src/HOL/Tools/lin_arith.ML
author wenzelm
Thu Jan 01 14:23:39 2009 +0100 (2009-01-01)
changeset 29288 253bcf2a5854
parent 28053 a2106c0d8c45
child 29528 00fe02648e5d
permissions -rw-r--r--
avoid polymorphic equality;
wenzelm@24092
     1
(*  Title:      HOL/Tools/lin_arith.ML
wenzelm@29288
     2
    Author:     Tjark Weber and Tobias Nipkow, TU Muenchen
wenzelm@24092
     3
wenzelm@24092
     4
HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).
wenzelm@24092
     5
*)
wenzelm@24092
     6
wenzelm@24092
     7
signature BASIC_LIN_ARITH =
wenzelm@24092
     8
sig
wenzelm@24092
     9
  type arith_tactic
wenzelm@24092
    10
  val mk_arith_tactic: string -> (Proof.context -> int -> tactic) -> arith_tactic
wenzelm@24092
    11
  val eq_arith_tactic: arith_tactic * arith_tactic -> bool
wenzelm@24092
    12
  val arith_split_add: attribute
wenzelm@24092
    13
  val arith_discrete: string -> Context.generic -> Context.generic
wenzelm@24092
    14
  val arith_inj_const: string * typ -> Context.generic -> Context.generic
wenzelm@24092
    15
  val arith_tactic_add: arith_tactic -> Context.generic -> Context.generic
wenzelm@24112
    16
  val fast_arith_split_limit: int Config.T
wenzelm@24112
    17
  val fast_arith_neq_limit: int Config.T
wenzelm@24092
    18
  val lin_arith_pre_tac: Proof.context -> int -> tactic
wenzelm@24092
    19
  val fast_arith_tac: Proof.context -> int -> tactic
wenzelm@24092
    20
  val fast_ex_arith_tac: Proof.context -> bool -> int -> tactic
wenzelm@24092
    21
  val trace_arith: bool ref
wenzelm@24092
    22
  val lin_arith_simproc: simpset -> term -> thm option
wenzelm@24092
    23
  val fast_nat_arith_simproc: simproc
wenzelm@24092
    24
  val simple_arith_tac: Proof.context -> int -> tactic
wenzelm@24092
    25
  val arith_tac: Proof.context -> int -> tactic
wenzelm@24092
    26
  val silent_arith_tac: Proof.context -> int -> tactic
wenzelm@24092
    27
end;
wenzelm@24092
    28
wenzelm@24092
    29
signature LIN_ARITH =
wenzelm@24092
    30
sig
wenzelm@24092
    31
  include BASIC_LIN_ARITH
wenzelm@24092
    32
  val map_data:
wenzelm@24092
    33
    ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
wenzelm@24092
    34
      lessD: thm list, neqE: thm list, simpset: Simplifier.simpset} ->
wenzelm@24092
    35
     {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
wenzelm@24092
    36
      lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}) ->
wenzelm@24092
    37
    Context.generic -> Context.generic
wenzelm@27017
    38
  val warning_count: int ref
wenzelm@24092
    39
  val setup: Context.generic -> Context.generic
wenzelm@24092
    40
end;
wenzelm@24092
    41
wenzelm@24092
    42
structure LinArith: LIN_ARITH =
wenzelm@24092
    43
struct
wenzelm@24092
    44
wenzelm@24092
    45
(* Parameters data for general linear arithmetic functor *)
wenzelm@24092
    46
wenzelm@24092
    47
structure LA_Logic: LIN_ARITH_LOGIC =
wenzelm@24092
    48
struct
wenzelm@24092
    49
wenzelm@24092
    50
val ccontr = ccontr;
wenzelm@24092
    51
val conjI = conjI;
wenzelm@24092
    52
val notI = notI;
wenzelm@24092
    53
val sym = sym;
wenzelm@24092
    54
val not_lessD = @{thm linorder_not_less} RS iffD1;
wenzelm@24092
    55
val not_leD = @{thm linorder_not_le} RS iffD1;
wenzelm@24092
    56
val le0 = thm "le0";
wenzelm@24092
    57
wenzelm@24092
    58
fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
wenzelm@24092
    59
wenzelm@24092
    60
val mk_Trueprop = HOLogic.mk_Trueprop;
wenzelm@24092
    61
wenzelm@24092
    62
fun atomize thm = case Thm.prop_of thm of
wenzelm@24092
    63
    Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
wenzelm@24092
    64
    atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
wenzelm@24092
    65
  | _ => [thm];
wenzelm@24092
    66
wenzelm@24092
    67
fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
wenzelm@24092
    68
  | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
wenzelm@24092
    69
  | neg_prop t = raise TERM ("neg_prop", [t]);
wenzelm@24092
    70
wenzelm@24092
    71
fun is_False thm =
wenzelm@24092
    72
  let val _ $ t = Thm.prop_of thm
wenzelm@24092
    73
  in t = Const("False",HOLogic.boolT) end;
wenzelm@24092
    74
wenzelm@24092
    75
fun is_nat(t) = fastype_of1 t = HOLogic.natT;
wenzelm@24092
    76
wenzelm@24092
    77
fun mk_nat_thm sg t =
wenzelm@24092
    78
  let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
wenzelm@24092
    79
  in instantiate ([],[(cn,ct)]) le0 end;
wenzelm@24092
    80
wenzelm@24092
    81
end;
wenzelm@24092
    82
wenzelm@24092
    83
wenzelm@24092
    84
(* arith context data *)
wenzelm@24092
    85
wenzelm@24092
    86
datatype arith_tactic =
wenzelm@24092
    87
  ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
wenzelm@24092
    88
wenzelm@24092
    89
fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
wenzelm@24092
    90
wenzelm@24092
    91
fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
wenzelm@24092
    92
wenzelm@24092
    93
structure ArithContextData = GenericDataFun
wenzelm@24092
    94
(
wenzelm@24092
    95
  type T = {splits: thm list,
wenzelm@24092
    96
            inj_consts: (string * typ) list,
wenzelm@24092
    97
            discrete: string list,
wenzelm@24092
    98
            tactics: arith_tactic list};
wenzelm@24092
    99
  val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
wenzelm@24092
   100
  val extend = I;
wenzelm@29288
   101
  fun merge _
wenzelm@29288
   102
   ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
wenzelm@29288
   103
    {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) : T =
wenzelm@24092
   104
   {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
wenzelm@24092
   105
    inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
wenzelm@24092
   106
    discrete = Library.merge (op =) (discrete1, discrete2),
wenzelm@24092
   107
    tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
wenzelm@24092
   108
);
wenzelm@24092
   109
wenzelm@24092
   110
val get_arith_data = ArithContextData.get o Context.Proof;
wenzelm@24092
   111
wenzelm@24092
