src/HOL/Integ/int_arith1.ML
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 14378 69c4d5997669
child 14390 55fe71faadda
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
     1 (*  Title:      HOL/Integ/int_arith1.ML
     2     ID:         $Id$
     3     Authors:    Larry Paulson and Tobias Nipkow
     4 
     5 Simprocs and decision procedure for linear arithmetic.
     6 *)
     7 
     8 (** Misc ML bindings **)
     9 
    10 val NCons_Pls = thm"NCons_Pls";
    11 val NCons_Min = thm"NCons_Min";
    12 val NCons_BIT = thm"NCons_BIT";
    13 val number_of_Pls = thm"number_of_Pls";
    14 val number_of_Min = thm"number_of_Min";
    15 val number_of_BIT = thm"number_of_BIT";
    16 val bin_succ_Pls = thm"bin_succ_Pls";
    17 val bin_succ_Min = thm"bin_succ_Min";
    18 val bin_succ_BIT = thm"bin_succ_BIT";
    19 val bin_pred_Pls = thm"bin_pred_Pls";
    20 val bin_pred_Min = thm"bin_pred_Min";
    21 val bin_pred_BIT = thm"bin_pred_BIT";
    22 val bin_minus_Pls = thm"bin_minus_Pls";
    23 val bin_minus_Min = thm"bin_minus_Min";
    24 val bin_minus_BIT = thm"bin_minus_BIT";
    25 val bin_add_Pls = thm"bin_add_Pls";
    26 val bin_add_Min = thm"bin_add_Min";
    27 val bin_mult_Pls = thm"bin_mult_Pls";
    28 val bin_mult_Min = thm"bin_mult_Min";
    29 val bin_mult_BIT = thm"bin_mult_BIT";
    30 
    31 val neg_def = thm "neg_def";
    32 val iszero_def = thm "iszero_def";
    33 
    34 val NCons_Pls_0 = thm"NCons_Pls_0";
    35 val NCons_Pls_1 = thm"NCons_Pls_1";
    36 val NCons_Min_0 = thm"NCons_Min_0";
    37 val NCons_Min_1 = thm"NCons_Min_1";
    38 val bin_succ_1 = thm"bin_succ_1";
    39 val bin_succ_0 = thm"bin_succ_0";
    40 val bin_pred_1 = thm"bin_pred_1";
    41 val bin_pred_0 = thm"bin_pred_0";
    42 val bin_minus_1 = thm"bin_minus_1";
    43 val bin_minus_0 = thm"bin_minus_0";
    44 val bin_add_BIT_11 = thm"bin_add_BIT_11";
    45 val bin_add_BIT_10 = thm"bin_add_BIT_10";
    46 val bin_add_BIT_0 = thm"bin_add_BIT_0";
    47 val bin_add_Pls_right = thm"bin_add_Pls_right";
    48 val bin_add_Min_right = thm"bin_add_Min_right";
    49 val bin_add_BIT_BIT = thm"bin_add_BIT_BIT";
    50 val bin_mult_1 = thm"bin_mult_1";
    51 val bin_mult_0 = thm"bin_mult_0";
    52 val number_of_NCons = thm"number_of_NCons";
    53 val number_of_succ = thm"number_of_succ";
    54 val number_of_pred = thm"number_of_pred";
    55 val number_of_minus = thm"number_of_minus";
    56 val number_of_add = thm"number_of_add";
    57 val diff_number_of_eq = thm"diff_number_of_eq";
    58 val number_of_mult = thm"number_of_mult";
    59 val double_number_of_BIT = thm"double_number_of_BIT";
    60 val numeral_0_eq_0 = thm"numeral_0_eq_0";
    61 val numeral_1_eq_1 = thm"numeral_1_eq_1";
    62 val numeral_m1_eq_minus_1 = thm"numeral_m1_eq_minus_1";
    63 val mult_minus1 = thm"mult_minus1";
    64 val mult_minus1_right = thm"mult_minus1_right";
    65 val minus_number_of_mult = thm"minus_number_of_mult";
    66 val zero_less_nat_eq = thm"zero_less_nat_eq";
    67 val eq_number_of_eq = thm"eq_number_of_eq";
    68 val iszero_number_of_Pls = thm"iszero_number_of_Pls";
    69 val nonzero_number_of_Min = thm"nonzero_number_of_Min";
    70 val iszero_number_of_BIT = thm"iszero_number_of_BIT";
    71 val iszero_number_of_0 = thm"iszero_number_of_0";
    72 val iszero_number_of_1 = thm"iszero_number_of_1";
    73 val less_number_of_eq_neg = thm"less_number_of_eq_neg";
    74 val le_number_of_eq = thm"le_number_of_eq";
    75 val not_neg_number_of_Pls = thm"not_neg_number_of_Pls";
    76 val neg_number_of_Min = thm"neg_number_of_Min";
    77 val neg_number_of_BIT = thm"neg_number_of_BIT";
    78 val le_number_of_eq_not_less = thm"le_number_of_eq_not_less";
    79 val abs_number_of = thm"abs_number_of";
    80 val number_of_reorient = thm"number_of_reorient";
    81 val add_number_of_left = thm"add_number_of_left";
    82 val mult_number_of_left = thm"mult_number_of_left";
