src/HOL/Tools/int_arith.ML
 author huffman Mon Mar 30 12:07:59 2009 -0700 (2009-03-30) changeset 30802 f9e9e800d27e parent 30732 afca5be252d6 child 31024 0fdf666e08bf permissions -rw-r--r--
simplify theorem references
```     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
```
```    30 (** Utilities **)
```
```    31
```
```    32 fun mk_number T n = HOLogic.number_of_const T \$ HOLogic.mk_numeral n;
```
```    33
```
```    34 fun find_first_numeral past (t::terms) =
```
```    35         ((snd (HOLogic.dest_number t), rev past @ terms)
```
```    36          handle TERM _ => find_first_numeral (t::past) terms)
```
```    37   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
```
```    38
```
```    39 val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
```
```    40
```
```    41 fun mk_minus t =
```
```    42   let val T = Term.fastype_of t
```
```    43   in Const (@{const_name HOL.uminus}, T --> T) \$ t end;
```
```    44
```
```    45 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
```
```    46 fun mk_sum T []        = mk_number T 0
```
```    47   | mk_sum T [t,u]     = mk_plus (t, u)
```
```    48   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
```
```    49
```
```    50 (*this version ALWAYS includes a trailing zero*)
```
```    51 fun long_mk_sum T []        = mk_number T 0
```
```    52   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
```
```    53
```
```    54 val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
```
```    55
```
```    56 (*decompose additions AND subtractions as a sum*)
```
```    57 fun dest_summing (pos, Const (@{const_name HOL.plus}, _) \$ t \$ u, ts) =
```
```    58         dest_summing (pos, t, dest_summing (pos, u, ts))
```
```    59   | dest_summing (pos, Const (@{const_name HOL.minus}, _) \$ t \$ u, ts) =
```
```    60         dest_summing (pos, t, dest_summing (not pos, u, ts))
```
```    61   | dest_summing (pos, t, ts) =
```
```    62         if pos then t::ts else mk_minus t :: ts;
```
```    63
```
```    64 fun dest_sum t = dest_summing (true, t, []);
```
```    65
```
```    66 val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
```
```    67 val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
```
```    68
```
```    69 val mk_times = HOLogic.mk_binop @{const_name HOL.times};
```
```    70
```
```    71 fun one_of T = Const(@{const_name HOL.one},T);
```
```    72
```
```    73 (* build product with trailing 1 rather than Numeral 1 in order to avoid the
```
```    74    unnecessary restriction to type class number_ring
```
```    75    which is not required for cancellation of common factors in divisions.
```
```    76 *)
```
```    77 fun mk_prod T =
```
```    78   let val one = one_of T
```
```    79   fun mk [] = one
```
```    80     | mk [t] = t
```
```    81     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
```
```    82   in mk end;
```
```    83
```
```    84 (*This version ALWAYS includes a trailing one*)
```
```    85 fun long_mk_prod T []        = one_of T
```
```    86   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
```
```    87
```
```    88 val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
```
```    89
```
```    90 fun dest_prod t =
```
```    91       let val (t,u) = dest_times t
```
```    92       in dest_prod t @ dest_prod u end
```
```    93       handle TERM _ => [t];
```
```    94
```
```    95 (*DON'T do the obvious simplifications; that would create special cases*)
```
```    96 fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
```
```    97
```
```    98 (*Express t as a product of (possibly) a numeral with other sorted terms*)
```
```    99 fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) \$ t) = dest_coeff (~sign) t
```
```   100   | dest_coeff sign t =
```
```   101     let val ts = sort TermOrd.term_ord (dest_prod t)
```
```   102         val (n, ts') = find_first_numeral [] ts
```
```   103                           handle TERM _ => (1, ts)
```
```   104     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
```
```   105
```
```   106 (*Find first coefficient-term THAT MATCHES u*)
```
```   107 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
```
```   108   | find_first_coeff past u (t::terms) =
```
```   109         let val (n,u') = dest_coeff 1 t
```
```   110         in if u aconv u' then (n, rev past @ terms)
```
```   111                          else find_first_coeff (t::past) u terms
```
```   112         end
```
```   113         handle TERM _ => find_first_coeff (t::past) u terms;
```
```   114
```
```   115 (*Fractions as pairs of ints. Can't use Rat.rat because the representation
```
```   116   needs to preserve negative values in the denominator.*)
```
```   117 fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
```
```   118
```
```   119 (*Don't reduce fractions; sums must be proved by rule add_frac_eq.
