src/HOL/Tools/int_arith.ML
author haftmann
Fri Mar 13 19:17:57 2009 +0100 (2009-03-13)
changeset 30518 07b45c1aa788
parent 30496 7cdcc9dd95cb
child 30685 dd5fe091ff04
permissions -rw-r--r--
moved some generic nonsense to arith_data.ML
     1 (* Authors: Larry Paulson and Tobias Nipkow
     2 
     3 Simprocs and decision procedure for numerals and linear arithmetic.
     4 *)
     5 
     6 structure Int_Numeral_Simprocs =
     7 struct
     8 
     9 (*reorientation simprules using ==, for the following simproc*)
    10 val meta_zero_reorient = @{thm zero_reorient} RS eq_reflection
    11 val meta_one_reorient = @{thm one_reorient} RS eq_reflection
    12 val meta_number_of_reorient = @{thm number_of_reorient} RS eq_reflection
    13 
    14 (*reorientation simplification procedure: reorients (polymorphic) 
    15   0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a Int.*)
    16 fun reorient_proc sg _ (_ $ t $ u) =
    17   case u of
    18       Const(@{const_name HOL.zero}, _) => NONE
    19     | Const(@{const_name HOL.one}, _) => NONE
    20     | Const(@{const_name Int.number_of}, _) $ _ => NONE
    21     | _ => SOME (case t of
    22         Const(@{const_name HOL.zero}, _) => meta_zero_reorient
    23       | Const(@{const_name HOL.one}, _) => meta_one_reorient
    24       | Const(@{const_name Int.number_of}, _) $ _ => meta_number_of_reorient)
    25 
    26 val reorient_simproc = 
    27   Arith_Data.prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc);
    28 
    29 (*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
    30 val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
    31 
    32 
    33 (** Utilities **)
    34 
    35 fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
    36 
    37 fun find_first_numeral past (t::terms) =
    38         ((snd (HOLogic.dest_number t), rev past @ terms)
    39          handle TERM _ => find_first_numeral (t::past) terms)
    40   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
    41 
    42 val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
    43 
    44 fun mk_minus t = 
    45   let val T = Term.fastype_of t
    46   in Const (@{const_name HOL.uminus}, T --> T) $ t end;
    47 
    48 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
    49 fun mk_sum T []        = mk_number T 0
    50   | mk_sum T [t,u]     = mk_plus (t, u)
    51   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
    52 
    53 (*this version ALWAYS includes a trailing zero*)
    54 fun long_mk_sum T []        = mk_number T 0
    55   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
    56 
    57 val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
    58 
    59 (*decompose additions AND subtractions as a sum*)
    60 fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
    61         dest_summing (pos, t, dest_summing (pos, u, ts))
    62   | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
    63         dest_summing (pos, t, dest_summing (not pos, u, ts))
    64   | dest_summing (pos, t, ts) =
    65         if pos then t::ts else mk_minus t :: ts;
    66 
    67 fun dest_sum t = dest_summing (true, t, []);
    68 
    69 val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
    70 val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
    71 
    72 val mk_times = HOLogic.mk_binop @{const_name HOL.times};
    73 
    74 fun one_of T = Const(@{const_name HOL.one},T);
    75 
    76 (* build product with trailing 1 rather than Numeral 1 in order to avoid the
    77    unnecessary restriction to type class number_ring
    78    which is not required for cancellation of common factors in divisions.
    79 *)
    80 fun mk_prod T = 
    81   let val one = one_of T
    82   fun mk [] = one
    83     | mk [t] = t
    84     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
    85   in mk end;
    86 
    87 (*This version ALWAYS includes a trailing one*)
    88 fun long_mk_prod T []        = one_of T
    89   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
    90 
    91 val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
    92 
    93 fun dest_prod t =
    94       let val (t,u) = dest_times t
    95       in dest_prod t @ dest_prod u end
    96       handle TERM _ => [t];
    97 
    98 (*DON'T do the obvious simplifications; that would create special cases*)
    99 fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
   100 
   101 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   102 fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t
   103   | dest_coeff sign t =
   104     let val ts = sort TermOrd.term_ord (dest_prod t)
   105         val (n, ts') = find_first_numeral [] ts
   106                           handle TERM _ => (1, ts)
   107     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
   108 
   109 (*Find first coefficient-term THAT MATCHES u*)
   110 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   111   | find_first_coeff past u (t::terms) =
   112         let val (n,u') = dest_coeff 1 t
   113         in if u aconv u' then (n, rev past @ terms)
   114                          else find_first_coeff (t::past) u terms
   115         end
   116         handle TERM _ => find_first_coeff (t::past) u terms;
   117 
   118 (*Fractions as pairs of ints. Can't use Rat.rat because the representation
   119   needs to preserve negative values in the denominator.*)
   120 fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
   121 
   122 (*Don't reduce fractions; sums must be proved by rule add_frac_eq.
