(*  Title:      HOL/Integ/int_arith1.ML

     ID:         $Id$

     Authors:    Larry Paulson and Tobias Nipkow

 Simprocs and decision procedure for linear arithmetic.

 *)

 (** Misc ML bindings **)

 val NCons_Pls = thm"NCons_Pls";

 val NCons_Min = thm"NCons_Min";

 val NCons_BIT = thm"NCons_BIT";

 val number_of_Pls = thm"number_of_Pls";

 val number_of_Min = thm"number_of_Min";

 val number_of_BIT = thm"number_of_BIT";

 val bin_succ_Pls = thm"bin_succ_Pls";

 val bin_succ_Min = thm"bin_succ_Min";

 val bin_succ_BIT = thm"bin_succ_BIT";

 val bin_pred_Pls = thm"bin_pred_Pls";

 val bin_pred_Min = thm"bin_pred_Min";

 val bin_pred_BIT = thm"bin_pred_BIT";

 val bin_minus_Pls = thm"bin_minus_Pls";

 val bin_minus_Min = thm"bin_minus_Min";

 val bin_minus_BIT = thm"bin_minus_BIT";

 val bin_add_Pls = thm"bin_add_Pls";

 val bin_add_Min = thm"bin_add_Min";

 val bin_mult_Pls = thm"bin_mult_Pls";

 val bin_mult_Min = thm"bin_mult_Min";

 val bin_mult_BIT = thm"bin_mult_BIT";

 val neg_def = thm "neg_def";

 val iszero_def = thm "iszero_def";

 val NCons_Pls_0 = thm"NCons_Pls_0";

 val NCons_Pls_1 = thm"NCons_Pls_1";

 val NCons_Min_0 = thm"NCons_Min_0";

 val NCons_Min_1 = thm"NCons_Min_1";

 val bin_succ_1 = thm"bin_succ_1";

 val bin_succ_0 = thm"bin_succ_0";

 val bin_pred_1 = thm"bin_pred_1";

 val bin_pred_0 = thm"bin_pred_0";

 val bin_minus_1 = thm"bin_minus_1";

 val bin_minus_0 = thm"bin_minus_0";

 val bin_add_BIT_11 = thm"bin_add_BIT_11";

 val bin_add_BIT_10 = thm"bin_add_BIT_10";

 val bin_add_BIT_0 = thm"bin_add_BIT_0";

 val bin_add_Pls_right = thm"bin_add_Pls_right";

 val bin_add_Min_right = thm"bin_add_Min_right";

 val bin_add_BIT_BIT = thm"bin_add_BIT_BIT";

 val bin_mult_1 = thm"bin_mult_1";

 val bin_mult_0 = thm"bin_mult_0";

 val number_of_NCons = thm"number_of_NCons";

 val number_of_succ = thm"number_of_succ";

 val number_of_pred = thm"number_of_pred";

 val number_of_minus = thm"number_of_minus";

 val number_of_add = thm"number_of_add";

 val diff_number_of_eq = thm"diff_number_of_eq";

 val number_of_mult = thm"number_of_mult";

 val double_number_of_BIT = thm"double_number_of_BIT";

 val numeral_0_eq_0 = thm"numeral_0_eq_0";

 val numeral_1_eq_1 = thm"numeral_1_eq_1";

 val numeral_m1_eq_minus_1 = thm"numeral_m1_eq_minus_1";

 val mult_minus1 = thm"mult_minus1";

 val mult_minus1_right = thm"mult_minus1_right";

 val minus_number_of_mult = thm"minus_number_of_mult";

 val zero_less_nat_eq = thm"zero_less_nat_eq";

 val eq_number_of_eq = thm"eq_number_of_eq";

 val iszero_number_of_Pls = thm"iszero_number_of_Pls";

 val nonzero_number_of_Min = thm"nonzero_number_of_Min";

 val iszero_number_of_BIT = thm"iszero_number_of_BIT";

 val iszero_number_of_0 = thm"iszero_number_of_0";

 val iszero_number_of_1 = thm"iszero_number_of_1";

 val less_number_of_eq_neg = thm"less_number_of_eq_neg";

 val le_number_of_eq = thm"le_number_of_eq";

 val not_neg_number_of_Pls = thm"not_neg_number_of_Pls";

 val neg_number_of_Min = thm"neg_number_of_Min";

 val neg_number_of_BIT = thm"neg_number_of_BIT";

 val le_number_of_eq_not_less = thm"le_number_of_eq_not_less";

 val abs_number_of = thm"abs_number_of";

