src/HOL/Tools/nat_arith.ML
author haftmann
Mon Feb 08 17:12:27 2010 +0100 (2010-02-08)
changeset 35047 1b2bae06c796
parent 34974 18b41bba42b5
child 35064 1bdef0c013d3
permissions -rw-r--r--
hide fact Nat.add_0_right; make add_0_right from Groups priority
     1 (* Author: Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
     2 
     3 Basic arithmetic for natural numbers.
     4 *)
     5 
     6 signature NAT_ARITH =
     7 sig
     8   val mk_sum: term list -> term
     9   val mk_norm_sum: term list -> term
    10   val dest_sum: term -> term list
    11 
    12   val nat_cancel_sums_add: simproc list
    13   val nat_cancel_sums: simproc list
    14   val setup: Context.generic -> Context.generic
    15 end;
    16 
    17 structure Nat_Arith: NAT_ARITH =
    18 struct
    19 
    20 (** abstract syntax of structure nat: 0, Suc, + **)
    21 
    22 val mk_plus = HOLogic.mk_binop @{const_name Algebras.plus};
    23 val dest_plus = HOLogic.dest_bin @{const_name Algebras.plus} HOLogic.natT;
    24 
    25 fun mk_sum [] = HOLogic.zero
    26   | mk_sum [t] = t
    27   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    28 
    29 (*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
    30 fun mk_norm_sum ts =
    31   let val (ones, sums) = List.partition (equal HOLogic.Suc_zero) ts in
    32     funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
    33   end;
    34 
    35 fun dest_sum tm =
    36   if HOLogic.is_zero tm then []
    37   else
    38     (case try HOLogic.dest_Suc tm of
    39       SOME t => HOLogic.Suc_zero :: dest_sum t
    40     | NONE =>
    41         (case try dest_plus tm of
    42           SOME (t, u) => dest_sum t @ dest_sum u
    43         | NONE => [tm]));
    44 
    45 
    46 (** cancel common summands **)
    47 
    48 structure CommonCancelSums =
    49 struct
    50   val mk_sum = mk_norm_sum;
    51   val dest_sum = dest_sum;
    52   val prove_conv = Arith_Data.prove_conv2;
    53   val norm_tac1 = Arith_Data.simp_all_tac [@{thm add_Suc}, @{thm add_Suc_right},
    54     @{thm add_0}, @{thm Nat.add_0_right}];
    55   val norm_tac2 = Arith_Data.simp_all_tac @{thms add_ac};
    56   fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss;
    57   fun gen_uncancel_tac rule = let val rule' = rule RS @{thm subst_equals}
    58     in fn ct => rtac (instantiate' [] [NONE, SOME ct] rule') 1 end;
    59 end;
    60 
    61 structure EqCancelSums = CancelSumsFun
    62 (struct
    63   open CommonCancelSums;
    64   val mk_bal = HOLogic.mk_eq;
    65   val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
    66   val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
    67 end);
    68 
    69 structure LessCancelSums = CancelSumsFun
    70 (struct
    71   open CommonCancelSums;
    72   val mk_bal = HOLogic.mk_binrel @{const_name Algebras.less};
    73   val dest_bal = HOLogic.dest_bin @{const_name Algebras.less} HOLogic.natT;
    74   val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
    75 end);
    76 
    77 structure LeCancelSums = CancelSumsFun
    78 (struct
    79   open CommonCancelSums;
    80   val mk_bal = HOLogic.mk_binrel @{const_name Algebras.less_eq};
    81   val dest_bal = HOLogic.dest_bin @{const_name Algebras.less_eq} HOLogic.natT;
    82   val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
    83 end);
    84 
    85 structure DiffCancelSums = CancelSumsFun
    86 (struct
    87   open CommonCancelSums;
    88   val mk_bal = HOLogic.mk_binop @{const_name Algebras.minus};
    89   val dest_bal = HOLogic.dest_bin @{const_name Algebras.minus} HOLogic.natT;
    90   val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
    91 end);
    92 
    93 val nat_cancel_sums_add =
    94   [Simplifier.simproc @{theory} "nateq_cancel_sums"
    95      ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"]
    96      (K EqCancelSums.proc),
    97    Simplifier.simproc @{theory} "natless_cancel_sums"
    98      ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"]
    99      (K LessCancelSums.proc),
   100    Simplifier.simproc @{theory} "natle_cancel_sums"
   101      ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"]
   102      (K LeCancelSums.proc)];
   103 
   104 val nat_cancel_sums = nat_cancel_sums_add @
   105   [Simplifier.simproc @{theory} "natdiff_cancel_sums"
   106     ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"]
   107     (K DiffCancelSums.proc)];
   108 
   109 val setup =
   110   Simplifier.map_ss (fn ss => ss addsimprocs nat_cancel_sums);
   111 
   112 end;