   123   Fractions are reduced later by the cancel_numeral_factor simproc.*)
   124 fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
   125 
   126 val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
   127 
   128 (*Build term (p / q) * t*)
   129 fun mk_fcoeff ((p, q), t) =
   130   let val T = Term.fastype_of t
   131   in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
   132 
   133 (*Express t as a product of a fraction with other sorted terms*)
   134 fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t
   135   | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) =
   136     let val (p, t') = dest_coeff sign t
   137         val (q, u') = dest_coeff 1 u
   138     in (mk_frac (p, q), mk_divide (t', u')) end
   139   | dest_fcoeff sign t =
   140     let val (p, t') = dest_coeff sign t
   141         val T = Term.fastype_of t
   142     in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
   143 
   144 
   145 (** New term ordering so that AC-rewriting brings numerals to the front **)
   146 
   147 (*Order integers by absolute value and then by sign. The standard integer
   148   ordering is not well-founded.*)
   149 fun num_ord (i,j) =
   150   (case int_ord (abs i, abs j) of
   151     EQUAL => int_ord (Int.sign i, Int.sign j) 
   152   | ord => ord);
   153 
   154 (*This resembles TermOrd.term_ord, but it puts binary numerals before other
   155   non-atomic terms.*)
   156 local open Term 
   157 in 
   158 fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
   159       (case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
   160   | numterm_ord
   161      (Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) =
   162      num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
   163   | numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS
   164   | numterm_ord (_, Const(@{const_name Int.number_of}, _) $ _) = GREATER
   165   | numterm_ord (t, u) =
   166       (case int_ord (size_of_term t, size_of_term u) of
   167         EQUAL =>
   168           let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
   169             (case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
   170           end
   171       | ord => ord)
   172 and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
   173 end;
   174 
   175 fun numtermless tu = (numterm_ord tu = LESS);
   176 
   177 val num_ss = HOL_ss settermless numtermless;
   178 
   179 
   180 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
   181 val add_0s =  thms "add_0s";
   182 val mult_1s = thms "mult_1s" @ [thm"mult_1_left", thm"mult_1_right", thm"divide_1"];
   183 
   184 (*Simplify inverse Numeral1, a/Numeral1*)
   185 val inverse_1s = [@{thm inverse_numeral_1}];
   186 val divide_1s = [@{thm divide_numeral_1}];
   187 
   188 (*To perform binary arithmetic.  The "left" rewriting handles patterns
   189   created by the Int_Numeral_Simprocs, such as 3 * (5 * x). *)
   190 val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
   191                  @{thm add_number_of_left}, @{thm mult_number_of_left}] @
   192                 @{thms arith_simps} @ @{thms rel_simps};
   193 
   194 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
   195   during re-arrangement*)
   196 val non_add_simps =
   197   subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
   198 
   199 (*To evaluate binary negations of coefficients*)
   200 val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
   201                    @{thms minus_bin_simps} @ @{thms pred_bin_simps};
   202 
   203 (*To let us treat subtraction as addition*)
   204 val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
   205 
   206 (*To let us treat division as multiplication*)
   207 val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
   208 
   209 (*push the unary minus down: - x * y = x * - y *)
   210 val minus_mult_eq_1_to_2 =
   211     [@{thm minus_mult_left} RS sym, @{thm minus_mult_right}] MRS trans |> standard;
   212 
   213 (*to extract again any uncancelled minuses*)
   214 val minus_from_mult_simps =
   215     [@{thm minus_minus}, @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym];
   216 
   217 (*combine unary minus with numeric literals, however nested within a product*)
   218 val mult_minus_simps =
   219     [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
   220 
   221 structure CancelNumeralsCommon =
   222   struct
   223   val mk_sum            = mk_sum
   224   val dest_sum          = dest_sum
   225   val mk_coeff          = mk_coeff
   226   val dest_coeff        = dest_coeff 1
   227   val find_first_coeff  = find_first_coeff []
   228   val trans_tac         = K Arith_Data.trans_tac
