply auto
+    done
+qed
+
+lemma partial_division_extend:
+  fixes t :: "'a::ordered_euclidean_space set"
+  assumes "p division_of s"
+    and "q division_of t"
+    and "s \<subseteq> t"
+  obtains r where "p \<subseteq> r" and "r division_of t"
+proof -
   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
-  obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
-  guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
-    apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
-  guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
-  then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
-    apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
-  { fix x assume x:"x\<in>t" "x\<notin>s"
-    hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
-    then guess r unfolding Union_iff .. note r=this moreover
-    have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
-      thus False using x by auto qed
-    ultimately have "x\<in>\<Union>(r1 - p)" by auto }
-  hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
-  show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
-    unfolding divp(6) apply(rule assms r2)+
-  proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
-    proof(rule inter_interior_unions_intervals)
-      show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
-      have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
-      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
-        fix m x assume as:"m\<in>r1-p"
-        have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
-          show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
-          show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
-        qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
-      qed qed        
-    thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
-  qed auto qed
+  obtain a b where ab: "t \<subseteq> {a..b}"
+    using elementary_subset_interval[OF assms(2)] by auto
+  obtain r1 where "p \<subseteq> r1" "r1 division_of {a..b}"
+    apply (rule partial_division_extend_interval)
+    apply (rule assms(1)[unfolded divp(6)[symmetric]])
+    apply (rule subset_trans)
+    apply (rule ab assms[unfolded divp(6)[symmetric]])+
+    apply assumption
+    done
+  note r1 = this division_ofD[OF this(2)]
+  obtain p' where "p' division_of \<Union>(r1 - p)"
+    apply (rule elementary_unions_intervals[of "r1 - p"])
+    using r1(3,6)
+    apply auto
+    done
+  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
+    apply -
+    apply (drule elementary_inter[OF _ assms(2)[unfolded divq(6)[symmetric]]])
+    apply auto
+    done
+  {
+    fix x
+    assume x: "x \<in> t" "x \<notin> s"
+    then have "x\<in>\<Union>r1"
+      unfolding r1 using ab by auto
+    then obtain r where r: "r \<in> r1" "x \<in> r"
+      unfolding Union_iff ..
+    moreover
+    have "r \<notin> p"
+    proof
+      assume "r \<in> p"
+      then have "x \<in> s" using divp(2) r by auto
+      then show False using x by auto
+    qed
+    ultimately have "x\<in>\<Union>(r1 - p)" by auto
+  }
+  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
+    unfolding divp divq using assms(3) by auto
+  show ?thesis
+    apply (rule that[of "p \<union> r2"])
+    unfolding *
+    defer
+    apply (rule division_disjoint_union)
+    unfolding divp(6)
+    apply(rule assms r2)+
+  proof -
+    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
+    proof (rule inter_interior_unions_intervals)
+      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}"
+        using r1 by auto
+      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
+        by auto
+      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
+      proof
+        fix m x
+        assume as: "m \<in> r1 - p"
+        have "interior m \<inter> interior (\<Union>p) = {}"
+        proof (rule inter_interior_unions_intervals)
+          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = {a..b}"
+            using divp by auto
+          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
+            apply (rule, rule r1(7))
+            using as
+            using r1 
+            apply auto
+            done
+        qed
+        then show "interior s \<inter> interior m = {}"
+          unfolding divp by auto
+      qed
+    qed
+    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
+      using interior_subset by auto
+  qed auto
+qed
+
 
 subsection {* Tagged (partial) divisions. *}
 
-definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
-  "(s tagged_partial_division_of i) \<equiv>
-        finite s \<and>
-        (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
-        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
-                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
-
-lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
-  shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
-  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
-  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> (x1,k1) \<noteq> (x2,k2) \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
-  using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
-
-definition tagged_division_of (infixr "tagged'_division'_of" 40) where
-  "(s tagged_division_of i) \<equiv>
-        (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
+  where "s tagged_partial_division_of i \<longleftrightarrow>
+    finite s \<and>
+    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
+      interior k1 \<inter> interior k2 = {})"
+
+lemma tagged_partial_division_ofD[dest]:
+  assumes "s tagged_partial_division_of i"
+  shows "finite s"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
+      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+  using assms unfolding tagged_partial_division_of_def by blast+
+
+definition tagged_division_of (infixr "tagged'_division'_of" 40)
+  where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
 
 lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
 
 lemma tagged_division_of:
- "(s tagged_division_of i) \<longleftrightarrow>
-        finite s \<and>
-        (\<forall>x k. (x,k) \<in> s
-               \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
-        (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
-                       \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
-        (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+  "s tagged_division_of i \<longleftrightarrow>
+    finite s \<and>
+    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
+      interior k1 \<inter> interior k2 = {}) \<and>
+    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
 
-lemma tagged_division_ofI: assumes
-  "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
-  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
-  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+lemma tagged_division_ofI:
+  assumes "finite s"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
+      interior k1 \<inter> interior k2 = {}"
+    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   shows "s tagged_division_of i"
-  unfolding tagged_division_of apply(rule) defer apply rule
-  apply(rule allI impI conjI assms)+ apply assumption
-  apply(rule, rule assms, assumption) apply(rule assms, assumption)
-  using assms(1,5-) apply- by blast+
-
-lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
-  shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
-  "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
-  "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
-
-lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
-proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
-  show "\<Union>(snd ` s) = i" "finite (snd ` s)" using assm by auto
-  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
-  thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastforce+
-  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
-  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+  unfolding tagged_division_of
+  apply rule
+  defer
+  apply rule
+  apply (rule allI impI conjI assms)+
+  apply assumption
+  apply rule
+  apply (rule assms)
+  apply assumption
+  apply (rule assms)
+  apply assumption
+  using assms(1,5-)
+  apply blast+
+  done
+
+lemma tagged_division_ofD[dest]:
+  assumes "s tagged_division_of i"
+  shows "finite s"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
+      interior k1 \<inter> interior k2 = {}"
+    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+  using assms unfolding tagged_division_of by blast+
+
+lemma division_of_tagged_division:
+  assumes "s tagged_division_of i"
+  shows "(snd ` s) division_of i"
+proof (rule division_ofI)
+  note assm = tagged_division_ofD[OF assms]
+  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
+    using assm by auto
+  fix k
+  assume k: "k \<in> snd ` s"
+  then obtain xk where xk: "(xk, k) \<in> s"
+    by auto
+  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}"
+    using assm by fastforce+
+  fix k'
+  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
+  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
+    by auto
+  then show "interior k \<inter> interior k' = {}"
+    apply -
+    apply (rule assm(5))
+    apply (rule xk xk')+
+    using k'
+    apply auto
+    done
 qed
 
-lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
+lemma partial_division_of_tagged_division:
+  assumes "s tagged_partial_division_of i"
   shows "(snd ` s) division_of \<Union>(snd ` s)"
-proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
-  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)" using assm by auto
-  fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
-  thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>(snd ` s)" using assm by auto
-  fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
-  thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+proof (rule division_ofI)
+  note assm = tagged_partial_division_ofD[OF assms]
+  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
+    using assm by auto
+  fix k
+  assume k: "k \<in> snd ` s"
+  then obtain xk where xk: "(xk, k) \<in> s"
+    by auto
+  then show "k \<noteq> {}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>(snd ` s)"
+    using assm by auto
+  fix k'
+  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
+  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
+    by auto
+  then show "interior k \<inter> interior k' = {}"
+    apply -
+    apply (rule assm(5))
+    apply(rule xk xk')+
+    using k'
+    apply auto
+    done
 qed
 
-lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
+lemma tagged_partial_division_subset:
+  assumes "s tagged_partial_division_of i"
+    and "t \<subseteq> s"
   shows "t tagged_partial_division_of i"
-  using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
-
-lemma setsum_over_tagged_division_lemma: fixes d::"('m::ordered_euclidean_space) set \<Rightarrow> 'a::real_normed_vector"
-  assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
+  using assms
+  unfolding tagged_partial_division_of_def
+  using finite_subset[OF assms(2)]
+  by blast
+
+lemma setsum_over_tagged_division_lemma:
+  fixes d :: "'m::ordered_euclidean_space set \<Rightarrow> 'a::real_normed_vector"
+  assumes "p tagged_division_of i"
+    and "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
-proof- note assm=tagged_division_ofD[OF assms(1)]
-  have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
-  show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
-    show "finite p" using assm by auto
-    fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
-    obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
-    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
-    hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
-    hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
-    hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto
-    thus "d (snd x) = 0" unfolding ab by auto qed qed
-
-lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
+proof -
+  note assm = tagged_division_ofD[OF assms(1)]
+  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
+    unfolding o_def by (rule ext) auto
+  show ?thesis
+    unfolding *
+    apply (subst eq_commute)
+  proof (rule setsum_reindex_nonzero)
+    show "finite p"
+      using assm by auto
+    fix x y
+    assume as: "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
+    obtain a b where ab: "snd x = {a..b}"
+      using assm(4)[of "fst x" "snd x"] as(1) by auto
+    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
+      unfolding as(4)[symmetric] using as(1-3) by auto
+    then have "interior (snd x) \<inter> interior (snd y) = {}"
+      apply -
+      apply (rule assm(5)[of "fst x" _ "fst y"])
+      using as
+      apply auto
+      done
+    then have "content {a..b} = 0"
+      unfolding as(4)[symmetric] ab content_eq_0_interior by auto
+    then have "d {a..b} = 0"
+      apply -
+      apply (rule assms(2))
+      using assm(2)[of "fst x" "snd x"] as(1)
+      unfolding ab[symmetric]
+      apply auto
+      done
+    then show "d (snd x) = 0"
+      unfolding ab by auto
+  qed
+qed
+
+lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
+  by auto
 
 lemma tagged_division_of_empty: "{} tagged_division_of {}"
   unfolding tagged_division_of by auto
 
-lemma tagged_partial_division_of_trivial[simp]:
- "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
+lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   unfolding tagged_partial_division_of_def by auto
 
-lemma tagged_division_of_trivial[simp]:
- "p tagged_division_of {} \<longleftrightarrow> p = {}"
+lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
   unfolding tagged_division_of by auto
 
-lemma tagged_division_of_self:
- "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
-  apply(rule tagged_division_ofI) by auto
+lemma tagged_division_of_self: "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
+  by (rule tagged_division_ofI) auto
 
 lemma tagged_division_union:
-  assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
+  assumes "p1 tagged_division_of s1"
+    and "p2 tagged_division_of s2"
+    and "interior s1 \<inter> interior s2 = {}"
   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
-proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
-  show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
-  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
-  fix x k assume xk:"(x,k)\<in>p1\<union>p2" show "x\<in>k" "\<exists>a b. k = {a..b}" using xk p1(2,4) p2(2,4) by auto
-  show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
-  fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
-  have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) interior_mono by blast
-  show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
-    apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
-    using p1(3) p2(3) using xk xk' by auto qed 
+proof (rule tagged_division_ofI)
+  note p1 = tagged_division_ofD[OF assms(1)]
+  note p2 = tagged_division_ofD[OF assms(2)]
+  show "finite (p1 \<union> p2)"
+    using p1(1) p2(1) by auto
+  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
+    using p1(6) p2(6) by blast
+  fix x k
+  assume xk: "(x, k) \<in> p1 \<union> p2"
+  show "x \<in> k" "\<exists>a b. k = {a..b}"
+    using xk p1(2,4) p2(2,4) by auto
+  show "k \<subseteq> s1 \<union> s2"
+    using xk p1(3) p2(3) by blast
+  fix x' k'
+  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
+  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
+    using assms(3) interior_mono by blast
+  show "interior k \<inter> interior k' = {}"
+    apply (cases "(x, k) \<in> p1")
+    apply (case_tac[!] "(x',k') \<in> p1")
+    apply (rule p1(5))
+    prefer 4
+    apply (rule *)
+    prefer 6
+    apply (subst Int_commute)
+    apply (rule *)
+    prefer 8
+    apply (rule p2(5))
+    using p1(3) p2(3)
+    using xk xk'
+    apply auto
+    done
+qed
 
 lemma tagged_division_unions:
-  assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
-  "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
+  assumes "finite iset"
+    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
+    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
-proof(rule tagged_division_ofI)
+proof (rule tagged_division_ofI)
   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
-  show "finite (\<Union>(pfn ` iset))" apply(rule finite_Union) using assms by auto
-  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)" by blast 
-  also have "\<dots> = \<Union>iset" using assm(6) by auto
-  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" . 
-  fix x k assume xk:"(x,k)\<in>\<Union>(pfn ` iset)" then obtain i where i:"i \<in> iset" "(x, k) \<in> pfn i" by auto
-  show "x\<in>k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset" using assm(2-4)[OF i] using i(1) by auto
-  fix x' k' assume xk':"(x',k')\<in>\<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')" then obtain i' where i':"i' \<in> iset" "(x', k') \<in> pfn i'" by auto
-  have *:"\<And>a b. i\<noteq>i' \<Longrightarrow> a\<subseteq> i \<Longrightarrow> b\<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" using i(1) i'(1)
-    using assms(3)[rule_format] interior_mono by blast
-  show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
-    using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
+  show "finite (\<Union>(pfn ` iset))"
+    apply (rule finite_Union)
+    using assms
+    apply auto
+    done
+  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
+    by blast
+  also have "\<dots> = \<Union>iset"
+    using assm(6) by auto
+  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
+  fix x k
+  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
+  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
+    by auto
+  show "x \<in> k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset"
+    using assm(2-4)[OF i] using i(1) by auto
+  fix x' k'
+  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
+  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
+    by auto
+  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
+    using i(1) i'(1)
+    using assms(3)[rule_format] interior_mono
+    by blast
+  show "interior k \<inter> interior k' = {}"
+    apply (cases "i = i'")
+    using assm(5)[OF i _ xk'(2)] i'(2)
+    using assm(3)[OF i] assm(3)[OF i']
+    defer
+    apply -
+    apply (rule *)
+    apply auto
+    done
 qed
 
 lemma tagged_partial_division_of_union_self:
-  assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
-  apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
-
-lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
+  assumes "p tagged_partial_division_of s"
   shows "p tagged_division_of (\<Union>(snd ` p))"
-  apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
+  apply (rule tagged_division_ofI)
+  using tagged_partial_division_ofD[OF assms]
+  apply auto
+  done
+
+lemma tagged_division_of_union_self:
+  assumes "p tagged_division_of s"
+  shows "p tagged_division_of (\<Union>(snd ` p))"
+  apply (rule tagged_division_ofI)
+  using tagged_division_ofD[OF assms]
+  apply auto
+  done
+
 
 subsection {* Fine-ness of a partition w.r.t. a gauge. *}
 
-definition fine (infixr "fine" 46) where
-  "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
-
-lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
-  shows "d fine s" using assms unfolding fine_def by auto
-
-lemma fineD[dest]: assumes "d fine s"
-  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
+definition fine  (infixr "fine" 46)
+  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
+
+lemma fineI:
+  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
+  shows "d fine s"
+  using assms unfolding fine_def by auto
+
+lemma fineD[dest]:
+  assumes "d fine s"
+  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
+  using assms unfolding fine_def by auto
 
 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   unfolding fine_def by auto
@@ -1393,570 +1911,1222 @@
  "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   unfolding fine_def by blast
 
-lemma fine_union:
-  "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
+lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   unfolding fine_def by blast
 
-lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
+lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   unfolding fine_def by auto
 
-lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
+lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   unfolding fine_def by blast
 
+
 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
 
-definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
-  "(f has_integral_compact_interval y) i \<equiv>
-        (\<forall>e>0. \<exists>d. gauge d \<and>
-          (\<forall>p. p tagged_division_of i \<and> d fine p
-                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
-
-definition has_integral (infixr "has'_integral" 46) where 
-"((f::('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
-        if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
-        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
-              \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
-                                       norm(z - y) < e))"
+definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
+  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
+    (\<forall>e>0. \<exists>d. gauge d \<and>
+      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
+        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
+
+definition has_integral ::
+    "('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
+  (infixr "has'_integral" 46)
+  where "(f has_integral y) i \<longleftrightarrow>
+    (if \<exists>a b. i = {a..b}
+     then (f has_integral_compact_interval y) i
+     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
+        norm (z - y) < e)))"
 
 lemma has_integral:
- "(f has_integral y) ({a..b}) \<longleftrightarrow>
-        (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
-                        \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
-  unfolding has_integral_def has_integral_compact_interval_def by auto
-
-lemma has_integralD[dest]: assumes
- "(f has_integral y) ({a..b})" "e>0"
-  obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
-                        \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
+  "(f has_integral y) {a..b} \<longleftrightarrow>
+    (\<forall>e>0. \<exists>d. gauge d \<and>
+      (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
+        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+  unfolding has_integral_def has_integral_compact_interval_def
+  by auto
+
+lemma has_integralD[dest]:
+  assumes "(f has_integral y) ({a..b})"
+    and "e > 0"
+  obtains d where "gauge d"
+    and "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p \<Longrightarrow>
+      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   using assms unfolding has_integral by auto
 
 lemma has_integral_alt:
- "(f has_integral y) i \<longleftrightarrow>
-      (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
-       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
-                               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
-                                        has_integral z) ({a..b}) \<and>
-                                       norm(z - y) < e)))"
-  unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
+  "(f has_integral y) i \<longleftrightarrow>
+    (if \<exists>a b. i = {a..b}
+     then (f has_integral y) i
+     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm (z - y) < e)))"
+  unfolding has_integral
+  unfolding has_integral_compact_interval_def has_integral_def
+  by auto
 
 lemma has_integral_altD:
-  assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
-  obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
-  using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
-
-definition integrable_on (infixr "integrable'_on" 46) where
-  "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
-
-definition "integral i f \<equiv> SOME y. (f has_integral y) i"
-
-lemma integrable_integral[dest]:
- "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
-  unfolding integrable_on_def integral_def by(rule someI_ex)
+  assumes "(f has_integral y) i"
+    and "\<not> (\<exists>a b. i = {a..b})"
+    and "e>0"
+  obtains B where "B > 0"
+    and "\<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
+  using assms
+  unfolding has_integral
+  unfolding has_integral_compact_interval_def has_integral_def
+  by auto
+
+definition integrable_on (infixr "integrable'_on" 46)
+  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
+
+definition "integral i f = (SOME y. (f has_integral y) i)"
+
+lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
+  unfolding integrable_on_def integral_def by (rule someI_ex)
 
 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   unfolding integrable_on_def by auto
 
-lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
+lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   by auto
 
 lemma setsum_content_null:
-  assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
+  assumes "content {a..b} = 0"
+    and "p tagged_division_of {a..b}"
   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
-proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
-  obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
+proof (rule setsum_0', rule)
+  fix y
+  assume y: "y \<in> p"
+  obtain x k where xk: "y = (x, k)"
+    using surj_pair[of y] by blast
   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
-  from this(2) guess c .. then guess d .. note c_d=this
-  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
-  also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
-    unfolding assms(1) c_d by auto
+  from this(2) obtain c d where k: "k = {c..d}" by blast
+  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
+    unfolding xk by auto
+  also have "\<dots> = 0"
+    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
+    unfolding assms(1) k
+    by auto
   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
 qed
 
+
 subsection {* Some basic combining lemmas. *}
 
 lemma tagged_division_unions_exists:
-  assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
-  "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
-   obtains p where "p tagged_division_of i" "d fine p"
-proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
-  show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
-    apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
-    apply(rule fine_unions) using pfn by auto
+  assumes "finite iset"
+    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
+    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
+    and "\<Union>iset = i"
+   obtains p where "p tagged_division_of i" and "d fine p"
+proof -
+  obtain pfn where pfn:
+    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
+    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
+    using bchoice[OF assms(2)] by auto
+  show thesis
+    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
+    unfolding assms(4)[symmetric]
+    apply (rule tagged_division_unions[OF assms(1) _ assms(3)])
+    defer
+    apply (rule fine_unions)
+    using pfn
+    apply auto
+    done
 qed
 
+
 subsection {* The set we're concerned with must be closed. *}
 
-lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::('n::ordered_euclidean_space) set)"
+lemma division_of_closed:
+  fixes i :: "'n::ordered_euclidean_space set"
+  shows "s division_of i \<Longrightarrow> closed i"
   unfolding division_of_def by fastforce
 
 subsection {* General bisection principle for intervals; might be useful elsewhere. *}
 
