src/HOL/Integ/int_arith1.ML
author paulson
Fri Mar 19 10:46:25 2004 +0100 (2004-03-19)
changeset 14474 00292f6f8d13
parent 14390 55fe71faadda
child 14738 83f1a514dcb4
permissions -rw-r--r--
New simplification ordering to move numerals together. Fixes a bug in the
nat cancellation simprocs
     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 (** New term ordering so that AC-rewriting brings numerals to the front **)
   161 
   162 (*Order integers by absolute value and then by sign. The standard integer
   163   ordering is not well-founded.*)
   164 fun num_ord (i,j) =
   165       (case Int.compare (abs i, abs j) of
   166             EQUAL => Int.compare (Int.sign i, Int.sign j) 
   167           | ord => ord);
   168 
   169 (*This resembles Term.term_ord, but it puts binary numerals before other
   170   non-atomic terms.*)
   171 local open Term 
   172 in 
   173 fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
   174       (case numterm_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
   175   | numterm_ord
   176      (Const("Numeral.number_of", _) $ v, Const("Numeral.number_of", _) $ w) =
   177      num_ord (HOLogic.dest_binum v, HOLogic.dest_binum w)
   178   | numterm_ord (Const("Numeral.number_of", _) $ _, _) = LESS
   179   | numterm_ord (_, Const("Numeral.number_of", _) $ _) = GREATER
   180   | numterm_ord (t, u) =
   181       (case Int.compare (size_of_term t, size_of_term u) of
   182         EQUAL =>
   183           let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
   184             (case hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
   185           end
   186       | ord => ord)
   187 and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
   188 end;
   189 
   190 fun numtermless tu = (numterm_ord tu = LESS);
   191 
   192 (*Defined in this file, but perhaps needed only for simprocs of type nat.*)
   193 val num_ss = HOL_ss settermless numtermless;
   194 
   195 
   196 (** Utilities **)
   197 
   198 fun mk_numeral T n = HOLogic.number_of_const T $ HOLogic.mk_bin n;
   199 
   200 (*Decodes a binary INTEGER*)
   201 fun dest_numeral (Const("0", _)) = 0
   202   | dest_numeral (Const("1", _)) = 1
   203   | dest_numeral (Const("Numeral.number_of", _) $ w) =
   204      (HOLogic.dest_binum w
   205       handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
   206   | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
   207 
   208 fun find_first_numeral past (t::terms) =
   209         ((dest_numeral t, rev past @ terms)
   210          handle TERM _ => find_first_numeral (t::past) terms)
   211   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
   212 
   213 val mk_plus = HOLogic.mk_binop "op +";
   214 
   215 fun mk_minus t = 
   216   let val T = Term.fastype_of t
   217   in Const ("uminus", T --> T) $ t
   218   end;
   219 
   220 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   221 fun mk_sum T []        = mk_numeral T 0
   222   | mk_sum T [t,u]     = mk_plus (t, u)
   223   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   224 
   225 (*this version ALWAYS includes a trailing zero*)
   226 fun long_mk_sum T []        = mk_numeral T 0
   227   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
   228 
   229 val dest_plus = HOLogic.dest_bin "op +" Term.dummyT;
   230 
   231 (*decompose additions AND subtractions as a sum*)
   232 fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
   233         dest_summing (pos, t, dest_summing (pos, u, ts))
   234   | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
   235         dest_summing (pos, t, dest_summing (not pos, u, ts))
   236   | dest_summing (pos, t, ts) =
   237         if pos then t::ts else mk_minus t :: ts;
   238 
   239 fun dest_sum t = dest_summing (true, t, []);
   240 
   241 val mk_diff = HOLogic.mk_binop "op -";
   242 val dest_diff = HOLogic.dest_bin "op -" Term.dummyT;
   243 
   244 val mk_times = HOLogic.mk_binop "op *";
   245 
   246 fun mk_prod T = 
   247   let val one = mk_numeral T 1
   248   fun mk [] = one
   249     | mk [t] = t
   250     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
   251   in mk end;
   252 
   253 (*This version ALWAYS includes a trailing one*)
   254 fun long_mk_prod T []        = mk_numeral T 1
   255   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
   256 
   257 val dest_times = HOLogic.dest_bin "op *" Term.dummyT;
   258 
   259 fun dest_prod t =
   260       let val (t,u) = dest_times t
   261       in  dest_prod t @ dest_prod u  end
   262       handle TERM _ => [t];
   263 
   264 (*DON'T do the obvious simplifications; that would create special cases*)
   265 fun mk_coeff (k, t) = mk_times (mk_numeral (Term.fastype_of t) k, t);
   266 
   267 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   268 fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
   269   | dest_coeff sign t =
