src/HOL/Real/rat_arith.ML

 (*FIXME DELETE*)
 val rat_mult_strict_left_mono =
     read_instantiate_sg(sign_of (the_context())) [("a","?a::rat")] mult_strict_left_mono;
 
 val rat_mult_left_mono =
-    read_instantiate_sg(sign_of (the_context())) [("a","?a::rat")] mult_left_mono;
-
-
-val rat_number_of_def = thm "rat_number_of_def";
-val diff_rat_def = thm "diff_rat_def";
-
-val rat_numeral_0_eq_0 = thm "rat_numeral_0_eq_0";
-val rat_numeral_1_eq_1 = thm "rat_numeral_1_eq_1";
-val add_rat_number_of = thm "add_rat_number_of";
-val minus_rat_number_of = thm "minus_rat_number_of";
-val diff_rat_number_of = thm "diff_rat_number_of";
-val mult_rat_number_of = thm "mult_rat_number_of";
-val rat_mult_2 = thm "rat_mult_2";
-val rat_mult_2_right = thm "rat_mult_2_right";
-val eq_rat_number_of = thm "eq_rat_number_of";
-val less_rat_number_of = thm "less_rat_number_of";
-val rat_minus_1_eq_m1 = thm "rat_minus_1_eq_m1";
-val rat_mult_minus1 = thm "rat_mult_minus1";
-val rat_mult_minus1_right = thm "rat_mult_minus1_right";
-val rat_add_number_of_left = thm "rat_add_number_of_left";
-val rat_mult_number_of_left = thm "rat_mult_number_of_left";
-val rat_add_number_of_diff1 = thm "rat_add_number_of_diff1";
-val rat_add_number_of_diff2 = thm "rat_add_number_of_diff2";
-
-val rat_add_0_left = thm "rat_add_0_left";
-val rat_add_0_right = thm "rat_add_0_right";
-val rat_mult_1_left = thm "rat_mult_1_left";
-val rat_mult_1_right = thm "rat_mult_1_right";
-
-(*Maps 0 to Numeral0 and 1 to Numeral1 and -(Numeral1) to -1*)
-val rat_numeral_ss =
-    HOL_ss addsimps [rat_numeral_0_eq_0 RS sym, rat_numeral_1_eq_1 RS sym,
-                     rat_minus_1_eq_m1];
-
-fun rename_numerals th =
-    asm_full_simplify rat_numeral_ss (Thm.transfer (the_context ()) th);
-
-
-structure Rat_Numeral_Simprocs =
-struct
-
-(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
-  isn't complicated by the abstract 0 and 1.*)
-val numeral_syms = [rat_numeral_0_eq_0 RS sym, rat_numeral_1_eq_1 RS sym];
-
-(*Utilities*)
-
-val ratT = Type("Rational.rat", []);
-
-fun mk_numeral n = HOLogic.number_of_const ratT $ HOLogic.mk_bin n;
-
-(*Decodes a binary rat constant, or 0, 1*)
-val dest_numeral = Int_Numeral_Simprocs.dest_numeral;
-val find_first_numeral = Int_Numeral_Simprocs.find_first_numeral;
-
-val zero = mk_numeral 0;
-val mk_plus = HOLogic.mk_binop "op +";
-
-val uminus_const = Const ("uminus", ratT --> ratT);
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum []        = zero
-  | mk_sum [t,u]     = mk_plus (t, u)
-  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum []        = zero
-  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
-
-val dest_plus = HOLogic.dest_bin "op +" ratT;
-
-(*decompose additions AND subtractions as a sum*)
-fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (pos, u, ts))
-  | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (not pos, u, ts))
-  | dest_summing (pos, t, ts) =
-        if pos then t::ts else uminus_const$t :: ts;
-
-fun dest_sum t = dest_summing (true, t, []);
-
-val mk_diff = HOLogic.mk_binop "op -";
-val dest_diff = HOLogic.dest_bin "op -" ratT;
-
-val one = mk_numeral 1;
-val mk_times = HOLogic.mk_binop "op *";
-
-fun mk_prod [] = one
-  | mk_prod [t] = t
-  | mk_prod (t :: ts) = if t = one then mk_prod ts