-lemma interval_bisection_step:  fixes type::"'a::ordered_euclidean_space"
-  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::'a})"
-  obtains c d where "~(P{c..d})"
-  "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
-proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
-  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" by (auto simp: interval_eq_empty not_le)
-  { fix f have "finite f \<Longrightarrow>
-        (\<forall>s\<in>f. P s) \<Longrightarrow>
-        (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
-        (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
-    proof(induct f rule:finite_induct)
-      case empty show ?case using assms(1) by auto
-    next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
-        apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
-        using insert by auto
-    qed } note * = this
-  let ?A = "{{c..d} | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or> (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
+lemma interval_bisection_step:
+  fixes type :: "'a::ordered_euclidean_space"
+  assumes "P {}"
+    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
+    and "\<not> P {a..b::'a}"
+  obtains c d where "\<not> P{c..d}"
+    and "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
+proof -
+  have "{a..b} \<noteq> {}"
+    using assms(1,3) by auto
+  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
+    by (auto simp: interval_eq_empty not_le)
+  {
+    fix f
+    have "finite f \<Longrightarrow>
+      \<forall>s\<in>f. P s \<Longrightarrow>
+      \<forall>s\<in>f. \<exists>a b. s = {a..b} \<Longrightarrow>
+      \<forall>s\<in>f.\<forall>t\<in>f. s \<noteq> t \<longrightarrow> interior s \<inter> interior t = {} \<Longrightarrow> P (\<Union>f)"
+    proof (induct f rule: finite_induct)
+      case empty
+      show ?case
+        using assms(1) by auto
+    next
+      case (insert x f)
+      show ?case
+        unfolding Union_insert
+        apply (rule assms(2)[rule_format])
+        apply rule
+        defer
+        apply rule
+        defer
+        apply (rule inter_interior_unions_intervals)
+        using insert
+        apply auto
+        done
+    qed
+  } note * = this
+  let ?A = "{{c..d} | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
+    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
   let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
-  { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
-    thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
-  assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
-  have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
+  {
+    presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
+    then show thesis
+      unfolding atomize_not not_all
+      apply -
+      apply (erule exE)+
+      apply (rule_tac c=x and d=xa in that)
+      apply auto
+      done
+  }
+  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
+  have "P (\<Union> ?A)"
+    apply (rule *)
+    apply (rule_tac[2-] ballI)
+    apply (rule_tac[4] ballI)
+    apply (rule_tac[4] impI)
+  proof -
     let ?B = "(\<lambda>s.{(\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i)::'a ..
       (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)}) ` {s. s \<subseteq> Basis}"
-    have "?A \<subseteq> ?B" proof case goal1
-      then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
-      have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
-      show "x\<in>?B" unfolding image_iff
-        apply(rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
-        unfolding c_d
-        apply(rule *)
+    have "?A \<subseteq> ?B"
+    proof
+      case goal1
+      then obtain c d where x: "x = {c..d}"
+        "\<And>i. i \<in> Basis \<Longrightarrow>
+          c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+          c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
+      have *: "\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}"
+        by auto
+      show "x \<in> ?B"
+        unfolding image_iff
+        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
+        unfolding x
+        apply (rule *)
         apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
-                        cong: ball_cong)
+          cong: ball_cong)
         apply safe
-      proof-
-        fix i :: 'a assume i: "i\<in>Basis"
-        thus " c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
-          "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
-          using c_d(2)[of i] ab[OF i] by(auto simp add:field_simps)
-      qed qed
-    thus "finite ?A" apply(rule finite_subset) by auto
-    fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
-    note c_d=this[rule_format]
-    show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 thus ?case 
-        using c_d(2)[of i] using ab[OF `i \<in> Basis`] by auto qed
-    show "\<exists>a b. s = {a..b}" unfolding c_d by auto
-    fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
-    note e_f=this[rule_format]
-    assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
-    then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i':"i\<in>Basis"
+      proof -
+        fix i :: 'a
+        assume i: "i \<in> Basis"
+        then show "c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
+          and "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
+          using x(2)[of i] ab[OF i] by (auto simp add:field_simps)
+      qed
+    qed
+    then show "finite ?A"
+      by (rule finite_subset) auto
+    fix s
+    assume "s \<in> ?A"
+    then obtain c d where s:
+      "s = {c..d}"
+      "\<And>i. i \<in> Basis \<Longrightarrow>
+         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
+      by blast
+    show "P s"
+      unfolding s
+      apply (rule as[rule_format])
+    proof -
+      case goal1
+      then show ?case
+        using s(2)[of i] using ab[OF `i \<in> Basis`] by auto
+    qed
+    show "\<exists>a b. s = {a..b}"
+      unfolding s by auto
+    fix t
+    assume "t \<in> ?A"
+    then obtain e f where t:
+      "t = {e..f}"
+      "\<And>i. i \<in> Basis \<Longrightarrow>
+        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
+      by blast
+    assume "s \<noteq> t"
+    then have "\<not> (c = e \<and> d = f)"
+      unfolding s t by auto
+    then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
       unfolding euclidean_eq_iff[where 'a='a] by auto
-    hence i:"c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i" apply- apply(erule_tac[!] disjE)
-    proof- assume "c\<bullet>i \<noteq> e\<bullet>i" thus "d\<bullet>i \<noteq> f\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
-    next   assume "d\<bullet>i \<noteq> f\<bullet>i" thus "c\<bullet>i \<noteq> e\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
-    qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
-    show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
-      fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
-      hence x:"c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i" unfolding mem_interval using i'
-        apply-apply(erule_tac[!] x=i in ballE)+ by auto
-      show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
-      proof(erule_tac[!] conjE) assume as:"c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
-        show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
-      next assume as:"c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
-        show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
-      qed qed qed
-  also have "\<Union> ?A = {a..b}" proof(rule set_eqI,rule)
-    fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
-    from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
-    note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
-    show "x\<in>{a..b}" unfolding mem_interval proof safe
-      fix i :: 'a assume i: "i\<in>Basis" thus "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
-        using c_d(1)[OF i] c_d(2)[unfolded mem_interval,THEN bspec, OF i] by auto qed
-  next fix x assume x:"x\<in>{a..b}"
-    have "\<forall>i\<in>Basis. \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
-      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d") unfolding mem_interval
+    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
+      apply -
+      apply(erule_tac[!] disjE)
+    proof -
+      assume "c\<bullet>i \<noteq> e\<bullet>i"
+      then show "d\<bullet>i \<noteq> f\<bullet>i"
+        using s(2)[OF i'] t(2)[OF i'] by fastforce
+    next
+      assume "d\<bullet>i \<noteq> f\<bullet>i"
+      then show "c\<bullet>i \<noteq> e\<bullet>i"
+        using s(2)[OF i'] t(2)[OF i'] by fastforce
+    qed
+    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
+      by auto
+    show "interior s \<inter> interior t = {}"
+      unfolding s t interior_closed_interval
+    proof (rule *)
+      fix x
+      assume "x \<in> {c<..<d}" "x \<in> {e<..<f}"
+      then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
+        unfolding mem_interval using i'
+        apply -
+        apply (erule_tac[!] x=i in ballE)+
+        apply auto
+        done
+      show False
+        using s(2)[OF i']
+        apply -
+        apply (erule_tac disjE)
+        apply (erule_tac[!] conjE)
+      proof -
+        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
+        show False
+          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
+      next
+        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
+        show False
+          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
+      qed
+    qed
+  qed
+  also have "\<Union> ?A = {a..b}"
+  proof (rule set_eqI,rule)
+    fix x
+    assume "x \<in> \<Union>?A"
+    then obtain c d where x:
+      "x \<in> {c..d}"
+      "\<And>i. i \<in> Basis \<Longrightarrow>
+        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
+    show "x\<in>{a..b}"
+      unfolding mem_interval
+    proof safe
+      fix i :: 'a
+      assume i: "i \<in> Basis"
+      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
+        using x(2)[OF i] x(1)[unfolded mem_interval,THEN bspec, OF i] by auto
+    qed
+  next
+    fix x
+    assume x: "x \<in> {a..b}"
+    have "\<forall>i\<in>Basis.
+      \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
+      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
+      unfolding mem_interval
     proof
-      fix i :: 'a assume i: "i \<in> Basis"
+      fix i :: 'a
+      assume i: "i \<in> Basis"
       have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
-        using x[unfolded mem_interval,THEN bspec, OF i] by auto thus "\<exists>c d. ?P i c d" by blast
+        using x[unfolded mem_interval,THEN bspec, OF i] by auto
+      then show "\<exists>c d. ?P i c d"
+        by blast
     qed
-    thus "x\<in>\<Union>?A"
+    then show "x\<in>\<Union>?A"
       unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
-      apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
-  qed finally show False using assms by auto qed
-
-lemma interval_bisection: fixes type::"'a::ordered_euclidean_space"
-  assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::'a}"
-  obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
-proof-
+      apply -
+      apply (erule exE)+
+      apply (rule_tac x="{xa..xaa}" in exI)
+      unfolding mem_interval
+      apply auto
+      done
+  qed
+  finally show False
+    using assms by auto
+qed
+
+lemma interval_bisection:
+  fixes type :: "'a::ordered_euclidean_space"
+  assumes "P {}"
+    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
+    and "\<not> P {a..b::'a}"
+  obtains x where "x \<in> {a..b}"
+    and "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
+proof -
   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
     (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
-                           2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" proof case goal1 thus ?case proof-
+       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
+  proof
+    case goal1
+    then show ?case
+    proof -
       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
-      thus ?thesis apply(cases "P {fst x..snd x}") by auto
-    next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
-      thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
-    qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
-  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
+      then show ?thesis by (cases "P {fst x..snd x}") auto
+    next
+      assume as: "\<not> P {fst x..snd x}"
+      obtain c d where "\<not> P {c..d}"
+        "\<forall>i\<in>Basis.
+           fst x \<bullet> i \<le> c \<bullet> i \<and>
+           c \<bullet> i \<le> d \<bullet> i \<and>
+           d \<bullet> i \<le> snd x \<bullet> i \<and>
+           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
+        by (rule interval_bisection_step[of P, OF assms(1-2) as])
+      then show ?thesis
+        apply -
+        apply (rule_tac x="(c,d)" in exI)
+        apply auto
+        done
+    qed
+  qed
+  then guess f
+    apply -
+    apply (drule choice)
+    apply (erule exE)
+    done
+  note f = this
+  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
+  def A \<equiv> "\<lambda>n. fst(AB n)"
+  def B \<equiv> "\<lambda>n. snd(AB n)"
+  note ab_def = A_def B_def AB_def
   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
-    (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and> 
+    (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and>
     2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
-  proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
-    case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
-    proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
-    next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
-    qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
-
-  have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
-  proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] .. note n=this
-    show ?case apply(rule_tac x=n in exI) proof(rule,rule)
-      fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
-      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis" unfolding dist_norm by(rule norm_le_l1)
+  proof -
+    show "A 0 = a" "B 0 = b"
+      unfolding ab_def by auto
+    case goal3
+    note S = ab_def funpow.simps o_def id_apply
+    show ?case
+    proof (induct n)
+      case 0
+      then show ?case
+        unfolding S
+        apply (rule f[rule_format]) using assms(3)
+        apply auto
+        done
+    next
+      case (Suc n)
+      show ?case
+        unfolding S
+        apply (rule f[rule_format])
+        using Suc
+        unfolding S
+        apply auto
+        done
+    qed
+  qed
+  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
+
+  have interv: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
+  proof -
+    case goal1
+    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
+      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
+    show ?case
+      apply (rule_tac x=n in exI)
+      apply rule
+      apply rule
+    proof -
+      fix x y
+      assume xy: "x\<in>{A n..B n}" "y\<in>{A n..B n}"
+      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
+        unfolding dist_norm by(rule norm_le_l1)
       also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
-      proof(rule setsum_mono)
-        fix i :: 'a assume i: "i \<in> Basis" show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
-          using xy[unfolded mem_interval,THEN bspec, OF i] by (auto simp: inner_diff_left) qed
-      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n" unfolding setsum_divide_distrib
-      proof(rule setsum_mono) case goal1 thus ?case
-        proof(induct n) case 0 thus ?case unfolding AB by auto
-        next case (Suc n) have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
+      proof (rule setsum_mono)
+        fix i :: 'a
+        assume i: "i \<in> Basis"
+        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
+          using xy[unfolded mem_interval,THEN bspec, OF i]
+          by (auto simp: inner_diff_left)
+      qed
+      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
+        unfolding setsum_divide_distrib
+      proof (rule setsum_mono)
+        case goal1
+        then show ?case
+        proof (induct n)
+          case 0
+          then show ?case
+            unfolding AB by auto
+        next
+          case (Suc n)
+          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
             using AB(4)[of i n] using goal1 by auto
-          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
-        qed qed
-      also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
-    qed qed
-  { fix n m :: nat assume "m \<le> n" then have "{A n..B n} \<subseteq> {A m..B m}"
-    proof(induct rule: inc_induct)
-      case (step i) show ?case
+          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
+            using Suc by (auto simp add:field_simps)
+          finally show ?case .
+        qed
+      qed
+      also have "\<dots> < e"
+        using n using goal1 by (auto simp add:field_simps)
+      finally show "dist x y < e" .
+    qed
+  qed
+  {
+    fix n m :: nat
+    assume "m \<le> n"
+    then have "{A n..B n} \<subseteq> {A m..B m}"
+    proof (induct rule: inc_induct)
+      case (step i)
+      show ?case
         using AB(4) by (intro order_trans[OF step(2)] subset_interval_imp) auto
-    qed simp } note ABsubset = this 
-  have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
-  proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
-  then guess x0 .. note x0=this[rule_format]
-  show thesis proof(rule that[rule_format,of x0])
-    show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
-    fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
+    qed simp
+  } note ABsubset = this
+  have "\<exists>a. \<forall>n. a\<in>{A n..B n}"
+    apply (rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
+  proof -
+    fix n
+    show "{A n..B n} \<noteq> {}"
+      apply (cases "0 < n")
+      using AB(3)[of "n - 1"] assms(1,3) AB(1-2)
+      apply auto
+      done
+  qed auto
+  then obtain x0 where x0: "\<And>n. x0 \<in> {A n..B n}"
+    by blast
+  show thesis
+  proof (rule that[rule_format, of x0])
+    show "x0\<in>{a..b}"
+      using x0[of 0] unfolding AB .
+    fix e :: real
+    assume "e > 0"
+    from interv[OF this] obtain n
+      where n: "\<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e" ..
     show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
-      apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
-    proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto
-      show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
-      show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
-    qed qed qed 
+      apply (rule_tac x="A n" in exI)
+      apply (rule_tac x="B n" in exI)
+      apply rule
+      apply (rule x0)
+      apply rule
+      defer
+      apply rule
+    proof -
+      show "\<not> P {A n..B n}"
+        apply (cases "0 < n")
+        using AB(3)[of "n - 1"] assms(3) AB(1-2)
+        apply auto
+        done
+      show "{A n..B n} \<subseteq> ball x0 e"
+        using n using x0[of n] by auto
+      show "{A n..B n} \<subseteq> {a..b}"
+        unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
+    qed
+  qed
+qed
+
 
 subsection {* Cousin's lemma. *}
 
-lemma fine_division_exists: assumes "gauge g" 
-  obtains p where "p tagged_division_of {a..b::'a::ordered_euclidean_space}" "g fine p"
-proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
-  then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
-next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
-  guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
-    apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
-  proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
-    fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"
-    thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule
-      apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
-  qed note x=this
-  obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
-  from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
-  have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
-  thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
+lemma fine_division_exists:
+  fixes a b :: "'a::ordered_euclidean_space"
+  assumes "gauge g"
+  obtains p where "p tagged_division_of {a..b}" "g fine p"
+proof -
+  presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
+  then obtain p where "p tagged_division_of {a..b}" "g fine p"
+    by blast
+  then show thesis ..
+next
+  assume as: "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
+  guess x
+    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
+    apply (rule_tac x="{}" in exI)
+    defer
+    apply (erule conjE exE)+
+  proof -
+    show "{} tagged_division_of {} \<and> g fine {}"
+      unfolding fine_def by auto
+    fix s t p p'
+    assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
+      "interior s \<inter> interior t = {}"
+    then show "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p"
+      apply -
+      apply (rule_tac x="p \<union> p'" in exI)
+      apply rule
+      apply (rule tagged_division_union)
+      prefer 4
+      apply (rule fine_union)
+      apply auto
+      done
+  qed note x = this
+  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
+    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
+  from x(2)[OF e(1)] obtain c d where c_d:
+    "x \<in> {c..d}"
+    "{c..d} \<subseteq> ball x e"
+    "{c..d} \<subseteq> {a..b}"
+    "\<not> (\<exists>p. p tagged_division_of {c..d} \<and> g fine p)"
+    by blast
+  have "g fine {(x, {c..d})}"
+    unfolding fine_def using e using c_d(2) by auto
+  then show False
+    using tagged_division_of_self[OF c_d(1)] using c_d by auto
+qed
+
 
 subsection {* Basic theorems about integrals. *}
 
-lemma has_integral_unique: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
-  assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
-proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
-  have lem:"\<And>f::'n \<Rightarrow> 'a.  \<And> a b k1 k2.
+lemma has_integral_unique:
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+  assumes "(f has_integral k1) i"
+    and "(f has_integral k2) i"
+  shows "k1 = k2"
+proof (rule ccontr)
+  let ?e = "norm(k1 - k2) / 2"
+  assume as:"k1 \<noteq> k2"
+  then have e: "?e > 0"
+    by auto
+  have lem: "\<And>f::'n \<Rightarrow> 'a.  \<And>a b k1 k2.
     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
-  proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
-    guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
-    guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
-    guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
-    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
-      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
+  proof -
+    case goal1
+    let ?e = "norm (k1 - k2) / 2"
+    from goal1(3) have e: "?e > 0" by auto
+    guess d1 by (rule has_integralD[OF goal1(1) e]) note d1=this
+    guess d2 by (rule has_integralD[OF goal1(2) e]) note d2=this
+    guess p by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
+    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
+    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
+      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
+      by (auto simp add:algebra_simps norm_minus_commute)
     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
-      apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
+      apply (rule add_strict_mono)
+      apply (rule_tac[!] d2(2) d1(2))
+      using p unfolding fine_def
+      apply auto
+      done
     finally show False by auto
-  qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
-    thus False apply-apply(cases "\<exists>a b. i = {a..b}")
-      using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
-  assume as:"\<not> (\<exists>a b. i = {a..b})"
-  guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
-  guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
-  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
-    using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
-  note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
-  guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
-  guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
-  have "z = w" using lem[OF w(1) z(1)] by auto
-  hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
-    using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
-  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
-  finally show False by auto qed
-
-lemma integral_unique[intro]:
-  "(f has_integral y) k \<Longrightarrow> integral k f = y"
-  unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
-
-lemma has_integral_is_0: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector" 
-  assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
-proof- have lem:"\<And>a b. \<And>f::'n \<Rightarrow> 'a.
-    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
-  proof(rule,rule) fix a b e and f::"'n \<Rightarrow> 'a"
-    assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
-    show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
-      apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
-    proof(rule,rule,erule conjE) case goal1
-      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
-        fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
-        thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
-      qed thus ?case using as by auto
-    qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
-    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
-      using assms by(auto simp add:has_integral intro:lem) }
-  have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
-  assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
-  apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
-  proof- fix e::real and a b assume "e>0"
-    thus "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
-      apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
-  qed auto qed
+  qed
+  {
+    presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
+    then show False
+      apply -
+      apply (cases "\<exists>a b. i = {a..b}")
+      using assms
+      apply (auto simp add:has_integral intro:lem[OF _ _ as])
+      done
+  }
+  assume as: "\<not> (\<exists>a b. i = {a..b})"
+  guess B1 by (rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
+  guess B2 by (rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
+  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}"
+    apply (rule bounded_subset_closed_interval)
+    using bounded_Un bounded_ball
+    apply auto
+    done
+  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> {a..b}" "ball 0 B2 \<subseteq> {a..b}"
+    by blast
+  obtain w where w:
+    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) {a..b}"
+    "norm (w - k1) < norm (k1 - k2) / 2"
+    using B1(2)[OF ab(1)] by blast
+  obtain z where z:
+    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) {a..b}"
+    "norm (z - k2) < norm (k1 - k2) / 2"
+    using B2(2)[OF ab(2)] by blast
+  have "z = w"
+    using lem[OF w(1) z(1)] by auto
+  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
+    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
+    by (auto simp add: norm_minus_commute)
+  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
+    apply (rule add_strict_mono)
+    apply (rule_tac[!] z(2) w(2))
+    done
+  finally show False by auto
+qed
+
+lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
+  unfolding integral_def
+  by (rule some_equality) (auto intro: has_integral_unique)
+
+lemma has_integral_is_0:
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+  assumes "\<forall>x\<in>s. f x = 0"
+  shows "(f has_integral 0) s"
+proof -
+  have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
+    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
+    unfolding has_integral
+    apply rule
+    apply rule
+  proof -
+    fix a b e
+    fix f :: "'n \<Rightarrow> 'a"
+    assume as: "\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
+    show "\<exists>d. gauge d \<and>
+      (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
+      apply (rule_tac x="\<lambda>x. ball x 1" in exI)
+      apply rule
+      apply (rule gaugeI)
+      unfolding centre_in_ball
+      defer
+      apply (rule open_ball)
+      apply rule
+      apply rule
+      apply (erule conjE)
+    proof -
+      case goal1
+      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
+      proof (rule setsum_0', rule)
+        fix x
+        assume x: "x \<in> p"
+        have "f (fst x) = 0"
+          using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
+        then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
+          apply (subst surjective_pairing[of x])
+          unfolding split_conv
+          apply auto
+          done
+      qed
+      then show ?case
+        using as by auto
+    qed auto
+  qed
+  {
+    presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+    then show ?thesis
+      apply -
+      apply (cases "\<exists>a b. s = {a..b}")
+      using assms
+      apply (auto simp add:has_integral intro: lem)
+      done
+  }
+  have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
+    apply (rule ext)
+    using assms
+    apply auto
+    done
+  assume "\<not> (\<exists>a b. s = {a..b})"
+  then show ?thesis
+    apply (subst has_integral_alt)
+    unfolding if_not_P *
+    apply rule
+    apply rule
+    apply (rule_tac x=1 in exI)
+    apply rule
+    defer
+    apply rule
+    apply rule
+    apply rule
+  proof -
+    fix e :: real
+    fix a b
+    assume "e > 0"
+    then show "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
+      apply (rule_tac x=0 in exI)
+      apply(rule,rule lem)
+      apply auto
+      done
+  qed auto
+qed
 
 lemma has_integral_0[simp]: "((\<lambda>x::'n::ordered_euclidean_space. 0) has_integral 0) s"
-  apply(rule has_integral_is_0) by auto 
+  by (rule has_integral_is_0) auto
 
 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
   using has_integral_unique[OF has_integral_0] by auto
 