   112
val arith_split_add = Thm.declaration_attribute (fn thm =>
wenzelm@24092
   113
  ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
wenzelm@24112
   114
    {splits = update Thm.eq_thm_prop thm splits,
wenzelm@24092
   115
     inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
wenzelm@24092
   116
wenzelm@24092
   117
fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
wenzelm@24092
   118
  {splits = splits, inj_consts = inj_consts,
wenzelm@24112
   119
   discrete = update (op =) d discrete, tactics = tactics});
wenzelm@24092
   120
wenzelm@24092
   121
fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
wenzelm@24112
   122
  {splits = splits, inj_consts = update (op =) c inj_consts,
wenzelm@24092
   123
   discrete = discrete, tactics= tactics});
wenzelm@24092
   124
wenzelm@24092
   125
fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
wenzelm@24092
   126
  {splits = splits, inj_consts = inj_consts, discrete = discrete,
wenzelm@24112
   127
   tactics = update eq_arith_tactic tac tactics});
wenzelm@24092
   128
wenzelm@24092
   129
wenzelm@24112
   130
val (fast_arith_split_limit, setup1) = Attrib.config_int "fast_arith_split_limit" 9;
wenzelm@24112
   131
val (fast_arith_neq_limit, setup2) = Attrib.config_int "fast_arith_neq_limit" 9;
wenzelm@24092
   132
val setup_options = setup1 #> setup2;
wenzelm@24092
   133
wenzelm@24092
   134
wenzelm@24092
   135
structure LA_Data_Ref =
wenzelm@24092
   136
struct
wenzelm@24092
   137
wenzelm@24092
   138
val fast_arith_neq_limit = fast_arith_neq_limit;
wenzelm@24092
   139
wenzelm@24092
   140
wenzelm@24092
   141
(* Decomposition of terms *)
wenzelm@24092
   142
wenzelm@24092
   143
(*internal representation of linear (in-)equations*)
wenzelm@26942
   144
type decomp =
webertj@24328
   145
  ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
wenzelm@24092
   146
wenzelm@24092
   147
fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
wenzelm@24092
   148
  | nT _                      = false;
wenzelm@24092
   149
wenzelm@24092
   150
fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
wenzelm@24092
   151
             (term * Rat.rat) list * Rat.rat =
webertj@24328
   152
  case AList.lookup (op =) p t of
webertj@24328
   153
      NONE   => ((t, m) :: p, i)
webertj@24328
   154
    | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
wenzelm@24092
   155
webertj@24328
   156
(* decompose nested multiplications, bracketing them to the right and combining
webertj@24328
   157
   all their coefficients
wenzelm@24092
   158
webertj@24328
   159
   inj_consts: list of constants to be ignored when encountered
webertj@24328
   160
               (e.g. arithmetic type conversions that preserve value)
wenzelm@24092
   161
webertj@24328
   162
   m: multiplicity associated with the entire product
wenzelm@24092
   163
webertj@24328
   164
   returns either (SOME term, associated multiplicity) or (NONE, constant)
webertj@24328
   165
*)
wenzelm@24092
   166
fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
wenzelm@24092
   167
let
webertj@24328
   168
  fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) =
webertj@24328
   169
      (case s of Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
webertj@24328
   170
        (* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)
wenzelm@24092
   171
        demult (mC $ s1 $ (mC $ s2 $ t), m)
webertj@24328
   172
      | _ =>
webertj@24328
   173
        (* product 's * t', where either factor can be 'NONE' *)
webertj@24328
   174
        (case demult (s, m) of
webertj@24328
   175
          (SOME s', m') =>
webertj@24328
   176
            (case demult (t, m') of
webertj@24328
   177
              (SOME t', m'') => (SOME (mC $ s' $ t'), m'')
webertj@24328
   178
            | (NONE,    m'') => (SOME s', m''))
webertj@24328
   179
        | (NONE,    m') => demult (t, m')))
webertj@24328
   180
    | demult ((mC as Const (@{const_name HOL.divide}, _)) $ s $ t, m) =
webertj@24328
   181
      (* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could
webertj@24328
   182
         become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ?   Note that
webertj@24328
   183
         if we choose to do so here, the simpset used by arith must be able to
webertj@24328
   184
         perform the same simplifications. *)
webertj@24328
   185
      (* FIXME: Currently we treat the numerator as atomic unless the
webertj@24328
   186
         denominator can be reduced to a numeric constant.  It might be better
webertj@24328
   187
         to demult the numerator in any case, and invent a new term of the form
webertj@24328
   188
         '1 / t' if the numerator can be reduced, but the denominator cannot. *)
webertj@24328
   189
      (* FIXME: Currently we even treat the whole fraction as atomic unless the
webertj@24328
   190
         denominator can be reduced to a numeric constant.  It might be better
webertj@25015
   191
         to use the partially reduced denominator (i.e. 's / (2*t)' could be
webertj@24328
   192
         demult'ed to 's / t' with multiplicity .5).   This would require a
webertj@24328
   193
         very simple change only below, but it breaks existing proofs. *)
webertj@24328
   194
      (* quotient 's / t', where the denominator t can be NONE *)
webertj@24328
   195
      (* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)
webertj@24328
   196
      (case demult (t, Rat.one) of
webertj@24328
   197
        (SOME _, _) => (SOME (mC $ s $ t), m)
webertj@24328
   198
      | (NONE,  m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))
webertj@24328
   199
    (* terms that evaluate to numeric constants *)
webertj@24328
   200
    | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
wenzelm@24092
   201
    | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
wenzelm@24092
   202
    | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
webertj@24328
   203
    (*Warning: in rare cases number_of encloses a non-numeral,
webertj@24328
   204
      in which case dest_numeral raises TERM; hence all the handles below.