   229 
   230   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   231     diff_simps @ minus_simps @ @{thms add_ac}
   232   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   233   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   234   fun norm_tac ss =
   235     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   236     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   237     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   238 
   239   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
   240   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   241   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
   242   end;
   243 
   244 
   245 structure EqCancelNumerals = CancelNumeralsFun
   246  (open CancelNumeralsCommon
   247   val prove_conv = Arith_Data.prove_conv
   248   val mk_bal   = HOLogic.mk_eq
   249   val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
   250   val bal_add1 = @{thm eq_add_iff1} RS trans
   251   val bal_add2 = @{thm eq_add_iff2} RS trans
   252 );
   253 
   254 structure LessCancelNumerals = CancelNumeralsFun
   255  (open CancelNumeralsCommon
   256   val prove_conv = Arith_Data.prove_conv
   257   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
   258   val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
   259   val bal_add1 = @{thm less_add_iff1} RS trans
   260   val bal_add2 = @{thm less_add_iff2} RS trans
   261 );
   262 
   263 structure LeCancelNumerals = CancelNumeralsFun
   264  (open CancelNumeralsCommon
   265   val prove_conv = Arith_Data.prove_conv
   266   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
   267   val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
   268   val bal_add1 = @{thm le_add_iff1} RS trans
   269   val bal_add2 = @{thm le_add_iff2} RS trans
   270 );
   271 
   272 val cancel_numerals =
   273   map Arith_Data.prep_simproc
   274    [("inteq_cancel_numerals",
   275      ["(l::'a::number_ring) + m = n",
   276       "(l::'a::number_ring) = m + n",
   277       "(l::'a::number_ring) - m = n",
   278       "(l::'a::number_ring) = m - n",
   279       "(l::'a::number_ring) * m = n",
   280       "(l::'a::number_ring) = m * n"],
   281      K EqCancelNumerals.proc),
   282     ("intless_cancel_numerals",
   283      ["(l::'a::{ordered_idom,number_ring}) + m < n",
   284       "(l::'a::{ordered_idom,number_ring}) < m + n",
   285       "(l::'a::{ordered_idom,number_ring}) - m < n",
   286       "(l::'a::{ordered_idom,number_ring}) < m - n",
   287       "(l::'a::{ordered_idom,number_ring}) * m < n",
   288       "(l::'a::{ordered_idom,number_ring}) < m * n"],
   289      K LessCancelNumerals.proc),
   290     ("intle_cancel_numerals",
   291      ["(l::'a::{ordered_idom,number_ring}) + m <= n",
   292       "(l::'a::{ordered_idom,number_ring}) <= m + n",
   293       "(l::'a::{ordered_idom,number_ring}) - m <= n",
   294       "(l::'a::{ordered_idom,number_ring}) <= m - n",
   295       "(l::'a::{ordered_idom,number_ring}) * m <= n",
   296       "(l::'a::{ordered_idom,number_ring}) <= m * n"],
   297      K LeCancelNumerals.proc)];
   298 
   299 
   300 structure CombineNumeralsData =
   301   struct
   302   type coeff            = int
   303   val iszero            = (fn x => x = 0)
   304   val add               = op +
   305   val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
   306   val dest_sum          = dest_sum
   307   val mk_coeff          = mk_coeff
   308   val dest_coeff        = dest_coeff 1
   309   val left_distrib      = @{thm combine_common_factor} RS trans
   310   val prove_conv        = Arith_Data.prove_conv_nohyps
   311   val trans_tac         = K Arith_Data.trans_tac
   312 
   313   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   314     diff_simps @ minus_simps @ @{thms add_ac}
   315   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   316   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   317   fun norm_tac ss =
   318     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   319     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   320     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   321 
   322   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
   323   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   324   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
   325   end;
   326 
   327 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   328 
   329 (*Version for fields, where coefficients can be fractions*)
   330 structure FieldCombineNumeralsData =
   331   struct
   332   type coeff            = int * int
   333   val iszero            = (fn (p, q) => p = 0)