-lemma has_integral_linear: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
-  assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
-proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
-  have lem:"\<And>f::'n \<Rightarrow> 'a. \<And> y a b.
-    (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
-  proof(subst has_integral,rule,rule) case goal1
-    from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
-    have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
-    guess g using has_integralD[OF goal1(1) *] . note g=this
-    show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
-    proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
-      have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto
+lemma has_integral_linear:
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+  assumes "(f has_integral y) s"
+    and "bounded_linear h"
+  shows "((h o f) has_integral ((h y))) s"
+proof -
+  interpret bounded_linear h
+    using assms(2) .
+  from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
+    by blast
+  have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
+    (f has_integral y) {a..b} \<Longrightarrow> ((h o f) has_integral h y) {a..b}"
+    apply (subst has_integral)
+    apply rule
+    apply rule
+  proof -
+    case goal1
+    from pos_bounded
+    obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
+      by blast
+    have *: "e / B > 0"
+      apply (rule divide_pos_pos)
+      using goal1(2) B
+      apply auto
+      done
+      thm has_integralD[OF goal1(1) *]
+    obtain g where g:
+      "gauge g"
+      "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> g fine p \<Longrightarrow>
+        norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
+      by (rule has_integralD[OF goal1(1) *]) blast
+    show ?case
+      apply (rule_tac x=g in exI)
+      apply rule
+      apply (rule g(1))
+      apply rule
+      apply rule
+      apply (erule conjE)
+    proof -
+      fix p
+      assume as: "p tagged_division_of {a..b}" "g fine p"
+      have *: "\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x"
+        by auto
       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
-        unfolding o_def unfolding scaleR[THEN sym] * by simp
-      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
-      finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
-      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
-        apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
-    qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
-    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
-  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
-  proof(rule,rule) fix e::real  assume e:"0<e"
-    have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
-    guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
-    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
-      apply(rule_tac x=M in exI) apply(rule,rule M(1))
-    proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
-      have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
-        unfolding o_def apply(rule ext) using zero by auto
-      show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
-        apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
-    qed qed qed
-
-lemma has_integral_cmul:
-  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
-  unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
-  by(rule bounded_linear_scaleR_right)
+        unfolding o_def unfolding scaleR[symmetric] * by simp
+      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
+        using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
+      finally have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
+      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
+        unfolding * diff[symmetric]
+        apply (rule le_less_trans[OF B(2)])
+        using g(2)[OF as] B(1)
+        apply (auto simp add: field_simps)
+        done
+    qed
+  qed
+  {
+    presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+    then show ?thesis
+      apply -
+      apply (cases "\<exists>a b. s = {a..b}")
+      using assms
+      apply (auto simp add:has_integral intro!:lem)
+      done
+  }
+  assume as: "\<not> (\<exists>a b. s = {a..b})"
+  then show ?thesis
+    apply (subst has_integral_alt)
+    unfolding if_not_P
+    apply rule
+    apply rule
+  proof -
+    fix e :: real
+    assume e: "e > 0"
+    have *: "0 < e/B"
+      by (rule divide_pos_pos,rule e,rule B(1))
+    obtain M where M:
+      "M > 0"
+      "\<And>a b. ball 0 M \<subseteq> {a..b} \<Longrightarrow>
+        \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) {a..b} \<and> norm (z - y) < e / B"
+      using has_integral_altD[OF assms(1) as *] by blast
+    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+      (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
+      apply (rule_tac x=M in exI)
+      apply rule
+      apply (rule M(1))
+      apply rule
+      apply rule
+      apply rule
+    proof -
+      case goal1
+      obtain z where z:
+        "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) {a..b}"
+        "norm (z - y) < e / B"
+        using M(2)[OF goal1(1)] by blast
+      have *: "(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
+        unfolding o_def
+        apply (rule ext)
+        using zero
+        apply auto
+        done
+      show ?case
+        apply (rule_tac x="h z" in exI)
+        apply rule
+        unfolding *
+        apply (rule lem[OF z(1)])
+        unfolding diff[symmetric]
+        apply (rule le_less_trans[OF B(2)])
+        using B(1) z(2)
+        apply (auto simp add: field_simps)
+        done
+    qed
+  qed
+qed
+
+lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
+  unfolding o_def[symmetric]
+  apply (rule has_integral_linear,assumption)
+  apply (rule bounded_linear_scaleR_right)
+  done
 
 lemma has_integral_cmult_real:
   fixes c :: real
   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
   shows "((\<lambda>x. c * f x) has_integral c * x) A"
-proof cases
-  assume "c \<noteq> 0"
+proof (cases "c = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
     unfolding real_scaleR_def .
-qed simp
-
-lemma has_integral_neg:
-  shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
-  apply(drule_tac c="-1" in has_integral_cmul) by auto
-
-lemma has_integral_add: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector" 
-  assumes "(f has_integral k) s" "(g has_integral l) s"
+qed
+
+lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
+  apply (drule_tac c="-1" in has_integral_cmul)
+  apply auto
+  done
+
+lemma has_integral_add:
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+  assumes "(f has_integral k) s"
+    and "(g has_integral l) s"
   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
-proof- have lem:"\<And>f g::'n \<Rightarrow> 'a. \<And>a b k l.
-    (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
-     ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
-    show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
-      guess d1 using has_integralD[OF goal1(1) *] . note d1=this
-      guess d2 using has_integralD[OF goal1(2) *] . note d2=this
-      show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
-        apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
-      proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
-        have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
-          unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
-          by(rule setsum_cong2,auto)
-        have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
-          unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
-        from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
-        have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
-          apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
-        finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
-      qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
-    thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
-  assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
-  proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
+proof -
+  have lem:"\<And>(f:: 'n \<Rightarrow> 'a) g a b k l.
+    (f has_integral k) {a..b} \<Longrightarrow>
+    (g has_integral l) {a..b} \<Longrightarrow>
+    ((\<lambda>x. f x + g x) has_integral (k + l)) {a..b}"
+  proof -
+    case goal1
+    show ?case
+      unfolding has_integral
+      apply rule
+      apply rule
+    proof -
+      fix e :: real
+      assume e: "e > 0"
+      then have *: "e/2 > 0"
+        by auto
+      obtain d1 where d1:
+        "gauge d1"
+        "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d1 fine p \<Longrightarrow>
+          norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) < e / 2"
+        using has_integralD[OF goal1(1) *] by blast
+      obtain d2 where d2:
+        "gauge d2"
+        "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d2 fine p \<Longrightarrow>
+          norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l) < e / 2"
+        using has_integralD[OF goal1(2) *] by blast
+      show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
+        norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
+        apply (rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI)
+        apply rule
+        apply (rule gauge_inter[OF d1(1) d2(1)])
+        apply (rule,rule,erule conjE)
+      proof -
+        fix p
+        assume as: "p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
+        have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) =
+          (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
+          unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,symmetric]
+          by (rule setsum_cong2) auto
+        have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) =
+          norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
+          unfolding * by (auto simp add: algebra_simps)
+        also
+        let ?res = "\<dots>"
+        from as have *: "d1 fine p" "d2 fine p"
+          unfolding fine_inter by auto
+        have "?res < e/2 + e/2"
+          apply (rule le_less_trans[OF norm_triangle_ineq])
+          apply (rule add_strict_mono)
+          using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)]
+          apply auto
+          done
+        finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e"
+          by auto
+      qed
+    qed
+  qed
+  {
+    presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+    then show ?thesis
+      apply -
+      apply (cases "\<exists>a b. s = {a..b}")
+      using assms
+      apply (auto simp add:has_integral intro!:lem)
+      done
+  }
+  assume as: "\<not> (\<exists>a b. s = {a..b})"
+  then show ?thesis
+    apply (subst has_integral_alt)
+    unfolding if_not_P
+    apply rule
+    apply rule
+  proof -
+    case goal1
+    then have *: "e/2 > 0"
+      by auto
     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
-    show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
-    proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
-      hence *:"ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}" by auto
-      guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
-      guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
-      have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto
+    show ?case
+      apply (rule_tac x="max B1 B2" in exI)
+      apply rule
+      apply (rule min_max.less_supI1)
+      apply (rule B1)
+      apply rule
+      apply rule
+      apply rule
+    proof -
+      fix a b
+      assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
+      then have *: "ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}"
+        by auto
+      obtain w where w:
+        "((\<lambda>x. if x \<in> s then f x else 0) has_integral w) {a..b}"
+        "norm (w - k) < e / 2"
+        using B1(2)[OF *(1)] by blast
+      obtain z where z:
+        "((\<lambda>x. if x \<in> s then g x else 0) has_integral z) {a..b}"
+        "norm (z - l) < e / 2"
+        using B2(2)[OF *(2)] by blast
+      have *: "\<And>x. (if x \<in> s then f x + g x else 0) =
+        (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)"
+        by auto
       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"
-        apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
-        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
-    qed qed qed
+        apply (rule_tac x="w + z" in exI)
+        apply rule
+        apply (rule lem[OF w(1) z(1), unfolded *[symmetric]])
+        using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
+        apply (auto simp add: field_simps)
+        done
+    qed
+  qed
+qed
 
 lemma has_integral_sub:
-  shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
-  using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
-
-lemma integral_0: "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
-  by(rule integral_unique has_integral_0)+
-
-lemma integral_add:
-  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
-   integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
-  apply(rule integral_unique) apply(drule integrable_integral)+
-  apply(rule has_integral_add) by assumption+
-
-lemma integral_cmul:
-  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
-  apply(rule integral_unique) apply(drule integrable_integral)+
-  apply(rule has_integral_cmul) by assumption+
-
-lemma integral_neg:
-  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
-  apply(rule integral_unique) apply(drule integrable_integral)+
-  apply(rule has_integral_neg) by assumption+
-
-lemma integral_sub:
-  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
-  apply(rule integral_unique) apply(drule integrable_integral)+
-  apply(rule has_integral_sub) by assumption+
+  "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow>
+    ((\<lambda>x. f x - g x) has_integral (k - l)) s"
+  using has_integral_add[OF _ has_integral_neg, of f k s g l]
+  unfolding algebra_simps
+  by auto
+
+lemma integral_0:
+  "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
+  by (rule integral_unique has_integral_0)+
+
+lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
+    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
+  apply (rule integral_unique)
+  apply (drule integrable_integral)+
+  apply (rule has_integral_add)
+  apply assumption+
+  done
+
+lemma integral_cmul: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
+  apply (rule integral_unique)
+  apply (drule integrable_integral)+
+  apply (rule has_integral_cmul)
+  apply assumption+
+  done
+
+lemma integral_neg: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
+  apply (rule integral_unique)
+  apply (drule integrable_integral)+
+  apply (rule has_integral_neg)
+  apply assumption+
+  done
+
+lemma integral_sub: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
+    integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
+  apply (rule integral_unique)
+  apply (drule integrable_integral)+
+  apply (rule has_integral_sub)
+  apply assumption+
+  done
 
 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
   unfolding integrable_on_def using has_integral_0 by auto
 
-lemma integrable_add:
-  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
+lemma integrable_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
   unfolding integrable_on_def by(auto intro: has_integral_add)
 
-lemma integrable_cmul:
-  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
+lemma integrable_cmul: "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
   unfolding integrable_on_def by(auto intro: has_integral_cmul)
 
 lemma integrable_on_cmult_iff:
-  fixes c :: real assumes "c \<noteq> 0"
+  fixes c :: real
+  assumes "c \<noteq> 0"
   shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
   using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] `c \<noteq> 0`
   by auto
 
-lemma integrable_neg:
-  shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
+lemma integrable_neg: "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
   unfolding integrable_on_def by(auto intro: has_integral_neg)
 
 lemma integrable_sub:
-  shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
+  "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
   unfolding integrable_on_def by(auto intro: has_integral_sub)
 
 lemma integrable_linear:
-  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
+  "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on s"
   unfolding integrable_on_def by(auto intro: has_integral_linear)
 
 lemma integral_linear:
-  shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
-  apply(rule has_integral_unique) defer unfolding has_integral_integral 
-  apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
-  apply(rule integrable_linear) by assumption+
-
-lemma integral_component_eq[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
-  assumes "f integrable_on s" shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
+  "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
+  apply (rule has_integral_unique)
+  defer
+  unfolding has_integral_integral
+  apply (drule (2) has_integral_linear)
+  unfolding has_integral_integral[symmetric]
+  apply (rule integrable_linear)
+  apply assumption+
+  done
+
+lemma integral_component_eq[simp]:
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
+  assumes "f integrable_on s"
+  shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
   unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
 
 lemma has_integral_setsum:
-  assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
+  assumes "finite t"
+    and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
-proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
-  case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
-    apply(rule has_integral_add) using insert assms by auto
-qed auto
-
-lemma integral_setsum:
-  shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
+  using assms(1) subset_refl[of t]
+proof (induct rule: finite_subset_induct)
+  case empty
+  then show ?case by auto
+next
+  case (insert x F)
+  show ?case
+    unfolding setsum_insert[OF insert(1,3)]
+    apply (rule has_integral_add)
+    using insert assms
+    apply auto
+    done
+qed
+
+lemma integral_setsum: "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
-  apply(rule integral_unique) apply(rule has_integral_setsum)
-  using integrable_integral by auto
+  apply (rule integral_unique)
+  apply (rule has_integral_setsum)
+  using integrable_integral
+  apply auto
+  done
 
 lemma integrable_setsum:
-  shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
-  unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
+  "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
+  unfolding integrable_on_def
+  apply (drule bchoice)
+  using has_integral_setsum[of t]
+  apply auto
+  done
 
 lemma has_integral_eq:
-  assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
+  assumes "\<forall>x\<in>s. f x = g x"
+    and "(f has_integral k) s"
+  shows "(g has_integral k) s"
   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
-  using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
-
-lemma integrable_eq:
-  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
-  unfolding integrable_on_def using has_integral_eq[of s f g] by auto
-
-lemma has_integral_eq_eq:
-  shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
-  using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
+  using has_integral_is_0[of s "\<lambda>x. f x - g x"]
+  using assms(1)
+  by auto
+
+lemma integrable_eq: "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
+  unfolding integrable_on_def
+  using has_integral_eq[of s f g]
+  by auto
+
+lemma has_integral_eq_eq: "\<forall>x\<in>s. f x = g x \<Longrightarrow> (f has_integral k) s \<longleftrightarrow> (g has_integral k) s"
+  using has_integral_eq[of s f g] has_integral_eq[of s g f]
+  by auto
 
 lemma has_integral_null[dest]:
-  assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
-  unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
-proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
-  fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
-  have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
-    using setsum_content_null[OF assms(1) p, of f] . 
-  thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
-
-lemma has_integral_null_eq[simp]:
-  shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
-  apply rule apply(rule has_integral_unique,assumption) 
-  apply(drule has_integral_null,assumption)
-  apply(drule has_integral_null) by auto
-
-lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
-  by(rule integral_unique,drule has_integral_null)
-
-lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
-  unfolding integrable_on_def apply(drule has_integral_null) by auto
-
-lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
-  unfolding empty_as_interval apply(rule has_integral_null) 
-  using content_empty unfolding empty_as_interval .
-
-lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
-  apply(rule,rule has_integral_unique,assumption) by auto
-
-lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
-
-lemma integral_empty[simp]: shows "integral {} f = 0"
-  apply(rule integral_unique) using has_integral_empty .
-
-lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a::'a::ordered_euclidean_space}"
-proof-
-  have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq_iff[where 'a='a]
-    apply safe prefer 3 apply(erule_tac x=b in ballE) by(auto simp add: field_simps)
-  show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
-    apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
-    unfolding interior_closed_interval using interval_sing by auto qed
-
-lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
-
-lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
+  assumes "content({a..b}) = 0"
+  shows "(f has_integral 0) ({a..b})"
+  unfolding has_integral
+  apply rule
+  apply rule
+  apply (rule_tac x="\<lambda>x. ball x 1" in exI)
+  apply rule
+  defer
+  apply rule
+  apply rule
+  apply (erule conjE)
+proof -
+  fix e :: real
+  assume e: "e > 0"
+  then show "gauge (\<lambda>x. ball x 1)"
+    by auto
+  fix p
+  assume p: "p tagged_division_of {a..b}"
+  have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0"
+    unfolding norm_eq_zero diff_0_right
+    using setsum_content_null[OF assms(1) p, of f] .
+  then show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
+    using e by auto
+qed
+
+lemma has_integral_null_eq[simp]: "content {a..b} = 0 \<Longrightarrow> (f has_integral i) {a..b} \<longleftrightarrow> i = 0"
+  apply rule
+  apply (rule has_integral_unique)
+  apply assumption
+  apply (drule (1) has_integral_null)
+  apply (drule has_integral_null)
+  apply auto
+  done
+
+lemma integral_null[dest]: "content {a..b} = 0 \<Longrightarrow> integral {a..b} f = 0"
+  apply (rule integral_unique)
+  apply (drule has_integral_null)
+  apply assumption
+  done
+
+lemma integrable_on_null[dest]: "content {a..b} = 0 \<Longrightarrow> f integrable_on {a..b}"
+  unfolding integrable_on_def
+  apply (drule has_integral_null)
+  apply auto
+  done
+
+lemma has_integral_empty[intro]: "(f has_integral 0) {}"
+  unfolding empty_as_interval
+  apply (rule has_integral_null)
+  using content_empty
+  unfolding empty_as_interval
+  apply assumption
+  done
+
+lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
+  apply rule
+  apply (rule has_integral_unique)
+  apply assumption
+  apply auto
+  done
+
+lemma integrable_on_empty[intro]: "f integrable_on {}"
+  unfolding integrable_on_def by auto
+
+lemma integral_empty[simp]: "integral {} f = 0"
+  by (rule integral_unique) (rule has_integral_empty)
+
+lemma has_integral_refl[intro]:
+  fixes a :: "'a::ordered_euclidean_space"
+  shows "(f has_integral 0) {a..a}"
+    and "(f has_integral 0) {a}"
+proof -
+  have *: "{a} = {a..a}"
+    apply (rule set_eqI)
+    unfolding mem_interval singleton_iff euclidean_eq_iff[where 'a='a]
+    apply safe
+    prefer 3
+    apply (erule_tac x=b in ballE)
+    apply (auto simp add: field_simps)
+    done
+  show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}"
+    unfolding *
+    apply (rule_tac[!] has_integral_null)
+    unfolding content_eq_0_interior
+    unfolding interior_closed_interval
+    using interval_sing
+    apply auto
+    done
+qed
+
+lemma integrable_on_refl[intro]: "f integrable_on {a..a}"
+  unfolding integrable_on_def by auto
+
+lemma integral_refl: "integral {a..a} f = 0"
+  by (rule integral_unique) auto
+
 
 subsection {* Cauchy-type criterion for integrability. *}
 
 (* XXXXXXX *)
-lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
+lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
   shows "f integrable_on {a..b} \<longleftrightarrow>
   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
                             p2 tagged_division_of {a..b} \<and> d fine p2
@@ -1985,15 +3155,15 @@
     proof(rule,rule,rule,rule) fix m n assume mn:"N \<le> m" "N \<le> n" have *:"N = (N - 1) + 1" using N by auto
       show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
-        using dp p(1) using mn by auto 
+        using dp p(1) using mn by auto
     qed qed
-  then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[THEN LIMSEQ_D]
+  then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
     guess N2 using y[OF *] .. note N2=this
     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
-      apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
+      apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
@@ -2019,12 +3189,12 @@
   have *:"Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
     using assms by auto
   have *:"\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
-    "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)" 
+    "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
   assume as:"a\<le>b" moreover have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c
     \<Longrightarrow> x* (b\<bullet>k - a\<bullet>k) = x*(max (a \<bullet> k) c - a \<bullet> k) + x*(b \<bullet> k - max (a \<bullet> k) c)"
     by  (auto simp add:field_simps)
-  moreover have **:"(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) = 
+  moreover have **:"(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
     "(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
@@ -2041,7 +3211,7 @@
 qed
 
 lemma division_split_left_inj: fixes type::"'a::ordered_euclidean_space"
-  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2" 
+  assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
   "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"and k:"k\<in>Basis"
   shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof- note d=division_ofD[OF assms(1)]
@@ -2052,7 +3222,7 @@
   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
- 
+
 lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
   "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" and k:"k\<in>Basis"
@@ -2067,7 +3237,7 @@
     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
 
 lemma tagged_division_split_left_inj: fixes x1::"'a::ordered_euclidean_space"
-  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}" 
+  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
   and k:"k\<in>Basis"
   shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
@@ -2075,7 +3245,7 @@
     apply(rule_tac[1-2] *) using assms(2-) by auto qed
 
 lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
-  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" 
+  assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
   and k:"k\<in>Basis"
   shows "content(k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
@@ -2084,10 +3254,10 @@
 
 lemma division_split: fixes a::"'a::ordered_euclidean_space"
   assumes "p division_of {a..b}" and k:"k\<in>Basis"
-  shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})} division_of({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is "?p1 division_of ?I1") and 
+  shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})} division_of({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is "?p1 division_of ?I1") and
         "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is "?p2 division_of ?I2")
 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms(1)]
-  show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
+  show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[symmetric] by auto
   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
@@ -2106,8 +3276,8 @@
   assumes "(f has_integral i) ({a..b} \<inter> {x. x\<bullet>k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" and k:"k\<in>Basis"
   shows "(f has_integral (i + j)) ({a..b})"
 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
-  guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[THEN sym,OF k]]
-  guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[THEN sym,OF k]]
+  guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[symmetric,OF k]]
+  guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[symmetric,OF k]]
   let ?d = "\<lambda>x. if x\<bullet>k = c then (d1 x \<inter> d2 x) else ball x (abs(x\<bullet>k - c)) \<inter> d1 x \<inter> d2 x"
   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
@@ -2119,7 +3289,7 @@
       proof(rule ccontr) case goal1
         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
-        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<le> c}" using goal1(1) by blast 
+        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<le> c}" using goal1(1) by blast
         then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c" apply-apply(rule le_less_trans)
           using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
@@ -2128,7 +3298,7 @@
       proof(rule ccontr) case goal1
         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
-        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<ge> c}" using goal1(1) by blast 
+        hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<ge> c}" using goal1(1) by blast
         then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c" apply-apply(rule le_less_trans)
           using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
@@ -2153,7 +3323,7 @@
     let ?M1 = "{(x,kk \<inter> {x. x\<bullet>k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
-    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x\<bullet>k \<le> c}" unfolding p(8)[THEN sym] by auto
+    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x\<bullet>k \<le> c}" unfolding p(8)[symmetric] by auto
       fix x l assume xl:"(x,l)\<in>?M1"
       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
@@ -2170,10 +3340,10 @@
         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
       qed qed moreover
 
-    let ?M2 = "{(x,kk \<inter> {x. x\<bullet>k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}" 
+    let ?M2 = "{(x,kk \<inter> {x. x\<bullet>k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
-    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x\<bullet>k \<ge> c}" unfolding p(8)[THEN sym] by auto
+    proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x\<bullet>k \<ge> c}" unfolding p(8)[symmetric] by auto
       fix x l assume xl:"(x,l)\<in>?M2"
       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
@@ -2198,15 +3368,15 @@
       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
-        defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
+        defer unfolding lem4[symmetric] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) using k by auto
       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) using k by auto
-      qed also note setsum_addf[THEN sym]
+      qed also note setsum_addf[symmetric]
       also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x
         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
         thus "content (b \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a = content b *\<^sub>R f a"
-          unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[OF k,of u v c] by auto
+          unfolding scaleR_left_distrib[symmetric] unfolding uv content_split[OF k,of u v c] by auto
       qed note setsum_cong2[OF this]
       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x \<bullet> k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x \<bullet> k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
@@ -2240,7 +3410,7 @@
     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
       have "b \<subseteq> {x. x\<bullet>k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastforce
-      moreover have "interior {x::'a. x \<bullet> k = c} = {}" 
+      moreover have "interior {x::'a. x \<bullet> k = c} = {}"
       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x\<bullet>k = c}" by auto
         then guess e unfolding mem_interior .. note e=this
         have x:"x\<bullet>k = c" using x interior_subset by fastforce
@@ -2248,7 +3418,7 @@
           = (if i = k then e/2 else 0)" using e k by (auto simp: inner_simps inner_not_same_Basis)
         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
           (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
-        also have "... < e" apply(subst setsum_delta) using e by auto 
+        also have "... < e" apply(subst setsum_delta) using e by auto
         finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
         hence "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}" using e by auto
@@ -2262,11 +3432,11 @@
 lemma integrable_split[intro]:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes "f integrable_on {a..b}" and k:"k\<in>Basis"
-  shows "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2) 
+  shows "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
 proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
   def b' \<equiv> "\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i::'a"
   def a' \<equiv> "\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i::'a"
-  show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
+  show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[symmetric,OF k]
   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
@@ -2280,7 +3450,7 @@
         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           using p using assms by(auto simp add:algebra_simps)
-      qed qed  
+      qed qed
     show "?P {x. x \<bullet> k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1
         \<and> p2 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p2"
@@ -2295,7 +3465,7 @@
 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
 