-                        else mk_times (t, mk_prod ts);
-
-val dest_times = HOLogic.dest_bin "op *" ratT;
-
-fun dest_prod t =
-      let val (t,u) = dest_times t
-      in  dest_prod t @ dest_prod u  end
-      handle TERM _ => [t];
-
-(*DON'T do the obvious simplifications; that would create special cases*)
-fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
-
-(*Express t as a product of (possibly) a numeral with other sorted terms*)
-fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
-  | dest_coeff sign t =
-    let val ts = sort Term.term_ord (dest_prod t)
-        val (n, ts') = find_first_numeral [] ts
-                          handle TERM _ => (1, ts)
-    in (sign*n, mk_prod ts') end;
-
-(*Find first coefficient-term THAT MATCHES u*)
-fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
-  | find_first_coeff past u (t::terms) =
-        let val (n,u') = dest_coeff 1 t
-        in  if u aconv u' then (n, rev past @ terms)
-                          else find_first_coeff (t::past) u terms
-        end
-        handle TERM _ => find_first_coeff (t::past) u terms;
-
-
-(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
-val add_0s  = map rename_numerals [rat_add_0_left, rat_add_0_right];
-val mult_1s = map rename_numerals [rat_mult_1_left, rat_mult_1_right] @
-              [rat_mult_minus1, rat_mult_minus1_right];
-
-(*To perform binary arithmetic*)
-val bin_simps =
-    [rat_numeral_0_eq_0 RS sym, rat_numeral_1_eq_1 RS sym,
-     add_rat_number_of, rat_add_number_of_left, minus_rat_number_of,
-     diff_rat_number_of, mult_rat_number_of, rat_mult_number_of_left] @
-    bin_arith_simps @ bin_rel_simps;
-
-(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
-  during re-arrangement*)
-val non_add_bin_simps = 
-    bin_simps \\ [rat_add_number_of_left, add_rat_number_of];
-
-(*To evaluate binary negations of coefficients*)
-val rat_minus_simps = NCons_simps @
-                   [rat_minus_1_eq_m1, minus_rat_number_of,
-                    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
-                    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
-
-(*To let us treat subtraction as addition*)
-val diff_simps = [diff_rat_def, minus_add_distrib, minus_minus];
-
-(*to extract again any uncancelled minuses*)
-val rat_minus_from_mult_simps =
-    [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
-
-(*combine unary minus with numeric literals, however nested within a product*)
-val rat_mult_minus_simps =
-    [mult_assoc, minus_mult_left, minus_mult_commute];
-
-(*Apply the given rewrite (if present) just once*)
-fun trans_tac None      = all_tac
-  | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
-
-(*Final simplification: cancel + and *  *)
-val simplify_meta_eq =
-    Int_Numeral_Simprocs.simplify_meta_eq
-         [add_0, add_0_right,
-          mult_zero_left, mult_zero_right, mult_1, mult_1_right];
-
-fun prep_simproc (name, pats, proc) =
-  Simplifier.simproc (Theory.sign_of (the_context ())) name pats proc;
-
-structure CancelNumeralsCommon =
-  struct
-  val mk_sum            = mk_sum
-  val dest_sum          = dest_sum
-  val mk_coeff          = mk_coeff
-  val dest_coeff        = dest_coeff 1
-  val find_first_coeff  = find_first_coeff []
-  val trans_tac         = trans_tac
-  val norm_tac =