    83 val add_number_of_diff1 = thm"add_number_of_diff1";
    84 val add_number_of_diff2 = thm"add_number_of_diff2";
    85 val less_iff_diff_less_0 = thm"less_iff_diff_less_0";
    86 val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0";
    87 val le_iff_diff_le_0 = thm"le_iff_diff_le_0";
    88 
    89 val NCons_simps = thms"NCons_simps";
    90 val bin_arith_extra_simps = thms"bin_arith_extra_simps";
    91 val bin_arith_simps = thms"bin_arith_simps";
    92 val bin_rel_simps = thms"bin_rel_simps";
    93 
    94 val zless_imp_add1_zle = thm "zless_imp_add1_zle";
    95 
    96 val combine_common_factor = thm"combine_common_factor";
    97 val eq_add_iff1 = thm"eq_add_iff1";
    98 val eq_add_iff2 = thm"eq_add_iff2";
    99 val less_add_iff1 = thm"less_add_iff1";
   100 val less_add_iff2 = thm"less_add_iff2";
   101 val le_add_iff1 = thm"le_add_iff1";
   102 val le_add_iff2 = thm"le_add_iff2";
   103 
   104 val arith_special = thms"arith_special";
   105 
   106 structure Bin_Simprocs =
   107   struct
   108   fun prove_conv tacs sg (hyps: thm list) xs (t, u) =
   109     if t aconv u then None
   110     else
   111       let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
   112       in Some (Tactic.prove sg xs [] eq (K (EVERY tacs))) end
   113 
   114   fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];
   115   fun prove_conv_nohyps_novars tacs sg = prove_conv tacs sg [] [];
   116 
   117   fun prep_simproc (name, pats, proc) =
   118     Simplifier.simproc (Theory.sign_of (the_context())) name pats proc;
   119 
   120   fun is_numeral (Const("Numeral.number_of", _) $ w) = true
   121     | is_numeral _ = false
   122 
   123   fun simplify_meta_eq f_number_of_eq f_eq =
   124       mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
   125 
   126   (*reorientation simprules using ==, for the following simproc*)
   127   val meta_zero_reorient = zero_reorient RS eq_reflection
   128   val meta_one_reorient = one_reorient RS eq_reflection
   129   val meta_number_of_reorient = number_of_reorient RS eq_reflection
   130 
   131   (*reorientation simplification procedure: reorients (polymorphic) 
   132     0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
   133   fun reorient_proc sg _ (_ $ t $ u) =
   134     case u of
   135 	Const("0", _) => None
   136       | Const("1", _) => None
   137       | Const("Numeral.number_of", _) $ _ => None
   138       | _ => Some (case t of
   139 		  Const("0", _) => meta_zero_reorient
   140 		| Const("1", _) => meta_one_reorient
   141 		| Const("Numeral.number_of", _) $ _ => meta_number_of_reorient)
   142 
   143   val reorient_simproc = 
   144       prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)
   145 
   146   end;
   147 
   148 
   149 Addsimps arith_special;
   150 Addsimprocs [Bin_Simprocs.reorient_simproc];
   151 
   152 
   153 structure Int_Numeral_Simprocs =
   154 struct
   155 
   156 (*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
   157   isn't complicated by the abstract 0 and 1.*)
   158 val numeral_syms = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];
   159 
   160 (*Utilities*)
   161 
   162 fun mk_numeral T n = HOLogic.number_of_const T $ HOLogic.mk_bin n;
   163 
   164 (*Decodes a binary INTEGER*)
   165 fun dest_numeral (Const("0", _)) = 0
   166   | dest_numeral (Const("1", _)) = 1
   167   | dest_numeral (Const("Numeral.number_of", _) $ w) =
   168      (HOLogic.dest_binum w
   169       handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
   170   | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
   171 
   172 fun find_first_numeral past (t::terms) =
   173         ((dest_numeral t, rev past @ terms)
   174          handle TERM _ => find_first_numeral (t::past) terms)
   175   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
   176 
   177 val mk_plus = HOLogic.mk_binop "op +";
   178 
   179 fun mk_minus t = 
   180   let val T = Term.fastype_of t
   181   in Const ("uminus", T --> T) $ t
