eliminated hard tabs;
authorwenzelm
Sat, 29 Aug 2009 12:01:25 +0200
changeset 32449 696d64ed85da
parent 32448 a89f876731c5
child 32450 375db037f4d2
eliminated hard tabs;
Admin/isatest/annomaly.ML
doc-src/rail.ML
src/FOL/fologic.ML
src/FOL/intprover.ML
src/FOLP/hypsubst.ML
src/FOLP/simp.ML
src/HOL/Algebra/abstract/Ring2.thy
--- a/Admin/isatest/annomaly.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/Admin/isatest/annomaly.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -20,7 +20,7 @@
   val isabelleHome =
       case OS.Process.getEnv "ISABELLE_HOME"
        of  NONE => OS.FileSys.getDir ()
-	 | SOME home => mkAbsolute home
+         | SOME home => mkAbsolute home
 
   fun noparent [] = []
     | noparent (n :: ns) =
@@ -33,12 +33,12 @@
 
   fun rewrite defPrefix name =
       let val abs = mkAbsolute name
-	  val rel = OS.Path.mkRelative { path = abs, relativeTo = isabelleHome }
-	  val exists = (OS.FileSys.access(abs, nil)
-			handle OS.SysErr _ => false)
+          val rel = OS.Path.mkRelative { path = abs, relativeTo = isabelleHome }
+          val exists = (OS.FileSys.access(abs, nil)
+                        handle OS.SysErr _ => false)
       in  if exists andalso rel <> ""
-	  then isabellePath (toArcs rel)
-	  else defPrefix @ noparent (toArcs name)
+          then isabellePath (toArcs rel)
+          else defPrefix @ noparent (toArcs name)
       end handle OS.Path.Path => defPrefix @ noparent (toArcs name)
 
 in
@@ -49,10 +49,10 @@
         (* should we have different files for different line numbers? *
         val arcs = if line <= 1 then arcs
                    else arcs @ [ "l." ^ Int.toString line ]
-	*)
-	val arcs = if t = "structure Isabelle =\nstruct\nend;"
-		      andalso name = "ML"
-		   then ["empty_Isabelle", "empty" ] else arcs
+        *)
+        val arcs = if t = "structure Isabelle =\nstruct\nend;"
+                      andalso name = "ML"
+                   then ["empty_Isabelle", "empty" ] else arcs
         val _    = AnnoMaLy.nameNextStream arcs
     in  smlnj_use_text tune str_of_pos name_space (line, name) p v t  end;
 
--- a/doc-src/rail.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/doc-src/rail.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -99,7 +99,7 @@
       |> (if markup = "" then I else enclose ("\\" ^ markup ^ "{") "}")
       |> (if ! ThyOutput.quotes then quote else I)
       |> (if ! ThyOutput.display then enclose "\\begin{isabelle}%\n" "%\n\\end{isabelle}"
-	  else hyper o enclose "\\mbox{\\isa{" "}}")), style)
+          else hyper o enclose "\\mbox{\\isa{" "}}")), style)
   else ("Bad " ^ kind ^ " " ^ name, false)
   end;
 end;
@@ -147,8 +147,8 @@
   ) >> (Identifier o enclose "\\isa{" "}" o Output.output o implode) ||
   scan_link >> (decode_link ctxt) >>
     (fn (txt, style) =>
-	if style then Special_Identifier(txt)
-	else Identifier(txt))
+        if style then Special_Identifier(txt)
+        else Identifier(txt))
 end;
 