-     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
-                                         rat_minus_simps@add_ac))
-     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@rat_mult_minus_simps))
-     THEN ALLGOALS
-              (simp_tac (HOL_ss addsimps rat_minus_from_mult_simps@
-                                         add_ac@mult_ac))
-  val numeral_simp_tac  = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
-  val simplify_meta_eq  = simplify_meta_eq
-  end;
-
-
-structure EqCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
-  val prove_conv = Bin_Simprocs.prove_conv
-  val mk_bal   = HOLogic.mk_eq
-  val dest_bal = HOLogic.dest_bin "op =" ratT
-  val bal_add1 = eq_add_iff1 RS trans
-  val bal_add2 = eq_add_iff2 RS trans
-);
-
-structure LessCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
-  val prove_conv = Bin_Simprocs.prove_conv
-  val mk_bal   = HOLogic.mk_binrel "op <"
-  val dest_bal = HOLogic.dest_bin "op <" ratT
-  val bal_add1 = less_add_iff1 RS trans
-  val bal_add2 = less_add_iff2 RS trans
-);
-
-structure LeCancelNumerals = CancelNumeralsFun
- (open CancelNumeralsCommon
-  val prove_conv = Bin_Simprocs.prove_conv
-  val mk_bal   = HOLogic.mk_binrel "op <="
-  val dest_bal = HOLogic.dest_bin "op <=" ratT
-  val bal_add1 = le_add_iff1 RS trans
-  val bal_add2 = le_add_iff2 RS trans
-);
-
-val cancel_numerals =
-  map prep_simproc
-   [("rateq_cancel_numerals",
-     ["(l::rat) + m = n", "(l::rat) = m + n",
-      "(l::rat) - m = n", "(l::rat) = m - n",
-      "(l::rat) * m = n", "(l::rat) = m * n"],
-     EqCancelNumerals.proc),
-    ("ratless_cancel_numerals",
-     ["(l::rat) + m < n", "(l::rat) < m + n",
-      "(l::rat) - m < n", "(l::rat) < m - n",
-      "(l::rat) * m < n", "(l::rat) < m * n"],
-     LessCancelNumerals.proc),
-    ("ratle_cancel_numerals",
-     ["(l::rat) + m <= n", "(l::rat) <= m + n",
-      "(l::rat) - m <= n", "(l::rat) <= m - n",
-      "(l::rat) * m <= n", "(l::rat) <= m * n"],
-     LeCancelNumerals.proc)];
-
-
-structure CombineNumeralsData =
-  struct
-  val add               = op + : int*int -> int
-  val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
-  val dest_sum          = dest_sum
-  val mk_coeff          = mk_coeff
-  val dest_coeff        = dest_coeff 1
-  val left_distrib      = combine_common_factor RS trans
-  val prove_conv        = Bin_Simprocs.prove_conv_nohyps
-  val trans_tac         = trans_tac
-  val norm_tac =
-     ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
-                                   diff_simps@rat_minus_simps@add_ac))
-     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@rat_mult_minus_simps))
-     THEN ALLGOALS (simp_tac (HOL_ss addsimps rat_minus_from_mult_simps@
-                                              add_ac@mult_ac))
-  val numeral_simp_tac  = ALLGOALS
-                    (simp_tac (HOL_ss addsimps add_0s@bin_simps))
-  val simplify_meta_eq  =
-        Int_Numeral_Simprocs.simplify_meta_eq (add_0s@mult_1s)
-  end;
-
-structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
-
-val combine_numerals =
-  prep_simproc ("rat_combine_numerals", ["(i::rat) + j", "(i::rat) - j"], CombineNumerals.proc);
-
-
-(** Declarations for ExtractCommonTerm **)
-
-(*this version ALWAYS includes a trailing one*)