 val number_of_reorient = thm"number_of_reorient";

 val add_number_of_left = thm"add_number_of_left";

 val mult_number_of_left = thm"mult_number_of_left";

 val add_number_of_diff1 = thm"add_number_of_diff1";

 val add_number_of_diff2 = thm"add_number_of_diff2";

 val less_iff_diff_less_0 = thm"less_iff_diff_less_0";

 val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0";

 val le_iff_diff_le_0 = thm"le_iff_diff_le_0";

 val NCons_simps = thms"NCons_simps";

 val bin_arith_extra_simps = thms"bin_arith_extra_simps";

 val bin_arith_simps = thms"bin_arith_simps";

 val bin_rel_simps = thms"bin_rel_simps";

 val zless_imp_add1_zle = thm "zless_imp_add1_zle";

 val combine_common_factor = thm"combine_common_factor";

 val eq_add_iff1 = thm"eq_add_iff1";

 val eq_add_iff2 = thm"eq_add_iff2";

 val less_add_iff1 = thm"less_add_iff1";

 val less_add_iff2 = thm"less_add_iff2";

 val le_add_iff1 = thm"le_add_iff1";

 val le_add_iff2 = thm"le_add_iff2";

 val arith_special = thms"arith_special";

 structure Bin_Simprocs =

   struct

   fun prove_conv tacs sg (hyps: thm list) xs (t, u) =

     if t aconv u then None

     else

       let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))

       in Some (Tactic.prove sg xs [] eq (K (EVERY tacs))) end

   fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];

   fun prove_conv_nohyps_novars tacs sg = prove_conv tacs sg [] [];

   fun prep_simproc (name, pats, proc) =

     Simplifier.simproc (Theory.sign_of (the_context())) name pats proc;

   fun is_numeral (Const("Numeral.number_of", _) $ w) = true

     | is_numeral _ = false

   fun simplify_meta_eq f_number_of_eq f_eq =

       mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)

   (*reorientation simprules using ==, for the following simproc*)

   val meta_zero_reorient = zero_reorient RS eq_reflection

   val meta_one_reorient = one_reorient RS eq_reflection

   val meta_number_of_reorient = number_of_reorient RS eq_reflection

   (*reorientation simplification procedure: reorients (polymorphic) 

     0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)

   fun reorient_proc sg _ (_ $ t $ u) =

     case u of

 	Const("0", _) => None

       | Const("1", _) => None

       | Const("Numeral.number_of", _) $ _ => None

       | _ => Some (case t of

 		  Const("0", _) => meta_zero_reorient

 		| Const("1", _) => meta_one_reorient

 		| Const("Numeral.number_of", _) $ _ => meta_number_of_reorient)

   val reorient_simproc = 

       prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)

   end;

 Addsimps arith_special;

 Addsimprocs [Bin_Simprocs.reorient_simproc];

 structure Int_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 = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];

 (** New term ordering so that AC-rewriting brings numerals to the front **)

 (*Order integers by absolute value and then by sign. The standard integer

   ordering is not well-founded.*)

 fun num_ord (i,j) =

       (case Int.compare (abs i, abs j) of

             EQUAL => Int.compare (Int.sign i, Int.sign j) 

           | ord => ord);

 (*This resembles Term.term_ord, but it puts binary numerals before other

   non-atomic terms.*)

 local open Term 

 in 

 fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =

       (case numterm_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)

   | numterm_ord

      (Const("Numeral.number_of", _) $ v, Const("Numeral.number_of", _) $ w) =

      num_ord (HOLogic.dest_binum v, HOLogic.dest_binum w)

   | numterm_ord (Const("Numeral.number_of", _) $ _, _) = LESS

   | numterm_ord (_, Const("Numeral.number_of", _) $ _) = GREATER

   | numterm_ord (t, u) =

       (case Int.compare (size_of_term t, size_of_term u) of

         EQUAL =>

           let val (f, ts) = strip_comb t and (g, us) = strip_comb u in

             (case hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)

           end

       | ord => ord)

 and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)

 end;

 fun numtermless tu = (numterm_ord tu = LESS);

 (*Defined in this file, but perhaps needed only for simprocs of type nat.*)

 val num_ss = HOL_ss settermless numtermless;

 (** Utilities **)

 fun mk_numeral T n = HOLogic.number_of_const T $ HOLogic.mk_bin n;