webertj@24328
   205
      Same for Suc-terms that turn out not to be numerals -
webertj@24328
   206
      although the simplifier should eliminate those anyway ...*)
haftmann@25919
   207
    | demult (t as Const ("Int.number_class.number_of", _) $ n, m) =
webertj@24328
   208
      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
webertj@24328
   209
        handle TERM _ => (SOME t, m))
webertj@24328
   210
    | demult (t as Const (@{const_name Suc}, _) $ _, m) =
webertj@24328
   211
      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
webertj@24328
   212
        handle TERM _ => (SOME t, m))
webertj@24328
   213
    (* injection constants are ignored *)
wenzelm@24092
   214
    | demult (t as Const f $ x, m) =
webertj@24328
   215
      if member (op =) inj_consts f then demult (x, m) else (SOME t, m)
webertj@24328
   216
    (* everything else is considered atomic *)
wenzelm@24092
   217
    | demult (atom, m) = (SOME atom, m)
wenzelm@24092
   218
in demult end;
wenzelm@24092
   219
wenzelm@24092
   220
fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
wenzelm@24092
   221
            ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
wenzelm@24092
   222
let
webertj@24328
   223
  (* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of
webertj@24328
   224
     summands and associated multiplicities, plus a constant 'i' (with implicit
webertj@24328
   225
     multiplicity 1) *)
wenzelm@24092
   226
  fun poly (Const (@{const_name HOL.plus}, _) $ s $ t,
wenzelm@24092
   227
        m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))
wenzelm@24092
   228
    | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
wenzelm@24092
   229
        if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
wenzelm@24092
   230
    | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
wenzelm@24092
   231
        if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
wenzelm@24092
   232
    | poly (Const (@{const_name HOL.zero}, _), _, pi) =
wenzelm@24092
   233
        pi
wenzelm@24092
   234
    | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
wenzelm@24092
   235
        (p, Rat.add i m)
wenzelm@24092
   236
    | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
wenzelm@24092
   237
        poly (t, m, (p, Rat.add i m))
wenzelm@24092
   238
    | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
wenzelm@24092
   239
        (case demult inj_consts (all, m) of
wenzelm@24092
   240
           (NONE,   m') => (p, Rat.add i m')
wenzelm@24092
   241
         | (SOME u, m') => add_atom u m' pi)
wenzelm@24092
   242
    | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
wenzelm@24092
   243
        (case demult inj_consts (all, m) of
wenzelm@24092
   244
           (NONE,   m') => (p, Rat.add i m')
wenzelm@24092
   245
         | (SOME u, m') => add_atom u m' pi)
haftmann@25919
   246
    | poly (all as Const ("Int.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
wenzelm@24092
   247
        (let val k = HOLogic.dest_numeral t
wenzelm@24092
   248
            val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
wenzelm@24092
   249
        in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
wenzelm@24092
   250
        handle TERM _ => add_atom all m pi)
wenzelm@24092
   251
    | poly (all as Const f $ x, m, pi) =
wenzelm@24092
   252
        if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
wenzelm@24092
   253
    | poly (all, m, pi) =
wenzelm@24092
   254
        add_atom all m pi
wenzelm@24092
   255
  val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
wenzelm@24092
   256
  val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
wenzelm@24092
   257
in
wenzelm@24092
   258
  case rel of
wenzelm@24092
   259
    @{const_name HOL.less}    => SOME (p, i, "<", q, j)
wenzelm@24092
   260
  | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
wenzelm@24092
   261
  | "op ="              => SOME (p, i, "=", q, j)
wenzelm@24092
   262
  | _                   => NONE
webertj@24328
   263
end handle Rat.DIVZERO => NONE;
wenzelm@24092
   264
wenzelm@24271
   265
fun of_lin_arith_sort thy U =
wenzelm@24271
   266
  Sign.of_sort thy (U, ["Ring_and_Field.ordered_idom"]);
wenzelm@24092
   267
wenzelm@24092
   268
fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
wenzelm@24092
   269
  if of_lin_arith_sort sg U then
wenzelm@24092
   270
    (true, D mem discrete)
wenzelm@24092
   271
  else (* special cases *)
wenzelm@24092
   272
    if D mem discrete then  (true, true)  else  (false, false)
wenzelm@24092
   273
  | allows_lin_arith sg discrete U =
wenzelm@24092
   274
  (of_lin_arith_sort sg U, false);
wenzelm@24092
   275
wenzelm@26942
   276
fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decomp option =
wenzelm@24092
   277
  case T of
wenzelm@24092
   278
    Type ("fun", [U, _]) =>
wenzelm@24092
   279
      (case allows_lin_arith thy discrete U of
wenzelm@24092
   280
        (true, d) =>
wenzelm@24092
   281
          (case decomp0 inj_consts xxx of
wenzelm@24092
   282
            NONE                   => NONE
wenzelm@24092
   283
          | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
wenzelm@24092
   284
      | (false, _) =>
wenzelm@24092
   285
          NONE)
wenzelm@24092
   286
  | _ => NONE;
wenzelm@24092
   287
wenzelm@24092
   288
fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
wenzelm@24092
   289
  | negate NONE                        = NONE;
wenzelm@24092
   290
wenzelm@24092
   291
fun decomp_negation data
wenzelm@26942
   292
  ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decomp option =
wenzelm@24092
   293
      decomp_typecheck data (T, (rel, lhs, rhs))
wenzelm@24092
   294
  | decomp_negation data ((Const ("Trueprop", _)) $
wenzelm@24092
   295
  (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
wenzelm@24092
   296
      negate (decomp_typecheck data (T, (rel, lhs, rhs)))
wenzelm@24092
   297
  | decomp_negation data _ =
wenzelm@24092
   298
      NONE;
wenzelm@24092
   299
wenzelm@26942
   300
fun decomp ctxt : term -> decomp option =
wenzelm@24092
   301
  let
wenzelm@24092
   302
    val thy = ProofContext.theory_of ctxt
wenzelm@24092
   303
    val {discrete, inj_consts, ...} = get_arith_data ctxt
wenzelm@24092
   304
  in decomp_negation (thy, discrete, inj_consts) end;
wenzelm@24092
   305
wenzelm@24092
   306
fun domain_is_nat (_ $ (Const (_, T) $ _ $ _))                      = nT T
wenzelm@24092
   307
  | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
wenzelm@24092
   308
  | domain_is_nat _                                                 = false;
wenzelm@24092
   309
wenzelm@24092
   310
fun number_of (n, T) = HOLogic.mk_number T n;
wenzelm@24092
   311
wenzelm@24092
   312
(*---------------------------------------------------------------------------*)
wenzelm@24092
   313
(* the following code performs splitting of certain constants (e.g. min,     *)
wenzelm@24092
   314
(* max) in a linear arithmetic problem; similar to what split_tac later does *)
wenzelm@24092
   315
(* to the proof state                                                        *)
wenzelm@24092
   316
(*---------------------------------------------------------------------------*)
wenzelm@24092
   317
wenzelm@24092
   318
(* checks if splitting with 'thm' is implemented                             *)
wenzelm@24092
   319
wenzelm@24092
   320
fun is_split_thm (thm : thm) : bool =
wenzelm@24092
   321
  case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
wenzelm@24092
   322
    (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
wenzelm@24092
   323
    case head_of lhs of
wenzelm@24092
   324
      Const (a, _) => member (op =) [@{const_name Orderings.max},
wenzelm@24092
   325
                                    @{const_name Orderings.min},
wenzelm@24092
   326
                                    @{const_name HOL.abs},
wenzelm@24092
   327
                                    @{const_name HOL.minus},
haftmann@25919
   328
                                    "Int.nat",
wenzelm@24092
   329
                                    "Divides.div_class.mod",
wenzelm@24092
   330
                                    "Divides.div_class.div"] a
wenzelm@24092
   331