-fun long_mk_prod []        = one
-  | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
-
-(*Find first term that matches u*)
-fun find_first past u []         = raise TERM("find_first", [])
-  | find_first past u (t::terms) =
-        if u aconv t then (rev past @ terms)
-        else find_first (t::past) u terms
-        handle TERM _ => find_first (t::past) u terms;
-
-(*Final simplification: cancel + and *  *)
-fun cancel_simplify_meta_eq cancel_th th =
-    Int_Numeral_Simprocs.simplify_meta_eq
-        [rat_mult_1_left, rat_mult_1_right]
-        (([th, cancel_th]) MRS trans);
-
-(*** Making constant folding work for 0 and 1 too ***)
-
-structure RatAbstractNumeralsData =
-  struct
-  val dest_eq         = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
-  val is_numeral      = Bin_Simprocs.is_numeral
-  val numeral_0_eq_0  = rat_numeral_0_eq_0
-  val numeral_1_eq_1  = rat_numeral_1_eq_1
-  val prove_conv      = Bin_Simprocs.prove_conv_nohyps_novars
-  fun norm_tac simps  = ALLGOALS (simp_tac (HOL_ss addsimps simps))
-  val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
-  end
-
-structure RatAbstractNumerals = AbstractNumeralsFun (RatAbstractNumeralsData)
-
-(*For addition, we already have rules for the operand 0.
-  Multiplication is omitted because there are already special rules for
-  both 0 and 1 as operands.  Unary minus is trivial, just have - 1 = -1.
-  For the others, having three patterns is a compromise between just having
-  one (many spurious calls) and having nine (just too many!) *)
-val eval_numerals =
-  map prep_simproc
-   [("rat_add_eval_numerals",
-     ["(m::rat) + 1", "(m::rat) + number_of v"],
-     RatAbstractNumerals.proc add_rat_number_of),
-    ("rat_diff_eval_numerals",
-     ["(m::rat) - 1", "(m::rat) - number_of v"],
-     RatAbstractNumerals.proc diff_rat_number_of),
-    ("rat_eq_eval_numerals",
-     ["(m::rat) = 0", "(m::rat) = 1", "(m::rat) = number_of v"],
-     RatAbstractNumerals.proc eq_rat_number_of),
-    ("rat_less_eval_numerals",
-     ["(m::rat) < 0", "(m::rat) < 1", "(m::rat) < number_of v"],
-     RatAbstractNumerals.proc less_rat_number_of),
-    ("rat_le_eval_numerals",
-     ["(m::rat) <= 0", "(m::rat) <= 1", "(m::rat) <= number_of v"],
-     RatAbstractNumerals.proc le_number_of_eq_not_less)]
-
-end;
-
-
-Addsimprocs Rat_Numeral_Simprocs.eval_numerals;
-Addsimprocs Rat_Numeral_Simprocs.cancel_numerals;
-Addsimprocs [Rat_Numeral_Simprocs.combine_numerals];
-
-
-
-(** Constant folding for rat plus and times **)
-
-(*We do not need
-    structure Rat_Plus_Assoc = Assoc_Fold (Rat_Plus_Assoc_Data);
-  because combine_numerals does the same thing*)
-
-structure Rat_Times_Assoc_Data : ASSOC_FOLD_DATA =
-struct
-  val ss                = HOL_ss
-  val eq_reflection     = eq_reflection
-  val sg_ref    = Sign.self_ref (Theory.sign_of (the_context ()))
-  val T      = Rat_Numeral_Simprocs.ratT
-  val plus   = Const ("op *", [Rat_Numeral_Simprocs.ratT,Rat_Numeral_Simprocs.ratT] ---> Rat_Numeral_Simprocs.ratT)
-  val add_ac = mult_ac
-end;
-
-structure Rat_Times_Assoc = Assoc_Fold (Rat_Times_Assoc_Data);
-
-Addsimprocs [Rat_Times_Assoc.conv];
+ read_instantiate_sg(sign_of (the_context())) [("a","?a::rat")] mult_left_mono;
+
+	val le_number_of_eq = thm"le_number_of_eq";
 
