src/HOL/int_arith1.ML
author wenzelm
Thu Oct 04 20:29:42 2007 +0200 (2007-10-04)
changeset 24850 0cfd722ab579
parent 24630 351a308ab58d
child 25481 aa16cd919dcc
permissions -rw-r--r--
Name.uu, Name.aT;
     1 (*  Title:      HOL/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 succ_Pls = thm "succ_Pls";
    11 val succ_Min = thm "succ_Min";
    12 val succ_1 = thm "succ_1";
    13 val succ_0 = thm "succ_0";
    14 
    15 val pred_Pls = thm "pred_Pls";
    16 val pred_Min = thm "pred_Min";
    17 val pred_1 = thm "pred_1";
    18 val pred_0 = thm "pred_0";
    19 
    20 val minus_Pls = thm "minus_Pls";
    21 val minus_Min = thm "minus_Min";
    22 val minus_1 = thm "minus_1";
    23 val minus_0 = thm "minus_0";
    24 
    25 val add_Pls = thm "add_Pls";
    26 val add_Min = thm "add_Min";
    27 val add_BIT_11 = thm "add_BIT_11";
    28 val add_BIT_10 = thm "add_BIT_10";
    29 val add_BIT_0 = thm "add_BIT_0";
    30 val add_Pls_right = thm "add_Pls_right";
    31 val add_Min_right = thm "add_Min_right";
    32 
    33 val mult_Pls = thm "mult_Pls";
    34 val mult_Min = thm "mult_Min";
    35 val mult_num1 = thm "mult_num1";
    36 val mult_num0 = thm "mult_num0";
    37 
    38 val neg_def = thm "neg_def";
    39 val iszero_def = thm "iszero_def";
    40 
    41 val number_of_succ = thm "number_of_succ";
    42 val number_of_pred = thm "number_of_pred";
    43 val number_of_minus = thm "number_of_minus";
    44 val number_of_add = thm "number_of_add";
    45 val diff_number_of_eq = thm "diff_number_of_eq";
    46 val number_of_mult = thm "number_of_mult";
    47 val double_number_of_BIT = thm "double_number_of_BIT";
    48 val numeral_0_eq_0 = thm "numeral_0_eq_0";
    49 val numeral_1_eq_1 = thm "numeral_1_eq_1";
    50 val numeral_m1_eq_minus_1 = thm "numeral_m1_eq_minus_1";
    51 val mult_minus1 = thm "mult_minus1";
    52 val mult_minus1_right = thm "mult_minus1_right";
    53 val minus_number_of_mult = thm "minus_number_of_mult";
    54 val zero_less_nat_eq = thm "zero_less_nat_eq";
    55 val eq_number_of_eq = thm "eq_number_of_eq";
    56 val iszero_number_of_Pls = thm "iszero_number_of_Pls";
    57 val nonzero_number_of_Min = thm "nonzero_number_of_Min";
    58 val iszero_number_of_BIT = thm "iszero_number_of_BIT";
    59 val iszero_number_of_0 = thm "iszero_number_of_0";
    60 val iszero_number_of_1 = thm "iszero_number_of_1";
    61 val less_number_of_eq_neg = thm "less_number_of_eq_neg";
    62 val le_number_of_eq = thm "le_number_of_eq";
    63 val not_neg_number_of_Pls = thm "not_neg_number_of_Pls";
    64 val neg_number_of_Min = thm "neg_number_of_Min";
    65 val neg_number_of_BIT = thm "neg_number_of_BIT";
    66 val le_number_of_eq_not_less = thm "le_number_of_eq_not_less";
    67 val abs_number_of = thm "abs_number_of";
    68 val number_of_reorient = thm "number_of_reorient";
    69 val add_number_of_left = thm "add_number_of_left";
    70 val mult_number_of_left = thm "mult_number_of_left";
    71 val add_number_of_diff1 = thm "add_number_of_diff1";
    72 val add_number_of_diff2 = thm "add_number_of_diff2";
    73 val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
    74 val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
    75 val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
    76 
    77 val arith_extra_simps = thms "arith_extra_simps";
    78 val arith_simps = thms "arith_simps";
    79 val rel_simps = thms "rel_simps";
    80 
    81 val zless_imp_add1_zle = thm "zless_imp_add1_zle";
    82 
    83 val combine_common_factor = thm "combine_common_factor";
    84 val eq_add_iff1 = thm "eq_add_iff1";
    85 val eq_add_iff2 = thm "eq_add_iff2";
    86 val less_add_iff1 = thm "less_add_iff1";
    87 val less_add_iff2 = thm "less_add_iff2";
    88 val le_add_iff1 = thm "le_add_iff1";
    89 val le_add_iff2 = thm "le_add_iff2";
    90 
    91 val arith_special = thms "arith_special";
    92 
    93 structure Int_Numeral_Base_Simprocs =
    94   struct
    95   fun prove_conv tacs ctxt (_: thm list) (t, u) =
    96     if t aconv u then NONE
    97     else
    98       let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
    99       in SOME (Goal.prove ctxt [] [] eq (K (EVERY tacs))) end
   100 
   101   fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];
   102 
   103   fun prep_simproc (name, pats, proc) =
   104     Simplifier.simproc (the_context()) name pats proc;
   105 
