src/HOL/Multivariate_Analysis/normarith.ML

-(* Title:      Library/normarith.ML
-   Author:     Amine Chaieb, University of Cambridge
-   Description: A simple decision procedure for linear problems in euclidean space
+(*  Title:      Library/normarith.ML
+    Author:     Amine Chaieb, University of Cambridge
+
+Simple decision procedure for linear problems in Euclidean space.
 *)
 
-  (* Now the norm procedure for euclidean spaces *)
-
-
-signature NORM_ARITH = 
+signature NORM_ARITH =
 sig
  val norm_arith : Proof.context -> conv
  val norm_arith_tac : Proof.context -> int -> tactic
 end
 
-structure NormArith : NORM_ARITH = 
+structure NormArith : NORM_ARITH =
 struct
 
  open Conv;
  val bool_eq = op = : bool *bool -> bool
   fun dest_ratconst t = case term_of t of
    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
  | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
  fun is_ratconst t = can dest_ratconst t
- fun augment_norm b t acc = case term_of t of 
+ fun augment_norm b t acc = case term_of t of
      Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,Thm.dest_arg t) acc
    | _ => acc
  fun find_normedterms t acc = case term_of t of
     @{term "op + :: real => _"}$_$_ =>
             find_normedterms (Thm.dest_arg1 t) (find_normedterms (Thm.dest_arg t) acc)
       | @{term "op * :: real => _"}$_$n =>
             if not (is_ratconst (Thm.dest_arg1 t)) then acc else
-            augment_norm (dest_ratconst (Thm.dest_arg1 t) >=/ Rat.zero) 
+            augment_norm (dest_ratconst (Thm.dest_arg1 t) >=/ Rat.zero)
                       (Thm.dest_arg t) acc
-      | _ => augment_norm true t acc 
+      | _ => augment_norm true t acc
 
  val cterm_lincomb_neg = FuncUtil.Ctermfunc.map Rat.neg
- fun cterm_lincomb_cmul c t = 
+ fun cterm_lincomb_cmul c t =
     if c =/ Rat.zero then FuncUtil.Ctermfunc.empty else FuncUtil.Ctermfunc.map (fn x => x */ c) t
  fun cterm_lincomb_add l r = FuncUtil.Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
  fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
  fun cterm_lincomb_eq l r = FuncUtil.Ctermfunc.is_empty (cterm_lincomb_sub l r)
 
  val int_lincomb_neg = FuncUtil.Intfunc.map Rat.neg
- fun int_lincomb_cmul c t = 
+ fun int_lincomb_cmul c t =
     if c =/ Rat.zero then FuncUtil.Intfunc.empty else FuncUtil.Intfunc.map (fn x => x */ c) t
  fun int_lincomb_add l r = FuncUtil.Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
  fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
  fun int_lincomb_eq l r = FuncUtil.Intfunc.is_empty (int_lincomb_sub l r)
 
-fun vector_lincomb t = case term_of t of 
+fun vector_lincomb t = case term_of t of
    Const(@{const_name plus}, _) $ _ $ _ =>
     cterm_lincomb_add (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
  | Const(@{const_name minus}, _) $ _ $ _ =>
     cterm_lincomb_sub (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
  | Const(@{const_name scaleR}, _)$_$_ =>
     cterm_lincomb_cmul (dest_ratconst (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
  | Const(@{const_name uminus}, _)$_ =>
      cterm_lincomb_neg (vector_lincomb (Thm.dest_arg t))
 (* FIXME: how should we handle numerals?
- | Const(@ {const_name vec},_)$_ => 
-   let 
-     val b = ((snd o HOLogic.dest_number o term_of o Thm.dest_arg) t = 0 
+ | Const(@ {const_name vec},_)$_ =>
+   let
+     val b = ((snd o HOLogic.dest_number o term_of o Thm.dest_arg) t = 0
                handle TERM _=> false)
    in if b then FuncUtil.Ctermfunc.onefunc (t,Rat.one)
       else FuncUtil.Ctermfunc.empty
    end
 *)
  | _ => FuncUtil.Ctermfunc.onefunc (t,Rat.one)
 
  fun vector_lincombs ts =
-  fold_rev 
+  fold_rev
    (fn t => fn fns => case AList.lookup (op aconvc) fns t of
-     NONE => 
-       let val f = vector_lincomb t 
+     NONE =>
+       let val f = vector_lincomb t
        in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
            SOME (_,f') => (t,f') :: fns
-         | NONE => (t,f) :: fns 
+         | NONE => (t,f) :: fns
        end
    | SOME _ => fns) ts []
 