 
 (****Common factor cancellation****)
 
 (*To quote from Provers/Arith/cancel_numeral_factor.ML:

 
 where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
 and d = gcd(m,m') and n=m/d and n'=m'/d.
 *)
 
-local
-  open Rat_Numeral_Simprocs
+val rel_number_of = [eq_number_of_eq, less_number_of_eq_neg, le_number_of_eq]
+
+local open Int_Numeral_Simprocs
 in
-
-val rel_rat_number_of = [eq_rat_number_of, less_rat_number_of,
-                          le_number_of_eq_not_less]
 
 structure CancelNumeralFactorCommon =
   struct
   val mk_coeff          = mk_coeff
   val dest_coeff        = dest_coeff 1
   val trans_tac         = trans_tac
   val norm_tac =
-     ALLGOALS (simp_tac (HOL_ss addsimps rat_minus_from_mult_simps @ mult_1s))
-     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@rat_mult_minus_simps))
+     ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps @ mult_1s))
+     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@mult_minus_simps))
      THEN ALLGOALS (simp_tac (HOL_ss addsimps mult_ac))
   val numeral_simp_tac  =
-         ALLGOALS (simp_tac (HOL_ss addsimps rel_rat_number_of@bin_simps))
-  val simplify_meta_eq  = simplify_meta_eq
+         ALLGOALS (simp_tac (HOL_ss addsimps rel_number_of@bin_simps))
+  val simplify_meta_eq  = 
+	Int_Numeral_Simprocs.simplify_meta_eq
+	     [add_0, add_0_right,
+	      mult_zero_left, mult_zero_right, mult_1, mult_1_right];
   end
 
 structure DivCancelNumeralFactor = CancelNumeralFactorFun
  (open CancelNumeralFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_binop "HOL.divide"
-  val dest_bal = HOLogic.dest_bin "HOL.divide" Rat_Numeral_Simprocs.ratT
+  val dest_bal = HOLogic.dest_bin "HOL.divide" Term.dummyT
   val cancel = mult_divide_cancel_left RS trans
   val neg_exchanges = false
 )
 
 structure EqCancelNumeralFactor = CancelNumeralFactorFun
  (open CancelNumeralFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_eq
-  val dest_bal = HOLogic.dest_bin "op =" Rat_Numeral_Simprocs.ratT
-  val cancel = mult_cancel_left RS trans
+  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+  val cancel = field_mult_cancel_left RS trans
   val neg_exchanges = false
 )
 
 structure LessCancelNumeralFactor = CancelNumeralFactorFun
  (open CancelNumeralFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_binrel "op <"
-  val dest_bal = HOLogic.dest_bin "op <" Rat_Numeral_Simprocs.ratT
+  val dest_bal = HOLogic.dest_bin "op <" Term.dummyT
   val cancel = mult_less_cancel_left RS trans
   val neg_exchanges = true
 )
 
 structure LeCancelNumeralFactor = CancelNumeralFactorFun
  (open CancelNumeralFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_binrel "op <="
-  val dest_bal = HOLogic.dest_bin "op <=" Rat_Numeral_Simprocs.ratT
+  val dest_bal = HOLogic.dest_bin "op <=" Term.dummyT
   val cancel = mult_le_cancel_left RS trans
   val neg_exchanges = true
 )
 
-val rat_cancel_numeral_factors_relations =
-  map prep_simproc
-   [("rateq_cancel_numeral_factor",
-     ["(l::rat) * m = n", "(l::rat) = m * n"],
+val field_cancel_numeral_factors_relations =
+  map Bin_Simprocs.prep_simproc
+   [("field_eq_cancel_numeral_factor",
+     ["(l::'a::{field,number_ring}) * m = n",
+      "(l::'a::{field,number_ring}) = m * n"],
      EqCancelNumeralFactor.proc),
-    ("ratless_cancel_numeral_factor",
-     ["(l::rat) * m < n", "(l::rat) < m * n"],
+    ("field_less_cancel_numeral_factor",
+     ["(l::'a::{ordered_field,number_ring}) * m < n",
+      "(l::'a::{ordered_field,number_ring}) < m * n"],
      LessCancelNumeralFactor.proc),
-    ("ratle_cancel_numeral_factor",
-     ["(l::rat) * m <= n", "(l::rat) <= m * n"],
+    ("field_le_cancel_numeral_factor",
+     ["(l::'a::{ordered_field,number_ring}) * m <= n",
+      "(l::'a::{ordered_field,number_ring}) <= m * n"],
      LeCancelNumeralFactor.proc)]
 