   106   fun is_numeral (Const(@{const_name Numeral.number_of}, _) $ w) = true
   107     | is_numeral _ = false
   108 
   109   fun simplify_meta_eq f_number_of_eq f_eq =
   110       mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
   111 
   112   (*reorientation simprules using ==, for the following simproc*)
   113   val meta_zero_reorient = @{thm zero_reorient} RS eq_reflection
   114   val meta_one_reorient = @{thm one_reorient} RS eq_reflection
   115   val meta_number_of_reorient = number_of_reorient RS eq_reflection
   116 
   117   (*reorientation simplification procedure: reorients (polymorphic) 
   118     0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
   119   fun reorient_proc sg _ (_ $ t $ u) =
   120     case u of
   121 	Const(@{const_name HOL.zero}, _) => NONE
   122       | Const(@{const_name HOL.one}, _) => NONE
   123       | Const(@{const_name Numeral.number_of}, _) $ _ => NONE
   124       | _ => SOME (case t of
   125 		  Const(@{const_name HOL.zero}, _) => meta_zero_reorient
   126 		| Const(@{const_name HOL.one}, _) => meta_one_reorient
   127 		| Const(@{const_name Numeral.number_of}, _) $ _ => meta_number_of_reorient)
   128 
   129   val reorient_simproc = 
   130       prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)
   131 
   132   end;
   133 
   134 
   135 Addsimprocs [Int_Numeral_Base_Simprocs.reorient_simproc];
   136 
   137 
   138 structure Int_Numeral_Simprocs =
   139 struct
   140 
   141 (*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in Int_Numeral_Base_Simprocs
   142   isn't complicated by the abstract 0 and 1.*)
   143 val numeral_syms = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];
   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 Term.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 => typ_ord (T, U) | ord => ord)
   160   | numterm_ord
   161      (Const(@{const_name Numeral.number_of}, _) $ v, Const(@{const_name Numeral.number_of}, _) $ w) =
   162      num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
   163   | numterm_ord (Const(@{const_name Numeral.number_of}, _) $ _, _) = LESS
   164   | numterm_ord (_, Const(@{const_name Numeral.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 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 (*Defined in this file, but perhaps needed only for Int_Numeral_Base_Simprocs of type nat.*)
   178 val num_ss = HOL_ss settermless numtermless;
   179 
   180 
   181 (** Utilities **)
   182 
   183 fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
   184 
   185 fun find_first_numeral past (t::terms) =
   186         ((snd (HOLogic.dest_number t), rev past @ terms)
   187          handle TERM _ => find_first_numeral (t::past) terms)
   188   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
   189 
   190 val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
   191 
   192 fun mk_minus t = 
   193   let val T = Term.fastype_of t
   194   in Const (@{const_name HOL.uminus}, T --> T) $ t end;
   195 
   196 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   197 fun mk_sum T []        = mk_number T 0
   198   | mk_sum T [t,u]     = mk_plus (t, u)
   199   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   200 
   201 (*this version ALWAYS includes a trailing zero*)
   202 fun long_mk_sum T []        = mk_number T 0
   203   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   204 
   205 val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
   206 
   207 (*decompose additions AND subtractions as a sum*)
   208 fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
   209         dest_summing (pos, t, dest_summing (pos, u, ts))
   210   | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
   211         dest_summing (pos, t, dest_summing (not pos, u, ts))
   212   | dest_summing (pos, t, ts) =
   213         if pos then t::ts else mk_minus t :: ts;
   214 
   215 fun dest_sum t = dest_summing (true, t, []);
   216 
   217 val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
   218 val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
   219 
   220 val mk_times = HOLogic.mk_binop @{const_name HOL.times};
   221 
   222 fun one_of T = Const(@{const_name HOL.one},T);
   223 
   224 (* build product with trailing 1 rather than Numeral 1 in order to avoid the
   225    unnecessary restriction to type class number_ring
   226    which is not required for cancellation of common factors in divisions.