```
```   120   Fractions are reduced later by the cancel_numeral_factor simproc.*)
```
```   121 fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
```
```   122
```
```   123 val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
```
```   124
```
```   125 (*Build term (p / q) * t*)
```
```   126 fun mk_fcoeff ((p, q), t) =
```
```   127   let val T = Term.fastype_of t
```
```   128   in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
```
```   129
```
```   130 (*Express t as a product of a fraction with other sorted terms*)
```
```   131 fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) \$ t) = dest_fcoeff (~sign) t
```
```   132   | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) \$ t \$ u) =
```
```   133     let val (p, t') = dest_coeff sign t
```
```   134         val (q, u') = dest_coeff 1 u
```
```   135     in (mk_frac (p, q), mk_divide (t', u')) end
```
```   136   | dest_fcoeff sign t =
```
```   137     let val (p, t') = dest_coeff sign t
```
```   138         val T = Term.fastype_of t
```
```   139     in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
```
```   140
```
```   141
```
```   142 (** New term ordering so that AC-rewriting brings numerals to the front **)
```
```   143
```
```   144 (*Order integers by absolute value and then by sign. The standard integer
```
```   145   ordering is not well-founded.*)
```
```   146 fun num_ord (i,j) =
```
```   147   (case int_ord (abs i, abs j) of
```
```   148     EQUAL => int_ord (Int.sign i, Int.sign j)
```
```   149   | ord => ord);
```
```   150
```
```   151 (*This resembles TermOrd.term_ord, but it puts binary numerals before other
```
```   152   non-atomic terms.*)
```
```   153 local open Term
```
```   154 in
```
```   155 fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
```
```   156       (case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
```
```   157   | numterm_ord
```
```   158      (Const(@{const_name Int.number_of}, _) \$ v, Const(@{const_name Int.number_of}, _) \$ w) =
```
```   159      num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
```
```   160   | numterm_ord (Const(@{const_name Int.number_of}, _) \$ _, _) = LESS
```
```   161   | numterm_ord (_, Const(@{const_name Int.number_of}, _) \$ _) = GREATER
```
```   162   | numterm_ord (t, u) =
```
```   163       (case int_ord (size_of_term t, size_of_term u) of
```
```   164         EQUAL =>
```
```   165           let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
```
```   166             (case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
```
```   167           end
```
```   168       | ord => ord)
```
```   169 and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
```
```   170 end;
```
```   171
```
```   172 fun numtermless tu = (numterm_ord tu = LESS);
```
```   173
```
```   174 val num_ss = HOL_ss settermless numtermless;
```
```   175
```
```   176 (*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
```
```   177 val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
```
```   178
```
```   179 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
```
```   180 val add_0s =  @{thms add_0s};
```
```   181 val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};
```
```   182
```
```   183 (*Simplify inverse Numeral1, a/Numeral1*)
```
```   184 val inverse_1s = [@{thm inverse_numeral_1}];
```
```   185 val divide_1s = [@{thm divide_numeral_1}];
```
```   186
```
```   187 (*To perform binary arithmetic.  The "left" rewriting handles patterns
```
```   188   created by the Int_Numeral_Simprocs, such as 3 * (5 * x). *)
```
```   189 val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
```
```   190                  @{thm add_number_of_left}, @{thm mult_number_of_left}] @
```
```   191                 @{thms arith_simps} @ @{thms rel_simps};
```
```   192
```
```   193 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
```
```   194   during re-arrangement*)
```
```   195 val non_add_simps =
```
```   196   subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
```
```   197
```
```   198 (*To evaluate binary negations of coefficients*)
```
```   199 val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
```
```   200                    @{thms minus_bin_simps} @ @{thms pred_bin_simps};
```
```   201
```
```   202 (*To let us treat subtraction as addition*)
```
```   203 val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
```
```   204
```
```   205 (*To let us treat division as multiplication*)
```
```   206 val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
```
```   207
```
```   208 (*push the unary minus down: - x * y = x * - y *)
```
```   209 val minus_mult_eq_1_to_2 =
```
```   210     [@{thm mult_minus_left}, @{thm minus_mult_right}] MRS trans |> standard;
```
```   211
```
```   212 (*to extract again any uncancelled minuses*)
```
```   213 val minus_from_mult_simps =
```
```   214     [@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
```
```   215
```
```   216 (*combine unary minus with numeric literals, however nested within a product*)
```
```   217 val mult_minus_simps =
```
```   218     [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
```
```   219
```
```   220 val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
```
```   221   diff_simps @ minus_simps @ @{thms add_ac}
```
```   222 val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
```
```   223 val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
```
```   224
```
```   225 structure CancelNumeralsCommon =
```
```   226   struct
```
```   227   val mk_sum            = mk_sum
```
```   228   val dest_sum          = dest_sum
```
```   229   val mk_coeff          = mk_coeff
```
```   230   val dest_coeff        = dest_coeff 1
```
```   231   val find_first_coeff  = find_first_coeff []
```
```   232   val trans_tac         = K Arith_Data.trans_tac
```
```   233
```