 fun scan_anot ctxt =
@@ -169,12 +169,12 @@
     val text_sq =
       Scan.repeat (
         Scan.one (fn s =>
-	  s <> "\n" andalso
-	  s <> "\t" andalso
-	  s <> "'" andalso
-	  s <> "\\" andalso
-	  Symbol.is_regular s) ||
-	($$ "\\" |-- $$ "'")
+          s <> "\n" andalso
+          s <> "\t" andalso
+          s <> "'" andalso
+          s <> "\\" andalso
+          Symbol.is_regular s) ||
+        ($$ "\\" |-- $$ "'")
       ) >> implode
   fun quoted scan = $$ "'" |-- scan --| $$ "'";
   in
@@ -305,9 +305,9 @@
   parse_body2 -- ($$$ "*" |-- !!! "body4e expected" (parse_body4e)) >>
     (fn (body1, body2) =>
       if is_empty body2 then
-	add_body(PLUS, new_empty_body, rev_body body1)
+        add_body(PLUS, new_empty_body, rev_body body1)
       else
-	add_body(BAR, new_empty_body, add_body (PLUS, body1, rev_body body2)) ) ||
+        add_body(BAR, new_empty_body, add_body (PLUS, body1, rev_body body2)) ) ||
   parse_body2 -- ($$$ "+" |-- !!! "body4e expected" (parse_body4e)) >>
     (fn (body1, body2) => new_body (PLUS, body1, rev_body body2) ) ||
   parse_body2e
@@ -365,36 +365,36 @@
 fun position_body (body as Body(kind, text, annot, id, bodies), ystart) =
   let fun max (x,y) = if x > y then x else y
     fun set_body_position (Body(kind, text, annot, id, bodies), ystart, ynext) =
-	  Body_Pos(kind, text, annot, id, bodies, ystart, ynext)
+          Body_Pos(kind, text, annot, id, bodies, ystart, ynext)
     fun pos_bodies_cat ([],_,ynext,liste) = (liste, ynext)
       | pos_bodies_cat (x::xs, ystart, ynext, liste) =
-	  if is_kind_of CR x then
-	      (case set_body_position(x, ystart, ynext+1) of
-		body as Body_Pos(_,_,_,_,_,_,ynext1) =>
-		  pos_bodies_cat(xs, ynext1, max(ynext,ynext1), liste@[body])
-	      )
-	  else
-	      (case position_body(x, ystart) of
-		body as Body_Pos(_,_,_,_,_,_,ynext1) =>
-		  pos_bodies_cat(xs, ystart, max(ynext,ynext1), liste@[body])
-	      )
+          if is_kind_of CR x then
+              (case set_body_position(x, ystart, ynext+1) of
+                body as Body_Pos(_,_,_,_,_,_,ynext1) =>
+                  pos_bodies_cat(xs, ynext1, max(ynext,ynext1), liste@[body])
+              )
+          else
+              (case position_body(x, ystart) of
+                body as Body_Pos(_,_,_,_,_,_,ynext1) =>
+                  pos_bodies_cat(xs, ystart, max(ynext,ynext1), liste@[body])
+              )
     fun pos_bodies_bar_plus ([],_,ynext,liste) = (liste, ynext)
       | pos_bodies_bar_plus (x::xs, ystart, ynext, liste) =
-	  (case position_body(x, ystart) of
-	    body as Body_Pos(_,_,_,_,_,_,ynext1) =>
-	      pos_bodies_bar_plus(xs, ynext1, max(ynext,ynext1), liste@[body])
-	  )
+          (case position_body(x, ystart) of
+            body as Body_Pos(_,_,_,_,_,_,ynext1) =>
+              pos_bodies_bar_plus(xs, ynext1, max(ynext,ynext1), liste@[body])
+          )
   in
   (case kind of
     CAT => (case pos_bodies_cat(bodies,ystart,ystart+1,[]) of
-	      (bodiesPos, ynext) =>
-		Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
+              (bodiesPos, ynext) =>
+                Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
   | BAR => (case pos_bodies_bar_plus(bodies,ystart,ystart+1,[]) of
-	      (bodiesPos, ynext) =>
-		Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
+              (bodiesPos, ynext) =>
+                Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
   | PLUS => (case pos_bodies_bar_plus(bodies,ystart,ystart+1,[]) of
-	      (bodiesPos, ynext) =>
-		Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
+              (bodiesPos, ynext) =>
+                Body_Pos(kind, text, annot, id, bodiesPos, ystart, ynext))
   | CR => set_body_position(body, ystart, ystart+3)
   | EMPTY => set_body_position(body, ystart, ystart+1)
   | ANNOTE => set_body_position(body, ystart, ystart+1)
@@ -406,15 +406,15 @@
 fun format_body (Body_Pos(EMPTY,_,_,_,_,_,_), _) = ""
   | format_body (Body_Pos(CAT,_,_,_,bodies,_,_), cent) =
     let fun format_bodies([]) = ""
-	  | format_bodies(x::xs) = format_body (x, "") ^ format_bodies(xs)
+          | format_bodies(x::xs) = format_body (x, "") ^ format_bodies(xs)
     in
       format_bodies(bodies)
     end
   | format_body (Body_Pos(BAR,_,_,_,bodies,_,_),cent) =
     let fun format_bodies([]) = "\\rail@endbar\n"
-	  | format_bodies(x::xs) =
-	      "\\rail@nextbar{" ^ string_of_int(getystart(x)) ^"}\n" ^
-	      format_body(x, "") ^ format_bodies(xs)
+          | format_bodies(x::xs) =
+              "\\rail@nextbar{" ^ string_of_int(getystart(x)) ^"}\n" ^
+              format_body(x, "") ^ format_bodies(xs)
     in
       "\\rail@bar\n" ^ format_body(hd(bodies), "") ^ format_bodies(tl(bodies))
     end
--- a/src/FOL/fologic.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/src/FOL/fologic.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -6,28 +6,28 @@
 