    | _            => (warning ("Lin. Arith.: wrong format for split rule " ^
wenzelm@24092
   332
                                 Display.string_of_thm thm);
wenzelm@24092
   333
                       false))
wenzelm@24092
   334
  | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
wenzelm@24092
   335
                   Display.string_of_thm thm);
wenzelm@24092
   336
          false);
wenzelm@24092
   337
wenzelm@24092
   338
(* substitute new for occurrences of old in a term, incrementing bound       *)
wenzelm@24092
   339
(* variables as needed when substituting inside an abstraction               *)
wenzelm@24092
   340
wenzelm@24092
   341
fun subst_term ([] : (term * term) list) (t : term) = t
wenzelm@24092
   342
  | subst_term pairs                     t          =
wenzelm@24092
   343
      (case AList.lookup (op aconv) pairs t of
wenzelm@24092
   344
        SOME new =>
wenzelm@24092
   345
          new
wenzelm@24092
   346
      | NONE     =>
wenzelm@24092
   347
          (case t of Abs (a, T, body) =>
wenzelm@24092
   348
            let val pairs' = map (pairself (incr_boundvars 1)) pairs
wenzelm@24092
   349
            in  Abs (a, T, subst_term pairs' body)  end
wenzelm@24092
   350
          | t1 $ t2                   =>
wenzelm@24092
   351
            subst_term pairs t1 $ subst_term pairs t2
wenzelm@24092
   352
          | _ => t));
wenzelm@24092
   353
wenzelm@24092
   354
(* approximates the effect of one application of split_tac (followed by NNF  *)
wenzelm@24092
   355
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)
wenzelm@24092
   356
(* list of new subgoals (each again represented by a typ list for bound      *)
wenzelm@24092
   357
(* variables and a term list for premises), or NONE if split_tac would fail  *)
wenzelm@24092
   358
(* on the subgoal                                                            *)
wenzelm@24092
   359
wenzelm@24092
   360
(* FIXME: currently only the effect of certain split theorems is reproduced  *)
wenzelm@24092
   361
(*        (which is why we need 'is_split_thm').  A more canonical           *)
wenzelm@24092
   362
(*        implementation should analyze the right-hand side of the split     *)
wenzelm@24092
   363
(*        theorem that can be applied, and modify the subgoal accordingly.   *)
wenzelm@24092
   364
(*        Or even better, the splitter should be extended to provide         *)
wenzelm@24092
   365
(*        splitting on terms as well as splitting on theorems (where the     *)
wenzelm@24092
   366
(*        former can have a faster implementation as it does not need to be  *)
wenzelm@24092
   367
(*        proof-producing).                                                  *)
wenzelm@24092
   368
wenzelm@24092
   369
fun split_once_items ctxt (Ts : typ list, terms : term list) :
wenzelm@24092
   370
                     (typ list * term list) list option =
wenzelm@24092
   371
let
wenzelm@24092
   372
  val thy = ProofContext.theory_of ctxt
wenzelm@24092
   373
  (* takes a list  [t1, ..., tn]  to the term                                *)
wenzelm@24092
   374
  (*   tn' --> ... --> t1' --> False  ,                                      *)
wenzelm@24092
   375
  (* where ti' = HOLogic.dest_Trueprop ti                                    *)
wenzelm@24092
   376
  fun REPEAT_DETERM_etac_rev_mp terms' =
wenzelm@24092
   377
    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
wenzelm@24092
   378
  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
wenzelm@24092
   379
  val cmap       = Splitter.cmap_of_split_thms split_thms
wenzelm@24092
   380
  val splits     = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
wenzelm@24112
   381
  val split_limit = Config.get ctxt fast_arith_split_limit
wenzelm@24092
   382
in
wenzelm@24092
   383
  if length splits > split_limit then
wenzelm@24092
   384
   (tracing ("fast_arith_split_limit exceeded (current value is " ^
wenzelm@24092
   385
      string_of_int split_limit ^ ")"); NONE)
wenzelm@24092
   386
  else (
wenzelm@24092
   387
  case splits of [] =>
wenzelm@24092
   388
    (* split_tac would fail: no possible split *)
wenzelm@24092
   389
    NONE
wenzelm@24092
   390
  | ((_, _, _, split_type, split_term) :: _) => (
wenzelm@24092
   391
    (* ignore all but the first possible split *)
wenzelm@24092
   392
    case strip_comb split_term of
wenzelm@24092
   393
    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
wenzelm@24092
   394
      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
wenzelm@24092
   395
      let
wenzelm@24092
   396
        val rev_terms     = rev terms
wenzelm@24092
   397
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   398
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
wenzelm@24092
   399
        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
wenzelm@24092
   400
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   401
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   402
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   403
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   404
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   405
      in
wenzelm@24092
   406
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   407
      end
wenzelm@24092
   408
    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
wenzelm@24092
   409
    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
wenzelm@24092
   410
      let
wenzelm@24092
   411
        val rev_terms     = rev terms
wenzelm@24092
   412
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   413
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
wenzelm@24092
   414
        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
wenzelm@24092
   415
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   416
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   417
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   418
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   419
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   420
      in
wenzelm@24092
   421
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   422
      end
wenzelm@24092
   423
    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
wenzelm@24092
   424
    | (Const (@{const_name HOL.abs}, _), [t1]) =>
wenzelm@24092
   425
      let
wenzelm@24092
   426
        val rev_terms   = rev terms
wenzelm@24092
   427
        val terms1      = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   428
        val terms2      = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
wenzelm@24092
   429
                            split_type --> split_type) $ t1)]) rev_terms
wenzelm@24092
   430
        val zero        = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   431
        val zero_leq_t1 = Const (@{const_name HOL.less_eq},
wenzelm@24092
   432
                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1
wenzelm@24092
   433
        val t1_lt_zero  = Const (@{const_name HOL.less},
wenzelm@24092
   434
                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
wenzelm@24092
   435
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   436
        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
wenzelm@24092
   437
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   438
      in
wenzelm@24092
   439
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   440
      end
wenzelm@24092
   441
    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
wenzelm@24092
   442
    | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
wenzelm@24092
   443
      let
wenzelm@24092
   444
        (* "d" in the above theorem becomes a new bound variable after NNF   *)
wenzelm@24092
   445
        (* transformation, therefore some adjustment of indices is necessary *)
wenzelm@24092
   446
        val rev_terms       = rev terms
wenzelm@24092
   447
        val zero            = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   448
        val d               = Bound 0
wenzelm@24092
   449
        val terms1          = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   450
        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])
wenzelm@24092
   451