-fun replacenegnorms cv t = case term_of t of 
+fun replacenegnorms cv t = case term_of t of
   @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
-| @{term "op * :: real => _"}$_$_ => 
+| @{term "op * :: real => _"}$_$_ =>
     if dest_ratconst (Thm.dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t
 | _ => reflexive t
-fun flip v eq = 
-  if FuncUtil.Ctermfunc.defined eq v 
+fun flip v eq =
+  if FuncUtil.Ctermfunc.defined eq v
   then FuncUtil.Ctermfunc.update (v, Rat.neg (FuncUtil.Ctermfunc.apply eq v)) eq else eq
-fun allsubsets s = case s of 
+fun allsubsets s = case s of
   [] => [[]]
 |(a::t) => let val res = allsubsets t in
                map (cons a) res @ res end
 fun evaluate env lin =
- FuncUtil.Intfunc.fold (fn (x,c) => fn s => s +/ c */ (FuncUtil.Intfunc.apply env x)) 
+ FuncUtil.Intfunc.fold (fn (x,c) => fn s => s +/ c */ (FuncUtil.Intfunc.apply env x))
    lin Rat.zero
 
 fun solve (vs,eqs) = case (vs,eqs) of
   ([],[]) => SOME (FuncUtil.Intfunc.onefunc (0,Rat.one))
- |(_,eq::oeqs) => 
+ |(_,eq::oeqs) =>
    (case filter (member (op =) vs) (FuncUtil.Intfunc.dom eq) of (*FIXME use find_first here*)
      [] => NONE
-    | v::_ => 
-       if FuncUtil.Intfunc.defined eq v 
-       then 
-        let 
+    | v::_ =>
+       if FuncUtil.Intfunc.defined eq v
+       then
+        let
          val c = FuncUtil.Intfunc.apply eq v
          val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq
-         fun eliminate eqn = if not (FuncUtil.Intfunc.defined eqn v) then eqn 
+         fun eliminate eqn = if not (FuncUtil.Intfunc.defined eqn v) then eqn
                              else int_lincomb_add (int_lincomb_cmul (FuncUtil.Intfunc.apply eqn v) vdef) eqn
         in (case solve (remove (op =) v vs, map eliminate oeqs) of
             NONE => NONE
           | SOME soln => SOME (FuncUtil.Intfunc.update (v, evaluate soln (FuncUtil.Intfunc.delete_safe v vdef)) soln))
         end
        else NONE)
 
 fun combinations k l = if k = 0 then [[]] else
- case l of 
+ case l of
   [] => []
 | h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
 
 
-fun forall2 p l1 l2 = case (l1,l2) of 
+fun forall2 p l1 l2 = case (l1,l2) of
    ([],[]) => true
  | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
  | _ => false;
 
 
 fun vertices vs eqs =
- let 
+ let
   fun vertex cmb = case solve(vs,cmb) of
     NONE => NONE
    | SOME soln => SOME (map (fn v => FuncUtil.Intfunc.tryapplyd soln v Rat.zero) vs)
   val rawvs = map_filter vertex (combinations (length vs) eqs)
-  val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs 
- in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset [] 
- end 
-
-fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m 
+  val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs
+ in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset []
+ end
+
+fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m
 
 fun subsume todo dun = case todo of
  [] => dun
-|v::ovs => 
+|v::ovs =>
    let val dun' = if exists (fn w => subsumes w v) dun then dun
-                  else v::(filter (fn w => not(subsumes v w)) dun) 
-   in subsume ovs dun' 
+                  else v::(filter (fn w => not(subsumes v w)) dun)
+   in subsume ovs dun'
    end;
 
 fun match_mp PQ P = P RS PQ;
 