 signature FOLOGIC =
 sig
-  val oT		: typ
-  val mk_Trueprop	: term -> term
-  val dest_Trueprop	: term -> term
-  val not		: term
-  val conj		: term
-  val disj		: term
-  val imp		: term
-  val iff		: term
-  val mk_conj		: term * term -> term
-  val mk_disj		: term * term -> term
-  val mk_imp		: term * term -> term
-  val dest_imp	       	: term -> term*term
-  val dest_conj         : term -> term list
-  val mk_iff		: term * term -> term
-  val dest_iff	       	: term -> term*term
-  val all_const		: typ -> term
-  val mk_all		: term * term -> term
-  val exists_const	: typ -> term
-  val mk_exists		: term * term -> term
-  val eq_const		: typ -> term
-  val mk_eq		: term * term -> term
-  val dest_eq 		: term -> term*term
+  val oT: typ
+  val mk_Trueprop: term -> term
+  val dest_Trueprop: term -> term
+  val not: term
+  val conj: term
+  val disj: term
+  val imp: term
+  val iff: term
+  val mk_conj: term * term -> term
+  val mk_disj: term * term -> term
+  val mk_imp: term * term -> term
+  val dest_imp: term -> term * term
+  val dest_conj: term -> term list
+  val mk_iff: term * term -> term
+  val dest_iff: term -> term * term
+  val all_const: typ -> term
+  val mk_all: term * term -> term
+  val exists_const: typ -> term
+  val mk_exists: term * term -> term
+  val eq_const: typ -> term
+  val mk_eq: term * term -> term
+  val dest_eq: term -> term * term
   val mk_binop: string -> term * term -> term
   val mk_binrel: string -> term * term -> term
   val dest_bin: string -> typ -> term -> term * term
@@ -46,7 +46,8 @@
 fun dest_Trueprop (Const ("Trueprop", _) $ P) = P
   | dest_Trueprop t = raise TERM ("dest_Trueprop", [t]);
 
-(** Logical constants **)
+
+(* Logical constants *)
 
 val not = Const ("Not", oT --> oT);
 val conj = Const("op &", [oT,oT]--->oT);
@@ -80,6 +81,7 @@
 fun exists_const T = Const ("Ex", [T --> oT] ---> oT);
 fun mk_exists (Free(x,T),P) = exists_const T $ (absfree (x,T,P));
 
+
 (* binary oprations and relations *)
 
 fun mk_binop c (t, u) =
@@ -97,5 +99,4 @@
       else raise TERM ("dest_bin " ^ c, [tm])
   | dest_bin c _ tm = raise TERM ("dest_bin " ^ c, [tm]);
 
-
 end;
--- a/src/FOL/intprover.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/src/FOL/intprover.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -79,8 +79,7 @@
 (*One safe or unsafe step. *)
 fun step_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_brls i];
 
-fun step_dup_tac i = FIRST [safe_tac, inst_step_tac i, 
-			    biresolve_tac haz_dup_brls i];
+fun step_dup_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_dup_brls i];
 
 (*Dumb but fast*)
 val fast_tac = SELECT_GOAL (DEPTH_SOLVE (step_tac 1));
--- a/src/FOLP/hypsubst.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/src/FOLP/hypsubst.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -27,7 +27,7 @@
   val inspect_pair        : bool -> term * term -> thm
   end;
 