                                (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   452
        val t1'             = incr_boundvars 1 t1
wenzelm@24092
   453
        val t2'             = incr_boundvars 1 t2
wenzelm@24092
   454
        val t1_lt_t2        = Const (@{const_name HOL.less},
wenzelm@24092
   455
                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   456
        val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   457
                                (Const (@{const_name HOL.plus},
wenzelm@24092
   458
                                  split_type --> split_type --> split_type) $ t2' $ d)
wenzelm@24092
   459
        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   460
        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
wenzelm@24092
   461
        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
wenzelm@24092
   462
      in
wenzelm@24092
   463
        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
wenzelm@24092
   464
      end
wenzelm@24092
   465
    (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
haftmann@25919
   466
    | (Const ("Int.nat", _), [t1]) =>
wenzelm@24092
   467
      let
wenzelm@24092
   468
        val rev_terms   = rev terms
wenzelm@24092
   469
        val zero_int    = Const (@{const_name HOL.zero}, HOLogic.intT)
wenzelm@24092
   470
        val zero_nat    = Const (@{const_name HOL.zero}, HOLogic.natT)
wenzelm@24092
   471
        val n           = Bound 0
wenzelm@24092
   472
        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])
wenzelm@24092
   473
                            (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   474
        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms
wenzelm@24092
   475
        val t1'         = incr_boundvars 1 t1
wenzelm@24092
   476
        val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
haftmann@24196
   477
                            (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
wenzelm@24092
   478
        val t1_lt_zero  = Const (@{const_name HOL.less},
wenzelm@24092
   479
                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
wenzelm@24092
   480
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   481
        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
wenzelm@24092
   482
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   483
      in
wenzelm@24092
   484
        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   485
      end
wenzelm@24092
   486
    (* "?P ((?n::nat) mod (number_of ?k)) =
wenzelm@24092
   487
         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
wenzelm@24092
   488
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   489
    | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   490
      let
wenzelm@24092
   491
        val rev_terms               = rev terms
wenzelm@24092
   492
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   493
        val i                       = Bound 1
wenzelm@24092
   494
        val j                       = Bound 0
wenzelm@24092
   495
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   496
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   497
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   498
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   499
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   500
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   501
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   502
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   503
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
wenzelm@24092
   504
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   505
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   506
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   507
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   508
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   509
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   510
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   511
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   512
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   513
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   514
                                          @ terms2 @ [not_false]
wenzelm@24092
   515
      in
wenzelm@24092
   516
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   517
      end
wenzelm@24092
   518
    (* "?P ((?n::nat) div (number_of ?k)) =
wenzelm@24092
   519
         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
wenzelm@24092
   520
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
wenzelm@24092
   521
    | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   522
      let
wenzelm@24092
   523
        val rev_terms               = rev terms
wenzelm@24092
   524
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   525
        val i                       = Bound 1
wenzelm@24092
   526
        val j                       = Bound 0
wenzelm@24092
   527
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   528
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   529
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   530
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   531
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   532
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   533
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   534
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   535
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
wenzelm@24092
   536
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   537
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   538
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   539
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   540
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   541
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   542
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   543
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   544
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   545
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   546
                                          @ terms2 @ [not_false]
wenzelm@24092
   547
      in
wenzelm@24092
   548
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   549
      end
wenzelm@24092
   550
    (* "?P ((?n::int) mod (number_of ?k)) =
wenzelm@24092
   551
         ((iszero (number_of ?k) --> ?P ?n) &
wenzelm@24092
   552
          (neg (number_of (uminus ?k)) -->
wenzelm@24092
   553
            (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
wenzelm@24092
   554
          (neg (number_of ?k) -->
wenzelm@24092
   555
            (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   556
    | (Const ("Divides.div_class.mod",
haftmann@25919
   557
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
wenzelm@24092
   558
      let
wenzelm@24092
   559
        val rev_terms               = rev terms
wenzelm@24092
   560
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   561
        val i                       = Bound 1
wenzelm@24092
   562
        val j                       = Bound 0
wenzelm@24092
   563
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   564
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   565
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   566
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   567
        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
haftmann@25919
   568
        val iszero_t2               = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
haftmann@25919
   569
        val neg_minus_k             = Const ("Int.neg", split_type --> HOLogic.boolT) $
wenzelm@24092
   570
                                        (number_of $
wenzelm@24092
   571
                                          (Const (@{const_name HOL.uminus},
wenzelm@24092
   572
                                            HOLogic.intT --> HOLogic.intT) $ k'))
wenzelm@24092
   573
        val zero_leq_j              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   574
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
wenzelm@24092
   575