-fun cterm_of_rat x = 
+fun cterm_of_rat x =
 let val (a, b) = Rat.quotient_of_rat x
-in 
+in
  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
-  else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
+  else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
                    (Numeral.mk_cnumber @{ctyp "real"} a))
         (Numeral.mk_cnumber @{ctyp "real"} b)
 end;
 
 fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
 
 fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm};
 
   (* I think here the static context should be sufficient!! *)
-fun inequality_canon_rule ctxt = 
- let 
+fun inequality_canon_rule ctxt =
+ let
   (* FIXME : Should be computed statically!! *)
-  val real_poly_conv = 
+  val real_poly_conv =
     Semiring_Normalizer.semiring_normalize_wrapper ctxt
      (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
  in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Numeral_Simprocs.field_comp_conv then_conv real_poly_conv)))
 end;
 
- fun absc cv ct = case term_of ct of 
- Abs (v,_, _) => 
+ fun absc cv ct = case term_of ct of
+ Abs (v,_, _) =>
   let val (x,t) = Thm.dest_abs (SOME v) ct
   in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
   end
  | _ => all_conv ct;
 
 fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct;
-fun botc1 conv ct = 
+fun botc1 conv ct =
   ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct;
 
  val apply_pth1 = rewr_conv @{thm pth_1};
  val apply_pth2 = rewr_conv @{thm pth_2};
  val apply_pth3 = rewr_conv @{thm pth_3};

  val apply_ptha = rewr_conv @{thm pth_a};
  val apply_pthb = rewrs_conv @{thms pth_b};
  val apply_pthc = rewrs_conv @{thms pth_c};
  val apply_pthd = try_conv (rewr_conv @{thm pth_d});
 
-fun headvector t = case t of 
+fun headvector t = case t of
   Const(@{const_name plus}, _)$
    (Const(@{const_name scaleR}, _)$l$v)$r => v
  | Const(@{const_name scaleR}, _)$l$v => v
  | _ => error "headvector: non-canonical term"
 
 fun vector_cmul_conv ct =
    ((apply_pth5 then_conv arg1_conv Numeral_Simprocs.field_comp_conv) else_conv
     (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
 
-fun vector_add_conv ct = apply_pth7 ct 
- handle CTERM _ => 
-  (apply_pth8 ct 
-   handle CTERM _ => 
-    (case term_of ct of 
+fun vector_add_conv ct = apply_pth7 ct
+ handle CTERM _ =>
+  (apply_pth8 ct
+   handle CTERM _ =>
+    (case term_of ct of
      Const(@{const_name plus},_)$lt$rt =>
-      let 
-       val l = headvector lt 
+      let
+       val l = headvector lt
        val r = headvector rt
       in (case Term_Ord.fast_term_ord (l,r) of
-         LESS => (apply_pthb then_conv arg_conv vector_add_conv 
+         LESS => (apply_pthb then_conv arg_conv vector_add_conv
                   then_conv apply_pthd) ct
-        | GREATER => (apply_pthc then_conv arg_conv vector_add_conv 
-                     then_conv apply_pthd) ct 
-        | EQUAL => (apply_pth9 then_conv 
-                ((apply_ptha then_conv vector_add_conv) else_conv 
+        | GREATER => (apply_pthc then_conv arg_conv vector_add_conv
+                     then_conv apply_pthd) ct
+        | EQUAL => (apply_pth9 then_conv
+                ((apply_ptha then_conv vector_add_conv) else_conv
               arg_conv vector_add_conv then_conv apply_pthd)) ct)
       end
      | _ => reflexive ct))
 
 fun vector_canon_conv ct = case term_of ct of
  Const(@{const_name plus},_)$_$_ =>
-  let 
+  let
    val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
-   val lth = vector_canon_conv l 
+   val lth = vector_canon_conv l
    val rth = vector_canon_conv r
-   val th = Drule.binop_cong_rule p lth rth 
+   val th = Drule.binop_cong_rule p lth rth
   in fconv_rule (arg_conv vector_add_conv) th end
 