-functor HypsubstFun(Data: HYPSUBST_DATA): HYPSUBST = 
+functor HypsubstFun(Data: HYPSUBST_DATA): HYPSUBST =
 struct
 
 local open Data in
@@ -43,13 +43,13 @@
     but how could we check for this?*)
 fun inspect_pair bnd (t,u) =
   case (Envir.eta_contract t, Envir.eta_contract u) of
-       (Bound i, _) => if loose(i,u) then raise Match 
+       (Bound i, _) => if loose(i,u) then raise Match
                        else sym         (*eliminates t*)
-     | (_, Bound i) => if loose(i,t) then raise Match 
+     | (_, Bound i) => if loose(i,t) then raise Match
                        else asm_rl      (*eliminates u*)
-     | (Free _, _) => if bnd orelse Logic.occs(t,u) then raise Match 
+     | (Free _, _) => if bnd orelse Logic.occs(t,u) then raise Match
                       else sym          (*eliminates t*)
-     | (_, Free _) => if bnd orelse Logic.occs(u,t) then raise Match 
+     | (_, Free _) => if bnd orelse Logic.occs(u,t) then raise Match
                       else asm_rl       (*eliminates u*)
      | _ => raise Match;
 
@@ -58,7 +58,7 @@
    the rule asm_rl (resp. sym). *)
 fun eq_var bnd =
   let fun eq_var_aux k (Const("all",_) $ Abs(_,_,t)) = eq_var_aux k t
-        | eq_var_aux k (Const("==>",_) $ A $ B) = 
+        | eq_var_aux k (Const("==>",_) $ A $ B) =
               ((k, inspect_pair bnd (dest_eq A))
                       (*Exception Match comes from inspect_pair or dest_eq*)
                handle Match => eq_var_aux (k+1) B)
@@ -70,13 +70,13 @@
 fun gen_hyp_subst_tac bnd = SUBGOAL(fn (Bi,i) =>
       let val n = length(Logic.strip_assums_hyp Bi) - 1
           val (k,symopt) = eq_var bnd Bi
-      in 
+      in
          DETERM
            (EVERY [REPEAT_DETERM_N k (etac rev_mp i),
-		   etac revcut_rl i,
-		   REPEAT_DETERM_N (n-k) (etac rev_mp i),
-		   etac (symopt RS subst) i,
-		   REPEAT_DETERM_N n (rtac imp_intr i)])
+                   etac revcut_rl i,
+                   REPEAT_DETERM_N (n-k) (etac rev_mp i),
+                   etac (symopt RS subst) i,
+                   REPEAT_DETERM_N n (rtac imp_intr i)])
       end
       handle THM _ => no_tac | EQ_VAR => no_tac);
 
--- a/src/FOLP/simp.ML	Sat Aug 29 10:50:04 2009 +0200
+++ b/src/FOLP/simp.ML	Sat Aug 29 12:01:25 2009 +0200
@@ -52,7 +52,7 @@
   val tracing   : bool ref
 end;
 
-functor SimpFun (Simp_data: SIMP_DATA) : SIMP = 
+functor SimpFun (Simp_data: SIMP_DATA) : SIMP =
 struct
 
 local open Simp_data in
@@ -74,12 +74,12 @@
   Similar to match_from_nat_tac, but the net does not contain numbers;
   rewrite rules are not ordered.*)
 fun net_tac net =
-  SUBGOAL(fn (prem,i) => 
+  SUBGOAL(fn (prem,i) =>
           resolve_tac (Net.unify_term net (Logic.strip_assums_concl prem)) i);
 
 (*match subgoal i against possible theorems indexed by lhs in the net*)
 fun lhs_net_tac net =
-  SUBGOAL(fn (prem,i) => 
+  SUBGOAL(fn (prem,i) =>
           biresolve_tac (Net.unify_term net
                        (lhs_of (Logic.strip_assums_concl prem))) i);
 
@@ -110,7 +110,7 @@
 
 (*Get the norm constants from norm_thms*)
 val norms =
-  let fun norm thm = 
+  let fun norm thm =
       case lhs_of(concl_of thm) of
           Const(n,_)$_ => n
         | _ => error "No constant in lhs of a norm_thm"
@@ -144,7 +144,7 @@
 (**** Adding "NORM" tags ****)
 