   227 *)
   228 fun mk_prod T = 
   229   let val one = one_of T
   230   fun mk [] = one
   231     | mk [t] = t
   232     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
   233   in mk end;
   234 
   235 (*This version ALWAYS includes a trailing one*)
   236 fun long_mk_prod T []        = one_of T
   237   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
   238 
   239 val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
   240 
   241 fun dest_prod t =
   242       let val (t,u) = dest_times t
   243       in dest_prod t @ dest_prod u end
   244       handle TERM _ => [t];
   245 
   246 (*DON'T do the obvious simplifications; that would create special cases*)
   247 fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
   248 
   249 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   250 fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t
   251   | dest_coeff sign t =
   252     let val ts = sort Term.term_ord (dest_prod t)
   253         val (n, ts') = find_first_numeral [] ts
   254                           handle TERM _ => (1, ts)
   255     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
   256 
   257 (*Find first coefficient-term THAT MATCHES u*)
   258 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   259   | find_first_coeff past u (t::terms) =
   260         let val (n,u') = dest_coeff 1 t
   261         in if u aconv u' then (n, rev past @ terms)
   262                          else find_first_coeff (t::past) u terms
   263         end
   264         handle TERM _ => find_first_coeff (t::past) u terms;
   265 
   266 (*Fractions as pairs of ints. Can't use Rat.rat because the representation
   267   needs to preserve negative values in the denominator.*)
   268 fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
   269 
   270 (*Don't reduce fractions; sums must be proved by rule add_frac_eq.
   271   Fractions are reduced later by the cancel_numeral_factor simproc.*)
   272 fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
   273 
   274 val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
   275 
   276 (*Build term (p / q) * t*)
   277 fun mk_fcoeff ((p, q), t) =
   278   let val T = Term.fastype_of t
   279   in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
   280 
   281 (*Express t as a product of a fraction with other sorted terms*)
   282 fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t
   283   | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) =
   284     let val (p, t') = dest_coeff sign t
   285         val (q, u') = dest_coeff 1 u
   286     in (mk_frac (p, q), mk_divide (t', u')) end
   287   | dest_fcoeff sign t =
   288     let val (p, t') = dest_coeff sign t
   289         val T = Term.fastype_of t
   290     in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
   291 
   292 
   293 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
   294 val add_0s =  thms "add_0s";
   295 val mult_1s = thms "mult_1s" @ [thm"mult_1_left", thm"mult_1_right", thm"divide_1"];
   296 
   297 (*Simplify inverse Numeral1, a/Numeral1*)
   298 val inverse_1s = [@{thm inverse_numeral_1}];
   299 val divide_1s = [@{thm divide_numeral_1}];
   300 
   301 (*To perform binary arithmetic.  The "left" rewriting handles patterns
   302   created by the Int_Numeral_Base_Simprocs, such as 3 * (5 * x). *)
   303 val simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
   304                  add_number_of_left, mult_number_of_left] @
   305                 arith_simps @ rel_simps;
   306 
   307 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
   308   during re-arrangement*)
   309 val non_add_simps =
   310   subtract Thm.eq_thm [add_number_of_left, number_of_add RS sym] simps;
   311 
   312 (*To evaluate binary negations of coefficients*)
   313 val minus_simps = [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
   314                    minus_1, minus_0, minus_Pls, minus_Min,
   315                    pred_1, pred_0, pred_Pls, pred_Min];
   316 
   317 (*To let us treat subtraction as addition*)
   318 val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
   319 
   320 (*To let us treat division as multiplication*)
   321 val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
   322 
   323 (*push the unary minus down: - x * y = x * - y *)
   324 val minus_mult_eq_1_to_2 =
   325     [@{thm minus_mult_left} RS sym, @{thm minus_mult_right}] MRS trans |> standard;
   326 
   327 (*to extract again any uncancelled minuses*)
   328 val minus_from_mult_simps =
   329     [@{thm minus_minus}, @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym];
   330 
   331 (*combine unary minus with numeric literals, however nested within a product*)
   332 val mult_minus_simps =
   333     [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
   334 
   335 (*Apply the given rewrite (if present) just once*)
   336 fun trans_tac NONE      = all_tac
   337   | trans_tac (SOME th) = ALLGOALS (rtac (th RS trans));