 | Const(@{const_name scaleR}, _)$_$_ =>
-  let 
+  let
    val (p,r) = Thm.dest_comb ct
-   val rth = Drule.arg_cong_rule p (vector_canon_conv r) 
+   val rth = Drule.arg_cong_rule p (vector_canon_conv r)
   in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth
   end
 
 | Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct
 
 | Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct
 
 (* FIXME
-| Const(@{const_name vec},_)$n => 
+| Const(@{const_name vec},_)$n =>
   let val n = Thm.dest_arg ct
-  in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero) 
+  in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero)
      then reflexive ct else apply_pth1 ct
   end
 *)
 | _ => apply_pth1 ct
 

 
 fun fold_rev2 f [] [] z = z
  | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z)
  | fold_rev2 f _ _ _ = raise UnequalLengths;
 
-fun int_flip v eq = 
-  if FuncUtil.Intfunc.defined eq v 
+fun int_flip v eq =
+  if FuncUtil.Intfunc.defined eq v
   then FuncUtil.Intfunc.update (v, Rat.neg (FuncUtil.Intfunc.apply eq v)) eq else eq;
 
 local
  val pth_zero = @{thm norm_zero}
  val tv_n = (ctyp_of_term o Thm.dest_arg o Thm.dest_arg1 o Thm.dest_arg o cprop_of)
              pth_zero
- val concl = Thm.dest_arg o cprop_of 
- fun real_vector_combo_prover ctxt translator (nubs,ges,gts) = 
-  let 
+ val concl = Thm.dest_arg o cprop_of
+ fun real_vector_combo_prover ctxt translator (nubs,ges,gts) =
+  let
    (* FIXME: Should be computed statically!!*)
-   val real_poly_conv = 
+   val real_poly_conv =
       Semiring_Normalizer.semiring_normalize_wrapper ctxt
        (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
    val sources = map (Thm.dest_arg o Thm.dest_arg1 o concl) nubs
-   val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) [] 
-   val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check" 
+   val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) []
+   val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
            else ()
    val dests = distinct (op aconvc) (map snd rawdests)
    val srcfuns = map vector_lincomb sources
-   val destfuns = map vector_lincomb dests 
+   val destfuns = map vector_lincomb dests
    val vvs = fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) (srcfuns @ destfuns) []
    val n = length srcfuns
    val nvs = 1 upto n
    val srccombs = srcfuns ~~ nvs
    fun consider d =
-    let 
+    let
      fun coefficients x =
-      let 
+      let
        val inp = if FuncUtil.Ctermfunc.defined d x then FuncUtil.Intfunc.onefunc (0, Rat.neg(FuncUtil.Ctermfunc.apply d x))
-                      else FuncUtil.Intfunc.empty 
-      in fold_rev (fn (f,v) => fn g => if FuncUtil.Ctermfunc.defined f x then FuncUtil.Intfunc.update (v, FuncUtil.Ctermfunc.apply f x) g else g) srccombs inp 
+                      else FuncUtil.Intfunc.empty
+      in fold_rev (fn (f,v) => fn g => if FuncUtil.Ctermfunc.defined f x then FuncUtil.Intfunc.update (v, FuncUtil.Ctermfunc.apply f x) g else g) srccombs inp
       end
      val equations = map coefficients vvs
      val inequalities = map (fn n => FuncUtil.Intfunc.onefunc (n,Rat.one)) nvs
      fun plausiblevertices f =
-      let 
+      let
        val flippedequations = map (fold_rev int_flip f) equations
        val constraints = flippedequations @ inequalities
        val rawverts = vertices nvs constraints
        fun check_solution v =
-        let 
+        let
           val f = fold_rev2 (curry FuncUtil.Intfunc.update) nvs v (FuncUtil.Intfunc.onefunc (0, Rat.one))
         in forall (fn e => evaluate f e =/ Rat.zero) flippedequations
         end
        val goodverts = filter check_solution rawverts
-       val signfixups = map (fn n => if member (op =) f n then ~1 else 1) nvs 
+       val signfixups = map (fn n => if member (op =) f n then ~1 else 1) nvs
       in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts
       end
-     val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) [] 
+     val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) []
     in subsume allverts []
     end
    fun compute_ineq v =
-    let 
-     val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE 
+    let
+     val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE
                                      else SOME(norm_cmul_rule v t))
-                            (v ~~ nubs) 
+                            (v ~~ nubs)
      fun end_itlist f xs = split_last xs |> uncurry (fold_rev f)
     in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
     end
    val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
                  map (inequality_canon_rule ctxt) nubs @ ges