   334   val add               = add_frac
   335   val mk_sum            = long_mk_sum
   336   val dest_sum          = dest_sum
   337   val mk_coeff          = mk_fcoeff
   338   val dest_coeff        = dest_fcoeff 1
   339   val left_distrib      = @{thm combine_common_factor} RS trans
   340   val prove_conv        = Arith_Data.prove_conv_nohyps
   341   val trans_tac         = K Arith_Data.trans_tac
   342 
   343   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   344     inverse_1s @ divide_simps @ diff_simps @ minus_simps @ @{thms add_ac}
   345   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   346   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   347   fun norm_tac ss =
   348     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   349     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   350     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   351 
   352   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
   353   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   354   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
   355   end;
   356 
   357 structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
   358 
   359 val combine_numerals =
   360   Arith_Data.prep_simproc
   361     ("int_combine_numerals", 
   362      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
   363      K CombineNumerals.proc);
   364 
   365 val field_combine_numerals =
   366   Arith_Data.prep_simproc
   367     ("field_combine_numerals", 
   368      ["(i::'a::{number_ring,field,division_by_zero}) + j",
   369       "(i::'a::{number_ring,field,division_by_zero}) - j"], 
   370      K FieldCombineNumerals.proc);
   371 
   372 (** Constant folding for multiplication in semirings **)
   373 
   374 (*We do not need folding for addition: combine_numerals does the same thing*)
   375 
   376 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
   377 struct
   378   val assoc_ss = HOL_ss addsimps @{thms mult_ac}
   379   val eq_reflection = eq_reflection
   380 end;
   381 
   382 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
   383 
   384 val assoc_fold_simproc =
   385   Arith_Data.prep_simproc
   386    ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
   387     K Semiring_Times_Assoc.proc);
   388 
   389 end;
   390 
   391 Addsimprocs [Int_Numeral_Simprocs.reorient_simproc];
   392 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
   393 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
   394 Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
   395 Addsimprocs [Int_Numeral_Simprocs.assoc_fold_simproc];
   396 
   397 (*examples:
   398 print_depth 22;
   399 set timing;
   400 set trace_simp;
   401 fun test s = (Goal s, by (Simp_tac 1));
   402 
   403 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
   404 
   405 test "2*u = (u::int)";
   406 test "(i + j + 12 + (k::int)) - 15 = y";
   407 test "(i + j + 12 + (k::int)) - 5 = y";
   408 
   409 test "y - b < (b::int)";
   410 test "y - (3*b + c) < (b::int) - 2*c";
   411 
   412 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
   413 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
   414 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
   415 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
   416 
   417 test "(i + j + 12 + (k::int)) = u + 15 + y";
   418 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
   419 
   420 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)";
   421 
   422 test "a + -(b+c) + b = (d::int)";
   423 test "a + -(b+c) - b = (d::int)";
   424 
   425 (*negative numerals*)
   426 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
   427 test "(i + j + -3 + (k::int)) < u + 5 + y";
   428 test "(i + j + 3 + (k::int)) < u + -6 + y";
   429 test "(i + j + -12 + (k::int)) - 15 = y";
   430 test "(i + j + 12 + (k::int)) - -15 = y";
   431 test "(i + j + -12 + (k::int)) - -15 = y";
   432 *)
   433 
   434 (*** decision procedure for linear arithmetic ***)
   435 
   436 (*---------------------------------------------------------------------------*)
   437 (* Linear arithmetic                                                         *)
   438 (*---------------------------------------------------------------------------*)
   439 
   440 (*
   441 Instantiation of the generic linear arithmetic package for int.
   442 *)
   443 
   444 structure Int_Arith =
   445 struct
   446 
   447 (* Update parameters of arithmetic prover *)
   448 
   449 (* reduce contradictory =/</<= to False *)
   450 
   451 (* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<",
   452    and m and n are ground terms over rings (roughly speaking).