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   576
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   577
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   578
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   579
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   580
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
haftmann@25919
   581
        val neg_t2                  = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
wenzelm@24092
   582
        val t2_lt_j                 = Const (@{const_name HOL.less},
wenzelm@24092
   583
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
wenzelm@24092
   584
        val j_leq_zero              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   585
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
wenzelm@24092
   586
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   587
        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
wenzelm@24092
   588
        val subgoal2                = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
wenzelm@24092
   589
                                        @ hd terms2_3
wenzelm@24092
   590
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   591
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   592
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   593
        val subgoal3                = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
wenzelm@24092
   594
                                        @ hd terms2_3
wenzelm@24092
   595
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   596
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   597
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   598
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   599
      in
wenzelm@24092
   600
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   601
      end
wenzelm@24092
   602
    (* "?P ((?n::int) div (number_of ?k)) =
wenzelm@24092
   603
         ((iszero (number_of ?k) --> ?P 0) &
wenzelm@24092
   604
          (neg (number_of (uminus ?k)) -->
wenzelm@24092
   605
            (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
wenzelm@24092
   606
          (neg (number_of ?k) -->
wenzelm@24092
   607
            (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
wenzelm@24092
   608
    | (Const ("Divides.div_class.div",
haftmann@25919
   609
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
wenzelm@24092
   610
      let
wenzelm@24092
   611
        val rev_terms               = rev terms
wenzelm@24092
   612
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   613
        val i                       = Bound 1
wenzelm@24092
   614
        val j                       = Bound 0
wenzelm@24092
   615
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   616
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   617
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   618
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   619
        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
haftmann@25919
   620
        val iszero_t2               = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
haftmann@25919
   621
        val neg_minus_k             = Const ("Int.neg", split_type --> HOLogic.boolT) $
wenzelm@24092
   622
                                        (number_of $
wenzelm@24092
   623
                                          (Const (@{const_name HOL.uminus},
wenzelm@24092
   624
                                            HOLogic.intT --> HOLogic.intT) $ k'))
wenzelm@24092
   625
        val zero_leq_j              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   626
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
wenzelm@24092
   627
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   628
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   629
        val t1_eq_t2_times_i_plus_j = Const ("op =",
wenzelm@24092
   630
                                        split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   631
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   632
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   633
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
haftmann@25919
   634
        val neg_t2                  = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
wenzelm@24092
   635
        val t2_lt_j                 = Const (@{const_name HOL.less},
wenzelm@24092
   636
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
wenzelm@24092
   637
        val j_leq_zero              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   638
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
wenzelm@24092
   639
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   640
        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
wenzelm@24092
   641
        val subgoal2                = (HOLogic.mk_Trueprop neg_minus_k)
wenzelm@24092
   642
                                        :: terms2_3
wenzelm@24092
   643
                                        @ not_false
wenzelm@24092
   644
                                        :: (map HOLogic.mk_Trueprop
wenzelm@24092
   645
                                             [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   646
        val subgoal3                = (HOLogic.mk_Trueprop neg_t2)
wenzelm@24092
   647
                                        :: terms2_3
wenzelm@24092
   648
                                        @ not_false
wenzelm@24092
   649
                                        :: (map HOLogic.mk_Trueprop
wenzelm@24092
   650
                                             [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   651
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   652
      in
wenzelm@24092
   653
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   654
      end
wenzelm@24092
   655
    (* this will only happen if a split theorem can be applied for which no  *)
wenzelm@24092
   656
    (* code exists above -- in which case either the split theorem should be *)
wenzelm@24092
   657
    (* implemented above, or 'is_split_thm' should be modified to filter it  *)
wenzelm@24092
   658
    (* out                                                                   *)
wenzelm@24092
   659
    | (t, ts) => (
wenzelm@24920
   660
      warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
wenzelm@24092
   661
               " (with " ^ string_of_int (length ts) ^
wenzelm@24092
   662
               " argument(s)) not implemented; proof reconstruction is likely to fail");
wenzelm@24092
   663
      NONE
wenzelm@24092
   664
    ))
wenzelm@24092
   665
  )
wenzelm@24092
   666
end;
wenzelm@24092
   667
wenzelm@24092
   668
(* remove terms that do not satisfy 'p'; change the order of the remaining   *)
wenzelm@24092
   669
(* terms in the same way as filter_prems_tac does                            *)
wenzelm@24092
   670
wenzelm@24092
   671
fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
wenzelm@24092
   672
let
wenzelm@24092
   673
  fun filter_prems (t, (left, right)) =
wenzelm@24092
   674
    if  p t  then  (left, right @ [t])  else  (left @ right, [])
wenzelm@24092
   675
  val (left, right) = foldl filter_prems ([], []) terms
wenzelm@24092
   676
in
wenzelm@24092
   677
  right @ left
wenzelm@24092
   678
end;
wenzelm@24092
   679
wenzelm@24092
   680
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)
wenzelm@24092
   681
(* subgoal that has 'terms' as premises                                      *)
wenzelm@24092
   682
wenzelm@24092
   683
fun negated_term_occurs_positively (terms : term list) : bool =
wenzelm@24092
   684
  List.exists
wenzelm@24092
   685
    (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
wenzelm@24092
   686
      | _                                   => false)
wenzelm@24092
   687
    terms;
wenzelm@24092
   688
wenzelm@24092
   689
fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
wenzelm@24092
   690
let
wenzelm@24092
   691
  (* repeatedly split (including newly emerging subgoals) until no further   *)
wenzelm@24092
   692
  (* splitting is possible                                                   *)
wenzelm@24092
   693
  fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