   in fst (RealArith.real_linear_prover translator
         (map (fn t => instantiate ([(tv_n, ctyp_of_term t)],[]) pth_zero)
             zerodests,
         map (fconv_rule (try_conv (Conv.top_sweep_conv (K norm_canon_conv) ctxt) then_conv
                        arg_conv (arg_conv real_poly_conv))) ges',
-        map (fconv_rule (try_conv (Conv.top_sweep_conv (K norm_canon_conv) ctxt) then_conv 
+        map (fconv_rule (try_conv (Conv.top_sweep_conv (K norm_canon_conv) ctxt) then_conv
                        arg_conv (arg_conv real_poly_conv))) gts))
   end
 in val real_vector_combo_prover = real_vector_combo_prover
 end;
 

  val pth = @{thm norm_imp_pos_and_ge}
  val norm_mp = match_mp pth
  val concl = Thm.dest_arg o cprop_of
  fun conjunct1 th = th RS @{thm conjunct1}
  fun conjunct2 th = th RS @{thm conjunct2}
-fun real_vector_ineq_prover ctxt translator (ges,gts) = 
- let 
+fun real_vector_ineq_prover ctxt translator (ges,gts) =
+ let
 (*   val _ = error "real_vector_ineq_prover: pause" *)
   val ntms = fold_rev find_normedterms (map (Thm.dest_arg o concl) (ges @ gts)) []
   val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
   val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
   fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)

        fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
               asl (map replace_rule ges)
   val gts' = map replace_rule gts
   val nubs = map (conjunct2 o norm_mp) asl
   val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
-  val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1)) 
+  val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1))
   val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1])
   val cps = map (swap o Thm.dest_equals) (cprems_of th11)
   val th12 = instantiate ([], cps) th11
   val th13 = fold Thm.elim_implies (map (reflexive o snd) cps) th12;
  in hd (Variable.export ctxt' ctxt [th13])
- end 
+ end
 in val real_vector_ineq_prover = real_vector_ineq_prover
 end;
 
 local
  val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
  fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS;
   (* FIXME: Lookup in the context every time!!! Fix this !!!*)
  fun splitequation ctxt th acc =
-  let 
+  let
    val real_poly_neg_conv = #neg
        (Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
         (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
    val (th1,th2) = conj_pair(rawrule th)
   in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc

 in fun real_vector_prover ctxt _ translator (eqs,ges,gts) =
      (real_vector_ineq_prover ctxt translator
          (fold_rev (splitequation ctxt) eqs ges,gts), RealArith.Trivial)
 end;
 
-  fun init_conv ctxt = 
-   Simplifier.rewrite (Simplifier.context ctxt 
+  fun init_conv ctxt =
+   Simplifier.rewrite (Simplifier.context ctxt
      (HOL_basic_ss addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
-   then_conv Numeral_Simprocs.field_comp_conv 
+   then_conv Numeral_Simprocs.field_comp_conv
    then_conv nnf_conv
 
  fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
- fun norm_arith ctxt ct = 
-  let 
+ fun norm_arith ctxt ct =
+  let
    val ctxt' = Variable.declare_term (term_of ct) ctxt
    val th = init_conv ctxt' ct
-  in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th)) 
+  in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th))
                 (pure ctxt' (Thm.rhs_of th))
  end
 
- fun norm_arith_tac ctxt = 
+ fun norm_arith_tac ctxt =
    clarify_tac HOL_cs THEN'
    Object_Logic.full_atomize_tac THEN'
    CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i);
 
 end;