```   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   fun norm_tac ss =
```
```   314     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
```
```   315     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
```
```   316     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
```
```   317
```
```   318   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
```
```   319   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
```
```   320   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
```
```   321   end;
```
```   322
```
```   323 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
```
```   324
```
```   325 (*Version for fields, where coefficients can be fractions*)
```
```   326 structure FieldCombineNumeralsData =
```
```   327   struct
```
```   328   type coeff            = int * int
```
```   329   val iszero            = (fn (p, q) => p = 0)
```
```   330   val add               = add_frac
```
```   331   val mk_sum            = long_mk_sum
```
```   332   val dest_sum          = dest_sum
```
```   333   val mk_coeff          = mk_fcoeff
```
```   334   val dest_coeff        = dest_fcoeff 1
```
```   335   val left_distrib      = @{thm combine_common_factor} RS trans
```
```   336   val prove_conv        = Arith_Data.prove_conv_nohyps
```
```   337   val trans_tac         = K Arith_Data.trans_tac
```
```   338
```
```   339   val norm_ss1a = norm_ss1 addsimps inverse_1s @ divide_simps
```
```   340   fun norm_tac ss =
```
```   341     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1a))
```
```   342     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
```
```   343     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
```
```   344
```
```   345   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
```
```   346   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
```
```   347   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
```
```   348   end;
```
```   349
```
```   350 structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
```
```   351
```
```   352 val combine_numerals =
```
```   353   Arith_Data.prep_simproc
```
```   354     ("int_combine_numerals",
```
```   355      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
```
```   356      K CombineNumerals.proc);
```
```   357
```
```   358 val field_combine_numerals =
```
```   359   Arith_Data.prep_simproc
```
```   360     ("field_combine_numerals",
```
```   361      ["(i::'a::{number_ring,field,division_by_zero}) + j",
```
```   362       "(i::'a::{number_ring,field,division_by_zero}) - j"],
```
```   363      K FieldCombineNumerals.proc);
```
```   364
```
```   365 (** Constant folding for multiplication in semirings **)
```
```   366
```
```   367 (*We do not need folding for addition: combine_numerals does the same thing*)
```
```   368
```
```   369 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
```
```   370 struct
```
```   371   val assoc_ss = HOL_ss addsimps @{thms mult_ac}
```
```   372   val eq_reflection = eq_reflection
```
```   373   fun is_numeral (Const(@{const_name Int.number_of}, _) \$ _) = true
```
```   374     | is_numeral _ = false;
```
```   375 end;
```
```   376
```
```   377 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
```
```   378
```
```   379 val assoc_fold_simproc =
```
```   380   Arith_Data.prep_simproc
```
```   381    ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
```
```   382     K Semiring_Times_Assoc.proc);
```
```   383
```
```   384 end;
```
```   385
```
```   386 Addsimprocs [Int_Numeral_Simprocs.reorient_simproc];
```
```   387 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
```
```   388 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
```
```   389 Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
```
```   390 Addsimprocs [Int_Numeral_Simprocs.assoc_fold_simproc];
```
```   391
```
```   392 (*examples:
```
```   393 print_depth 22;
```
```   394 set timing;
```
```   395 set trace_simp;
```
```   396 fun test s = (Goal s, by (Simp_tac 1));
```
```   397
```
```   398 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
```
```   399
```
```   400 test "2*u = (u::int)";
```
```   401 test "(i + j + 12 + (k::int)) - 15 = y";
```
```   402 test "(i + j + 12 + (k::int)) - 5 = y";
```
```   403
```
```   404 test "y - b < (b::int)";
```
```   405 test "y - (3*b + c) < (b::int) - 2*c";
```
```   406
```
```   407 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
```
```   408 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
```
```   409 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
```
```   410 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
```
```   411
```
```   412 test "(i + j + 12 + (k::int)) = u + 15 + y";
```
```   413 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
```
```   414
```
```   415 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)";
```
```   416
```
```   417 test "a + -(b+c) + b = (d::int)";
```
```   418 test "a + -(b+c) - b = (d::int)";
```
```   419
```
```   420 (*negative numerals*)
```
```   421 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
```
```   422 test "(i + j + -3 + (k::int)) < u + 5 + y";
```
```   423 test "(i + j + 3 + (k::int)) < u + -6 + y";
```
```   424 test "(i + j + -12 + (k::int)) - 15 = y";
```
```   425 test "(i + j + 12 + (k::int)) - -15 = y";
```
```   426 test "(i + j + -12 + (k::int)) - -15 = y";
```
```   427 *)
```
```   428
```
```   429 (*** decision procedure for linear arithmetic ***)
```
```   430
```
```   431 (*---------------------------------------------------------------------------*)
```
```   432 (* Linear arithmetic                                                         *)
```
```   433 (*---------------------------------------------------------------------------*)
```
```   434
```
```   435 (*
```
```   436 Instantiation of the generic linear arithmetic package for int.
