(*  Title:      HOL/Real/rat_arith0.ML

     ID:         $Id$

     Author:     Lawrence C Paulson

     Copyright   2004 University of Cambridge

 Simprocs for common factor cancellation & Rational coefficient handling

 Instantiation of the generic linear arithmetic package for type rat.

 *)

 (*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];

 (****Common factor cancellation****)

 (*To quote from Provers/Arith/cancel_numeral_factor.ML:

 This simproc Cancels common coefficients in balanced expressions:

      u*#m ~~ u'*#m'  ==  #n*u ~~ #n'*u'

 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

 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))

      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

   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 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 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 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 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"],

      EqCancelNumeralFactor.proc),

     ("ratless_cancel_numeral_factor",

      ["(l::rat) * m < n", "(l::rat) < m * n"],

      LessCancelNumeralFactor.proc),

     ("ratle_cancel_numeral_factor",

      ["(l::rat) * m <= n", "(l::rat) <= 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)"],

          DivCancelNumeralFactor.proc)

 val rat_cancel_numeral_factors =

     rat_cancel_numeral_factors_relations @

     [rat_cancel_numeral_factors_divide]

 end;

 Addsimprocs rat_cancel_numeral_factors;

 (*examples:

 print_depth 22;

 set timing;

 set trace_simp;

 fun test s = (Goal s; by (Simp_tac 1));

 test "0 <= (y::rat) * -2";

 test "9*x = 12 * (y::rat)";

 test "(9*x) / (12 * (y::rat)) = z";

 test "9*x < 12 * (y::rat)";

 test "9*x <= 12 * (y::rat)";

 test "-99*x = 132 * (y::rat)";

 test "(-99*x) / (132 * (y::rat)) = z";

 test "-99*x < 132 * (y::rat)";

 test "-99*x <= 132 * (y::rat)";

 test "999*x = -396 * (y::rat)";

 test "(999*x) / (-396 * (y::rat)) = z";

 test "999*x < -396 * (y::rat)";

 test "999*x <= -396 * (y::rat)";

 test  "(- ((2::rat) * x) <= 2 * y)";

 test "-99*x = -81 * (y::rat)";

 test "(-99*x) / (-81 * (y::rat)) = z";

 test "-99*x <= -81 * (y::rat)";

 test "-99*x < -81 * (y::rat)";

 test "-2 * x = -1 * (y::rat)";

 test "-2 * x = -(y::rat)";

 test "(-2 * x) / (-1 * (y::rat)) = z";

 test "-2 * x < -(y::rat)";

 test "-2 * x <= -1 * (y::rat)";

 test "-x < -23 * (y::rat)";

 test "-x <= -23 * (y::rat)";

 *)

 (** Declarations for ExtractCommonTerm **)

 local

   open Rat_Numeral_Simprocs

 in

 structure CancelFactorCommon =

   struct

   val mk_sum            = long_mk_prod

   val dest_sum          = dest_prod

   val mk_coeff          = mk_coeff

   val dest_coeff        = dest_coeff

   val find_first        = find_first []

   val trans_tac         = trans_tac

   val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@mult_ac))

   end;

 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

 );

 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 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)"],

      DivideCancelFactor.proc)];

 end;

 Addsimprocs rat_cancel_factor;

 (*examples:

 print_depth 22;

 set timing;

 set trace_simp;

 fun test s = (Goal s; by (Asm_simp_tac 1));

 test "x*k = k*(y::rat)";

 test "k = k*(y::rat)";

 test "a*(b*c) = (b::rat)";

 test "a*(b*c) = d*(b::rat)*(x*a)";

 test "(x*k) / (k*(y::rat)) = (uu::rat)";

 test "(k) / (k*(y::rat)) = (uu::rat)";

 test "(a*(b*c)) / ((b::rat)) = (uu::rat)";

 test "(a*(b*c)) / (d*(b::rat)*(x*a)) = (uu::rat)";

 (*FIXME: what do we do about this?*)

 test "a*(b*c)/(y*z) = d*(b::rat)*(x*a)/z";

 *)

 (****Instantiation of the generic linear arithmetic package****)

 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 mono_ss = simpset() addsimps

                 [add_mono,add_strict_mono,add_less_le_mono,add_le_less_mono];

 val add_mono_thms_ordered_field =

   map (fn s => prove_goal (the_context ()) s

                  (fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1]))

     ["(i < j) & (k = l)   ==> i + k < j + (l::'a::ordered_field)",

      "(i = j) & (k < l)   ==> i + k < j + (l::'a::ordered_field)",

      "(i < j) & (k <= l)  ==> i + k < j + (l::'a::ordered_field)",

      "(i <= j) & (k < l)  ==> i + k < j + (l::'a::ordered_field)",

      "(i < j) & (k < l)   ==> i + k < j + (l::'a::ordered_field)"];

 fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var;

 val rat_mult_mono_thms =

  [(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,

              inst "a" "(number_of ?v)" right_distrib,

              divide_1, 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];

 in

 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 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

                       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_discrete ("Rational.rat",false),

   Simplifier.change_simpset_of (op addsimprocs) [fast_rat_arith_simproc]];

 end;