   453    That is, m and n consist only of 1s combined with "+", "-" and "*".
   454 *)
   455 
   456 val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0};
   457 
   458 val lhss0 = [@{cpat "0::?'a::ring"}];
   459 
   460 fun proc0 phi ss ct =
   461   let val T = ctyp_of_term ct
   462   in if typ_of T = @{typ int} then NONE else
   463      SOME (instantiate' [SOME T] [] zeroth)
   464   end;
   465 
   466 val zero_to_of_int_zero_simproc =
   467   make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc",
   468   proc = proc0, identifier = []};
   469 
   470 val oneth = (symmetric o mk_meta_eq) @{thm of_int_1};
   471 
   472 val lhss1 = [@{cpat "1::?'a::ring_1"}];
   473 
   474 fun proc1 phi ss ct =
   475   let val T = ctyp_of_term ct
   476   in if typ_of T = @{typ int} then NONE else
   477      SOME (instantiate' [SOME T] [] oneth)
   478   end;
   479 
   480 val one_to_of_int_one_simproc =
   481   make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc",
   482   proc = proc1, identifier = []};
   483 
   484 val allowed_consts =
   485   [@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"},
   486    @{const_name "HOL.minus"}, @{const_name "HOL.plus"},
   487    @{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"},
   488    @{const_name "HOL.less_eq"}];
   489 
   490 fun check t = case t of
   491    Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int})
   492                 else s mem_string allowed_consts
   493  | a$b => check a andalso check b
   494  | _ => false;
   495 
   496 val conv =
   497   Simplifier.rewrite
   498    (HOL_basic_ss addsimps
   499      ((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult},
   500              @{thm of_int_diff},  @{thm of_int_minus}])@
   501       [@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}])
   502      addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]);
   503 
   504 fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE
   505 
   506 val lhss' =
   507   [@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
   508    @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
   509    @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
   510 
   511 val zero_one_idom_simproc =
   512   make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
   513   proc = sproc, identifier = []}
   514 
   515 val add_rules =
   516     simp_thms @ @{thms arith_simps} @ @{thms rel_simps} @ @{thms arith_special} @
   517     [@{thm neg_le_iff_le}, @{thm numeral_0_eq_0}, @{thm numeral_1_eq_1},
   518      @{thm minus_zero}, @{thm diff_minus}, @{thm left_minus}, @{thm right_minus},
   519      @{thm mult_zero_left}, @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right},
   520      @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym,
   521      @{thm minus_add_distrib}, @{thm minus_minus}, @{thm mult_assoc},
   522      @{thm of_nat_0}, @{thm of_nat_1}, @{thm of_nat_Suc}, @{thm of_nat_add},
   523      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, @{thm of_int_add},
   524      @{thm of_int_mult}]
   525 
   526 val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2]
   527 
   528 val int_numeral_base_simprocs = Int_Numeral_Simprocs.assoc_fold_simproc :: zero_one_idom_simproc
   529   :: Int_Numeral_Simprocs.combine_numerals
   530   :: Int_Numeral_Simprocs.cancel_numerals;
   531 
   532 val setup =
   533   LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
   534    {add_mono_thms = add_mono_thms,
   535     mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
   536     inj_thms = nat_inj_thms @ inj_thms,
   537     lessD = lessD @ [@{thm zless_imp_add1_zle}],
   538     neqE = neqE,
   539     simpset = simpset addsimps add_rules
   540                       addsimprocs int_numeral_base_simprocs
   541                       addcongs [if_weak_cong]}) #>
   542   arith_inj_const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) #>
   543   arith_discrete @{type_name Int.int}
   544 
   545 val fast_int_arith_simproc =
   546   Simplifier.simproc (the_context ())
   547   "fast_int_arith" 
   548      ["(m::'a::{ordered_idom,number_ring}) < n",
   549       "(m::'a::{ordered_idom,number_ring}) <= n",
   550       "(m::'a::{ordered_idom,number_ring}) = n"] (K LinArith.lin_arith_simproc);
   551 
   552 end;
   553 
   554 Addsimprocs [Int_Arith.fast_int_arith_simproc];