 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
-  "operative opp f \<equiv> 
+  "operative opp f \<equiv>
     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
     (\<forall>a b c. \<forall>k\<in>Basis. f({a..b}) =
                    opp (f({a..b} \<inter> {x. x\<bullet>k \<le> c}))
@@ -2311,7 +3481,7 @@
   unfolding operative_def by auto
 
 lemma property_empty_interval:
- "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
+ "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
   using content_empty unfolding empty_as_interval by auto
 
 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
@@ -2395,10 +3565,10 @@
   unfolding support_def by auto
 
 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
-  unfolding iterate_def fold'_def by auto 
+  unfolding iterate_def fold'_def by auto
 
 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
-  shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
+  shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
   show ?thesis unfolding iterate_def if_P[OF True] * by auto
 next case False note x=this
@@ -2408,7 +3578,7 @@
       unfolding True monoidal_simps[OF assms(1)] by auto
   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
       apply(subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
-      using `finite s` unfolding support_def using False x by auto qed qed 
+      using `finite s` unfolding support_def using False x by auto qed qed
 
 lemma iterate_some:
   assumes "monoidal opp"  "finite s"
@@ -2419,19 +3589,19 @@
 subsection {* Two key instances of additivity. *}
 
 lemma neutral_add[simp]:
-  "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
+  "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
 
-lemma operative_content[intro]: "operative (op +) content" 
-  unfolding operative_def neutral_add apply safe 
-  unfolding content_split[THEN sym] ..
+lemma operative_content[intro]: "operative (op +) content"
+  unfolding operative_def neutral_add apply safe
+  unfolding content_split[symmetric] ..
 
 lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
   by (rule neutral_add) (* FIXME: duplicate *)
 
 lemma monoidal_monoid[intro]:
   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
-  unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps) 
+  unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps)
 
 lemma operative_integral: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
@@ -2442,25 +3612,25 @@
   show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
     lifted op + (if f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c} then Some (integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f) else None)
     (if f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k} then Some (integral ({a..b} \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
-  proof(cases "f integrable_on {a..b}") 
+  proof(cases "f integrable_on {a..b}")
     case True show ?thesis unfolding if_P[OF True] using k apply-
       unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
-      unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k]) 
+      unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k])
       apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
   next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k}))"
     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
         apply(rule_tac x="integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f + integral ({a..b} \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
         apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
       thus False using False by auto
-    qed thus ?thesis using False by auto 
-  qed next 
+    qed thus ?thesis using False by auto
+  qed next
   fix a b assume as:"content {a..b::'a} = 0"
   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
 
 subsection {* Points of division of a partition. *}
 
-definition "division_points (k::('a::ordered_euclidean_space) set) d = 
+definition "division_points (k::('a::ordered_euclidean_space) set) d =
     {(j,x). j\<in>Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
            (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
 
@@ -2502,7 +3672,7 @@
     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
     have *:"\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i"
       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
-    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[symmetric] by auto
     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
       apply (rule bexI[OF _ `l \<in> d`])
       using as(1-3,5) fstx
@@ -2520,12 +3690,12 @@
     apply(erule exE conjE)+
   proof
     fix i l x assume as:"(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
-      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" 
+      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
       "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" and fstx:"fst x \<in> Basis"
     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
     have *:"\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i"
       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
-    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+    have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[symmetric] by auto
     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
       apply (rule bexI[OF _ `l \<in> d`])
       using as(1-3,5) fstx
@@ -2540,9 +3710,9 @@
   assumes "d division_of {a..b}"  "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
   "l \<in> d" "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c" and k:"k\<in>Basis"
   shows "division_points ({a..b} \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}
-              \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
+              \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
         "division_points ({a..b} \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}
-              \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 
+              \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D")
 proof- have ab:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" using assms(2) by(auto intro!:less_imp_le)
   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
   have uv:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i"
@@ -2555,7 +3725,7 @@
   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
     apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
     apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
-    unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
+    unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
 
   have *:"interval_lowerbound ({a..b} \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
@@ -2565,7 +3735,7 @@
   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
     apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
     apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
-    unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
+    unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4) k]) by auto qed
 
 subsection {* Preservation by divisions and tagged divisions. *}
@@ -2578,7 +3748,7 @@
 
 lemma iterate_expand_cases:
   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
-  apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
+  apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
 
 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
@@ -2587,14 +3757,14 @@
   proof- case goal1 show ?case using goal1
     proof(induct s) case empty thus ?case using assms(1) by auto
     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
-        unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
+        unfolding if_not_P[OF insert(2)] apply(subst insert(3)[symmetric])
         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
         apply(rule finite_imageI insert)+ apply(subst if_not_P)
         unfolding image_iff o_def using insert(2,4) by auto
     qed qed
-  show ?thesis 
+  show ?thesis
     apply(cases "finite (support opp g (f ` s))")
-    apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
+    apply(subst (1) iterate_support[symmetric],subst (2) iterate_support[symmetric])
     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
     apply(rule subset_inj_on[OF assms(2) support_subset])+
     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
@@ -2610,16 +3780,16 @@
   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
     unfolding support_def using assms(3) by auto
   show ?thesis unfolding *
-    apply(subst iterate_support[THEN sym]) unfolding support_clauses
+    apply(subst iterate_support[symmetric]) unfolding support_clauses
     apply(subst iterate_image[OF assms(1)]) defer
-    apply(subst(2) iterate_support[THEN sym]) apply(subst **)
+    apply(subst(2) iterate_support[symmetric]) apply(subst **)
     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
 
 lemma iterate_eq_neutral:
   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
   shows "(iterate opp s f = neutral opp)"
 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
-  show ?thesis apply(subst iterate_support[THEN sym]) 
+  show ?thesis apply(subst iterate_support[symmetric])
     unfolding * using assms(1) by auto qed
 
 lemma iterate_op: assumes "monoidal opp" "finite s"
@@ -2637,11 +3807,11 @@
     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
       unfolding * by auto
   next def su \<equiv> "support opp f s"
-    case True note support_subset[of opp f s] 
-    thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
+    case True note support_subset[of opp f s]
+    thus ?thesis apply- apply(subst iterate_support[symmetric],subst(2) iterate_support[symmetric]) unfolding * using True
       unfolding su_def[symmetric]
     proof(induct su) case empty show ?case by auto
-    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
+    next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
         unfolding if_not_P[OF insert(2)] apply(subst insert(3))
         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
 
@@ -2659,11 +3829,11 @@
         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
         proof fix x assume x:"x\<in>d"
           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
-          thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
+          thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
             using operativeD(1)[OF assms(2)] x by auto
         qed qed }
-    assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
-    hence ab':"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" by (auto intro!: less_imp_le) show ?case 
+    assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
+    hence ab':"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" by (auto intro!: less_imp_le) show ?case
     proof(cases "division_points {a..b} d = {}")
       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
         (\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)"
@@ -2677,7 +3847,7 @@
           "(j, v\<bullet>j) \<notin> division_points {a..b} d" using True by auto
         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
-        moreover have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" using division_ofD(2,2,3)[OF goal1(4) as] 
+        moreover have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" using division_ofD(2,2,3)[OF goal1(4) as]
           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
           unfolding interval_ne_empty mem_interval using j by auto
         ultimately show "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j"
@@ -2685,7 +3855,7 @@
       qed
       have "(1/2) *\<^sub>R (a+b) \<in> {a..b}"
         unfolding mem_interval using ab by(auto intro!: less_imp_le simp: inner_simps)
-      note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
+      note this[unfolded division_ofD(6)[OF goal1(4),symmetric] Union_iff]
       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
       have "{a..b} \<in> d"
       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
@@ -2700,12 +3870,12 @@
       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
         then guess u v apply-by(erule exE conjE)+ note uv=this
-        have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
+        have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
         then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j:"j\<in>Basis" unfolding euclidean_eq_iff[where 'a='a] by auto
         hence "u\<bullet>j = v\<bullet>j" using uv(2)[rule_format,OF j] by auto
         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in bexI) using j by auto
         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
-      qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
+      qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
@@ -2723,32 +3893,32 @@
         unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
         using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
-        unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto 
+        unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto
       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<le> c}))"
         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
-        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
-      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y" 
+        unfolding empty_as_interval[symmetric] apply(rule content_empty)
+      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
-        show "f (l \<inter> {x. x \<bullet> k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
-          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
-          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
+        show "f (l \<inter> {x. x \<bullet> k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
+          apply(rule operativeD(1) goal1)+ unfolding interval_split[symmetric,OF kc(4)] apply(rule division_split_left_inj)
+          apply(rule goal1) unfolding l[symmetric] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<ge> c}))"
         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
-        unfolding empty_as_interval[THEN sym] apply(rule content_empty)
-      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y" 
+        unfolding empty_as_interval[symmetric] apply(rule content_empty)
+      proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
-        show "f (l \<inter> {x. x \<bullet> k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
-          apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
-          apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
+        show "f (l \<inter> {x. x \<bullet> k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
+          apply(rule operativeD(1) goal1)+ unfolding interval_split[symmetric,OF kc(4)] apply(rule division_split_right_inj)
+          apply(rule goal1) unfolding l[symmetric] apply(rule as(1),rule as(2)) by(rule as kc(4))+
       qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x \<bullet> k \<le> c})) (f (x \<inter> {x. c \<le> x \<bullet> k}))"
-        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto 
+        unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto
       have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x \<bullet> k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x \<bullet> k})))
         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
-        apply(rule iterate_op[THEN sym]) using goal1 by auto
+        apply(rule iterate_op[symmetric]) using goal1 by auto
       finally show ?thesis by auto
-    qed qed qed 
+    qed qed qed
 
 lemma iterate_image_nonzero: assumes "monoidal opp"
   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
@@ -2763,20 +3933,20 @@
     apply(subst iterate_insert[OF assms(1) goal2(1)])
     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
-    using goal2 unfolding o_def by auto qed 
+    using goal2 unfolding o_def by auto qed
 
 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
-    apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
+    apply(rule iterate_image_nonzero[symmetric,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
-      unfolding as(4)[THEN sym] uv by auto
-  qed also have "\<dots> = f {a..b}" 
+      unfolding as(4)[symmetric] uv by auto
+  qed also have "\<dots> = f {a..b}"
     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
   finally show ?thesis . qed
 
@@ -2794,13 +3964,13 @@
 
 lemma additive_content_division: assumes "d division_of {a..b}"
   shows "setsum content d = content({a..b})"
-  unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
+  unfolding operative_division[OF monoidal_monoid operative_content assms,symmetric]
   apply(subst setsum_iterate) using assms by auto
 
 lemma additive_content_tagged_division:
   assumes "d tagged_division_of {a..b}"
   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
-  unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
+  unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,symmetric]
   apply(subst setsum_iterate) using assms by auto
 
 subsection {* Finally, the integral of a constant *}
@@ -2809,7 +3979,7 @@
   "((\<lambda>x. c) has_integral (content({a..b::'a::ordered_euclidean_space}) *\<^sub>R c)) ({a..b})"
   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
-  unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
+  unfolding split_def apply(subst scaleR_left.setsum[symmetric, unfolded o_def])
   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
 
 lemma integral_const[simp]:
@@ -2821,7 +3991,7 @@
 
 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
-  apply(rule order_trans,rule norm_setsum) unfolding norm_scaleR setsum_left_distrib[THEN sym]
+  apply(rule order_trans,rule norm_setsum) unfolding norm_scaleR setsum_left_distrib[symmetric]
   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
   apply(subst mult_commute) apply(rule mult_left_mono)
   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
@@ -2838,11 +4008,11 @@
 next case False show ?thesis
     apply(rule order_trans,rule norm_setsum) unfolding split_def norm_scaleR
     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
-    unfolding setsum_left_distrib[THEN sym] apply(subst mult_commute) apply(rule mult_left_mono)
+    unfolding setsum_left_distrib[symmetric] apply(subst mult_commute) apply(rule mult_left_mono)
     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
     apply(subst o_def, rule abs_of_nonneg)
   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
-      unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
+      unfolding additive_content_tagged_division[OF assms(1),symmetric] split_def by auto
     guess w using nonempty_witness[OF False] .
     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
@@ -2855,7 +4025,7 @@
   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le> e * content({a..b})"
   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
-  unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR_diff_right by auto
+  unfolding setsum_subtractf[symmetric] apply(rule setsum_cong2) unfolding scaleR_diff_right by auto
 
 lemma has_integral_bound: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
@@ -2863,7 +4033,7 @@
 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
     thus ?thesis proof(cases ?P) case False
       hence *:"content {a..b} = 0" using content_lt_nz by auto
-      hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
+      hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[symmetric]) by auto
       show ?thesis unfolding * ** using assms(1) by auto
     qed auto } assume ab:?P
   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
@@ -2893,7 +4063,7 @@
   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
-  have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow> 
+  have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
     (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)\<bullet>k \<le> (g x)\<bullet>k \<Longrightarrow> i\<bullet>k \<le> j\<bullet>k"
   proof (rule ccontr)
     case goal1
@@ -2935,7 +4105,7 @@
   apply(rule has_integral_component_le) using integrable_integral assms by auto
 
 lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> i\<bullet>k" 
+  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> i\<bullet>k"
   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)] using assms(3-) by auto
 
 lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
@@ -2943,7 +4113,7 @@
   apply(rule has_integral_component_nonneg) using assms by auto
 
 lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"shows "i\<bullet>k \<le> 0" 
+  assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"shows "i\<bullet>k \<le> 0"
   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-) by auto
 
 lemma has_integral_component_lbound:
@@ -2966,7 +4136,7 @@
   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
 
 lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
-  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\<bullet>k \<le> B" "k\<in>Basis" 
+  assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\<bullet>k \<le> B" "k\<in>Basis"
   shows "(integral({a..b}) f)\<bullet>k \<le> B * content({a..b})"
   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
 
@@ -2982,7 +4152,7 @@
   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
-  
+
   have "Cauchy i" unfolding Cauchy_def
   proof(rule,rule) fix e::real assume "e>0"
     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
@@ -3003,10 +4173,10 @@
         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
-      proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
+      proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
           using M as by(auto simp add:field_simps)
         fix x assume x:"x \<in> {a..b}"
-        have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
+        have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
             using g(1)[OF x, of n] g(1)[OF x, of m] by auto
         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
@@ -3015,10 +4185,10 @@
           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
           by(auto simp add:algebra_simps simp add:norm_minus_commute)
       qed qed qed
-  from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
+  from this[unfolded convergent_eq_cauchy[symmetric]] guess s .. note s=this
 
   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
-  proof(rule,rule)  
+  proof(rule,rule)
     case goal1 hence *:"e/3 > 0" by auto
     from LIMSEQ_D [OF s this] guess N1 .. note N1=this
     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
@@ -3038,7 +4208,7 @@
       proof- have "content {a..b} < e / 3 * (real N2)"
           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
-          apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
+          apply-apply(rule less_le_trans,assumption) using `e>0` by auto
         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
           unfolding inverse_eq_divide by(auto simp add:field_simps)
         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format],auto)
@@ -3050,17 +4220,17 @@
 
 subsection {* Negligibility of hyperplane. *}
 
-lemma vsum_nonzero_image_lemma: 
+lemma vsum_nonzero_image_lemma:
   assumes "finite s" "g(a) = 0"
   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
-  unfolding assms using neutral_add unfolding neutral_add using assms by auto 
+  unfolding assms using neutral_add unfolding neutral_add using assms by auto
 
 lemma interval_doublesplit:  fixes a::"'a::ordered_euclidean_space" assumes "k\<in>Basis"
-  shows "{a..b} \<inter> {x . abs(x\<bullet>k - c) \<le> (e::real)} = 
-  {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) .. 
+  shows "{a..b} \<inter> {x . abs(x\<bullet>k - c) \<le> (e::real)} =
+  {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) ..
    (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)}"
 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
@@ -3071,7 +4241,7 @@
 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
-  note division_split(2)[OF this, where c="c-e" and k=k,OF k] 
+  note division_split(2)[OF this, where c="c-e" and k=k,OF k]
   thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
     apply(rule set_eqI) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI) apply rule defer apply rule
@@ -3082,17 +4252,17 @@
 proof(cases "content {a..b} = 0")
   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
-    unfolding interval_doublesplit[THEN sym,OF k] using assms by auto 
+    unfolding interval_doublesplit[symmetric,OF k] using assms by auto
 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})"
   note False[unfolded content_eq_0 not_ex not_le, rule_format]
   hence "\<And>x. x\<in>Basis \<Longrightarrow> b\<bullet>x > a\<bullet>x" by(auto simp add:not_le)
   hence prod0:"0 < setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
   proof(rule that[of d]) have *:"Basis = insert k (Basis - {k})" using k by auto
-    have **:"{a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
+    have **:"{a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
       (\<Prod>i\<in>Basis - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i
       - interval_lowerbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i)
-      = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)" apply(rule setprod_cong,rule refl) 
+      = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)" apply(rule setprod_cong,rule refl)
       unfolding interval_doublesplit[OF k] apply(subst interval_bounds) defer apply(subst interval_bounds)
       unfolding interval_eq_empty not_ex not_less by auto
     show "content ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
@@ -3109,10 +4279,10 @@
   qed
 qed
 
-lemma negligible_standard_hyperplane[intro]: 
+lemma negligible_standard_hyperplane[intro]:
   fixes k :: "'a::ordered_euclidean_space"
   assumes k: "k \<in> Basis"
-  shows "negligible {x. x\<bullet>k = c}" 
+  shows "negligible {x. x\<bullet>k = c}"
   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
 proof-
   case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
@@ -3136,30 +4306,30 @@
       prefer 2 apply(subst(asm) eq_commute) apply assumption
       apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
     proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
-        apply(rule setsum_mono) unfolding split_paired_all split_conv 
+        apply(rule setsum_mono) unfolding split_paired_all split_conv
         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k])
       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content {u..v}"
-          unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
+          unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[symmetric,OF k] by auto
         thus ?case unfolding goal1 unfolding interval_doublesplit[OF k]
           by (blast intro: antisym)
       next have *:"setsum content {l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
-          apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
+          apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
         proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
-        note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
+        note dsum_bound[OF this **,unfolded interval_doublesplit[symmetric,OF k]]
         note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
         from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e"
-          apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
+          apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,symmetric])
           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
           assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           have "({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
           note interior_mono[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
           hence "interior ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
-          thus "content ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
+          thus "content ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[symmetric] .
         qed qed
       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
     qed qed qed
@@ -3177,7 +4347,7 @@
     presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
   } fix p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set" assume as:"finite p"
-  show "?P p" apply(rule,rule) using as proof(induct p) 
+  show "?P p" apply(rule,rule) using as proof(induct p)
     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
     note tagged_partial_division_subset[OF insert(4) subset_insertI]
@@ -3186,19 +4356,19 @@
     note p = tagged_partial_division_ofD[OF insert(4)]
     from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
 
-    have "finite {k. \<exists>x. (x, k) \<in> p}" 
+    have "finite {k. \<exists>x. (x, k) \<in> p}"
       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
     hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
       apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
-      unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
+      unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
       using insert(2) unfolding uv xk by auto
 
     show ?case proof(cases "{u..v} \<subseteq> d x")
       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
-        apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
+        apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int)
         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
@@ -3214,7 +4384,7 @@
 
 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
-proof(induct) case (insert x s) 
+proof(induct) case (insert x s)
   have *:"{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} = (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
   show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
 
@@ -3241,16 +4411,16 @@
       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
       apply(rule,rule P) using assms(2) by auto
   qed
-next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
+next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
   show "(f has_integral 0) {a..b}" unfolding has_integral
   proof(safe) case goal1
-    hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
+    hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
-    note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
+    note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
-    show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
+    show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
-      fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
+      fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
@@ -3258,7 +4428,7 @@
       have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
-      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe) 
+      have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe)
         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
       have **:"\<And>f g s t. finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow> (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t"
       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
@@ -3266,7 +4436,7 @@
       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {0..N+1}"
         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
-        apply(rule order_trans,rule norm_setsum) apply(subst sum_sum_product) prefer 3 
+        apply(rule order_trans,rule norm_setsum) apply(subst sum_sum_product) prefer 3
       proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto
         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
@@ -3286,11 +4456,11 @@
         qed ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le> (real y + 1) * (content k *\<^sub>R indicator s x)"
           apply(rule_tac x=n in exI,safe) apply(rule_tac x=n in exI,rule_tac x="(x,k)" in exI,safe) by auto
       qed(insert as, auto)
-      also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
-      proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
+      also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
+      proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[symmetric])
           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
-      qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[THEN sym]
-        apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
+      qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[symmetric]
+        apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[symmetric]
         apply(subst sumr_geometric) using goal1 by auto
       finally show "?goal" by auto qed qed qed
 
@@ -3323,7 +4493,7 @@
 
 subsection {* Some other trivialities about negligible sets. *}
 
-lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
+lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
     using assms(2) unfolding indicator_def by auto qed
@@ -3332,7 +4502,7 @@
 
 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
 
-lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
+lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
     defer apply assumption unfolding indicator_def by auto qed
@@ -3340,8 +4510,8 @@
 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
   using negligible_union by auto
 
-lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}" 
-  using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto 
+lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}"
+  using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto
 
 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
@@ -3352,7 +4522,7 @@
   using assms apply(induct s) by auto
 
 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
-  using assms by(induct,auto) 
+  using assms by(induct,auto)
 
 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
   apply safe defer apply(subst negligible_def)
@@ -3377,7 +4547,7 @@
 
 subsection {* Finite case of the spike theorem is quite commonly needed. *}
 
-lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
+lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
   "(f has_integral y) t" shows "(g has_integral y) t"
   apply(rule has_integral_spike) using assms by auto
 