   338 
   339 fun simplify_meta_eq rules =
   340   let val ss0 = HOL_basic_ss addeqcongs [eq_cong2] addsimps rules
   341   in fn ss => simplify (Simplifier.inherit_context ss ss0) o mk_meta_eq end
   342 
   343 structure CancelNumeralsCommon =
   344   struct
   345   val mk_sum            = mk_sum
   346   val dest_sum          = dest_sum
   347   val mk_coeff          = mk_coeff
   348   val dest_coeff        = dest_coeff 1
   349   val find_first_coeff  = find_first_coeff []
   350   val trans_tac         = fn _ => trans_tac
   351 
   352   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   353     diff_simps @ minus_simps @ @{thms add_ac}
   354   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   355   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   356   fun norm_tac ss =
   357     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   358     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   359     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   360 
   361   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
   362   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   363   val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
   364   end;
   365 
   366 
   367 structure EqCancelNumerals = CancelNumeralsFun
   368  (open CancelNumeralsCommon
   369   val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
   370   val mk_bal   = HOLogic.mk_eq
   371   val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
   372   val bal_add1 = eq_add_iff1 RS trans
   373   val bal_add2 = eq_add_iff2 RS trans
   374 );
   375 
   376 structure LessCancelNumerals = CancelNumeralsFun
   377  (open CancelNumeralsCommon
   378   val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
   379   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
   380   val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
   381   val bal_add1 = less_add_iff1 RS trans
   382   val bal_add2 = less_add_iff2 RS trans
   383 );
   384 
   385 structure LeCancelNumerals = CancelNumeralsFun
   386  (open CancelNumeralsCommon
   387   val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
   388   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
   389   val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
   390   val bal_add1 = le_add_iff1 RS trans
   391   val bal_add2 = le_add_iff2 RS trans
   392 );
   393 
   394 val cancel_numerals =
   395   map Int_Numeral_Base_Simprocs.prep_simproc
   396    [("inteq_cancel_numerals",
   397      ["(l::'a::number_ring) + m = n",
   398       "(l::'a::number_ring) = m + n",
   399       "(l::'a::number_ring) - m = n",
   400       "(l::'a::number_ring) = m - n",
   401       "(l::'a::number_ring) * m = n",
   402       "(l::'a::number_ring) = m * n"],
   403      K EqCancelNumerals.proc),
   404     ("intless_cancel_numerals",
   405      ["(l::'a::{ordered_idom,number_ring}) + m < n",
   406       "(l::'a::{ordered_idom,number_ring}) < m + n",
   407       "(l::'a::{ordered_idom,number_ring}) - m < n",
   408       "(l::'a::{ordered_idom,number_ring}) < m - n",
   409       "(l::'a::{ordered_idom,number_ring}) * m < n",
   410       "(l::'a::{ordered_idom,number_ring}) < m * n"],
   411      K LessCancelNumerals.proc),
   412     ("intle_cancel_numerals",
   413      ["(l::'a::{ordered_idom,number_ring}) + m <= n",
   414       "(l::'a::{ordered_idom,number_ring}) <= m + n",
   415       "(l::'a::{ordered_idom,number_ring}) - m <= n",
   416       "(l::'a::{ordered_idom,number_ring}) <= m - n",
   417       "(l::'a::{ordered_idom,number_ring}) * m <= n",
   418       "(l::'a::{ordered_idom,number_ring}) <= m * n"],
   419      K LeCancelNumerals.proc)];
   420 
   421 
   422 structure CombineNumeralsData =
   423   struct
   424   type coeff            = int
   425   val iszero            = (fn x => x = 0)
   426   val add               = op +
   427   val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
   428   val dest_sum          = dest_sum
   429   val mk_coeff          = mk_coeff
   430   val dest_coeff        = dest_coeff 1
   431   val left_distrib      = combine_common_factor RS trans
   432   val prove_conv        = Int_Numeral_Base_Simprocs.prove_conv_nohyps
   433   val trans_tac         = fn _ => trans_tac
   434 
   435   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   436     diff_simps @ minus_simps @ @{thms add_ac}
   437   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   438   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   439   fun norm_tac ss =
   440     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   441     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   442     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   443 
   444   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
   445   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   446   val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
   447   end;