-val rat_cancel_numeral_factors_divide = prep_simproc
-        ("ratdiv_cancel_numeral_factor",
-         ["((l::rat) * m) / n", "(l::rat) / (m * n)",
-          "((number_of v)::rat) / (number_of w)"],
+val field_cancel_numeral_factors_divide = 
+    Bin_Simprocs.prep_simproc
+        ("field_cancel_numeral_factor",
+         ["((l::'a::{field,number_ring}) * m) / n",
+          "(l::'a::{field,number_ring}) / (m * n)",
+          "((number_of v)::'a::{field,number_ring}) / (number_of w)"],
          DivCancelNumeralFactor.proc)
 
-val rat_cancel_numeral_factors =
-    rat_cancel_numeral_factors_relations @
-    [rat_cancel_numeral_factors_divide]
+val field_cancel_numeral_factors =
+    field_cancel_numeral_factors_relations @
+    [field_cancel_numeral_factors_divide]
 
 end;
 
-Addsimprocs rat_cancel_numeral_factors;
+Addsimprocs field_cancel_numeral_factors;
 
 
 (*examples:
 print_depth 22;
 set timing;

 *)
 
 
 (** Declarations for ExtractCommonTerm **)
 
-local
-  open Rat_Numeral_Simprocs
+local open Int_Numeral_Simprocs
 in
 
 structure CancelFactorCommon =
   struct
   val mk_sum            = long_mk_prod

   val find_first        = find_first []
   val trans_tac         = trans_tac
   val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@mult_ac))
   end;
 
+(*This version works for all fields, including unordered ones (complex).
+  The version declared in int_factor_simprocs.ML is for integers.*)
 structure EqCancelFactor = ExtractCommonTermFun
  (open CancelFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_eq
-  val dest_bal = HOLogic.dest_bin "op =" Rat_Numeral_Simprocs.ratT
-  val simplify_meta_eq  = cancel_simplify_meta_eq mult_cancel_left
+  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
+  val simplify_meta_eq  = cancel_simplify_meta_eq field_mult_cancel_left
 );
 
 
+(*This version works for fields, with the generic divides operator (/).
+  The version declared in int_factor_simprocs.ML for integers with div.*)
 structure DivideCancelFactor = ExtractCommonTermFun
  (open CancelFactorCommon
   val prove_conv = Bin_Simprocs.prove_conv
   val mk_bal   = HOLogic.mk_binop "HOL.divide"
-  val dest_bal = HOLogic.dest_bin "HOL.divide" Rat_Numeral_Simprocs.ratT
+  val dest_bal = HOLogic.dest_bin "HOL.divide" Term.dummyT
   val simplify_meta_eq  = cancel_simplify_meta_eq mult_divide_cancel_eq_if
 );
 
-val rat_cancel_factor =
-  map prep_simproc
-   [("rat_eq_cancel_factor", ["(l::rat) * m = n", "(l::rat) = m * n"], EqCancelFactor.proc),
-    ("rat_divide_cancel_factor", ["((l::rat) * m) / n", "(l::rat) / (m * n)"],
+val field_cancel_factor =
+  map Bin_Simprocs.prep_simproc
+   [("field_eq_cancel_factor",
+     ["(l::'a::field) * m = n", "(l::'a::field) = m * n"], 
+     EqCancelFactor.proc),
+    ("field_divide_cancel_factor",
+     ["((l::'a::field) * m) / n", "(l::'a::field) / (m * n)"],
      DivideCancelFactor.proc)];
 
 end;
 
-Addsimprocs rat_cancel_factor;
+Addsimprocs field_cancel_factor;
 