   182   end;
   183 
   184 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   185 fun mk_sum T []        = mk_numeral T 0
   186   | mk_sum T [t,u]     = mk_plus (t, u)
   187   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   188 
   189 (*this version ALWAYS includes a trailing zero*)
   190 fun long_mk_sum T []        = mk_numeral T 0
   191   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   192 
   193 val dest_plus = HOLogic.dest_bin "op +" Term.dummyT;
   194 
   195 (*decompose additions AND subtractions as a sum*)
   196 fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
   197         dest_summing (pos, t, dest_summing (pos, u, ts))
   198   | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
   199         dest_summing (pos, t, dest_summing (not pos, u, ts))
   200   | dest_summing (pos, t, ts) =
   201         if pos then t::ts else mk_minus t :: ts;
   202 
   203 fun dest_sum t = dest_summing (true, t, []);
   204 
   205 val mk_diff = HOLogic.mk_binop "op -";
   206 val dest_diff = HOLogic.dest_bin "op -" Term.dummyT;
   207 
   208 val mk_times = HOLogic.mk_binop "op *";
   209 
   210 fun mk_prod T = 
   211   let val one = mk_numeral T 1
   212   fun mk [] = one
   213     | mk [t] = t
   214     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
   215   in mk end;
   216 
   217 (*This version ALWAYS includes a trailing one*)
   218 fun long_mk_prod T []        = mk_numeral T 1
   219   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
   220 
   221 val dest_times = HOLogic.dest_bin "op *" Term.dummyT;
   222 
   223 fun dest_prod t =
   224       let val (t,u) = dest_times t
   225       in  dest_prod t @ dest_prod u  end
   226       handle TERM _ => [t];
   227 
   228 (*DON'T do the obvious simplifications; that would create special cases*)
   229 fun mk_coeff (k, t) = mk_times (mk_numeral (Term.fastype_of t) k, t);
   230 
   231 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   232 fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
   233   | dest_coeff sign t =
   234     let val ts = sort Term.term_ord (dest_prod t)
   235         val (n, ts') = find_first_numeral [] ts
   236                           handle TERM _ => (1, ts)
   237     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
   238 
   239 (*Find first coefficient-term THAT MATCHES u*)
   240 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   241   | find_first_coeff past u (t::terms) =
   242         let val (n,u') = dest_coeff 1 t
   243         in  if u aconv u' then (n, rev past @ terms)
   244                           else find_first_coeff (t::past) u terms
   245         end
   246         handle TERM _ => find_first_coeff (t::past) u terms;
   247 
   248 
   249 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
   250 val add_0s =  thms "add_0s";
   251 val mult_1s = thms "mult_1s";
   252 
   253 (*To perform binary arithmetic.  The "left" rewriting handles patterns
   254   created by the simprocs, such as 3 * (5 * x). *)
   255 val bin_simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
   256                  add_number_of_left, mult_number_of_left] @
   257                 bin_arith_simps @ bin_rel_simps;
   258 
   259 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
   260   during re-arrangement*)
   261 val non_add_bin_simps = 
   262     bin_simps \\ [add_number_of_left, number_of_add RS sym];
   263 
   264 (*To evaluate binary negations of coefficients*)
   265 val minus_simps = NCons_simps @
   266                    [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
   267                     bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
   268                     bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
   269 
   270 (*To let us treat subtraction as addition*)
   271 val diff_simps = [diff_minus, minus_add_distrib, minus_minus];
   272 
   273 (*push the unary minus down: - x * y = x * - y *)