wenzelm@24092
   694
    | split_loop (subgoal::subgoals)                = (
wenzelm@24092
   695
        case split_once_items ctxt subgoal of
wenzelm@24092
   696
          SOME new_subgoals => split_loop (new_subgoals @ subgoals)
wenzelm@24092
   697
        | NONE              => subgoal :: split_loop subgoals
wenzelm@24092
   698
      )
wenzelm@24092
   699
  fun is_relevant t  = isSome (decomp ctxt t)
wenzelm@24092
   700
  (* filter_prems_tac is_relevant: *)
wenzelm@24092
   701
  val relevant_terms = filter_prems_tac_items is_relevant terms
wenzelm@24092
   702
  (* split_tac, NNF normalization: *)
wenzelm@24092
   703
  val split_goals    = split_loop [(Ts, relevant_terms)]
wenzelm@24092
   704
  (* necessary because split_once_tac may normalize terms: *)
wenzelm@24092
   705
  val beta_eta_norm  = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
wenzelm@24092
   706
  (* TRY (etac notE) THEN eq_assume_tac: *)
wenzelm@24092
   707
  val result         = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
wenzelm@24092
   708
in
wenzelm@24092
   709
  result
wenzelm@24092
   710
end;
wenzelm@24092
   711
wenzelm@24092
   712
(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)
wenzelm@24092
   713
(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)
wenzelm@24092
   714
(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)
wenzelm@24092
   715
(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)
wenzelm@24092
   716
(* disjunctions and existential quantifiers from the premises, possibly (in  *)
wenzelm@24092
   717
(* the case of disjunctions) resulting in several new subgoals, each of the  *)
wenzelm@24092
   718
(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)
wenzelm@24092
   719
(* !fast_arith_split_limit splits are possible.                              *)
wenzelm@24092
   720
wenzelm@24092
   721
local
wenzelm@24092
   722
  val nnf_simpset =
wenzelm@24092
   723
    empty_ss setmkeqTrue mk_eq_True
wenzelm@24092
   724
    setmksimps (mksimps mksimps_pairs)
wenzelm@24092
   725
    addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
wenzelm@24092
   726
      not_all, not_ex, not_not]
wenzelm@24092
   727
  fun prem_nnf_tac i st =
wenzelm@24092
   728
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
wenzelm@24092
   729
in
wenzelm@24092
   730
wenzelm@24092
   731
fun split_once_tac ctxt split_thms =
wenzelm@24092
   732
  let
wenzelm@24092
   733
    val thy = ProofContext.theory_of ctxt
wenzelm@24092
   734
    val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
wenzelm@24092
   735
      let
wenzelm@24092
   736
        val Ts = rev (map snd (Logic.strip_params subgoal))
wenzelm@24092
   737
        val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
wenzelm@24092
   738
        val cmap = Splitter.cmap_of_split_thms split_thms
wenzelm@24092
   739
        val splits = Splitter.split_posns cmap thy Ts concl
wenzelm@24112
   740
        val split_limit = Config.get ctxt fast_arith_split_limit
wenzelm@24092
   741
      in
wenzelm@24092
   742
        if length splits > split_limit then no_tac
wenzelm@24092
   743
        else split_tac split_thms i
wenzelm@24092
   744
      end)
wenzelm@24092
   745
  in
wenzelm@24092
   746
    EVERY' [
wenzelm@24092
   747
      REPEAT_DETERM o etac rev_mp,
wenzelm@24092
   748
      cond_split_tac,
wenzelm@24092
   749
      rtac ccontr,
wenzelm@24092
   750
      prem_nnf_tac,
wenzelm@24092
   751
      TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
wenzelm@24092
   752
    ]
wenzelm@24092
   753
  end;
wenzelm@24092
   754
wenzelm@24092
   755
end;  (* local *)
wenzelm@24092
   756
wenzelm@24092
   757
(* remove irrelevant premises, then split the i-th subgoal (and all new      *)
wenzelm@24092
   758
(* subgoals) by using 'split_once_tac' repeatedly.  Beta-eta-normalize new   *)
wenzelm@24092
   759
(* subgoals and finally attempt to solve them by finding an immediate        *)
wenzelm@24092
   760
(* contradiction (i.e. a term and its negation) in their premises.           *)
wenzelm@24092
   761
wenzelm@24092
   762
fun pre_tac ctxt i =
wenzelm@24092
   763
let
wenzelm@24092
   764
  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
wenzelm@24092
   765
  fun is_relevant t = isSome (decomp ctxt t)
wenzelm@24092
   766
in
wenzelm@24092
   767
  DETERM (
wenzelm@24092
   768
    TRY (filter_prems_tac is_relevant i)
wenzelm@24092
   769
      THEN (
wenzelm@24092
   770
        (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
wenzelm@24092
   771
          THEN_ALL_NEW
wenzelm@24092
   772
            (CONVERSION Drule.beta_eta_conversion
wenzelm@24092
   773
              THEN'
wenzelm@24092
   774
            (TRY o (etac notE THEN' eq_assume_tac)))
wenzelm@24092
   775
      ) i
wenzelm@24092
   776
  )
wenzelm@24092
   777
end;
wenzelm@24092
   778
wenzelm@24092
   779
end;  (* LA_Data_Ref *)
wenzelm@24092
   780
wenzelm@24092
   781
wenzelm@24092
   782
val lin_arith_pre_tac = LA_Data_Ref.pre_tac;
wenzelm@24092
   783
wenzelm@27017
   784
structure Fast_Arith = Fast_Lin_Arith(structure LA_Logic = LA_Logic and LA_Data = LA_Data_Ref);
wenzelm@24092
   785
wenzelm@24092
   786
val map_data = Fast_Arith.map_data;
wenzelm@24092
   787
wenzelm@27017
   788
fun fast_arith_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
wenzelm@27017
   789
val fast_ex_arith_tac = Fast_Arith.lin_arith_tac;
wenzelm@27017
   790
val trace_arith = Fast_Arith.trace;
wenzelm@27017
   791
val warning_count = Fast_Arith.warning_count;
wenzelm@24092
   792
wenzelm@24092
   793
(* reduce contradictory <= to False.
wenzelm@24092
   794
   Most of the work is done by the cancel tactics. *)
wenzelm@24092
   795
wenzelm@24092
   796
val init_arith_data =
wenzelm@24092
   797
 Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
wenzelm@24092
   798
   {add_mono_thms = add_mono_thms @
wenzelm@24092
   799
    @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
wenzelm@24092
   800
    mult_mono_thms = mult_mono_thms,
wenzelm@24092
   801
    inj_thms = inj_thms,
wenzelm@24092
   802
    lessD = lessD @ [thm "Suc_leI"],
wenzelm@24092
   803
    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
wenzelm@24092
   804
    simpset = HOL_basic_ss
wenzelm@24092
   805
      addsimps
haftmann@28053
   806
       [@{thm "monoid_add_class.add_0_left"},
haftmann@28053
   807
        @{thm "monoid_add_class.add_0_right"},
wenzelm@24092
   808
        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
wenzelm@24092
   809
        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
wenzelm@24092
   810
        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
wenzelm@24092
   811
        @{thm "not_one_less_zero"}]
wenzelm@24092
   812
      addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
wenzelm@24092
   813
       (*abel_cancel helps it work in abstract algebraic domains*)
haftmann@26101
   814
      addsimprocs ArithData.nat_cancel_sums_add}) #>
wenzelm@24092
   815
  arith_discrete "nat";
wenzelm@24092
   816
wenzelm@24092
   817
val lin_arith_simproc = Fast_Arith.lin_arith_simproc;
wenzelm@24092
   818
wenzelm@24092
   819
val fast_nat_arith_simproc =
wenzelm@24092
   820
  Simplifier.simproc (the_context ()) "fast_nat_arith"
wenzelm@24092
   821
    ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
wenzelm@24092
   822
wenzelm@24092
   823
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
wenzelm@24092
   824
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@24092
   825
*not* themselves (in)equalities, because the latter activate
wenzelm@24092
   826
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@24092
   827
solver all the time rather than add the additional check. *)
wenzelm@24092
   828
wenzelm@24092
   829
haftmann@26110
   830
(* generic refutation procedure *)
haftmann@26110
   831
haftmann@26110
   832
(* parameters:
haftmann@26110
   833
haftmann@26110
   834
   test: term -> bool
haftmann@26110
   835
   tests if a term is at all relevant to the refutation proof;
haftmann@26110
   836
   if not, then it can be discarded. Can improve performance,
haftmann@26110
   837
   esp. if disjunctions can be discarded (no case distinction needed!).