   448 
   449 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   450 
   451 (*Version for fields, where coefficients can be fractions*)
   452 structure FieldCombineNumeralsData =
   453   struct
   454   type coeff            = int * int
   455   val iszero            = (fn (p, q) => p = 0)
   456   val add               = add_frac
   457   val mk_sum            = long_mk_sum
   458   val dest_sum          = dest_sum
   459   val mk_coeff          = mk_fcoeff
   460   val dest_coeff        = dest_fcoeff 1
   461   val left_distrib      = combine_common_factor RS trans
   462   val prove_conv        = Int_Numeral_Base_Simprocs.prove_conv_nohyps
   463   val trans_tac         = fn _ => trans_tac
   464 
   465   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
   466     inverse_1s @ divide_simps @ diff_simps @ minus_simps @ @{thms add_ac}
   467   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
   468   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
   469   fun norm_tac ss =
   470     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
   471     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
   472     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
   473 
   474   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
   475   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   476   val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
   477   end;
   478 
   479 structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
   480 
   481 val combine_numerals =
   482   Int_Numeral_Base_Simprocs.prep_simproc
   483     ("int_combine_numerals", 
   484      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
   485      K CombineNumerals.proc);
   486 
   487 val field_combine_numerals =
   488   Int_Numeral_Base_Simprocs.prep_simproc
   489     ("field_combine_numerals", 
   490      ["(i::'a::{number_ring,field,division_by_zero}) + j",
   491       "(i::'a::{number_ring,field,division_by_zero}) - j"], 
   492      K FieldCombineNumerals.proc);
   493 
   494 end;
   495 
   496 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
   497 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
   498 Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
   499 
   500 (*examples:
   501 print_depth 22;
   502 set timing;
   503 set trace_simp;
   504 fun test s = (Goal s, by (Simp_tac 1));
   505 
   506 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
   507 
   508 test "2*u = (u::int)";
   509 test "(i + j + 12 + (k::int)) - 15 = y";
   510 test "(i + j + 12 + (k::int)) - 5 = y";
   511 
   512 test "y - b < (b::int)";
   513 test "y - (3*b + c) < (b::int) - 2*c";
   514 
   515 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
   516 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
   517 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
   518 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
   519 
   520 test "(i + j + 12 + (k::int)) = u + 15 + y";
   521 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
   522 
   523 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)";
   524 
   525 test "a + -(b+c) + b = (d::int)";
   526 test "a + -(b+c) - b = (d::int)";
   527 
   528 (*negative numerals*)
   529 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
   530 test "(i + j + -3 + (k::int)) < u + 5 + y";
   531 test "(i + j + 3 + (k::int)) < u + -6 + y";
   532 test "(i + j + -12 + (k::int)) - 15 = y";
   533 test "(i + j + 12 + (k::int)) - -15 = y";
   534 test "(i + j + -12 + (k::int)) - -15 = y";
   535 *)
   536 
   537 
   538 (** Constant folding for multiplication in semirings **)
   539 
   540 (*We do not need folding for addition: combine_numerals does the same thing*)
   541 
   542 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
   543 struct
   544   val assoc_ss = HOL_ss addsimps @{thms mult_ac}
   545   val eq_reflection = eq_reflection
   546 end;
   547 
   548 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
   549 
   550 val assoc_fold_simproc =
   551   Int_Numeral_Base_Simprocs.prep_simproc
   552    ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
   553     K Semiring_Times_Assoc.proc);
   554 
   555 Addsimprocs [assoc_fold_simproc];
   556 
   557 
   558 
   559 
   560 (*** decision procedure for linear arithmetic ***)
   561 
   562 (*---------------------------------------------------------------------------*)
   563 (* Linear arithmetic                                                         *)
   564 (*---------------------------------------------------------------------------*)
   565 
   566 (*
   567 Instantiation of the generic linear arithmetic package for int.
   568 *)
   569 
   570 (* Update parameters of arithmetic prover *)
   571 local
   572 
   573 (* reduce contradictory =/</<= to False *)
   574 
   575 (* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<",
   576    and m and n are ground terms over rings (roughly speaking).