 (*get name of the constant from conclusion of a congruence rule*)
-fun cong_const cong = 
+fun cong_const cong =
     case head_of (lhs_of (concl_of cong)) of
         Const(c,_) => c
       | _ => ""                 (*a placeholder distinct from const names*);
@@ -156,9 +156,9 @@
 fun add_hidden_vars ccs =
   let fun add_hvars tm hvars = case tm of
               Abs(_,_,body) => OldTerm.add_term_vars(body,hvars)
-            | _$_ => let val (f,args) = strip_comb tm 
+            | _$_ => let val (f,args) = strip_comb tm
                      in case f of
-                            Const(c,T) => 
+                            Const(c,T) =>
                                 if member (op =) ccs c
                                 then fold_rev add_hvars args hvars
                                 else OldTerm.add_term_vars (tm, hvars)
@@ -202,13 +202,13 @@
     val hvs = map (#1 o dest_Var) hvars
     val refl1_tac = refl_tac 1
     fun norm_step_tac st = st |>
-	 (case head_of(rhs_of_eq 1 st) of
-	    Var(ixn,_) => if ixn mem hvs then refl1_tac
-			  else resolve_tac normI_thms 1 ORELSE refl1_tac
-	  | Const _ => resolve_tac normI_thms 1 ORELSE
-		       resolve_tac congs 1 ORELSE refl1_tac
-	  | Free _ => resolve_tac congs 1 ORELSE refl1_tac
-	  | _ => refl1_tac)
+         (case head_of(rhs_of_eq 1 st) of
+            Var(ixn,_) => if ixn mem hvs then refl1_tac
+                          else resolve_tac normI_thms 1 ORELSE refl1_tac
+          | Const _ => resolve_tac normI_thms 1 ORELSE
+                       resolve_tac congs 1 ORELSE refl1_tac
+          | Free _ => resolve_tac congs 1 ORELSE refl1_tac
+          | _ => refl1_tac)
     val add_norm_tac = DEPTH_FIRST (has_fewer_prems nops) norm_step_tac
     val SOME(thm'',_) = Seq.pull(add_norm_tac thm')
 in thm'' end;
@@ -246,9 +246,9 @@
 (** Insertion of congruences and rewrites **)
 
 (*insert a thm in a thm net*)
-fun insert_thm_warn th net = 
+fun insert_thm_warn th net =
   Net.insert_term Thm.eq_thm_prop (concl_of th, th) net
-  handle Net.INSERT => 
+  handle Net.INSERT =>
     (writeln ("Duplicate rewrite or congruence rule:\n" ^
         Display.string_of_thm_without_context th); net);
 
@@ -272,9 +272,9 @@
 (** Deletion of congruences and rewrites **)
 
 (*delete a thm from a thm net*)
-fun delete_thm_warn th net = 
+fun delete_thm_warn th net =
   Net.delete_term Thm.eq_thm_prop (concl_of th, th) net
-  handle Net.DELETE => 
+  handle Net.DELETE =>
     (writeln ("No such rewrite or congruence rule:\n" ^
         Display.string_of_thm_without_context th); net);
 
@@ -337,17 +337,17 @@
     in find_if(tm,0) end;
 
 fun IF1_TAC cong_tac i =
-    let fun seq_try (ifth::ifths,ifc::ifcs) thm = 
+    let fun seq_try (ifth::ifths,ifc::ifcs) thm =
                 (COND (if_rewritable ifc i) (DETERM(rtac ifth i))
                         (seq_try(ifths,ifcs))) thm
               | seq_try([],_) thm = no_tac thm
         and try_rew thm = (seq_try(case_rews,case_consts) ORELSE one_subt) thm
         and one_subt thm =
                 let val test = has_fewer_prems (nprems_of thm + 1)
-                    fun loop thm = 
-			COND test no_tac
+                    fun loop thm =
+                        COND test no_tac
                           ((try_rew THEN DEPTH_FIRST test (refl_tac i))
-			   ORELSE (refl_tac i THEN loop)) thm
+                           ORELSE (refl_tac i THEN loop)) thm
                 in (cong_tac THEN loop) thm end
     in COND (may_match(case_consts,i)) try_rew no_tac end;
 