   270     let val ts = sort Term.term_ord (dest_prod t)
   271         val (n, ts') = find_first_numeral [] ts
   272                           handle TERM _ => (1, ts)
   273     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
   274 
   275 (*Find first coefficient-term THAT MATCHES u*)
   276 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   277   | find_first_coeff past u (t::terms) =
   278         let val (n,u') = dest_coeff 1 t
   279         in  if u aconv u' then (n, rev past @ terms)
   280                           else find_first_coeff (t::past) u terms
   281         end
   282         handle TERM _ => find_first_coeff (t::past) u terms;
   283 
   284 
   285 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
   286 val add_0s =  thms "add_0s";
   287 val mult_1s = thms "mult_1s";
   288 
   289 (*To perform binary arithmetic.  The "left" rewriting handles patterns
   290   created by the simprocs, such as 3 * (5 * x). *)
   291 val bin_simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
   292                  add_number_of_left, mult_number_of_left] @
   293                 bin_arith_simps @ bin_rel_simps;
   294 
   295 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
   296   during re-arrangement*)
   297 val non_add_bin_simps = 
   298     bin_simps \\ [add_number_of_left, number_of_add RS sym];
   299 
   300 (*To evaluate binary negations of coefficients*)
   301 val minus_simps = NCons_simps @
   302                    [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
   303                     bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
   304                     bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
   305 
   306 (*To let us treat subtraction as addition*)
   307 val diff_simps = [diff_minus, minus_add_distrib, minus_minus];
   308 
   309 (*push the unary minus down: - x * y = x * - y *)
   310 val minus_mult_eq_1_to_2 =
   311     [minus_mult_left RS sym, minus_mult_right] MRS trans |> standard;
   312 
   313 (*to extract again any uncancelled minuses*)
   314 val minus_from_mult_simps =
   315     [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
   316 
   317 (*combine unary minus with numeric literals, however nested within a product*)
   318 val mult_minus_simps =
   319     [mult_assoc, minus_mult_left, minus_mult_eq_1_to_2];
   320 
   321 (*Apply the given rewrite (if present) just once*)
   322 fun trans_tac None      = all_tac
   323   | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
   324 
   325 fun simplify_meta_eq rules =
   326     simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
   327     o mk_meta_eq;
   328 
   329 structure CancelNumeralsCommon =
   330   struct
   331   val mk_sum            = mk_sum
   332   val dest_sum          = dest_sum
   333   val mk_coeff          = mk_coeff
   334   val dest_coeff        = dest_coeff 1
   335   val find_first_coeff  = find_first_coeff []
   336   val trans_tac         = trans_tac
   337   val norm_tac =
   338      ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
   339                                          diff_simps@minus_simps@add_ac))
   340      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
   341      THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
   342                                               add_ac@mult_ac))
   343   val numeral_simp_tac  = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   344   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   345   end;
   346 
   347 
   348 structure EqCancelNumerals = CancelNumeralsFun
   349  (open CancelNumeralsCommon
   350   val prove_conv = Bin_Simprocs.prove_conv
   351   val mk_bal   = HOLogic.mk_eq
   352   val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
   353   val bal_add1 = eq_add_iff1 RS trans
   354   val bal_add2 = eq_add_iff2 RS trans
   355 );
   356 
   357 structure LessCancelNumerals = CancelNumeralsFun
   358  (open CancelNumeralsCommon
   359   val prove_conv = Bin_Simprocs.prove_conv
   360   val mk_bal   = HOLogic.mk_binrel "op <"
   361   val dest_bal = HOLogic.dest_bin "op <" Term.dummyT
   362   val bal_add1 = less_add_iff1 RS trans
   363   val bal_add2 = less_add_iff2 RS trans
   364 );
   365 
   366 structure LeCancelNumerals = CancelNumeralsFun
   367  (open CancelNumeralsCommon
   368   val prove_conv = Bin_Simprocs.prove_conv
   369   val mk_bal   = HOLogic.mk_binrel "op <="
   370   val dest_bal = HOLogic.dest_bin "op <=" Term.dummyT
   371   val bal_add1 = le_add_iff1 RS trans
   372   val bal_add2 = le_add_iff2 RS trans
   373 );
   374 
   375 val cancel_numerals =
   376   map Bin_Simprocs.prep_simproc
   377    [("inteq_cancel_numerals",
   378      ["(l::'a::number_ring) + m = n",
   379       "(l::'a::number_ring) = m + n",
   380       "(l::'a::number_ring) - m = n",
   381       "(l::'a::number_ring) = m - n",