   274 val minus_mult_eq_1_to_2 =
   275     [minus_mult_left RS sym, minus_mult_right] MRS trans |> standard;
   276 
   277 (*to extract again any uncancelled minuses*)
   278 val minus_from_mult_simps =
   279     [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
   280 
   281 (*combine unary minus with numeric literals, however nested within a product*)
   282 val mult_minus_simps =
   283     [mult_assoc, minus_mult_left, minus_mult_eq_1_to_2];
   284 
   285 (*Apply the given rewrite (if present) just once*)
   286 fun trans_tac None      = all_tac
   287   | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
   288 
   289 fun simplify_meta_eq rules =
   290     simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
   291     o mk_meta_eq;
   292 
   293 structure CancelNumeralsCommon =
   294   struct
   295   val mk_sum            = mk_sum
   296   val dest_sum          = dest_sum
   297   val mk_coeff          = mk_coeff
   298   val dest_coeff        = dest_coeff 1
   299   val find_first_coeff  = find_first_coeff []
   300   val trans_tac         = trans_tac
   301   val norm_tac =
   302      ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
   303                                          diff_simps@minus_simps@add_ac))
   304      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
   305      THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
   306                                               add_ac@mult_ac))
   307   val numeral_simp_tac  = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   308   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   309   end;
   310 
   311 
   312 structure EqCancelNumerals = CancelNumeralsFun
   313  (open CancelNumeralsCommon
   314   val prove_conv = Bin_Simprocs.prove_conv
   315   val mk_bal   = HOLogic.mk_eq
   316   val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
   317   val bal_add1 = eq_add_iff1 RS trans
   318   val bal_add2 = eq_add_iff2 RS trans
   319 );
   320 
   321 structure LessCancelNumerals = CancelNumeralsFun
   322  (open CancelNumeralsCommon
   323   val prove_conv = Bin_Simprocs.prove_conv
   324   val mk_bal   = HOLogic.mk_binrel "op <"
   325   val dest_bal = HOLogic.dest_bin "op <" Term.dummyT
   326   val bal_add1 = less_add_iff1 RS trans
   327   val bal_add2 = less_add_iff2 RS trans
   328 );
   329 
   330 structure LeCancelNumerals = CancelNumeralsFun
   331  (open CancelNumeralsCommon
   332   val prove_conv = Bin_Simprocs.prove_conv
   333   val mk_bal   = HOLogic.mk_binrel "op <="
   334   val dest_bal = HOLogic.dest_bin "op <=" Term.dummyT
   335   val bal_add1 = le_add_iff1 RS trans
   336   val bal_add2 = le_add_iff2 RS trans
   337 );
   338 
   339 val cancel_numerals =
   340   map Bin_Simprocs.prep_simproc
   341    [("inteq_cancel_numerals",
   342      ["(l::'a::number_ring) + m = n",
   343       "(l::'a::number_ring) = m + n",
   344       "(l::'a::number_ring) - m = n",
   345       "(l::'a::number_ring) = m - n",
   346       "(l::'a::number_ring) * m = n",
   347       "(l::'a::number_ring) = m * n"],
   348      EqCancelNumerals.proc),
   349     ("intless_cancel_numerals",
   350      ["(l::'a::{ordered_ring,number_ring}) + m < n",
   351       "(l::'a::{ordered_ring,number_ring}) < m + n",
   352       "(l::'a::{ordered_ring,number_ring}) - m < n",
   353       "(l::'a::{ordered_ring,number_ring}) < m - n",
   354       "(l::'a::{ordered_ring,number_ring}) * m < n",
   355       "(l::'a::{ordered_ring,number_ring}) < m * n"],
   356      LessCancelNumerals.proc),
   357     ("intle_cancel_numerals",
   358      ["(l::'a::{ordered_ring,number_ring}) + m <= n",
   359       "(l::'a::{ordered_ring,number_ring}) <= m + n",
   360       "(l::'a::{ordered_ring,number_ring}) - m <= n",
   361       "(l::'a::{ordered_ring,number_ring}) <= m - n",
   362       "(l::'a::{ordered_ring,number_ring}) * m <= n",
   363       "(l::'a::{ordered_ring,number_ring}) <= m * n"],