   577    That is, m and n consist only of 1s combined with "+", "-" and "*".
   578 *)
   579 local
   580 val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0};
   581 val lhss0 = [@{cpat "0::?'a::ring"}];
   582 fun proc0 phi ss ct =
   583   let val T = ctyp_of_term ct
   584   in if typ_of T = @{typ int} then NONE else
   585      SOME (instantiate' [SOME T] [] zeroth)
   586   end;
   587 val zero_to_of_int_zero_simproc =
   588   make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc",
   589   proc = proc0, identifier = []};
   590 
   591 val oneth = (symmetric o mk_meta_eq) @{thm of_int_1};
   592 val lhss1 = [@{cpat "1::?'a::ring_1"}];
   593 fun proc1 phi ss ct =
   594   let val T = ctyp_of_term ct
   595   in if typ_of T = @{typ int} then NONE else
   596      SOME (instantiate' [SOME T] [] oneth)
   597   end;
   598 val one_to_of_int_one_simproc =
   599   make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc",
   600   proc = proc1, identifier = []};
   601 
   602 val allowed_consts =
   603   [@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"},
   604    @{const_name "HOL.minus"}, @{const_name "HOL.plus"},
   605    @{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"},
   606    @{const_name "HOL.less_eq"}];
   607 
   608 fun check t = case t of
   609    Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int})
   610                 else s mem_string allowed_consts
   611  | a$b => check a andalso check b
   612  | _ => false;
   613 
   614 val conv =
   615   Simplifier.rewrite
   616    (HOL_basic_ss addsimps
   617      ((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult},
   618              @{thm of_int_diff},  @{thm of_int_minus}])@
   619       [@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}])
   620      addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]);
   621 
   622 fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE
   623 val lhss' =
   624   [@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
   625    @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
   626    @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
   627 in
   628 val zero_one_idom_simproc =
   629   make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
   630   proc = sproc, identifier = []}
   631 end;
   632 
   633 val add_rules =
   634     simp_thms @ arith_simps @ rel_simps @ arith_special @
   635     [@{thm neg_le_iff_le}, @{thm numeral_0_eq_0}, @{thm numeral_1_eq_1},
   636      @{thm minus_zero}, @{thm diff_minus}, @{thm left_minus}, @{thm right_minus},
   637      @{thm mult_zero_left}, @{thm mult_zero_right}, @{thm mult_num1}, @{thm mult_1_right},
   638      @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym,
   639      @{thm minus_add_distrib}, @{thm minus_minus}, @{thm mult_assoc},
   640      @{thm of_nat_0}, @{thm of_nat_1}, @{thm of_nat_Suc}, @{thm of_nat_add},
   641      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, @{thm of_int_add},
   642      @{thm of_int_mult}]
   643 
   644 val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2]
   645 
   646 val Int_Numeral_Base_Simprocs = assoc_fold_simproc :: zero_one_idom_simproc
   647   :: Int_Numeral_Simprocs.combine_numerals
   648   :: Int_Numeral_Simprocs.cancel_numerals;
   649 
   650 in
   651 
   652 val int_arith_setup =
   653   LinArith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
   654    {add_mono_thms = add_mono_thms,
   655     mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
   656     inj_thms = nat_inj_thms @ inj_thms,
   657     lessD = lessD @ [zless_imp_add1_zle],
   658     neqE = neqE,
   659     simpset = simpset addsimps add_rules
   660                       addsimprocs Int_Numeral_Base_Simprocs
   661                       addcongs [if_weak_cong]}) #>
   662   arith_inj_const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) #>
   663   arith_discrete "IntDef.int"
   664 
   665 end;
   666 
   667 val fast_int_arith_simproc =
   668   Simplifier.simproc @{theory}
   669   "fast_int_arith" 
   670      ["(m::'a::{ordered_idom,number_ring}) < n",
   671       "(m::'a::{ordered_idom,number_ring}) <= n",
   672       "(m::'a::{ordered_idom,number_ring}) = n"] (K LinArith.lin_arith_simproc);
   673 
   674 Addsimprocs [fast_int_arith_simproc];