 (*Decodes a binary INTEGER*)

 fun dest_numeral (Const("0", _)) = 0

   | dest_numeral (Const("1", _)) = 1

   | dest_numeral (Const("Numeral.number_of", _) $ w) =

      (HOLogic.dest_binum w

       handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))

   | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);

 fun find_first_numeral past (t::terms) =

         ((dest_numeral t, rev past @ terms)

          handle TERM _ => find_first_numeral (t::past) terms)

   | find_first_numeral past [] = raise TERM("find_first_numeral", []);

 val mk_plus = HOLogic.mk_binop "op +";

 fun mk_minus t = 

   let val T = Term.fastype_of t

   in Const ("uminus", T --> T) $ t

   end;

 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)

 fun mk_sum T []        = mk_numeral T 0

   | mk_sum T [t,u]     = mk_plus (t, u)

   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);

 (*this version ALWAYS includes a trailing zero*)

 fun long_mk_sum T []        = mk_numeral T 0

   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);

 val dest_plus = HOLogic.dest_bin "op +" Term.dummyT;

 (*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 mk_minus t :: ts;

 fun dest_sum t = dest_summing (true, t, []);

 val mk_diff = HOLogic.mk_binop "op -";

 val dest_diff = HOLogic.dest_bin "op -" Term.dummyT;

 val mk_times = HOLogic.mk_binop "op *";

 fun mk_prod T = 

   let val one = mk_numeral T 1

   fun mk [] = one

     | mk [t] = t

     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)

   in mk end;

 (*This version ALWAYS includes a trailing one*)

 fun long_mk_prod T []        = mk_numeral T 1

   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);

 val dest_times = HOLogic.dest_bin "op *" Term.dummyT;

 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, t) = mk_times (mk_numeral (Term.fastype_of t) k, t);

 (*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 (Term.fastype_of t) 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 =  thms "add_0s";

 val mult_1s = thms "mult_1s";

 (*To perform binary arithmetic.  The "left" rewriting handles patterns

   created by the simprocs, such as 3 * (5 * x). *)

 val bin_simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,

                  add_number_of_left, 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 \\ [add_number_of_left, number_of_add RS sym];

 (*To evaluate binary negations of coefficients*)

 val minus_simps = NCons_simps @

                    [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,

                     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_minus, minus_add_distrib, minus_minus];

 (*push the unary minus down: - x * y = x * - y *)

 val minus_mult_eq_1_to_2 =

     [minus_mult_left RS sym, minus_mult_right] MRS trans |> standard;

 (*to extract again any uncancelled minuses*)

 val 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 mult_minus_simps =

     [mult_assoc, minus_mult_left, minus_mult_eq_1_to_2];

 (*Apply the given rewrite (if present) just once*)

 fun trans_tac None      = all_tac

   | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));

 fun simplify_meta_eq rules =

     simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)

     o mk_meta_eq;

 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 numeral_syms@add_0s@mult_1s@

                                          diff_simps@minus_simps@add_ac))

      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))

      THEN ALLGOALS (simp_tac (HOL_ss addsimps 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 (add_0s@mult_1s)

   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 =" Term.dummyT

   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 <" Term.dummyT

   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 <=" Term.dummyT

   val bal_add1 = le_add_iff1 RS trans

   val bal_add2 = le_add_iff2 RS trans

 );

 val cancel_numerals =

   map Bin_Simprocs.prep_simproc

    [("inteq_cancel_numerals",

      ["(l::'a::number_ring) + m = n",

       "(l::'a::number_ring) = m + n",

       "(l::'a::number_ring) - m = n",

       "(l::'a::number_ring) = m - n",

       "(l::'a::number_ring) * m = n",

       "(l::'a::number_ring) = m * n"],

      EqCancelNumerals.proc),

     ("intless_cancel_numerals",

      ["(l::'a::{ordered_idom,number_ring}) + m < n",

       "(l::'a::{ordered_idom,number_ring}) < m + n",

       "(l::'a::{ordered_idom,number_ring}) - m < n",

       "(l::'a::{ordered_idom,number_ring}) < m - n",

       "(l::'a::{ordered_idom,number_ring}) * m < n",

       "(l::'a::{ordered_idom,number_ring}) < m * n"],

      LessCancelNumerals.proc),

     ("intle_cancel_numerals",

      ["(l::'a::{ordered_idom,number_ring}) + m <= n",

       "(l::'a::{ordered_idom,number_ring}) <= m + n",

       "(l::'a::{ordered_idom,number_ring}) - m <= n",

       "(l::'a::{ordered_idom,number_ring}) <= m - n",

       "(l::'a::{ordered_idom,number_ring}) * m <= n",

       "(l::'a::{ordered_idom,number_ring}) <= 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@minus_simps@add_ac))