```
```   437 *)
```
```   438
```
```   439 structure Int_Arith =
```
```   440 struct
```
```   441
```
```   442 (* Update parameters of arithmetic prover *)
```
```   443
```
```   444 (* reduce contradictory =/</<= to False *)
```
```   445
```
```   446 (* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<",
```
```   447    and m and n are ground terms over rings (roughly speaking).
```
```   448    That is, m and n consist only of 1s combined with "+", "-" and "*".
```
```   449 *)
```
```   450
```
```   451 val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0};
```
```   452
```
```   453 val lhss0 = [@{cpat "0::?'a::ring"}];
```
```   454
```
```   455 fun proc0 phi ss ct =
```
```   456   let val T = ctyp_of_term ct
```
```   457   in if typ_of T = @{typ int} then NONE else
```
```   458      SOME (instantiate' [SOME T] [] zeroth)
```
```   459   end;
```
```   460
```
```   461 val zero_to_of_int_zero_simproc =
```
```   462   make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc",
```
```   463   proc = proc0, identifier = []};
```
```   464
```
```   465 val oneth = (symmetric o mk_meta_eq) @{thm of_int_1};
```
```   466
```
```   467 val lhss1 = [@{cpat "1::?'a::ring_1"}];
```
```   468
```
```   469 fun proc1 phi ss ct =
```
```   470   let val T = ctyp_of_term ct
```
```   471   in if typ_of T = @{typ int} then NONE else
```
```   472      SOME (instantiate' [SOME T] [] oneth)
```
```   473   end;
```
```   474
```
```   475 val one_to_of_int_one_simproc =
```
```   476   make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc",
```
```   477   proc = proc1, identifier = []};
```
```   478
```
```   479 val allowed_consts =
```
```   480   [@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"},
```
```   481    @{const_name "HOL.minus"}, @{const_name "HOL.plus"},
```
```   482    @{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"},
```
```   483    @{const_name "HOL.less_eq"}];
```
```   484
```
```   485 fun check t = case t of
```
```   486    Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int})
```
```   487                 else s mem_string allowed_consts
```
```   488  | a\$b => check a andalso check b
```
```   489  | _ => false;
```
```   490
```
```   491 val conv =
```
```   492   Simplifier.rewrite
```
```   493    (HOL_basic_ss addsimps
```
```   494      ((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult},
```
```   495              @{thm of_int_diff},  @{thm of_int_minus}])@
```
```   496       [@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}])
```
```   497      addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]);
```
```   498
```
```   499 fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE
```
```   500
```
```   501 val lhss' =
```
```   502   [@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
```
```   503    @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
```
```   504    @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
```
```   505
```
```   506 val zero_one_idom_simproc =
```
```   507   make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
```
```   508   proc = sproc, identifier = []}
```
```   509
```
```   510 val add_rules =
```
```   511     simp_thms @ @{thms arith_simps} @ @{thms rel_simps} @ @{thms arith_special} @
```
```   512     @{thms int_arith_rules}
```
```   513
```
```   514 val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2]
```
```   515
```
```   516 val int_numeral_base_simprocs = Int_Numeral_Simprocs.assoc_fold_simproc :: zero_one_idom_simproc
```
```   517   :: Int_Numeral_Simprocs.combine_numerals
```
```   518   :: Int_Numeral_Simprocs.cancel_numerals;
```
```   519
```
```   520 val setup =
```
```   521   Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
```
```   522    {add_mono_thms = add_mono_thms,
```
```   523     mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
```
```   524     inj_thms = nat_inj_thms @ inj_thms,
```
```   525     lessD = lessD @ [@{thm zless_imp_add1_zle}],
```
```   526     neqE = neqE,
```
```   527     simpset = simpset addsimps add_rules
```
```   528                       addsimprocs int_numeral_base_simprocs
```
```   529                       addcongs [if_weak_cong]}) #>
```
```   530   arith_inj_const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) #>
```
```   531   arith_discrete @{type_name Int.int}
```
```   532
```
```   533 val fast_int_arith_simproc =
```
```   534   Simplifier.simproc (the_context ())
```
```   535   "fast_int_arith"
```
```   536      ["(m::'a::{ordered_idom,number_ring}) < n",
```
```   537       "(m::'a::{ordered_idom,number_ring}) <= n",
```
```   538       "(m::'a::{ordered_idom,number_ring}) = n"] (K Lin_Arith.lin_arith_simproc);
```
```   539
```
```   540 end;
```
```   541
```
```   542 Addsimprocs [Int_Arith.fast_int_arith_simproc];
```