   364      LeCancelNumerals.proc)];
   365 
   366 
   367 structure CombineNumeralsData =
   368   struct
   369   val add               = op + : int*int -> int
   370   val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
   371   val dest_sum          = dest_sum
   372   val mk_coeff          = mk_coeff
   373   val dest_coeff        = dest_coeff 1
   374   val left_distrib      = combine_common_factor RS trans
   375   val prove_conv        = Bin_Simprocs.prove_conv_nohyps
   376   val trans_tac          = trans_tac
   377   val norm_tac =
   378      ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
   379                                          diff_simps@minus_simps@add_ac))
   380      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
   381      THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
   382                                               add_ac@mult_ac))
   383   val numeral_simp_tac  = ALLGOALS
   384                     (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   385   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   386   end;
   387 
   388 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   389 
   390 val combine_numerals =
   391   Bin_Simprocs.prep_simproc
   392     ("int_combine_numerals", 
   393      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
   394      CombineNumerals.proc);
   395 
   396 end;
   397 
   398 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
   399 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
   400 
   401 (*examples:
   402 print_depth 22;
   403 set timing;
   404 set trace_simp;
   405 fun test s = (Goal s, by (Simp_tac 1));
   406 
   407 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
   408 
   409 test "2*u = (u::int)";
   410 test "(i + j + 12 + (k::int)) - 15 = y";
   411 test "(i + j + 12 + (k::int)) - 5 = y";
   412 
   413 test "y - b < (b::int)";
   414 test "y - (3*b + c) < (b::int) - 2*c";
   415 
   416 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
   417 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
   418 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
   419 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
   420 
   421 test "(i + j + 12 + (k::int)) = u + 15 + y";
   422 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
   423 
   424 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)";
   425 
   426 test "a + -(b+c) + b = (d::int)";
   427 test "a + -(b+c) - b = (d::int)";
   428 
   429 (*negative numerals*)
   430 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
   431 test "(i + j + -3 + (k::int)) < u + 5 + y";
   432 test "(i + j + 3 + (k::int)) < u + -6 + y";
   433 test "(i + j + -12 + (k::int)) - 15 = y";
   434 test "(i + j + 12 + (k::int)) - -15 = y";
   435 test "(i + j + -12 + (k::int)) - -15 = y";
   436 *)
   437 
   438 
   439 (** Constant folding for multiplication in semirings **)
   440 
   441 (*We do not need folding for addition: combine_numerals does the same thing*)
   442 
   443 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
   444 struct
   445   val ss                = HOL_ss
   446   val eq_reflection     = eq_reflection
   447   val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
   448   val add_ac = mult_ac
   449 end;
   450 
   451 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
   452 
   453 val assoc_fold_simproc =
   454   Bin_Simprocs.prep_simproc
   455    ("semiring_assoc_fold", ["(a::'a::semiring) * b"],
   456     Semiring_Times_Assoc.proc);
   457 
   458 Addsimprocs [assoc_fold_simproc];
   459 
   460 
   461 
   462 
   463 (*** decision procedure for linear arithmetic ***)
   464 
   465 (*---------------------------------------------------------------------------*)
   466 (* Linear arithmetic                                                         *)
   467 (*---------------------------------------------------------------------------*)