 
 (*examples:
 print_depth 22;
 set timing;

 test "a*(b*c)/(y*z) = d*(b::rat)*(x*a)/z";
 *)
 
 
 
-(****Instantiation of the generic linear arithmetic package****)
+(****Instantiation of the generic linear arithmetic package for fields****)
 
 
 local
 
-(* reduce contradictory <= to False *)
-val add_rules = 
-    [order_less_irrefl, rat_numeral_0_eq_0, rat_numeral_1_eq_1,
-     rat_minus_1_eq_m1, 
-     add_rat_number_of, minus_rat_number_of, diff_rat_number_of,
-     mult_rat_number_of, eq_rat_number_of, less_rat_number_of];
-
-val simprocs = [Rat_Times_Assoc.conv, Rat_Numeral_Simprocs.combine_numerals,
-                rat_cancel_numeral_factors_divide]@
-               Rat_Numeral_Simprocs.cancel_numerals @
-               Rat_Numeral_Simprocs.eval_numerals;
+val simprocs = [field_cancel_numeral_factors_divide]
 
 val mono_ss = simpset() addsimps
                 [add_mono,add_strict_mono,add_less_le_mono,add_le_less_mono];
 
 val add_mono_thms_ordered_field =

  [(rat_mult_strict_left_mono,
    cvar(rat_mult_strict_left_mono, hd(tl(prems_of rat_mult_strict_left_mono)))),
   (rat_mult_left_mono,
    cvar(rat_mult_left_mono, hd(tl(prems_of rat_mult_left_mono))))]
 
-val simps = [True_implies_equals,
+val simps = [order_less_irrefl, True_implies_equals,
              inst "a" "(number_of ?v)" right_distrib,
-             divide_1, times_divide_eq_right, times_divide_eq_left,
+             divide_1, divide_zero_left,
+             times_divide_eq_right, times_divide_eq_left,
 	     of_int_0, of_int_1, of_int_add, of_int_minus, of_int_diff,
-	     of_int_mult, of_int_of_nat_eq, rat_number_of_def];
+	     of_int_mult, of_int_of_nat_eq];
 
 in
 
-val fast_rat_arith_simproc = Simplifier.simproc (Theory.sign_of(the_context()))
+val fast_rat_arith_simproc = 
+ Simplifier.simproc (Theory.sign_of(the_context()))
   "fast_rat_arith" ["(m::rat) < n","(m::rat) <= n", "(m::rat) = n"]
   Fast_Arith.lin_arith_prover;
 
 val nat_inj_thms = [of_nat_le_iff RS iffD2, of_nat_less_iff RS iffD2,
                     of_nat_eq_iff RS iffD2];
 
 val int_inj_thms = [of_int_le_iff RS iffD2, of_int_less_iff RS iffD2,
                     of_int_eq_iff RS iffD2];
+
+val ratT = Type("Rational.rat", []);
 
 val rat_arith_setup =
  [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
    {add_mono_thms = add_mono_thms @ add_mono_thms_ordered_field,
     mult_mono_thms = mult_mono_thms @ rat_mult_mono_thms,
     inj_thms = int_inj_thms @ inj_thms,
     lessD = lessD,  (*Can't change LA_Data_Ref.lessD: the rats are dense!*)
-    simpset = simpset addsimps add_rules
-                      addsimps simps
+    simpset = simpset addsimps simps
                       addsimprocs simprocs}),
-  arith_inj_const("IntDef.of_nat", HOLogic.natT --> Rat_Numeral_Simprocs.ratT),
-  arith_inj_const("IntDef.of_int", HOLogic.intT --> Rat_Numeral_Simprocs.ratT),
+  arith_inj_const("IntDef.of_nat", HOLogic.natT --> ratT),
+  arith_inj_const("IntDef.of_int", HOLogic.intT --> ratT),
   arith_discrete ("Rational.rat",false),
   Simplifier.change_simpset_of (op addsimprocs) [fast_rat_arith_simproc]];
 
-
 end;
-
-