@@ -3438,7 +4608,7 @@
 proof safe
   fix a b::"'b"
   { assume "content {a..b} = 0"
-    thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
+    thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
   { fix c g and k :: 'b
     assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k\<in>Basis"
@@ -3452,7 +4622,7 @@
   show "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" apply(rule_tac x="?g" in exI)
   proof safe case goal1 thus ?case apply- apply(cases "x\<bullet>k=c", case_tac "x\<bullet>k < c") using as assms by auto
   next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
-    then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k] 
+    then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k]
     show ?case unfolding integrable_on_def by auto
   next show "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
@@ -3472,7 +4642,7 @@
   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
   note p' = tagged_division_ofD[OF p(1)]
   have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
-  proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
+  proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
     from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l" apply(rule_tac x="\<lambda>y. f x" in exI)
     proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
@@ -3480,11 +4650,11 @@
       note d(2)[OF _ _ this[unfolded mem_ball]]
       thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce qed qed
   from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
-  thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
+  thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
 
 subsection {* Specialization of additivity to one dimension. *}
 
-lemma 
+lemma
   shows real_inner_1_left: "inner 1 x = x"
   and real_inner_1_right: "inner x 1 = x"
   by simp_all
@@ -3510,9 +4680,9 @@
     qed
   next case True hence *:"min (b) c = c" "max a c = c" by auto
     have **: "(1::real) \<in> Basis" by simp
-    have ***:"\<And>P Q. (\<Sum>i\<in>Basis. (if i = 1 then P i else Q i) *\<^sub>R i) = (P 1::real)" 
+    have ***:"\<And>P Q. (\<Sum>i\<in>Basis. (if i = 1 then P i else Q i) *\<^sub>R i) = (P 1::real)"
       by simp
-    show ?thesis 
+    show ?thesis
       unfolding interval_split[OF **, unfolded real_inner_1_right] unfolding *** *
     proof(cases "c = a \<or> c = b")
       case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
@@ -3540,7 +4710,7 @@
       proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
         thus ?thesis using assms unfolding * by auto
       next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
-        thus ?thesis using assms unfolding * by auto qed qed qed 
+        thus ?thesis using assms unfolding * by auto qed qed qed
 
 subsection {* Special case of additivity we need for the FCT. *}
 
@@ -3554,8 +4724,8 @@
   have ***:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" using assms by auto
   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty by auto
   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
-  note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],THEN sym]
-  show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
+  note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
+  show ?thesis unfolding * apply(subst setsum_iterate[symmetric]) defer
     apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
 
 subsection {* A useful lemma allowing us to factor out the content size. *}
@@ -3565,10 +4735,10 @@
     \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
 proof(cases "content {a..b} = 0")
   case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
-    apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
+    apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
     apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
     apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
-next case False note F = this[unfolded content_lt_nz[THEN sym]]
+next case False note F = this[unfolded content_lt_nz[symmetric]]
   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
   show ?thesis apply(subst has_integral)
   proof safe fix e::real assume e:"e>0"
@@ -3599,10 +4769,10 @@
     apply(rule_tac x="\<lambda>x. ball x (d x)" in exI,safe)
     apply(rule gauge_ball_dependent,rule,rule d(1))
   proof- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. ball x (d x)) fine p"
-    show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}" 
-      unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,THEN sym]
-      unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",THEN sym]
-      unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
+    show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
+      unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,symmetric]
+      unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",symmetric]
+      unfolding setsum_right_distrib defer unfolding setsum_subtractf[symmetric]
     proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
       note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
       have *:"u \<le> v" using xk unfolding k by auto
@@ -3615,8 +4785,8 @@
       also have "... \<le> e * norm (u - x) + e * norm (v - x)"
         apply(rule add_mono) apply(rule d(2)[of "x" "u",unfolded o_def]) prefer 4
         apply(rule d(2)[of "x" "v",unfolded o_def])
-        using ball[rule_format,of u] ball[rule_format,of v] 
-        using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def) 
+        using ball[rule_format,of u] ball[rule_format,of v]
+        using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def)
       also have "... \<le> e * (interval_upperbound k - interval_lowerbound k)"
         unfolding k interval_bounds_real[OF *] using xk(1) unfolding k by(auto simp add:dist_real_def field_simps)
       finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
@@ -3638,7 +4808,7 @@
   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
 proof(induct "card s" arbitrary:s rule:nat_less_induct)
   fix s::"'a set set" assume assm:"s division_of {a..b}"
-    "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}" 
+    "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}"
   note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
   { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
     show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
@@ -3651,12 +4821,12 @@
     apply safe apply(rule closed_interval) using assm(1) by auto
   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
-    from k(2)[unfolded k content_eq_0] guess i .. 
+    from k(2)[unfolded k content_eq_0] guess i ..
     hence i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
     hence xi:"x\<bullet>i = d\<bullet>i" using as unfolding k mem_interval by (metis antisym)
     def y \<equiv> "(\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i) / 2 then c\<bullet>i +
       min e (b\<bullet>i - c\<bullet>i) / 2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i) / 2 else x\<bullet>j) *\<^sub>R j)::'a"
-    show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
+    show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
     proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastforce simp add: not_less)
       hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN bspec[where x=i]]
       hence xyi:"y\<bullet>i \<noteq> x\<bullet>i"
@@ -3677,7 +4847,7 @@
         using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i unfolding s mem_interval y_def
         by (auto simp: field_simps elim!: ballE[of _ _ i])
       ultimately show "y \<in> \<Union>(s - {k})" by auto
-    qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
+    qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[symmetric] by auto
   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
     apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
   moreover have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}" using k by auto ultimately show ?thesis by auto qed
@@ -3690,10 +4860,10 @@
   unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption,assumption)+
   unfolding integrable_on_def by(auto intro!: has_integral_split)
 
-lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach" 
-  assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
+lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
+  assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
   apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
-  using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
+  using operative_division_and[OF operative_integrable,symmetric,of _ _ _ f] assms(1) by auto
 
 subsection {* Combining adjacent intervals in 1 dimension. *}
 
@@ -3710,7 +4880,7 @@
 lemma integral_combine: fixes f::"real \<Rightarrow> 'a::banach"
   assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
   shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
-  apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
+  apply(rule integral_unique[symmetric]) apply(rule has_integral_combine[OF assms(1-2)])
   apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
 
 lemma integrable_combine: fixes f::"real \<Rightarrow> 'a::banach"
@@ -3725,7 +4895,7 @@
 proof- have "\<forall>x. \<exists>d. x\<in>{a..b} \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v})"
     using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
   guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
-  note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
+  note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,symmetric,of f]
   show ?thesis unfolding * apply safe unfolding snd_conv
   proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
     thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
@@ -3765,10 +4935,10 @@
       hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
         using True using assms(2) goal1 by auto
       have **:"norm (y - x) = content {y..x}" apply(subst content_real) using True unfolding not_less by auto
-      have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto 
+      have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto
       show ?thesis apply(subst ***) unfolding norm_minus_cancel **
         apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
-        defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
+        defer apply(rule has_integral_sub) apply(subst minus_minus[symmetric]) unfolding minus_minus
         apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
       proof- show "{y..x} \<subseteq> {a..b}" using goal1 assms(2) by auto
         have *:"x - y = norm(y - x)" using True by auto
@@ -3813,8 +4983,8 @@
     def d' \<equiv> "\<lambda>x. {y. g y \<in> d (g x)}" have d':"\<And>x. d' x = {y. g y \<in> (d (g x))}" unfolding d'_def ..
     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
     proof(rule_tac x=d' in exI,safe) show "gauge d'" using d(1) unfolding gauge_def d' using continuous_open_preimage_univ[OF assms(4)] by auto
-      fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
-      have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of 
+      fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
+      have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of
       proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
         fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
@@ -3852,12 +5022,12 @@
 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
   apply(rule setprod_cong) using assms by auto
 
-lemma content_image_affinity_interval: 
+lemma content_image_affinity_interval:
  "content((\<lambda>x::'a::ordered_euclidean_space. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ DIM('a) * content {a..b}" (is "?l = ?r")
 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
       unfolding not_not using content_empty by auto }
-  assume as: "{a..b}\<noteq>{}" 
-  show ?thesis 
+  assume as: "{a..b}\<noteq>{}"
+  show ?thesis
   proof (cases "m \<ge> 0")
     case True
     with as have "{m *\<^sub>R a + c..m *\<^sub>R b + c} \<noteq> {}"
@@ -3903,10 +5073,10 @@
 lemma image_stretch_interval:
   "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k)) *\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} =
   (if {a..b} = {} then {} else
-    {(\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)::'a .. 
+    {(\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)::'a ..
      (\<Sum>k\<in>Basis. (max (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)})"
 proof cases
-  assume *: "{a..b} \<noteq> {}" 
+  assume *: "{a..b} \<noteq> {}"
   show ?thesis
     unfolding interval_ne_empty if_not_P[OF *]
     apply (simp add: interval image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
@@ -3929,14 +5099,14 @@
           "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (b \<bullet> i) else m i * (a \<bullet> i))"
         using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
       with False show ?thesis using a_le_b
-        unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps) 
+        unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps)
     qed
   qed
 qed simp
 
-lemma interval_image_stretch_interval: 
+lemma interval_image_stretch_interval:
     "\<exists>u v. (\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
-  unfolding image_stretch_interval by auto 
+  unfolding image_stretch_interval by auto
 
 lemma content_image_stretch_interval:
   "content((\<lambda>x::'a::ordered_euclidean_space. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b}) = abs(setprod m Basis) * content({a..b})"
@@ -3944,12 +5114,12 @@
     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
 next case False hence "(\<lambda>x. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b} \<noteq> {}" by auto
   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
-    unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff
+    unfolding abs_setprod setprod_timesf[symmetric] apply(rule setprod_cong2) unfolding lessThan_iff
   proof (simp only: inner_setsum_left_Basis)
     fix i :: 'a assume i:"i\<in>Basis" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
-    thus "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) - min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = 
+    thus "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) - min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) =
         \<bar>m i\<bar> * (b \<bullet> i - a \<bullet> i)"
-      apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i 
+      apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i
       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
 
 lemma has_integral_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
@@ -3966,7 +5136,7 @@
 lemma integrable_stretch:  fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
   assumes "f integrable_on {a..b}" "\<forall>k\<in>Basis. ~(m k = 0)"
   shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` {a..b})"
-  using assms unfolding integrable_on_def apply-apply(erule exE) 
+  using assms unfolding integrable_on_def apply-apply(erule exE)
   apply(drule has_integral_stretch,assumption) by auto
 
 subsection {* even more special cases. *}
@@ -4001,13 +5171,13 @@
   unfolding split_def by(rule refl)
 
 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
-  apply(subst(asm)(2) norm_minus_cancel[THEN sym])
+  apply(subst(asm)(2) norm_minus_cancel[symmetric])
   apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
 
 lemma fundamental_theorem_of_calculus_interior: fixes f::"real => 'a::real_normed_vector"
   assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
   shows "(f' has_integral (f b - f a)) {a..b}"
-proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
+proof- { presume *:"a < b \<Longrightarrow> ?thesis"
     show ?thesis proof(cases,rule *,assumption)
       assume "\<not> a < b" hence "a = b" using assms(1) by auto
       hence *:"{a .. b} = {b}" "f b - f a = 0" by(auto simp add:  order_antisym)
@@ -4034,15 +5204,15 @@
     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
     proof(cases "f' a = 0") case True
-      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
+      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
     next case False thus ?thesis
-        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps) 
+        apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps)
     qed then guess l .. note l = conjunctD2[OF this]
     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
-    proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
+    proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
       have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)" by(rule norm_triangle_ineq4)
-      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
+      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
       proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
       next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
@@ -4060,16 +5230,16 @@
     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
     proof(cases "f' b = 0") case True
-      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
-    next case False thus ?thesis 
+      thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
+    next case False thus ?thesis
         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
         using ab e by(auto simp add:field_simps)
     qed then guess l .. note l = conjunctD2[OF this]
     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
-    proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
+    proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
       have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)" by(rule norm_triangle_ineq4)
-      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
+      also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
       proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
       next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
@@ -4083,11 +5253,11 @@
   proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
   next case goal2 note as=this let ?A = "{t. fst t \<in> {a, b}}" note p = tagged_division_ofD[OF goal2(1)]
     have pA:"p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"  using goal2 by auto
-    note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
+    note * = additive_tagged_division_1'[OF assms(1) goal2(1), symmetric]
     have **:"\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2" by arith
-    show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
+    show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[symmetric] split_minus
       unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
-    proof(rule norm_triangle_le,rule **) 
+    proof(rule norm_triangle_le,rule **)
       case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) defer apply(subst setsum_divide_distrib)
       proof(rule order_refl,safe,unfold not_le o_def split_conv fst_conv,rule ccontr) fix x k assume as:"(x,k) \<in> p"
           "e * (interval_upperbound k -  interval_lowerbound k) / 2
@@ -4099,8 +5269,8 @@
         assume as':"x \<noteq> a" "x \<noteq> b" hence "x \<in> {a<..<b}" using p(2-3)[OF as(1)] by auto
         note  * = d(2)[OF this]
         have "norm ((v - u) *\<^sub>R f' (x) - (f (v) - f (u))) =
-          norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))" 
-          apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
+          norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))"
+          apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
         also have "... \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)" apply(rule norm_triangle_le_sub)
           apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
           apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp:dist_real_def)
@@ -4110,7 +5280,7 @@
 
     next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
       case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
-        defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
+        defer unfolding setsum_Un_disjoint[OF pA(2-),symmetric] pA(1)[symmetric] unfolding setsum_right_distrib[symmetric]
         apply(subst additive_tagged_division_1[OF _ as(1)]) apply(rule assms)
       proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}}" note xk=IntD1[OF this]
         from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
@@ -4119,7 +5289,7 @@
           unfolding uv using e by(auto simp add:field_simps)
       next have *:"\<And>s f t e. setsum f s = setsum f t \<Longrightarrow> norm(setsum f t) \<le> e \<Longrightarrow> norm(setsum f s) \<le> e" by auto
         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
-          (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2" 
+          (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2"
           apply(rule *[where t="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
           apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
         proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
@@ -4127,7 +5297,7 @@
           have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk
             unfolding uv content_eq_0 interval_eq_empty by auto
           thus "content k *\<^sub>R (f' (x)) - (f ((interval_upperbound k)) - f ((interval_lowerbound k))) = 0" using xk unfolding uv by auto
-        next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} = 
+        next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
             {t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}" by blast
           have **:"\<And>s f. \<And>e::real. (\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y) \<Longrightarrow> (\<forall>x. x \<in> s \<longrightarrow> norm(f x) \<le> e)
             \<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
@@ -4135,22 +5305,22 @@
             thus ?case using `x\<in>s` goal2(2) by auto
           qed auto
           case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4
-            apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"]) 
+            apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
             apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
           proof- let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
-            have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v" 
+            have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v"
             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
-              have u:"u = a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
+              have u:"u = a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
                 have "u \<ge> a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>a" ultimately
                 have "u > a" by auto
                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
               qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
             qed
-            have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v" 
+            have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v"
             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
-              have u:"v =  b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
+              have u:"v =  b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
                 have "v \<le>  b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq> b" ultimately
                 have "v <  b" by auto
                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
@@ -4168,7 +5338,7 @@
               ultimately have " ((a + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
-            qed 
+            qed
             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
               unfolding mem_Collect_eq fst_conv snd_conv apply safe
             proof- fix x k k' assume k:"( b, k) \<in> p" "( b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
@@ -4184,7 +5354,7 @@
             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
             show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) - (f ((interval_upperbound k)) -
               f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4" apply(rule,rule) unfolding mem_Collect_eq
-              unfolding split_paired_all fst_conv snd_conv 
+              unfolding split_paired_all fst_conv snd_conv
             proof safe case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
               have " ?a\<in>{ ?a..v}" using v(2) by auto hence "v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
               moreover have "{?a..v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
@@ -4195,7 +5365,7 @@
             qed
             show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) -
               (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4"
-              apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv 
+              apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv
             proof safe case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
               have " ?b\<in>{v.. ?b}" using v(2) by auto hence "v \<ge> ?a" using p(3)[OF goal1(1)]
                 unfolding subset_eq v by auto
@@ -4213,7 +5383,7 @@
 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
   assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
   "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
-  shows "(f' has_integral (f b - f a)) {a..b}" using assms apply- 
+  shows "(f' has_integral (f b - f a)) {a..b}" using assms apply-
 proof(induct "card s" arbitrary:s a b)
   case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
@@ -4249,10 +5419,10 @@
       hence "c - t < e / 3 / norm (f c)" by auto
       hence "norm (c - t) < e / 3 / norm (f c)" using as by auto
       thus "norm (f c) * norm (c - t) < e / 3" using False apply-
-        apply(subst mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
+        apply(subst mult_commute) apply(subst pos_less_divide_eq[symmetric]) by auto
     qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
   qed then guess w .. note w = conjunctD2[OF this,rule_format]
-  
+
   have *:"e / 3 > 0" using assms by auto
   have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
   from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
@@ -4281,7 +5451,7 @@
     have pt:"\<forall>(x,k)\<in>p. x \<le> t" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
-    
+
     have *:"{a..c} \<inter> {x. x \<bullet> 1 \<le> t} = {a..t}" "{a..c} \<inter> {x. x \<bullet> 1 \<ge> t} = {t..c}"
       using assms(2-3) as by(auto simp add:field_simps)
     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
@@ -4290,30 +5460,30 @@
     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
     next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real_def
-        using as(1) by(auto simp add:field_simps) 
+        using as(1) by(auto simp add:field_simps)
       thus "x \<in> d1 c" using k(2) unfolding d_def by auto
     qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
 
     have *:"integral{a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
-        integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c" 
+        integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c"
       "e = (e/3 + e/3) + e/3" by auto
     have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
     proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
       have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
         have "c \<in> {a..t}" by auto thus False using `t<c` by auto
       qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
-        unfolding split_conv defer apply(subst content_real) using as(2) by auto qed 
+        unfolding split_conv defer apply(subst content_real) using as(2) by auto qed
 
     have ***:"c - w < t \<and> t < c"
     proof- have "c - k < t" using `k>0` as(1) by(auto simp add:field_simps)
-      moreover have "k \<le> w" apply(rule ccontr) using k(2) 
+      moreover have "k \<le> w" apply(rule ccontr) using k(2)
         unfolding subset_eq apply(erule_tac x="c + ((k + w)/2)" in ballE)
         unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real_def)
       ultimately show  ?thesis using `t<c` by(auto simp add:field_simps) qed
 
     show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
       unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
-      using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed 
+      using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed
 
 lemma indefinite_integral_continuous_right: fixes f::"real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
@@ -4327,9 +5497,9 @@
       "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
       apply(rule_tac[!] integral_combine) using assms as by auto
     have "(- c) - d < (- t) \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
-    thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding * 
+    thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
       unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
-   
+
 lemma indefinite_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
 proof(unfold continuous_on_iff, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
@@ -4359,7 +5529,7 @@
       thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute)
         apply(cases "y < x") unfolding dist_norm apply(rule d1(2)[rule_format]) defer
         apply(rule d2(2)[rule_format]) unfolding not_less by(auto simp add:field_simps)
-    qed qed qed 
+    qed qed qed
 
 subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
 
@@ -4372,7 +5542,7 @@
   have "((\<lambda>x. 0\<Colon>'a) has_integral f x - f a) {a..x}"
     apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
     apply(rule continuous_on_subset[OF assms(2)]) defer
-    apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
+    apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[symmetric])
     apply assumption apply(rule open_interval) apply(rule has_derivative_within_subset[where s="{a..b}"])
     using assms(4) assms(5) by auto note this[unfolded *]
   note has_integral_unique[OF has_integral_0 this]
@@ -4385,16 +5555,16 @@
   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
   shows "f x = y"
 proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
-      unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
+      unfolding assms(5)[symmetric] by auto } assume "x\<noteq>c"
   note conv = assms(1)[unfolded convex_alt,rule_format]
   have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
     apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
     apply safe apply(rule conv) using assms(4,7) by auto
   have *:"\<And>t xa. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x \<Longrightarrow> t = xa"
-  proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c" 
+  proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
       unfolding scaleR_simps by(auto simp add:algebra_simps)
     thus ?case using `x\<noteq>c` by auto qed
-  have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2) 
+  have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
     apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
     apply safe unfolding image_iff apply rule defer apply assumption
     apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
@@ -4402,7 +5572,7 @@
     apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
     unfolding o_def using assms(5) defer apply-apply(rule)
   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
-    have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps]) 
+    have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
       using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
     have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
       apply(rule diff_chain_within) apply(rule has_derivative_add)
@@ -4414,7 +5584,7 @@
     thus "((\<lambda>xa. f ((1 - xa) *\<^sub>R c + xa *\<^sub>R x)) has_derivative (\<lambda>h. 0)) (at t within {0..1})" unfolding o_def .
   qed auto thus ?thesis by auto qed
 
-subsection {* Also to any open connected set with finite set of exceptions. Could 
+subsection {* Also to any open connected set with finite set of exceptions. Could
  generalize to locally convex set with limpt-free set of exceptions. *}
 
 lemma has_derivative_zero_unique_strong_connected: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
@@ -4425,7 +5595,7 @@
     apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
     apply(rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
     apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
-  proof safe fix x assume "x\<in>s" 
+  proof safe fix x assume "x\<in>s"
     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
     proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
@@ -4444,12 +5614,12 @@
 proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
   { presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
     show ?thesis apply(cases,rule *,assumption)
-    proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto 
+    proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto
       show ?thesis using assms(1) unfolding * using goal1 by auto
     qed } assume "{c..d}\<noteq>{}"
   from partial_division_extend_1[OF assms(2) this] guess p . note p=this
-  note mon = monoidal_lifted[OF monoidal_monoid] 
-  note operat = operative_division[OF this operative_integral p(1), THEN sym]
+  note mon = monoidal_lifted[OF monoidal_monoid]
+  note operat = operative_division[OF this operative_integral p(1), symmetric]
   let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
   { presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
       apply- apply(cases,subst(asm) if_P,assumption) by auto
@@ -4476,13 +5646,13 @@
     unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
 
 lemma has_integral_restrict_closed_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
-  assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}" 
+  assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}"
   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
 proof- note has_integral_restrict_open_subinterval[OF assms]
   note * = has_integral_spike[OF negligible_frontier_interval _ this]
   show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
 
-lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}" 
+lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}"
   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b} \<longleftrightarrow> (f has_integral i) {c..d}" (is "?l = ?r")
 proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
   show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
@@ -4512,38 +5682,38 @@
         apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
         by(auto simp add:dist_norm)
     qed(insert B `e>0`, auto)
-  next assume as:"\<forall>e>0. ?r e" 
+  next assume as:"\<forall>e>0. ?r e"
     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
-    def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space" 
+    def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
     def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
     proof
       case goal1 thus ?case using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def
         by(auto simp add:field_simps setsum_negf)
     qed
-    have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
+    have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
     proof
       case goal1 thus ?case
         using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def by (auto simp: setsum_negf)
     qed
     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
-      unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
+      unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,symmetric] unfolding s by auto
     then guess y .. note y=this
 
     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
-      def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space" 
+      def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
       def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
       proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def
           by(auto simp add:field_simps setsum_negf) qed
-      have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
+      have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
       proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by (auto simp: setsum_negf) qed
       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
       thus False by auto qed
-    thus ?l using y unfolding s by auto qed qed 
+    thus ?l using y unfolding s by auto qed qed
 
 lemma has_integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x) \<le> (g x)"
@@ -4556,12 +5726,12 @@
   using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] .
 
 lemma has_integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
-  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" 
+  assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
   using has_integral_component_nonneg[of 1 f i s]
   unfolding o_def using assms by auto
 
 lemma integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
-  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" 
+  assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
   using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
 
 subsection {* Hence a general restriction property. *}
@@ -4574,20 +5744,20 @@
 lemma has_integral_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
   "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
 
-lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
+lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
   shows "(f has_integral i) t"
 proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
     apply(rule) using assms(1-2) by auto
-  thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
-  apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
-
-lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
+  thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[symmetric])
+  apply- apply(subst(asm) has_integral_restrict_univ[symmetric]) by auto qed
+
+lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
   shows "f integrable_on t"
   using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
 
-lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
+lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
   apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
 
@@ -4600,9 +5770,9 @@
   proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
       unfolding indicator_def by auto qed qed auto
 
-lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
+lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
-  unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (auto split: split_if_asm)
+  unfolding has_integral_restrict_univ[symmetric,of f] apply(rule has_integral_spike_eq[OF assms]) by (auto split: split_if_asm)
 
 lemma has_integral_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
@@ -4611,7 +5781,7 @@
 
 lemma integrable_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
-  shows "f integrable_on t" using assms(2) unfolding integrable_on_def 
+  shows "f integrable_on t" using assms(2) unfolding integrable_on_def
   unfolding has_integral_spike_set_eq[OF assms(1)] .
 
 lemma integrable_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
@@ -4656,7 +5826,7 @@
 lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
   assumes k: "k\<in>Basis" and as: "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
-proof- note has_integral_restrict_univ[THEN sym, of f]
+proof- note has_integral_restrict_univ[symmetric, of f]
   note as(2-3)[unfolded this] note * = has_integral_component_le[OF k this]
   show ?thesis apply(rule *) using as(1,4) by auto qed
 
@@ -4701,12 +5871,12 @@
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
                     norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
     proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
-      from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed 
+      from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed
 
 
 subsection {* Continuity of the integral (for a 1-dimensional interval). *}
 
-lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows 
+lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
   "f integrable_on s \<longleftrightarrow>
           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
           (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
@@ -4718,7 +5888,7 @@
     show ?case apply(rule,rule,rule B)
     proof safe case goal1 show ?case apply(rule norm_triangle_half_l)
         using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
-        
+
 next assume ?r note as = conjunctD2[OF this,rule_format]
   let ?cube = "\<lambda>n. {(\<Sum>i\<in>Basis. - real n *\<^sub>R i)::'n .. (\<Sum>i\<in>Basis. real n *\<^sub>R i)} :: 'n set"
   have "Cauchy (\<lambda>n. integral (?cube n) (\<lambda>x. if x \<in> s then f x else 0))"
@@ -4730,7 +5900,7 @@
       proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
           using n N by(auto simp add:field_simps setsum_negf) qed }
     thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto
-  qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
+  qed from this[unfolded convergent_eq_cauchy[symmetric]] guess i ..
   note i = this[THEN LIMSEQ_D]
 
   show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
@@ -4747,7 +5917,7 @@
         apply(rule N[of n])
       proof safe show "N \<le> n" using n by auto
         fix x::"'n::ordered_euclidean_space" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
-        thus "x\<in>{a..b}" using ab by blast 
+        thus "x\<in>{a..b}" using ab by blast
         show "x\<in>?cube n" using x unfolding mem_interval mem_ball dist_norm apply-
         proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
             using n by(auto simp add:field_simps setsum_negf) qed qed qed qed qed
@@ -4777,31 +5947,31 @@
   from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
   show ?case apply(rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI) apply(rule conjI gauge_inter d1 d2)+ unfolding fine_inter
   proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
-      abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow>  abs(g1 - i) < e / 3 \<Longrightarrow> 
+      abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow>  abs(g1 - i) < e / 3 \<Longrightarrow>
       abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith
     case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)]
 
     have "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
-      "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)" 
+      "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
       "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
-      "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)" 
-      unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg)
-      apply safe unfolding real_scaleR_def right_diff_distrib[THEN sym]
+      "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
+      unfolding setsum_subtractf[symmetric] apply- apply(rule_tac[!] setsum_nonneg)
+      apply safe unfolding real_scaleR_def right_diff_distrib[symmetric]
       apply(rule_tac[!] mult_nonneg_nonneg)
     proof- fix a b assume ab:"(a,b) \<in> p1"
       show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
       show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(1-2)[OF ab] using obt(4)[rule_format,of a] by auto
     next fix a b assume ab:"(a,b) \<in> p2"
       show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
-      show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto qed 
+      show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto qed
 
     thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
-      unfolding real_norm_def[THEN sym] apply(rule obt(3))
+      unfolding real_norm_def[symmetric] apply(rule obt(3))
       apply(rule d1(2)[OF conjI[OF goal1(4,5)]])
       apply(rule d1(2)[OF conjI[OF goal1(1,2)]])
       apply(rule d2(2)[OF conjI[OF goal1(4,6)]])
-      apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed 
-     
+      apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed
+
 lemma integrable_straddle: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
   assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
   norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))"
@@ -4822,7 +5992,7 @@
       case goal2 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by auto qed
     have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow>
       norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e"
-      using obt(3) unfolding real_norm_def by arith 
+      using obt(3) unfolding real_norm_def by arith
     show ?case apply(rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
                apply(rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI)
@@ -4836,7 +6006,7 @@
                    integral {a..b} (\<lambda>x. if x \<in> s then g x else 0))
            \<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) -
                    integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))"
-        unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
+        unfolding integral_sub[OF h g,symmetric] real_norm_def apply(subst **) defer apply(subst **) defer
         apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+
       proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
           apply - apply rule apply(erule_tac x=i in ballE) by auto
@@ -4856,30 +6026,30 @@
         abs(hc - j) < e / 3 \<and> abs(i - j) < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc\<Longrightarrow> abs(fa - fc) < e"
         by (simp add: abs_real_def split: split_if_asm)
       show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - integral {c..d} (\<lambda>x. if x \<in> s then f x else 0)) < e"
-        unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
-        apply(rule B1(2),rule order_trans,rule **,rule as(1)) 
-        apply(rule B1(2),rule order_trans,rule **,rule as(2)) 
-        apply(rule B2(2),rule order_trans,rule **,rule as(1)) 
-        apply(rule B2(2),rule order_trans,rule **,rule as(2)) 
+        unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[symmetric]
+        apply(rule B1(2),rule order_trans,rule **,rule as(1))
+        apply(rule B1(2),rule order_trans,rule **,rule as(2))
+        apply(rule B2(2),rule order_trans,rule **,rule as(1))
+        apply(rule B2(2),rule order_trans,rule **,rule as(2))
         apply(rule obt) apply(rule_tac[!] integral_le) using obt
-        by(auto intro!: h g interv) qed qed qed 
+        by(auto intro!: h g interv) qed qed qed
 
 subsection {* Adding integrals over several sets. *}
 
 lemma has_integral_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)"
   shows "(f has_integral (i + j)) (s \<union> t)"
-proof- note * = has_integral_restrict_univ[THEN sym, of f]
+proof- note * = has_integral_restrict_univ[symmetric, of f]
   show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)])
     defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed
 
 lemma has_integral_unions: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "finite t" "\<forall>s\<in>t. (f has_integral (i s)) s"  "\<forall>s\<in>t. \<forall>s'\<in>t. ~(s = s') \<longrightarrow> negligible(s \<inter> s')"
   shows "(f has_integral (setsum i t)) (\<Union>t)"
-proof- note * = has_integral_restrict_univ[THEN sym, of f]
+proof- note * = has_integral_restrict_univ[symmetric, of f]
   have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))"
-    apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer 
-    apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto 
+    apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer
+    apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto
   note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this]
   thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption
   proof safe case goal1 thus ?case
@@ -4895,7 +6065,7 @@
   assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k"
   shows "(f has_integral (setsum i d)) s"
 proof- note d = division_ofD[OF assms(1)]
-  show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions)
+  show ?thesis unfolding d(6)[symmetric] apply(rule has_integral_unions)
     apply(rule d assms)+ apply(rule,rule,rule)
   proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)]
     guess a c b d apply-by(erule exE)+ note obt=this
@@ -4913,7 +6083,7 @@
   assumes "f integrable_on s" "d division_of k" "k \<subseteq> s"
   shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
   apply(rule has_integral_combine_division[OF assms(2)])
-  apply safe unfolding has_integral_integral[THEN sym]
+  apply safe unfolding has_integral_integral[symmetric]
 proof- case goal1 from division_ofD(2,4)[OF assms(2) this]
   show ?case apply safe apply(rule integrable_on_subinterval)
     apply(rule assms) using assms(3) by auto qed
@@ -4944,7 +6114,7 @@
   shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
 proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
     apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)])
-    using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto)
+    using assms(2) unfolding has_integral_integral[symmetric] by(safe,auto)
   thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption
     apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute)
     apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption)
@@ -4998,22 +6168,22 @@
 
   let ?p = "p \<union> \<Union>(qq ` r)" have "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - integral {a..b} f) < e"
     apply(rule assms(4)[rule_format])
-  proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto 
+  proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto
     note * = tagged_partial_division_of_union_self[OF p(1)]
     have "p \<union> \<Union>(qq ` r) tagged_division_of \<Union>(snd ` p) \<union> \<Union>r"
     proof(rule tagged_division_union[OF * tagged_division_unions])
       show "finite r" by fact case goal2 thus ?case using qq by auto
     next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto
     next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule)
-        apply(rule,rule q') defer apply(rule,subst Int_commute) 
+        apply(rule,rule q') defer apply(rule,subst Int_commute)
         apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer
         apply(rule,rule q') using q(1) p' unfolding r_def by auto qed
     moreover have "\<Union>(snd ` p) \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r"
-      unfolding Union_Un_distrib[THEN sym] r_def using q by auto
+      unfolding Union_Un_distrib[symmetric] r_def using q by auto
     ultimately show "?p tagged_division_of {a..b}" by fastforce qed
 
   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>\<Union>(qq ` r). content k *\<^sub>R f x) -
-    integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3 
+    integral {a..b} f) < e" apply(subst setsum_Un_zero[symmetric]) apply(rule p') prefer 3
     apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq)
   proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r"
     note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
@@ -5021,7 +6191,7 @@
     have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
     hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto
     note q'(5)[OF this] hence "interior l = {}" using interior_mono[OF `l \<subseteq> k`] by blast
-    thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto
+    thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto qed auto
 
   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
     (qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero)
@@ -5032,23 +6202,23 @@
     from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this
     have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q')
       using as unfolding r_def by auto
-    have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym]
+    have "interior m = {}" unfolding subset_empty[symmetric] unfolding *[symmetric]
       apply(rule interior_mono) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto
-    thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto 
+    thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto
   qed(insert qq, auto)
 
   hence **:"norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x) \<circ> qq) r -
     integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact
     apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption
   proof- fix k l x m assume as:"k\<in>r" "l\<in>r" "k\<noteq>l" "qq k = qq l" "(x,m)\<in>qq k"
-    note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)] 
+    note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)]
     show "content m *\<^sub>R f x = 0"  using as(3) unfolding as by auto qed
-  
-  have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow> 
-    ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k" 
-  proof- case goal1 thus ?case  using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]  
-      unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
-  
+
+  have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow>
+    ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k"
+  proof- case goal1 thus ?case  using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
+      unfolding goal1(3)[symmetric] norm_minus_cancel by(auto simp add:algebra_simps) qed
+
   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
     unfolding split_def setsum_subtractf ..
   also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"])
@@ -5059,15 +6229,15 @@
       from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this
       show "integral l f = 0" unfolding uv apply(rule integral_unique)
         apply(rule has_integral_null) unfolding content_eq_0_interior
-        using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto
-    qed auto 
+        using p'(5)[OF as(1-3)] unfolding uv as(4)[symmetric] by auto
+    qed auto
     show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
-      apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto
+      apply(rule setsum_Un_disjoint'[symmetric]) using q(1) q'(1) p'(1) by auto
   next  case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps)
     show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
-      unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le)
-      apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[THEN sym]
-      unfolding divide_inverse[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat)
+      unfolding setsum_subtractf[symmetric] apply(rule setsum_norm_le)
+      apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[symmetric]
+      unfolding divide_inverse[symmetric] using * by(auto simp add:field_simps real_eq_of_nat)
   qed finally show "?x \<le> e + k" . qed
 
 lemma henstock_lemma_part2: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
@@ -5075,12 +6245,12 @@
   "\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p -
           integral({a..b}) f) < e"    "p tagged_partial_division_of {a..b}" "d fine p"
   shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
-  unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer 
+  unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer
   apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
   apply safe apply(rule assms[rule_format,unfolded split_def]) defer
   apply(rule tagged_partial_division_subset,rule assms,assumption)
   apply(rule fine_subset,assumption,rule assms) using assms(5) by auto
-  
+
 lemma henstock_lemma: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
   assumes "f integrable_on {a..b}" "e>0"
   obtains d where "gauge d"
@@ -5201,7 +6371,7 @@
         unfolding dist_real_def using fg[rule_format,OF goal1]
         by (auto simp add:field_simps) qed
     from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
-    def d \<equiv> "\<lambda>x. c (m x) x" 
+    def d \<equiv> "\<lambda>x. c (m x) x"
 
     show ?case apply(rule_tac x=d in exI)
     proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
@@ -5211,7 +6381,7 @@
         by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
       then guess s .. note s=this
       have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and>
-            norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e" 
+            norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e"
       proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
           norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
           by(auto simp add:algebra_simps) qed
@@ -5219,17 +6389,17 @@
           b="\<Sum>(x, k)\<in>p. content k *\<^sub>R f (m x) x" and c="\<Sum>(x, k)\<in>p. integral k (f (m x))"])
       proof safe case goal1
          show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"])
-           unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule norm_setsum)
+           unfolding setsum_subtractf[symmetric] apply(rule order_trans,rule norm_setsum)
            apply(rule setsum_mono) unfolding split_paired_all split_conv
-           unfolding split_def setsum_left_distrib[THEN sym] scaleR_diff_right[THEN sym]
+           unfolding split_def setsum_left_distrib[symmetric] scaleR_diff_right[symmetric]
            unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
          proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto
            from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this
            show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))"
-             unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le] 
+             unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le]
              apply(rule mult_left_mono) using m(2)[OF x,of "m x"] by auto
          qed(insert ab,auto)
-         
+
        next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
            \<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
            apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer
@@ -5240,7 +6410,7 @@
              apply(rule setsum_norm_le)
            proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
                unfolding power_add divide_inverse inverse_mult_distrib
-               unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym]
+               unfolding setsum_right_distrib[symmetric] setsum_left_distrib[symmetric]
                unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e])
                unfolding power2_eq_square by auto
              fix t assume "t\<in>{0..s}"
@@ -5259,22 +6429,22 @@
        next case goal3
          note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
          have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i\<bullet>1 - kr\<bullet>1
-           \<and> i\<bullet>1 - kr\<bullet>1 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto 
+           \<and> i\<bullet>1 - kr\<bullet>1 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto
          show ?case unfolding real_norm_def apply(rule *[rule_format],safe)
-           apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded real_inner_1_right]) 
+           apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded real_inner_1_right])
            apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
            apply(rule_tac[1-2] integral_le[OF ])
          proof safe show "0 \<le> i\<bullet>1 - (integral {a..b} (f r))\<bullet>1" using r(1) by auto
            show "i\<bullet>1 - (integral {a..b} (f r))\<bullet>1 < e / 4" using r(2) by auto
            fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
-           show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k" 
+           show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k"
              unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
-             using p'(3)[OF xk] unfolding uv by auto 
+             using p'(3)[OF xk] unfolding uv by auto
            fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto
            hence *:"\<And>m. \<forall>n\<ge>m. (f m y) \<le> (f n y)" apply-apply(rule transitive_stepwise_le) using assms(2) by auto
            show "(f r y) \<le> (f (m x) y)" "(f (m x) y) \<le> (f s y)"
              apply(rule_tac[!] *[rule_format]) using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk] by auto
-         qed qed qed qed note * = this 
+         qed qed qed qed note * = this
 
    have "integral {a..b} g = i" apply(rule integral_unique) using * .
    thus ?thesis using i * by auto qed
@@ -5300,13 +6470,13 @@
       apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
       apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
       apply simp
-      apply(rule goal1(2)[rule_format])+ by auto 
+      apply(rule goal1(2)[rule_format])+ by auto
 
     note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
     have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) =
       (\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto
-    have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym])
-      apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int .
+    have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[symmetric])
+      apply(subst ifif[symmetric]) apply(subst integrable_restrict_univ) using int .
     have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and>
       ((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) --->
       integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
@@ -5320,7 +6490,7 @@
         unfolding real_norm_def apply(subst abs_of_nonneg) apply(rule *[OF int])
         apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3
         apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]])
-        apply(subst integral_restrict_univ[THEN sym,OF int]) 
+        apply(subst integral_restrict_univ[symmetric,OF int])
         unfolding ifif unfolding integral_restrict_univ[OF int']
         apply(rule integral_subset_le[OF _ int' assms(2)]) using assms(1) by auto
       thus ?case using assms(5) unfolding bounded_iff apply safe
@@ -5341,7 +6511,7 @@
           apply-defer apply(subst norm_minus_commute) by auto
         have *:"\<And>f1 f2 g. abs(f1 - i) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i
           \<longrightarrow> abs(g - i) < e" unfolding real_inner_1_right by arith
-        show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e" 
+        show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e"
           unfolding real_norm_def apply(rule *[rule_format])
           apply(rule **[unfolded real_norm_def])
           apply(rule M[rule_format,of "M + N",unfolded real_norm_def]) apply(rule le_add1)
@@ -5349,10 +6519,10 @@
           apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded real_inner_1_right]])
         proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)\<bullet>1 \<le> (f n x)\<bullet>1"
             apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto
-        next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int]) 
+        next case goal1 show ?case apply(subst integral_restrict_univ[symmetric,OF int])
             unfolding ifif integral_restrict_univ[OF int']
             apply(rule integral_subset_le[OF _ int']) using assms by auto
-        qed qed qed 
+        qed qed qed
     thus ?case apply safe defer apply(drule integral_unique) using i by auto qed
 
   have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
@@ -5364,7 +6534,7 @@
   proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
   next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
   next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
-  next case goal4 thus ?case apply-apply(rule tendsto_diff) 
+  next case goal4 thus ?case apply-apply(rule tendsto_diff)
       using seq_offset[OF assms(3)[rule_format],of x 1] by auto
   next case goal5 thus ?case using assms(4) unfolding bounded_iff
       apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
@@ -5390,7 +6560,7 @@
   note * = conjunctD2[OF this]
   show ?thesis apply rule using integrable_neg[OF *(1)] defer
     using tendsto_minus[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
-    unfolding integral_neg[OF *(1),THEN sym] by auto qed
+    unfolding integral_neg[OF *(1),symmetric] by auto qed
 
 subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
 
@@ -5415,9 +6585,9 @@
 proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr)
     apply(erule_tac x="x - y" in allE) by auto
   have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
-    \<longrightarrow> norm(ig) < dia + e" 
+    \<longrightarrow> norm(ig) < dia + e"
   proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
-      apply(subst real_sum_of_halves[of e,THEN sym]) unfolding add_assoc[symmetric]
+      apply(subst real_sum_of_halves[of e,symmetric]) unfolding add_assoc[symmetric]
       apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
       apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
   qed note norm=this[rule_format]
@@ -5440,7 +6610,7 @@
         apply(rule mult_left_mono) using goal1(3) as by auto
     qed(insert p[unfolded fine_inter],auto) qed
 
-  { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e" 
+  { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
     thus ?thesis apply-apply(rule *[rule_format]) by auto }
   fix e::real assume "e>0" hence e:"e/2 > 0" by auto
   note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
@@ -5505,7 +6675,7 @@
   apply(drule absolutely_integrable_norm) unfolding real_norm_def .
 
 lemma absolutely_integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
-  "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}" 
+  "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}"
   unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
 
 lemma absolutely_integrable_bounded_variation: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
@@ -5520,14 +6690,14 @@
     apply(subst integral_combine_division_topdown[OF _ goal1(2)])
     using integrable_on_subdivision[OF goal1(2)] using assms by auto
   also have "... \<le> integral UNIV (\<lambda>x. norm (f x))"
-    apply(rule integral_subset_le) 
+    apply(rule integral_subset_le)
     using integrable_on_subdivision[OF goal1(2)] using assms by auto
   finally show ?case . qed
 
 lemma helplemma:
   assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s"
   shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e"
-  unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs])
+  unfolding setsum_subtractf[symmetric] apply(rule le_less_trans[OF setsum_abs])
   apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
   using norm_triangle_ineq3 .
 