   468 
   469 (*
   470 Instantiation of the generic linear arithmetic package for int.
   471 *)
   472 
   473 (* Update parameters of arithmetic prover *)
   474 local
   475 
   476 (* reduce contradictory <= to False *)
   477 val add_rules =
   478     simp_thms @ bin_arith_simps @ bin_rel_simps @ arith_special @
   479     [numeral_0_eq_0, numeral_1_eq_1,
   480      minus_zero, diff_minus, left_minus, right_minus,
   481      mult_zero_left, mult_zero_right, mult_1, mult_1_right,
   482      minus_mult_left RS sym, minus_mult_right RS sym,
   483      minus_add_distrib, minus_minus, mult_assoc,
   484      of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,
   485      of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat,
   486      zero_neq_one, zero_less_one, zero_le_one, 
   487      zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];
   488 
   489 val simprocs = [assoc_fold_simproc, Int_Numeral_Simprocs.combine_numerals]@
   490                Int_Numeral_Simprocs.cancel_numerals;
   491 
   492 in
   493 
   494 val int_arith_setup =
   495  [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
   496    {add_mono_thms = add_mono_thms,
   497     mult_mono_thms = mult_mono_thms,
   498     inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,
   499     lessD = lessD @ [zless_imp_add1_zle],
   500     simpset = simpset addsimps add_rules
   501                       addsimprocs simprocs
   502                       addcongs [if_weak_cong]}),
   503   arith_inj_const ("IntDef.of_nat", HOLogic.natT --> HOLogic.intT),
   504   arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),
   505   arith_discrete ("IntDef.int", true)];
   506 
   507 end;
   508 
   509 val fast_int_arith_simproc =
   510   Simplifier.simproc (Theory.sign_of (the_context()))
   511   "fast_int_arith" 
   512      ["(m::'a::{ordered_ring,number_ring}) < n",
   513       "(m::'a::{ordered_ring,number_ring}) <= n",
   514       "(m::'a::{ordered_ring,number_ring}) = n"] Fast_Arith.lin_arith_prover;
   515 
   516 Addsimprocs [fast_int_arith_simproc]
   517 
   518 
   519 (* Some test data
   520 Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
   521 by (fast_arith_tac 1);
   522 Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";
   523 by (fast_arith_tac 1);
   524 Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";
   525 by (fast_arith_tac 1);
   526 Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
   527 by (fast_arith_tac 1);
   528 Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \
   529 \     ==> a+a <= j+j";
   530 by (fast_arith_tac 1);
   531 Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \
   532 \     ==> a+a - - -1 < j+j - 3";
   533 by (fast_arith_tac 1);
   534 Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
   535 by (arith_tac 1);
   536 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   537 \     ==> a <= l";
   538 by (fast_arith_tac 1);
   539 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   540 \     ==> a+a+a+a <= l+l+l+l";
   541 by (fast_arith_tac 1);
   542 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   543 \     ==> a+a+a+a+a <= l+l+l+l+i";
   544 by (fast_arith_tac 1);
   545 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   546 \     ==> a+a+a+a+a+a <= l+l+l+l+i+l";
   547 by (fast_arith_tac 1);
   548 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   549 \     ==> 6*a <= 5*l+i";
   550 by (fast_arith_tac 1);
   551 *)