@@ -381,12 +381,12 @@
 
 (*print lhs of conclusion of subgoal i*)
 fun pr_goal_lhs i st =
-    writeln (Syntax.string_of_term_global (Thm.theory_of_thm st) 
+    writeln (Syntax.string_of_term_global (Thm.theory_of_thm st)
              (lhs_of (prepare_goal i st)));
 
 (*print conclusion of subgoal i*)
 fun pr_goal_concl i st =
-    writeln (Syntax.string_of_term_global (Thm.theory_of_thm st) (prepare_goal i st)) 
+    writeln (Syntax.string_of_term_global (Thm.theory_of_thm st) (prepare_goal i st))
 
 (*print subgoals i to j (inclusive)*)
 fun pr_goals (i,j) st =
@@ -439,7 +439,7 @@
         then writeln (cat_lines
           ("Adding rewrites:" :: map Display.string_of_thm_without_context new_rws))
         else ();
-        (ss,thm,anet',anet::ats,cs) 
+        (ss,thm,anet',anet::ats,cs)
     end;
 
 fun rew(seq,thm,ss,anet,ats,cs, more) = case Seq.pull seq of
@@ -492,7 +492,7 @@
 
 fun EXEC_TAC(ss,fl) (SS{auto_tac,cong_net,simp_net,...}) =
 let val cong_tac = net_tac cong_net
-in fn i => 
+in fn i =>
     (fn thm =>
      if i <= 0 orelse nprems_of thm < i then Seq.empty
      else Seq.single(execute(ss,fl,auto_tac,cong_tac,simp_net,i,thm)))
--- a/src/HOL/Algebra/abstract/Ring2.thy	Sat Aug 29 10:50:04 2009 +0200
+++ b/src/HOL/Algebra/abstract/Ring2.thy	Sat Aug 29 12:01:25 2009 +0200
@@ -241,7 +241,7 @@
 proof (induct n)
   case 0 show ?case by simp
 next
-  case Suc thus ?case by (simp add: add_assoc) 
+  case Suc thus ?case by (simp add: add_assoc)
 qed
 
 lemma natsum_cong [cong]:
@@ -269,21 +269,21 @@
 
 ML {*
   local
-    val lhss = 
+    val lhss =
         ["t + u::'a::ring",
-	 "t - u::'a::ring",
-	 "t * u::'a::ring",
-	 "- t::'a::ring"];
-    fun proc ss t = 
+         "t - u::'a::ring",
+         "t * u::'a::ring",
+         "- t::'a::ring"];
+    fun proc ss t =
       let val rew = Goal.prove (Simplifier.the_context ss) [] []
             (HOLogic.mk_Trueprop
               (HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
                 (fn _ => simp_tac (Simplifier.inherit_context ss ring_ss) 1)
             |> mk_meta_eq;
           val (t', u) = Logic.dest_equals (Thm.prop_of rew);
-      in if t' aconv u 
+      in if t' aconv u
         then NONE
-        else SOME rew 
+        else SOME rew
     end;
   in
     val ring_simproc = Simplifier.simproc @{theory} "ring" lhss (K proc);
@@ -305,7 +305,7 @@
 declare one_not_zero [simp]
 
 lemma zero_not_one [simp]:
-  "0 ~= (1::'a::domain)" 
+  "0 ~= (1::'a::domain)"
 by (rule not_sym) simp
 
 lemma integral_iff: (* not by default a simp rule! *)
@@ -322,7 +322,7 @@
 *)
 (*
 lemma bug: "(b::'a::ring) - (b - a) = a" by simp
-   simproc for rings cannot prove "(a::'a::ring) - (a - b) = b" 
+   simproc for rings cannot prove "(a::'a::ring) - (a - b) = b"
 *)
 lemma m_lcancel:
   assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"
@@ -330,8 +330,8 @@
   assume eq: "a * b = a * c"
   then have "a * (b - c) = 0" by simp
   then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)
-  with prem have "b - c = 0" by auto 
-  then have "b = b - (b - c)" by simp 
+  with prem have "b - c = 0" by auto
+  then have "b = b - (b - c)" by simp
   also have "b - (b - c) = c" by simp
   finally show "b = c" .
 next
@@ -386,7 +386,7 @@
 qed
 
 
-lemma unit_mult: 
+lemma unit_mult:
   "!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"
   apply (unfold dvd_def)
   apply clarify