@@ -5542,7 +6712,7 @@
   show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i])
   proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) `
         {d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format]
-      unfolding setge_def ubs_def by auto 
+      unfolding setge_def ubs_def by auto
     hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))"
       unfolding mem_Collect_eq isUb_def setle_def by(simp add:not_le) then guess d .. note d=conjunctD2[OF this]
     note d' = division_ofD[OF this(1)]
@@ -5567,7 +6737,7 @@
       have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto
       have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI)
       proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l))
-          ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def 
+          ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def
           defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
           apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto
         fix x k assume "(x,k)\<in>p'"
@@ -5590,15 +6760,15 @@
         show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least)
           unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe
         proof- fix y assume y:"y\<in>{a..b}"
-          hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto
+          hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[symmetric] by auto
           then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this]
-          hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto
+          hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[symmetric] by auto
           then guess i .. note i = conjunctD2[OF this]
           have "x\<in>i" using fineD[OF p(3) xl(1)] using k(2)[OF i(1), of x] using i(2) xl(2) by auto
           thus "y\<in>\<Union>{k. \<exists>x. (x, k) \<in> p'}" unfolding p'_def Union_iff apply(rule_tac x="i \<inter> l" in bexI)
             defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI)
-            apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto 
-        qed qed 
+            apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto
+        qed qed
 
       hence "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
         apply-apply(rule g(2)[rule_format]) unfolding tagged_division_of_def apply safe using gp' .
@@ -5625,7 +6795,7 @@
 
       have *:"\<And>sni sni' sf sf'. abs(sf' - sni') < e / 2 \<longrightarrow> i - e / 2 < sni \<and> sni' \<le> i \<and>
         sni \<le> sni' \<and> sf' = sf \<longrightarrow> abs(sf - i) < e" by arith
-      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e" 
+      show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e"
         unfolding real_norm_def apply(rule *[rule_format,OF **],safe) apply(rule d(2))
       proof- case goal1 show ?case unfolding sum_p'
           apply(rule isLubD2[OF i]) using division_of_tagged_division[OF p''] by auto
@@ -5635,7 +6805,7 @@
         proof(rule setsum_mono) case goal1 note k=this
           from d'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
           def d' \<equiv> "{{u..v} \<inter> l |l. l \<in> snd ` p \<and>  ~({u..v} \<inter> l = {})}" note uvab = d'(2)[OF k[unfolded uv]]
-          have "d' division_of {u..v}" apply(subst d'_def) apply(rule division_inter_1) 
+          have "d' division_of {u..v}" apply(subst d'_def) apply(rule division_inter_1)
             apply(rule division_of_tagged_division[OF p(1)]) using uvab .
           hence "norm (integral k f) \<le> setsum (\<lambda>k. norm (integral k f)) d'"
             unfolding uv apply(subst integral_combine_division_topdown[of _ _ d'])
@@ -5653,18 +6823,18 @@
               apply(rule Int_greatest) defer apply(subst goal1(4)) by auto
             hence *:"interior (k \<inter> l) = {}" using snd_p(5)[OF goal1(1-3)] by auto
             from d'(4)[OF k] snd_p(4)[OF goal1(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this
-            show ?case using * unfolding uv inter_interval content_eq_0_interior[THEN sym] by auto
+            show ?case using * unfolding uv inter_interval content_eq_0_interior[symmetric] by auto
           qed finally show ?case .
         qed also have "... = (\<Sum>(i,l)\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (i\<inter>l) f))"
-          apply(subst sum_sum_product[THEN sym],fact) using p'(1) by auto
+          apply(subst sum_sum_product[symmetric],fact) using p'(1) by auto
         also have "... = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (split op \<inter> x) f))"
           unfolding split_def ..
         also have "... = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
-          unfolding * apply(rule setsum_reindex_nonzero[THEN sym,unfolded o_def])
+          unfolding * apply(rule setsum_reindex_nonzero[symmetric,unfolded o_def])
           apply(rule finite_product_dependent) apply(fact,rule finite_imageI,rule p')
           unfolding split_paired_all mem_Collect_eq split_conv o_def
         proof- note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)]
-          fix l1 l2 k1 k2 assume as:"(l1, k1) \<noteq> (l2, k2)"  "l1 \<inter> k1 = l2 \<inter> k2" 
+          fix l1 l2 k1 k2 assume as:"(l1, k1) \<noteq> (l2, k2)"  "l1 \<inter> k1 = l2 \<inter> k2"
             "\<exists>i l. (l1, k1) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
             "\<exists>i l. (l2, k2) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
           hence "l1 \<in> d" "k1 \<in> snd ` p" by auto from d'(4)[OF this(1)] *(1)[OF this(2)]
@@ -5676,7 +6846,7 @@
           moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" using as(2) by auto
           ultimately have "interior(l1 \<inter> k1) = {}" by auto
           thus "norm (integral (l1 \<inter> k1) f) = 0" unfolding uv inter_interval
-            unfolding content_eq_0_interior[THEN sym] by auto
+            unfolding content_eq_0_interior[symmetric] by auto
         qed also have "... = (\<Sum>(x, k)\<in>p'. norm (integral k f))" unfolding sum_p'
           apply(rule setsum_mono_zero_right) apply(subst *)
           apply(rule finite_imageI[OF finite_product_dependent]) apply fact
@@ -5684,7 +6854,7 @@
         proof- case goal2 have "ia \<inter> b = {}" using goal2 unfolding p'alt image_iff Bex_def not_ex
             apply(erule_tac x="(a,ia\<inter>b)" in allE) by auto thus ?case by auto
         next case goal1 thus ?case unfolding p'_def apply safe
-            apply(rule_tac x=i in exI,rule_tac x=l in exI) unfolding snd_conv image_iff 
+            apply(rule_tac x=i in exI,rule_tac x=l in exI) unfolding snd_conv image_iff
             apply safe apply(rule_tac x="(a,l)" in bexI) by auto
         qed finally show ?case .
 
@@ -5705,15 +6875,15 @@
             "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> k1 = k2)"
           from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this
           from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2" by auto
-          hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})" 
+          hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})"
             apply-apply(erule disjE) apply(rule disjI2) defer apply(rule disjI1)
             apply(rule d'(5)[OF as(3-4)],assumption) apply(rule p'(5)[OF as(1-2)]) by auto
           moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" unfolding  as ..
           ultimately have "interior (l1 \<inter> k1) = {}" by auto
           thus "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0" unfolding uv inter_interval
-            unfolding content_eq_0_interior[THEN sym] by auto
+            unfolding content_eq_0_interior[symmetric] by auto
         qed safe also have "... = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))" unfolding Sigma_alt
-          apply(subst sum_sum_product[THEN sym]) apply(rule p', rule,rule d')
+          apply(subst sum_sum_product[symmetric]) apply(rule p', rule,rule d')
           apply(rule setsum_cong2) unfolding split_paired_all split_conv
         proof- fix x l assume as:"(x,l)\<in>p"
           note xl = p'(2-4)[OF this] from this(3) guess u v apply-by(erule exE)+ note uv=this
@@ -5721,7 +6891,7 @@
             apply(rule setsum_cong2) apply(drule d'(4),safe) apply(subst Int_commute)
             unfolding inter_interval uv apply(subst abs_of_nonneg) by auto
           also have "... = setsum content {k\<inter>{u..v}| k. k\<in>d}" unfolding simple_image
-            apply(rule setsum_reindex_nonzero[unfolded o_def,THEN sym]) apply(rule d')
+            apply(rule setsum_reindex_nonzero[unfolded o_def,symmetric]) apply(rule d')
           proof- case goal1 from d'(4)[OF this(1)] d'(4)[OF this(2)]
             guess u1 v1 u2 v2 apply- by(erule exE)+ note uv=this
             have "{} = interior ((k \<inter> y) \<inter> {u..v})" apply(subst interior_inter)
@@ -5738,11 +6908,11 @@
               unfolding ab inter_interval content_eq_0_interior by auto
             thus ?case using goal1(1) using interior_subset[of "k \<inter> {u..v}"] by auto
           qed finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
-            unfolding setsum_left_distrib[THEN sym] real_scaleR_def apply -
+            unfolding setsum_left_distrib[symmetric] real_scaleR_def apply -
             apply(subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
             using xl(2)[unfolded uv] unfolding uv by auto
-        qed finally show ?case . 
-      qed qed qed qed 
+        qed finally show ?case .
+      qed qed qed qed
 
 lemma bounded_variation_absolutely_integrable:  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
   assumes "f integrable_on UNIV" "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
@@ -5755,7 +6925,7 @@
   have f_int:"\<And>a b. f absolutely_integrable_on {a..b}"
     apply(rule bounded_variation_absolutely_integrable_interval[where B=B])
     apply(rule integrable_on_subinterval[OF assms(1)]) defer apply safe
-    apply(rule assms(2)[rule_format]) by auto 
+    apply(rule assms(2)[rule_format]) by auto
   show "((\<lambda>x. norm (f x)) has_integral i) UNIV" apply(subst has_integral_alt',safe)
   proof- case goal1 show ?case using f_int[of a b] by auto
   next case goal2 have "\<exists>y\<in>setsum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> i - e"
@@ -5775,11 +6945,11 @@
       proof- case goal1 have "(\<Sum>k\<in>d. norm (integral k f)) \<le> setsum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
           apply(rule setsum_mono) apply(rule absolutely_integrable_le)
           apply(drule d'(4),safe) by(rule f_int)
-        also have "... = integral (\<Union>d) (\<lambda>x. norm(f x))" 
-          apply(rule integral_combine_division_bottomup[THEN sym])
+        also have "... = integral (\<Union>d) (\<lambda>x. norm(f x))"
+          apply(rule integral_combine_division_bottomup[symmetric])
           apply(rule d) unfolding forall_in_division[OF d(1)] using f_int by auto
-        also have "... \<le> integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)" 
-        proof- case goal1 have "\<Union>d \<subseteq> {a..b}" apply rule apply(drule K(2)[rule_format]) 
+        also have "... \<le> integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)"
+        proof- case goal1 have "\<Union>d \<subseteq> {a..b}" apply rule apply(drule K(2)[rule_format])
             apply(rule ab[unfolded subset_eq,rule_format]) by(auto simp add:dist_norm)
           thus ?case apply- apply(subst if_P,rule) apply(rule integral_subset_le) defer
             apply(rule integrable_on_subdivision[of _ _ _ "{a..b}"])
@@ -5795,7 +6965,7 @@
         have *:"\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> i \<longrightarrow> abs(sf - si) < e / 2
           \<longrightarrow> abs(sf' - di) < e / 2 \<longrightarrow> di < i + e" by arith
         show "integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < i + e" apply(subst if_P,rule)
-        proof(rule *[rule_format]) 
+        proof(rule *[rule_format])
           show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e / 2"
             unfolding split_def apply(rule helplemma) using d2(2)[rule_format,of p]
             using p(1,3) unfolding tagged_division_of_def split_def by auto
@@ -5810,7 +6980,7 @@
             unfolding image_iff apply(rule_tac x="snd ` p" in bexI) unfolding mem_Collect_eq defer
             apply(rule partial_division_of_tagged_division[of _ "{a..b}"])
             using p(1) unfolding tagged_division_of_def by auto
-        qed qed qed(insert K,auto) qed qed 
+        qed qed qed(insert K,auto) qed qed
 
 lemma absolutely_integrable_restrict_univ:
  "(\<lambda>x. if x \<in> s then f x else (0::'a::banach)) absolutely_integrable_on UNIV \<longleftrightarrow> f absolutely_integrable_on s"
@@ -5821,12 +6991,12 @@
   shows "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on s"
 proof- let ?P = "\<And>f g::'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space. f absolutely_integrable_on UNIV \<Longrightarrow>
     g absolutely_integrable_on UNIV \<Longrightarrow> (\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV"
-  { presume as:"PROP ?P" note a = absolutely_integrable_restrict_univ[THEN sym]
+  { presume as:"PROP ?P" note a = absolutely_integrable_restrict_univ[symmetric]
     have *:"\<And>x. (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)
       = (if x \<in> s then f x + g x else 0)" by auto
     show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]] unfolding * . }
   fix f g::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assume assms:"f absolutely_integrable_on UNIV"
-    "g absolutely_integrable_on UNIV" 
+    "g absolutely_integrable_on UNIV"
   note absolutely_integrable_bounded_variation
   from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
   show "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV"
@@ -5837,7 +7007,7 @@
       apply(rule_tac[!] integrable_on_subinterval[of _ UNIV]) using assms by auto
     hence "(\<Sum>k\<in>d. norm (integral k (\<lambda>x. f x + g x))) \<le>
       (\<Sum>k\<in>d. norm (integral k f)) + (\<Sum>k\<in>d. norm (integral k g))" apply-
-      unfolding setsum_addf[THEN sym] apply(rule setsum_mono)
+      unfolding setsum_addf[symmetric] apply(rule setsum_mono)
       apply(subst integral_add) prefer 3 apply(rule norm_triangle_ineq) by auto
     also have "... \<le> B1 + B2" using B(1)[OF goal1] B(2)[OF goal1] by auto
     finally show ?case .
@@ -5852,18 +7022,18 @@
 lemma absolutely_integrable_linear: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space" and h::"'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space"
   assumes "f absolutely_integrable_on s" "bounded_linear h"
   shows "(h o f) absolutely_integrable_on s"
-proof- { presume as:"\<And>f::'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space. \<And>h::'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space. 
+proof- { presume as:"\<And>f::'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space. \<And>h::'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space.
     f absolutely_integrable_on UNIV \<Longrightarrow> bounded_linear h \<Longrightarrow>
-    (h o f) absolutely_integrable_on UNIV" note a = absolutely_integrable_restrict_univ[THEN sym]
+    (h o f) absolutely_integrable_on UNIV" note a = absolutely_integrable_restrict_univ[symmetric]
     show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]]
       unfolding o_def if_distrib linear_simps[OF assms(2)] . }
   fix f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space" and h::"'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space"
-  assume assms:"f absolutely_integrable_on UNIV" "bounded_linear h" 
+  assume assms:"f absolutely_integrable_on UNIV" "bounded_linear h"
   from absolutely_integrable_bounded_variation[OF assms(1)] guess B . note B=this
   from bounded_linear.pos_bounded[OF assms(2)] guess b .. note b=conjunctD2[OF this]
   show "(h o f) absolutely_integrable_on UNIV"
     apply(rule bounded_variation_absolutely_integrable[of _ "B * b"])
-    apply(rule integrable_linear[OF _ assms(2)]) 
+    apply(rule integrable_linear[OF _ assms(2)])
   proof safe case goal2
     have "(\<Sum>k\<in>d. norm (integral k (h \<circ> f))) \<le> setsum (\<lambda>k. norm(integral k f)) d * b"
       unfolding setsum_left_distrib apply(rule setsum_mono)
@@ -5953,14 +7123,14 @@
 proof
   assume ?l thus ?r apply-apply rule defer
     apply(drule absolutely_integrable_vector_abs) by auto
-next 
+next
   assume ?r
   { presume lem:"\<And>f::'n \<Rightarrow> 'm. f integrable_on UNIV \<Longrightarrow>
       (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
     have *:"\<And>x. (\<Sum>i\<in>Basis. \<bar>(if x \<in> s then f x else 0) \<bullet> i\<bar> *\<^sub>R i) =
         (if x\<in>s then (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) else (0::'m))"
       unfolding euclidean_eq_iff[where 'a='m] by auto
-    show ?l apply(subst absolutely_integrable_restrict_univ[THEN sym]) apply(rule lem)
+    show ?l apply(subst absolutely_integrable_restrict_univ[symmetric]) apply(rule lem)
       unfolding integrable_restrict_univ * using `?r` by auto }
   fix f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
   assume assms:"f integrable_on UNIV" "(\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV"
@@ -5976,7 +7146,7 @@
       from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this
       show "\<bar>integral k f \<bullet> i\<bar> \<le> integral k (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) \<bullet> i"
         apply (rule abs_leI)
-        unfolding inner_minus_left[THEN sym] defer apply(subst integral_neg[THEN sym])
+        unfolding inner_minus_left[symmetric] defer apply(subst integral_neg[symmetric])
         defer apply(rule_tac[1-2] integral_component_le[OF i]) apply(rule integrable_neg)
         using integrable_on_subinterval[OF assms(1),of a b]
           integrable_on_subinterval[OF assms(2),of a b] i unfolding ab by auto
@@ -6009,7 +7179,7 @@
   shows "f absolutely_integrable_on s"
 proof- { presume *:"\<And>f::'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space. \<And> g. \<forall>x. norm(f x) \<le> g x \<Longrightarrow> f integrable_on UNIV
     \<Longrightarrow> g integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
-    show ?thesis apply(subst absolutely_integrable_restrict_univ[THEN sym])
+    show ?thesis apply(subst absolutely_integrable_restrict_univ[symmetric])
       apply(rule *[of _ "\<lambda>x. if x\<in>s then g x else 0"])
       using assms unfolding integrable_restrict_univ by auto }
   fix g and f :: "'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
@@ -6018,9 +7188,9 @@
     apply(rule bounded_variation_absolutely_integrable[OF assms(2),where B="integral UNIV g"])
   proof safe case goal1 note d=this and d'=division_ofD[OF this]
     have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>k\<in>d. integral k g)"
-      apply(rule setsum_mono) apply(rule integral_norm_bound_integral) apply(drule_tac[!] d'(4),safe) 
+      apply(rule setsum_mono) apply(rule integral_norm_bound_integral) apply(drule_tac[!] d'(4),safe)
       apply(rule_tac[1-2] integrable_on_subinterval) using assms by auto
-    also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[THEN sym])
+    also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[symmetric])
       apply(rule d,safe) apply(drule d'(4),safe)
       apply(rule integrable_on_subinterval[OF assms(3)]) by auto
     also have "... \<le> integral UNIV g" apply(rule integral_subset_le) defer
@@ -6161,7 +7331,7 @@
       qed
       then guess y .. note y=this[unfolded not_le]
       from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this]
-      
+
       show ?case
         apply (rule_tac x=N in exI)
       proof safe
@@ -6247,7 +7417,7 @@
         case goal1
         thus ?case using assms(3)[rule_format,OF x, of j] by auto
       qed auto
-      
+
       have "\<exists>y\<in>?S. \<not> y \<le> i - r"
       proof (rule ccontr)
         case goal1
@@ -6262,7 +7432,7 @@
       qed
       then guess y .. note y=this[unfolded not_le]
       from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this]
-      
+
       show ?case
         apply (rule_tac x=N in exI)
       proof safe
@@ -6291,7 +7461,7 @@
   have "g integrable_on s \<and>
     ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) ---> integral s g) sequentially"
     apply (rule monotone_convergence_increasing,safe)
-    apply fact 
+    apply fact
   proof -
     show "bounded {integral s (\<lambda>x. Inf {f j x |j. k \<le> j}) |k. True}"
       unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
@@ -6418,7 +7588,7 @@
         show "integral s (\<lambda>x. Inf {f j x |j. n \<le> j}) \<le> integral s (f n)"
         proof (rule integral_le[OF dec1(1) assms(1)], safe)
           fix x
-          assume x: "x \<in> s" 
+          assume x: "x \<in> s"
           have *: "\<And>x y::real. x \<ge> - y \<Longrightarrow> - x \<le> y" by auto
           show "Inf {f j x |j. n \<le> j} \<le> f n x"
             apply (rule cInf_lower[where z="- h x"])
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Fri Sep 06 10:57:27 2013 +0200
@@ -17,11 +17,15 @@
 
 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
 proof -
-  have "(x + 1/2)\<^sup>2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
-  then show ?thesis by (simp add: field_simps power2_eq_square)
+  have "(x + 1/2)\<^sup>2 + 3/4 > 0"
+    using zero_le_power2[of "x+1/2"] by arith
+  then show ?thesis
+    by (simp add: field_simps power2_eq_square)
 qed
 
-lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
+lemma square_continuous:
+  fixes e :: real
+  shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)"
   using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
   apply (auto simp add: power2_eq_square)
   apply (rule_tac x="s" in exI)
@@ -30,7 +34,7 @@
   apply auto
   done
 
-lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y\<^sup>2 ==> sqrt x <= y"
+lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
   using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
 
 lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
@@ -41,46 +45,49 @@
 
 lemma sqrt_even_pow2:
   assumes n: "even n"
-  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
+  shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
 proof -
-  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
-  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
+  from n obtain m where m: "n = 2 * m"
+    unfolding even_mult_two_ex ..
+  from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
     by (simp only: power_mult[symmetric] mult_commute)
-  then show ?thesis  using m by simp
+  then show ?thesis
+    using m by simp
 qed
 
-lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
-  apply (cases "x = 0", simp_all)
+lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
+  apply (cases "x = 0")
+  apply simp_all
   using sqrt_divide_self_eq[of x]
   apply (simp add: inverse_eq_divide field_simps)
   done
 
 text{* Hence derive more interesting properties of the norm. *}
 
-lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
+lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
   by simp (* TODO: delete *)
 
-lemma norm_cauchy_schwarz: "inner x y <= norm x * norm y"
+lemma norm_cauchy_schwarz: "x \<bullet> y \<le> norm x * norm y"
   (* TODO: move to Inner_Product.thy *)
   using Cauchy_Schwarz_ineq2[of x y] by auto
 
 lemma norm_triangle_sub:
   fixes x y :: "'a::real_normed_vector"
-  shows "norm x \<le> norm y  + norm (x - y)"
+  shows "norm x \<le> norm y + norm (x - y)"
   using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
 
-lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
-  by (simp add: norm_eq_sqrt_inner) 
-
-lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
   by (simp add: norm_eq_sqrt_inner)
 
-lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+  by (simp add: norm_eq_sqrt_inner)
+
+lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   apply (subst order_eq_iff)
   apply (auto simp: norm_le)
   done
 
-lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
   by (simp add: norm_eq_sqrt_inner)
 
 text{* Squaring equations and inequalities involving norms.  *}
@@ -88,7 +95,7 @@
 lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
   by (simp only: power2_norm_eq_inner) (* TODO: move? *)
 
-lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a\<^sup>2"
+lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
   by (auto simp add: norm_eq_sqrt_inner)
 
 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
@@ -102,13 +109,13 @@
   then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
 qed
 
-lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a\<^sup>2"
+lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
   done
 
-lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a\<^sup>2"
+lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
@@ -116,16 +123,17 @@
 
 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
   by (metis not_le norm_ge_square)
+
 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
   by (metis norm_le_square not_less)
 
 text{* Dot product in terms of the norm rather than conversely. *}
 
-lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left 
+lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
   inner_scaleR_left inner_scaleR_right
 
 lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
-  unfolding power2_norm_eq_inner inner_simps inner_commute by auto 
+  unfolding power2_norm_eq_inner inner_simps inner_commute by auto
 
 lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
   unfolding power2_norm_eq_inner inner_simps inner_commute
@@ -133,32 +141,37 @@
 