      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))

      THEN ALLGOALS (simp_tac (HOL_ss addsimps 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 (add_0s@mult_1s)

   end;

 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

 val combine_numerals =

   Bin_Simprocs.prep_simproc

     ("int_combine_numerals", 

      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 

      CombineNumerals.proc);

 end;

 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;

 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];

 (*examples:

 print_depth 22;

 set timing;

 set trace_simp;

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

 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";

 test "2*u = (u::int)";

 test "(i + j + 12 + (k::int)) - 15 = y";

 test "(i + j + 12 + (k::int)) - 5 = y";

 test "y - b < (b::int)";

 test "y - (3*b + c) < (b::int) - 2*c";

 test "(2*x - (u*v) + y) - v*3*u = (w::int)";

 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";

 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";

 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";

 test "(i + j + 12 + (k::int)) = u + 15 + y";

 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";

 test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";

 test "a + -(b+c) + b = (d::int)";

 test "a + -(b+c) - b = (d::int)";

 (*negative numerals*)

 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";

 test "(i + j + -3 + (k::int)) < u + 5 + y";

 test "(i + j + 3 + (k::int)) < u + -6 + y";

 test "(i + j + -12 + (k::int)) - 15 = y";

 test "(i + j + 12 + (k::int)) - -15 = y";

 test "(i + j + -12 + (k::int)) - -15 = y";

 *)

 (** Constant folding for multiplication in semirings **)

 (*We do not need folding for addition: combine_numerals does the same thing*)

 structure Semiring_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 add_ac = mult_ac

 end;

 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);

 val assoc_fold_simproc =

   Bin_Simprocs.prep_simproc

    ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],

     Semiring_Times_Assoc.proc);

 Addsimprocs [assoc_fold_simproc];

 (*** decision procedure for linear arithmetic ***)

 (*---------------------------------------------------------------------------*)

 (* Linear arithmetic                                                         *)

 (*---------------------------------------------------------------------------*)

 (*

 Instantiation of the generic linear arithmetic package for int.

 *)

 (* Update parameters of arithmetic prover *)

 local

 (* reduce contradictory <= to False *)

 val add_rules =

     simp_thms @ bin_arith_simps @ bin_rel_simps @ arith_special @

     [neg_le_iff_le, numeral_0_eq_0, numeral_1_eq_1,

      minus_zero, diff_minus, left_minus, right_minus,

      mult_zero_left, mult_zero_right, mult_1, mult_1_right,

      minus_mult_left RS sym, minus_mult_right RS sym,

      minus_add_distrib, minus_minus, mult_assoc,

      of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,

      of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat,

      zero_neq_one, zero_less_one, zero_le_one, 

      zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];

 val simprocs = [assoc_fold_simproc, Int_Numeral_Simprocs.combine_numerals]@

                Int_Numeral_Simprocs.cancel_numerals;

 in

 val int_arith_setup =

  [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>

    {add_mono_thms = add_mono_thms,

     mult_mono_thms = mult_mono_thms,

     inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,

     lessD = lessD @ [zless_imp_add1_zle],

     simpset = simpset addsimps add_rules

                       addsimprocs simprocs

                       addcongs [if_weak_cong]}),

   arith_inj_const ("IntDef.of_nat", HOLogic.natT --> HOLogic.intT),

   arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),

   arith_discrete ("IntDef.int", true)];

 end;

 val fast_int_arith_simproc =

   Simplifier.simproc (Theory.sign_of (the_context()))

   "fast_int_arith" 

      ["(m::'a::{ordered_idom,number_ring}) < n",

       "(m::'a::{ordered_idom,number_ring}) <= n",

       "(m::'a::{ordered_idom,number_ring}) = n"] Fast_Arith.lin_arith_prover;

 Addsimprocs [fast_int_arith_simproc]

 (* Some test data

 Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \

 \     ==> a+a <= j+j";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \

 \     ==> a+a - - -1 < j+j - 3";

 by (fast_arith_tac 1);

 Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";

 by (arith_tac 1);

 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

 \     ==> a <= l";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

 \     ==> a+a+a+a <= l+l+l+l";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

 \     ==> a+a+a+a+a <= l+l+l+l+i";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

 \     ==> a+a+a+a+a+a <= l+l+l+l+i+l";

 by (fast_arith_tac 1);

 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

 \     ==> 6*a <= 5*l+i";

 by (fast_arith_tac 1);

 *)