   382       "(l::'a::number_ring) * m = n",
   383       "(l::'a::number_ring) = m * n"],
   384      EqCancelNumerals.proc),
   385     ("intless_cancel_numerals",
   386      ["(l::'a::{ordered_ring,number_ring}) + m < n",
   387       "(l::'a::{ordered_ring,number_ring}) < m + n",
   388       "(l::'a::{ordered_ring,number_ring}) - m < n",
   389       "(l::'a::{ordered_ring,number_ring}) < m - n",
   390       "(l::'a::{ordered_ring,number_ring}) * m < n",
   391       "(l::'a::{ordered_ring,number_ring}) < m * n"],
   392      LessCancelNumerals.proc),
   393     ("intle_cancel_numerals",
   394      ["(l::'a::{ordered_ring,number_ring}) + m <= n",
   395       "(l::'a::{ordered_ring,number_ring}) <= m + n",
   396       "(l::'a::{ordered_ring,number_ring}) - m <= n",
   397       "(l::'a::{ordered_ring,number_ring}) <= m - n",
   398       "(l::'a::{ordered_ring,number_ring}) * m <= n",
   399       "(l::'a::{ordered_ring,number_ring}) <= m * n"],
   400      LeCancelNumerals.proc)];
   401 
   402 
   403 structure CombineNumeralsData =
   404   struct
   405   val add               = op + : int*int -> int
   406   val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
   407   val dest_sum          = dest_sum
   408   val mk_coeff          = mk_coeff
   409   val dest_coeff        = dest_coeff 1
   410   val left_distrib      = combine_common_factor RS trans
   411   val prove_conv        = Bin_Simprocs.prove_conv_nohyps
   412   val trans_tac          = trans_tac
   413   val norm_tac =
   414      ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
   415                                          diff_simps@minus_simps@add_ac))
   416      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
   417      THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
   418                                               add_ac@mult_ac))
   419   val numeral_simp_tac  = ALLGOALS
   420                     (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   421   val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
   422   end;
   423 
   424 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   425 
   426 val combine_numerals =
   427   Bin_Simprocs.prep_simproc
   428     ("int_combine_numerals", 
   429      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
   430      CombineNumerals.proc);
   431 
   432 end;
   433 
   434 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
   435 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
   436 
   437 (*examples:
   438 print_depth 22;
   439 set timing;
   440 set trace_simp;
   441 fun test s = (Goal s, by (Simp_tac 1));
   442 
   443 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
   444 
   445 test "2*u = (u::int)";
   446 test "(i + j + 12 + (k::int)) - 15 = y";
   447 test "(i + j + 12 + (k::int)) - 5 = y";
   448 
   449 test "y - b < (b::int)";
   450 test "y - (3*b + c) < (b::int) - 2*c";
   451 
   452 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
   453 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
   454 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
   455 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
   456 
   457 test "(i + j + 12 + (k::int)) = u + 15 + y";
   458 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
   459 
   460 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)";
   461 
   462 test "a + -(b+c) + b = (d::int)";
   463 test "a + -(b+c) - b = (d::int)";
   464 
   465 (*negative numerals*)
   466 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
   467 test "(i + j + -3 + (k::int)) < u + 5 + y";
   468 test "(i + j + 3 + (k::int)) < u + -6 + y";
   469 test "(i + j + -12 + (k::int)) - 15 = y";
   470 test "(i + j + 12 + (k::int)) - -15 = y";
   471 test "(i + j + -12 + (k::int)) - -15 = y";
   472 *)
   473 
   474 
   475 (** Constant folding for multiplication in semirings **)
   476 
   477 (*We do not need folding for addition: combine_numerals does the same thing*)
   478 
   479 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
   480 struct
   481   val ss                = HOL_ss
   482   val eq_reflection     = eq_reflection
   483   val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
   484   val add_ac = mult_ac
   485 end;
   486 
   487 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
   488 
   489 val assoc_fold_simproc =
   490   Bin_Simprocs.prep_simproc
   491    ("semiring_assoc_fold", ["(a::'a::semiring) * b"],
   492     Semiring_Times_Assoc.proc);
   493 
   494 Addsimprocs [assoc_fold_simproc];
   495 
   496 
   497 
   498 
   499 (*** decision procedure for linear arithmetic ***)
   500 
   501 (*---------------------------------------------------------------------------*)
   502 (* Linear arithmetic                                                         *)
   503 (*---------------------------------------------------------------------------*)