 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
 
-lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
+  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then show ?rhs by simp
 next
   assume ?rhs
-  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
-  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_diff inner_commute)
-  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_diff inner_commute)
-  then show "x = y" by (simp)
+  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
+    by simp
+  then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
+    by (simp add: inner_diff inner_commute)
+  then have "(x - y) \<bullet> (x - y) = 0"
+    by (simp add: field_simps inner_diff inner_commute)
+  then show "x = y" by simp
 qed
 
 lemma norm_triangle_half_r:
-  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
-  using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
+  "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
+  using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
 
 lemma norm_triangle_half_l:
-  assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
+  assumes "norm (x - y) < e / 2"
+    and "norm (x' - (y)) < e / 2"
   shows "norm (x - x') < e"
-  using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
-  unfolding dist_norm[THEN sym] .
-
-lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
+  using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
+  unfolding dist_norm[symmetric] .
+
+lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
   by (rule norm_triangle_ineq [THEN order_trans])
 
-lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
+lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
   by (rule norm_triangle_ineq [THEN le_less_trans])
 
 lemma setsum_clauses:
@@ -191,7 +204,8 @@
 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
 proof
   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
-  then have "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_diff)
+  then have "\<forall>x. x \<bullet> (y - z) = 0"
+    by (simp add: inner_diff)
   then have "(y - z) \<bullet> (y - z) = 0" ..
   then show "y = z" by simp
 qed simp
@@ -199,7 +213,8 @@
 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
 proof
   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
-  then have "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_diff)
+  then have "\<forall>z. (x - y) \<bullet> z = 0"
+    by (simp add: inner_diff)
   then have "(x - y) \<bullet> (x - y) = 0" ..
   then show "x = y" by simp
 qed simp
@@ -237,31 +252,35 @@
   where "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
 
 lemma linearI:
-  assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+  assumes "\<And>x y. f (x + y) = f x + f y"
+    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
   shows "linear f"
   using assms unfolding linear_def by auto
 
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
+lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
   by (simp add: linear_def)
 
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
   by (simp add: linear_def algebra_simps)
 
-lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
+lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
   by (simp add: linear_def)
 
-lemma linear_id: "linear id" by (simp add: linear_def id_def)
-
-lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
+lemma linear_id: "linear id"
+  by (simp add: linear_def id_def)
+
+lemma linear_zero: "linear (\<lambda>x. 0)"
+  by (simp add: linear_def)
 
 lemma linear_compose_setsum:
-  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
+  assumes fS: "finite S"
+    and lS: "\<forall>a \<in> S. linear (f a)"
   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
   using lS
   apply (induct rule: finite_induct[OF fS])
@@ -275,88 +294,100 @@
   apply simp
   done
 
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x"
+lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
   by (simp add: linear_def)
 
-lemma linear_neg: "linear f ==> f (-x) = - f x"
+lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
   using linear_cmul [where c="-1"] by simp
 
-lemma linear_add: "linear f ==> f(x + y) = f x + f y"
+lemma linear_add: "linear f \<Longrightarrow> f(x + y) = f x + f y"
   by (metis linear_def)
 
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+lemma linear_sub: "linear f \<Longrightarrow> f(x - y) = f x - f y"
   by (simp add: diff_minus linear_add linear_neg)
 
 lemma linear_setsum:
-  assumes lf: "linear f" and fS: "finite S"
-  shows "f (setsum g S) = setsum (f o g) S"
-  using fS
-proof (induct rule: finite_induct)
+  assumes lin: "linear f"
+    and fin: "finite S"
+  shows "f (setsum g S) = setsum (f \<circ> g) S"
+  using fin
+proof induct
   case empty
-  then show ?case by (simp add: linear_0[OF lf])
+  then show ?case
+    by (simp add: linear_0[OF lin])
 next
   case (insert x F)
-  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using insert.hyps
-    by simp
-  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
-  also have "\<dots> = setsum (f o g) (insert x F)" using insert.hyps by simp
+  have "f (setsum g (insert x F)) = f (g x + setsum g F)"
+    using insert.hyps by simp
+  also have "\<dots> = f (g x) + f (setsum g F)"
+    using linear_add[OF lin] by simp
+  also have "\<dots> = setsum (f \<circ> g) (insert x F)"
+    using insert.hyps by simp
   finally show ?case .
 qed
 
 lemma linear_setsum_mul:
-  assumes lf: "linear f" and fS: "finite S"
+  assumes lin: "linear f"
+    and fin: "finite S"
   shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
-  using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lf]
+  using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
   by simp
 
 lemma linear_injective_0:
-  assumes lf: "linear f"
+  assumes lin: "linear f"
   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
 proof -
-  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
-  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
+  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
+    by (simp add: inj_on_def)
+  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
+    by simp
   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
-    by (simp add: linear_sub[OF lf])
-  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
+    by (simp add: linear_sub[OF lin])
+  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
+    by auto
   finally show ?thesis .
 qed
 
 
 subsection {* Bilinear functions. *}
 
-definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
-
-lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
+definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
+
+lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
+lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
+lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
+lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_def)
 
-lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
+lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
   by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
 
-lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
+lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
   by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
 
-lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
+lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
   using add_imp_eq[of x y 0] by auto
 
-lemma bilinear_lzero: assumes "bilinear h" shows "h 0 x = 0"
+lemma bilinear_lzero:
+  assumes "bilinear h"
+  shows "h 0 x = 0"
   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
 
-lemma bilinear_rzero: assumes "bilinear h" shows "h x 0 = 0"
+lemma bilinear_rzero:
+  assumes "bilinear h"
+  shows "h x 0 = 0"
   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
 
-lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
+lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
   by (simp  add: diff_minus bilinear_ladd bilinear_lneg)
 
-lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
+lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
   by (simp  add: diff_minus bilinear_radd bilinear_rneg)
 
 lemma bilinear_setsum:
@@ -367,7 +398,8 @@
 proof -
   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
     apply (rule linear_setsum[unfolded o_def])
-    using bh fS apply (auto simp add: bilinear_def)
+    using bh fS
+    apply (auto simp add: bilinear_def)
     done
   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
     apply (rule setsum_cong, simp)
@@ -375,7 +407,8 @@
     using bh fT
     apply (auto simp add: bilinear_def)
     done
-  finally show ?thesis unfolding setsum_cartesian_product .
+  finally show ?thesis
+    unfolding setsum_cartesian_product .
 qed
 
 
@@ -388,13 +421,19 @@
   shows "adjoint f = g"
   unfolding adjoint_def
 proof (rule some_equality)
-  show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
+  show "\<forall>x y. inner (f x) y = inner x (g y)"
+    by (rule assms)
 next
-  fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
-  then have "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
-  then have "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
-  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
-  then have "\<forall>y. h y = g y" by simp
+  fix h
+  assume "\<forall>x y. inner (f x) y = inner x (h y)"
+  then have "\<forall>x y. inner x (g y) = inner x (h y)"
+    using assms by simp
+  then have "\<forall>x y. inner x (g y - h y) = 0"
+    by (simp add: inner_diff_right)
+  then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
+    by simp
+  then have "\<forall>y. h y = g y"
+    by simp
   then show "h = g" by (simp add: ext)
 qed
 
@@ -418,7 +457,7 @@
       unfolding linear_setsum[OF lf finite_Basis]
       by (simp add: linear_cmul[OF lf])
     finally show "f x \<bullet> y = x \<bullet> ?w"
-        by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+      by (simp add: inner_setsum_left inner_setsum_right mult_commute)
   qed
   then show ?thesis
     unfolding adjoint_def choice_iff
@@ -445,18 +484,22 @@
   shows "adjoint (adjoint f) = f"
   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
 
+
 subsection {* Interlude: Some properties of real sets *}
 
-lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
+lemma seq_mono_lemma:
+  assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
+    and "\<forall>n \<ge> m. e n \<le> e m"
   shows "\<forall>n \<ge> m. d n < e m"
-  using assms apply auto
+  using assms
+  apply auto
   apply (erule_tac x="n" in allE)
   apply (erule_tac x="n" in allE)
   apply auto
   done
 
-
-lemma infinite_enumerate: assumes fS: "infinite S"
+lemma infinite_enumerate:
+  assumes fS: "infinite S"
   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
   unfolding subseq_def
   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
@@ -467,53 +510,57 @@
   apply auto
   done
 
-
 lemma triangle_lemma:
-  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x\<^sup>2 <= y\<^sup>2 + z\<^sup>2"
-  shows "x <= y + z"
+  fixes x y z :: real
+  assumes x: "0 \<le> x"
+    and y: "0 \<le> y"
+    and z: "0 \<le> z"
+    and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+  shows "x \<le> y + z"
 proof -
-  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2*y*z + z\<^sup>2" using z y by (simp add: mult_nonneg_nonneg)
-  with xy have th: "x\<^sup>2 \<le> (y+z)\<^sup>2" by (simp add: power2_eq_square field_simps)
-  from y z have yz: "y + z \<ge> 0" by arith
+  have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
+    using z y by (simp add: mult_nonneg_nonneg)
+  with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+    by (simp add: power2_eq_square field_simps)
+  from y z have yz: "y + z \<ge> 0"
+    by arith
   from power2_le_imp_le[OF th yz] show ?thesis .
 qed
 
 
 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
 
-definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
-  where "S hull s = Inter {t. S t \<and> s \<subseteq> t}"
+definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
+  where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
 
 lemma hull_same: "S s \<Longrightarrow> S hull s = s"
   unfolding hull_def by auto
 
-lemma hull_in: "(\<And>T. Ball T S ==> S (Inter T)) ==> S (S hull s)"
+lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
   unfolding hull_def Ball_def by auto
 
-lemma hull_eq: "(\<And>T. Ball T S ==> S (Inter T)) ==> (S hull s) = s \<longleftrightarrow> S s"
+lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
   using hull_same[of S s] hull_in[of S s] by metis
 
-
 lemma hull_hull: "S hull (S hull s) = S hull s"
   unfolding hull_def by blast
 
 lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
   unfolding hull_def by blast
 
-lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
+lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
   unfolding hull_def by blast
 
-lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x ==> (T hull s) \<subseteq> (S hull s)"
+lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
   unfolding hull_def by blast
 
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t ==> (S hull s) \<subseteq> t"
+lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
   unfolding hull_def by blast
 
-lemma subset_hull: "S t ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
+lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
   unfolding hull_def by blast
 
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow>
-    (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
+lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
   unfolding hull_def by auto
 
 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
@@ -527,7 +574,7 @@
   unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
 
 lemma hull_union:
-  assumes T: "\<And>T. Ball T S ==> S (Inter T)"
+  assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
   apply rule
   apply (rule hull_mono)
@@ -541,13 +588,13 @@
 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
   unfolding hull_def by blast
 
-lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
+lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
   by (metis hull_redundant_eq)
 
 
 subsection {* Archimedean properties and useful consequences *}
 
-lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
+lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
   unfolding real_of_nat_def by (rule ex_le_of_nat)
 
 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
@@ -558,60 +605,77 @@
   apply simp
   done
 
-lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
+lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
-  then have h: "1 + real n * x \<le> (1 + x) ^ n" by simp
-  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
-  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
-  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
+  then have h: "1 + real n * x \<le> (1 + x) ^ n"
+    by simp
+  from h have p: "1 \<le> (1 + x) ^ n"
+    using Suc.prems by simp
+  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
+    by simp
+  also have "\<dots> \<le> (1 + x) ^ Suc n"
+    apply (subst diff_le_0_iff_le[symmetric])
     apply (simp add: field_simps)
-    using mult_left_mono[OF p Suc.prems] apply simp
+    using mult_left_mono[OF p Suc.prems]
+    apply simp
     done
-  finally show ?case  by (simp add: real_of_nat_Suc field_simps)
+  finally show ?case
+    by (simp add: real_of_nat_Suc field_simps)
 qed
 
-lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
+lemma real_arch_pow:
+  fixes x :: real
+  assumes x: "1 < x"
+  shows "\<exists>n. y < x^n"
 proof -
-  from x have x0: "x - 1 > 0" by arith
+  from x have x0: "x - 1 > 0"
+    by arith
   from reals_Archimedean3[OF x0, rule_format, of y]
-  obtain n::nat where n:"y < real n * (x - 1)" by metis
+  obtain n :: nat where n: "y < real n * (x - 1)" by metis
   from x0 have x00: "x- 1 \<ge> 0" by arith
   from real_pow_lbound[OF x00, of n] n
   have "y < x^n" by auto
   then show ?thesis by metis
 qed
 
-lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
+lemma real_arch_pow2:
+  fixes x :: real
+  shows "\<exists>n. x < 2^ n"
   using real_arch_pow[of 2 x] by simp
 
 lemma real_arch_pow_inv:
-  assumes y: "(y::real) > 0" and x1: "x < 1"
+  fixes x y :: real
+  assumes y: "y > 0"
+    and x1: "x < 1"
   shows "\<exists>n. x^n < y"
-proof -
-  { assume x0: "x > 0"
-    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
-    from real_arch_pow[OF ix, of "1/y"]
-    obtain n where n: "1/y < (1/x)^n" by blast
-    then have ?thesis using y x0
-      by (auto simp add: field_simps power_divide) }
-  moreover
-  { assume "\<not> x > 0"
-    with y x1 have ?thesis apply auto by (rule exI[where x=1], auto) }
-  ultimately show ?thesis by metis
+proof (cases "x > 0")
+  case True
+  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
+  from real_arch_pow[OF ix, of "1/y"]
+  obtain n where n: "1/y < (1/x)^n" by blast
+  then show ?thesis using y `x > 0`
+    by (auto simp add: field_simps power_divide)
+next
+  case False
+  with y x1 show ?thesis
+    apply auto
+    apply (rule exI[where x=1])
+    apply auto
+    done
 qed
 
 lemma forall_pos_mono:
-  "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
-    (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
   by (metis real_arch_inv)
 
 lemma forall_pos_mono_1:
-  "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
-    (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+  "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+    (\<And>n. P(inverse(real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
   apply (rule forall_pos_mono)
   apply auto
   apply (atomize)
@@ -620,15 +684,20 @@
   done
 
 lemma real_archimedian_rdiv_eq_0:
-  assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
+  assumes x0: "x \<ge> 0"
+    and c: "c \<ge> 0"
+    and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
   shows "x = 0"
-proof -
-  { assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
-    from reals_Archimedean3[OF xp, rule_format, of c]
-    obtain n::nat where n: "c < real n * x" by blast
-    with xc[rule_format, of n] have "n = 0" by arith
-    with n c have False by simp }
-  then show ?thesis by blast
+proof (rule ccontr)
+  assume "x \<noteq> 0"
+  with x0 have xp: "x > 0" by arith
+  from reals_Archimedean3[OF xp, rule_format, of c]
+  obtain n :: nat where n: "c < real n * x"
+    by blast
+  with xc[rule_format, of n] have "n = 0"
+    by arith
+  with n c show False
+    by simp
 qed
 
 
@@ -639,15 +708,17 @@
 
 definition (in real_vector) "span S = (subspace hull S)"
 definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
-abbreviation (in real_vector) "independent s == ~(dependent s)"
+abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
 
 text {* Closure properties of subspaces. *}
 
-lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+lemma subspace_UNIV[simp]: "subspace UNIV"
+  by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
+  by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
   by (metis subspace_def)
 
 lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
@@ -660,7 +731,8 @@
   by (metis diff_minus subspace_add subspace_neg)
 
 lemma (in real_vector) subspace_setsum:
-  assumes sA: "subspace A" and fB: "finite B"
+  assumes sA: "subspace A"
+    and fB: "finite B"
     and f: "\<forall>x\<in> B. f x \<in> A"
   shows "setsum f B \<in> A"
   using  fB f sA
@@ -668,36 +740,39 @@
     (simp add: subspace_def sA, auto simp add: sA subspace_add)
 
 lemma subspace_linear_image:
-  assumes lf: "linear f" and sS: "subspace S"
-  shows "subspace(f ` S)"
+  assumes lf: "linear f"
+    and sS: "subspace S"
+  shows "subspace (f ` S)"
   using lf sS linear_0[OF lf]
   unfolding linear_def subspace_def
   apply (auto simp add: image_iff)
-  apply (rule_tac x="x + y" in bexI, auto)
-  apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
+  apply (rule_tac x="x + y" in bexI)
+  apply auto
+  apply (rule_tac x="c *\<^sub>R x" in bexI)
+  apply auto
   done
 
 lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
   by (auto simp add: subspace_def linear_def linear_0[of f])
 
-lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
+lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
   by (auto simp add: subspace_def linear_def linear_0[of f])
 
 lemma subspace_trivial: "subspace {0}"
   by (simp add: subspace_def)
 
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
   by (simp add: subspace_def)
 
-lemma subspace_Times: "\<lbrakk>subspace A; subspace B\<rbrakk> \<Longrightarrow> subspace (A \<times> B)"
+lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
   unfolding subspace_def zero_prod_def by simp
 
 text {* Properties of span. *}
 
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
   by (metis span_def hull_mono)
 
-lemma (in real_vector) subspace_span: "subspace(span S)"
+lemma (in real_vector) subspace_span: "subspace (span S)"
   unfolding span_def
   apply (rule hull_in)
   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
@@ -705,12 +780,11 @@
   done
 
 lemma (in real_vector) span_clauses:
-  "a \<in> S ==> a \<in> span S"
+  "a \<in> S \<Longrightarrow> a \<in> span S"
   "0 \<in> span S"
-  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+  "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
   "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
-  by (metis span_def hull_subset subset_eq)
-     (metis subspace_span subspace_def)+
+  by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
 
 lemma span_unique:
   "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
@@ -722,12 +796,14 @@
 lemma (in real_vector) span_induct:
   assumes x: "x \<in> span S"
     and P: "subspace P"
-    and SP: "\<And>x. x \<in> S ==> x \<in> P"
+    and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
   shows "x \<in> P"
 proof -
-  from SP have SP': "S \<subseteq> P" by (simp add: subset_eq)
+  from SP have SP': "S \<subseteq> P"
+    by (simp add: subset_eq)
   from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
-  show "x \<in> P" by (metis subset_eq)
+  show "x \<in> P"
+    by (metis subset_eq)
 qed
 
 lemma span_empty[simp]: "span {} = {0}"
@@ -742,7 +818,7 @@
 lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
   unfolding dependent_def by auto
 
-lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
+lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
   apply (clarsimp simp add: dependent_def span_mono)
   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
   apply force
@@ -760,34 +836,46 @@
   using span_induct SP P by blast
 
 inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
-  where
+where
   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
 | span_induct_alt_help_S:
-    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
+    "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
+      (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
 
 lemma span_induct_alt':
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+  assumes h0: "h 0"
+    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
   shows "\<forall>x \<in> span S. h x"
 proof -
-  { fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
+  {
+    fix x :: 'a
+    assume x: "x \<in> span_induct_alt_help S"
     have "h x"
       apply (rule span_induct_alt_help.induct[OF x])
       apply (rule h0)
-      apply (rule hS, assumption, assumption)
-      done }
+      apply (rule hS)
+      apply assumption
+      apply assumption
+      done
+  }
   note th0 = this
-  { fix x assume x: "x \<in> span S"
+  {
+    fix x
+    assume x: "x \<in> span S"
     have "x \<in> span_induct_alt_help S"
     proof (rule span_induct[where x=x and S=S])
-      show "x \<in> span S" using x .
+      show "x \<in> span S" by (rule x)
     next
-      fix x assume xS : "x \<in> S"
-        from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
-        show "x \<in> span_induct_alt_help S" by simp
+      fix x
+      assume xS: "x \<in> S"
+      from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+      show "x \<in> span_induct_alt_help S"
+        by simp
     next
       have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
       moreover
-      { fix x y
+      {
+        fix x y
         assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
         from h have "(x + y) \<in> span_induct_alt_help S"
           apply (induct rule: span_induct_alt_help.induct)
@@ -796,9 +884,11 @@
           apply (rule span_induct_alt_help_S)
           apply assumption
           apply simp
-          done }
+          done
+      }
       moreover
-      { fix c x
+      {
+        fix c x
         assume xt: "x \<in> span_induct_alt_help S"
         then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
           apply (induct rule: span_induct_alt_help.induct)
@@ -808,15 +898,17 @@
           apply assumption
           apply simp
           done }
-      ultimately
-      show "subspace (span_induct_alt_help S)"
+      ultimately show "subspace (span_induct_alt_help S)"
         unfolding subspace_def Ball_def by blast
-    qed }
+    qed
+  }
   with th0 show ?thesis by blast
 qed
 
 lemma span_induct_alt:
-  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
+  assumes h0: "h 0"
+    and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+    and x: "x \<in> span S"
   shows "h x"
   using span_induct_alt'[of h S] h0 hS x by blast
 
@@ -825,35 +917,43 @@
 lemma span_span: "span (span A) = span A"
   unfolding span_def hull_hull ..
 
-lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
-
-lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
+lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
+  by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S"
+  by (metis subspace_span subspace_0)
 
 lemma span_inc: "S \<subseteq> span S"
   by (metis subset_eq span_superset)
 
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
-  unfolding dependent_def apply(rule_tac x=0 in bexI)
-  using assms span_0 by auto
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+lemma (in real_vector) dependent_0:
+  assumes "0 \<in> A"
+  shows "dependent A"
+  unfolding dependent_def
+  apply (rule_tac x=0 in bexI)
+  using assms span_0
+  apply auto
+  done
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
   by (metis subspace_add subspace_span)
 
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
   by (metis subspace_span subspace_mul)
 
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
   by (metis subspace_neg subspace_span)
 
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
   by (metis subspace_span subspace_sub)
 
-lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
+lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
   by (rule subspace_setsum, rule subspace_span)
 
 lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
   apply (auto simp only: span_add span_sub)
-  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
+  apply (subgoal_tac "(x + y) - x \<in> span S")
+  apply simp
   apply (simp only: span_add span_sub)
   done
 
@@ -871,7 +971,8 @@
   show "subspace (f ` span S)"
     using lf subspace_span by (rule subspace_linear_image)
 next
-  fix T assume "f ` S \<subseteq> T" and "subspace T"
+  fix T
+  assume "f ` S \<subseteq> T" and "subspace T"
   then show "f ` span S \<subseteq> T"
     unfolding image_subset_iff_subset_vimage
     by (intro span_minimal subspace_linear_vimage lf)
@@ -904,7 +1005,10 @@
   show "subspace (range (\<lambda>k. k *\<^sub>R x))"
     unfolding subspace_def
     by (auto intro: scaleR_add_left [symmetric])
-  fix T assume "{x} \<subseteq> T" and "subspace T" then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
+next
+  fix T
+  assume "{x} \<subseteq> T" and "subspace T"
+  then show "range (\<lambda>k. k *\<^sub>R x)