haftmann@26110
   838
haftmann@26110
   839
   prep_tac: int -> tactic
haftmann@26110
   840
   A preparation tactic to be applied to the goal once all relevant premises
haftmann@26110
   841
   have been moved to the conclusion.
haftmann@26110
   842
haftmann@26110
   843
   ref_tac: int -> tactic
haftmann@26110
   844
   the actual refutation tactic. Should be able to deal with goals
haftmann@26110
   845
   [| A1; ...; An |] ==> False
haftmann@26110
   846
   where the Ai are atomic, i.e. no top-level &, | or EX
haftmann@26110
   847
*)
haftmann@26110
   848
haftmann@26110
   849
local
haftmann@26110
   850
  val nnf_simpset =
haftmann@26110
   851
    empty_ss setmkeqTrue mk_eq_True
haftmann@26110
   852
    setmksimps (mksimps mksimps_pairs)
haftmann@26110
   853
    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
haftmann@26110
   854
      @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];
haftmann@26110
   855
  fun prem_nnf_tac i st =
haftmann@26110
   856
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
haftmann@26110
   857
in
haftmann@26110
   858
fun refute_tac test prep_tac ref_tac =
haftmann@26110
   859
  let val refute_prems_tac =
haftmann@26110
   860
        REPEAT_DETERM
haftmann@26110
   861
              (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
haftmann@26110
   862
               filter_prems_tac test 1 ORELSE
haftmann@26110
   863
               etac @{thm disjE} 1) THEN
haftmann@26110
   864
        (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
haftmann@26110
   865
         ref_tac 1);
haftmann@26110
   866
  in EVERY'[TRY o filter_prems_tac test,
haftmann@26110
   867
            REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
haftmann@26110
   868
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
haftmann@26110
   869
  end;
haftmann@26110
   870
end;
haftmann@26110
   871
haftmann@26110
   872
wenzelm@24092
   873
(* arith proof method *)
wenzelm@24092
   874
wenzelm@24092
   875
local
wenzelm@24092
   876
wenzelm@24092
   877
fun raw_arith_tac ctxt ex =
wenzelm@24092
   878
  (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
wenzelm@24092
   879
     decomp sg"? -- but note that the test is applied to terms already before
wenzelm@24092
   880
     they are split/normalized) to speed things up in case there are lots of
wenzelm@24092
   881
     irrelevant terms involved; elimination of min/max can be optimized:
wenzelm@24092
   882
     (max m n + k <= r) = (m+k <= r & n+k <= r)
wenzelm@24092
   883
     (l <= min m n + k) = (l <= m+k & l <= n+k)
wenzelm@24092
   884
  *)
wenzelm@24092
   885
  refute_tac (K true)
wenzelm@24092
   886
    (* Splitting is also done inside fast_arith_tac, but not completely --   *)
wenzelm@24092
   887
    (* split_tac may use split theorems that have not been implemented in    *)
wenzelm@24092
   888
    (* fast_arith_tac (cf. pre_decomp and split_once_items above), and       *)
wenzelm@24092
   889
    (* fast_arith_split_limit may trigger.                                   *)
wenzelm@24092
   890
    (* Therefore splitting outside of fast_arith_tac may allow us to prove   *)
wenzelm@24092
   891
    (* some goals that fast_arith_tac alone would fail on.                   *)
wenzelm@24092
   892
    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
wenzelm@24092
   893
    (fast_ex_arith_tac ctxt ex);
wenzelm@24092
   894
wenzelm@24092
   895
fun more_arith_tacs ctxt =
wenzelm@24092
   896
  let val tactics = #tactics (get_arith_data ctxt)
wenzelm@24092
   897
  in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
wenzelm@24092
   898
wenzelm@24092
   899
in
wenzelm@24092
   900
wenzelm@24092
   901
fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   902
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
wenzelm@24092
   903
wenzelm@24092
   904
fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   905
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
wenzelm@24092
   906
  more_arith_tacs ctxt];
wenzelm@24092
   907
wenzelm@24092
   908
fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   909
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
wenzelm@24092
   910
  more_arith_tacs ctxt];
wenzelm@24092
   911
wenzelm@24092
   912
fun arith_method src =
wenzelm@24092
   913
  Method.syntax Args.bang_facts src
wenzelm@24092
   914
  #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
wenzelm@24092
   915
      HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
wenzelm@24092
   916
wenzelm@24092
   917
end;
wenzelm@24092
   918
wenzelm@24092
   919
wenzelm@24092
   920
(* context setup *)
wenzelm@24092
   921
wenzelm@24092
   922
val setup =
wenzelm@24092
   923
  init_arith_data #>
wenzelm@24092
   924
  Simplifier.map_ss (fn ss => ss addsimprocs [fast_nat_arith_simproc]
wenzelm@24092
   925
    addSolver (mk_solver' "lin_arith" Fast_Arith.cut_lin_arith_tac)) #>
wenzelm@24092
   926
  Context.mapping
wenzelm@24092
   927
   (setup_options #>
wenzelm@24092
   928
    Method.add_methods
huffman@26061
   929
      [("arith", arith_method, "decide linear arithmetic")] #>
wenzelm@24092
   930
    Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
wenzelm@24092
   931
      "declaration of split rules for arithmetic procedure")]) I;
wenzelm@24092
   932
wenzelm@24092
   933
end;
wenzelm@24092
   934
wenzelm@24092
   935
structure BasicLinArith: BASIC_LIN_ARITH = LinArith;
wenzelm@24092
   936
open BasicLinArith;