   504 
   505 (*
   506 Instantiation of the generic linear arithmetic package for int.
   507 *)
   508 
   509 (* Update parameters of arithmetic prover *)
   510 local
   511 
   512 (* reduce contradictory <= to False *)
   513 val add_rules =
   514     simp_thms @ bin_arith_simps @ bin_rel_simps @ arith_special @
   515     [neg_le_iff_le, numeral_0_eq_0, numeral_1_eq_1,
   516      minus_zero, diff_minus, left_minus, right_minus,
   517      mult_zero_left, mult_zero_right, mult_1, mult_1_right,
   518      minus_mult_left RS sym, minus_mult_right RS sym,
   519      minus_add_distrib, minus_minus, mult_assoc,
   520      of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,
   521      of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat,
   522      zero_neq_one, zero_less_one, zero_le_one, 
   523      zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];
   524 
   525 val simprocs = [assoc_fold_simproc, Int_Numeral_Simprocs.combine_numerals]@
   526                Int_Numeral_Simprocs.cancel_numerals;
   527 
   528 in
   529 
   530 val int_arith_setup =
   531  [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
   532    {add_mono_thms = add_mono_thms,
   533     mult_mono_thms = mult_mono_thms,
   534     inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,
   535     lessD = lessD @ [zless_imp_add1_zle],
   536     simpset = simpset addsimps add_rules
   537                       addsimprocs simprocs
   538                       addcongs [if_weak_cong]}),
   539   arith_inj_const ("IntDef.of_nat", HOLogic.natT --> HOLogic.intT),
   540   arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),
   541   arith_discrete ("IntDef.int", true)];
   542 
   543 end;
   544 
   545 val fast_int_arith_simproc =
   546   Simplifier.simproc (Theory.sign_of (the_context()))
   547   "fast_int_arith" 
   548      ["(m::'a::{ordered_ring,number_ring}) < n",
   549       "(m::'a::{ordered_ring,number_ring}) <= n",
   550       "(m::'a::{ordered_ring,number_ring}) = n"] Fast_Arith.lin_arith_prover;
   551 
   552 Addsimprocs [fast_int_arith_simproc]
   553 
   554 
   555 (* Some test data
   556 Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
   557 by (fast_arith_tac 1);
   558 Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";
   559 by (fast_arith_tac 1);
   560 Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";
   561 by (fast_arith_tac 1);
   562 Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
   563 by (fast_arith_tac 1);
   564 Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \
   565 \     ==> a+a <= j+j";
   566 by (fast_arith_tac 1);
   567 Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \
   568 \     ==> a+a - - -1 < j+j - 3";
   569 by (fast_arith_tac 1);
   570 Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
   571 by (arith_tac 1);
   572 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   573 \     ==> a <= l";
   574 by (fast_arith_tac 1);
   575 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   576 \     ==> a+a+a+a <= l+l+l+l";
   577 by (fast_arith_tac 1);
   578 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   579 \     ==> a+a+a+a+a <= l+l+l+l+i";
   580 by (fast_arith_tac 1);
   581 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   582 \     ==> a+a+a+a+a+a <= l+l+l+l+i+l";
   583 by (fast_arith_tac 1);
   584 Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
   585 \     ==> 6*a <= 5*l+i";
   586 by (fast_arith_tac 1);
   587 *)