src/Provers/Arith/fast_lin_arith.ML
author wenzelm
Thu Aug 26 17:37:26 2010 +0200 (2010-08-26)
changeset 38762 996afaa9254a
parent 38052 04a8de29e8f7
child 38763 283f1f9969ba
permissions -rw-r--r--
slightly more abstract data handling in Fast_Lin_Arith;
nipkow@5982
     1
(*  Title:      Provers/Arith/fast_lin_arith.ML
boehmes@31510
     2
    Author:     Tobias Nipkow and Tjark Weber and Sascha Boehme
nipkow@6102
     3
wenzelm@24076
     4
A generic linear arithmetic package.  It provides two tactics
wenzelm@24076
     5
(cut_lin_arith_tac, lin_arith_tac) and a simplification procedure
wenzelm@24076
     6
(lin_arith_simproc).
nipkow@6102
     7
wenzelm@24076
     8
Only take premises and conclusions into account that are already
wenzelm@24076
     9
(negated) (in)equations. lin_arith_simproc tries to prove or disprove
wenzelm@24076
    10
the term.
nipkow@5982
    11
*)
nipkow@5982
    12
nipkow@5982
    13
(*** Data needed for setting up the linear arithmetic package ***)
nipkow@5982
    14
nipkow@6102
    15
signature LIN_ARITH_LOGIC =
nipkow@6102
    16
sig
webertj@20276
    17
  val conjI       : thm  (* P ==> Q ==> P & Q *)
webertj@20276
    18
  val ccontr      : thm  (* (~ P ==> False) ==> P *)
webertj@20276
    19
  val notI        : thm  (* (P ==> False) ==> ~ P *)
webertj@20276
    20
  val not_lessD   : thm  (* ~(m < n) ==> n <= m *)
webertj@20276
    21
  val not_leD     : thm  (* ~(m <= n) ==> n < m *)
webertj@20276
    22
  val sym         : thm  (* x = y ==> y = x *)
boehmes@31510
    23
  val trueI       : thm  (* True *)
webertj@20276
    24
  val mk_Eq       : thm -> thm
webertj@20276
    25
  val atomize     : thm -> thm list
webertj@20276
    26
  val mk_Trueprop : term -> term
webertj@20276
    27
  val neg_prop    : term -> term
webertj@20276
    28
  val is_False    : thm -> bool
webertj@20276
    29
  val is_nat      : typ list * term -> bool
webertj@20276
    30
  val mk_nat_thm  : theory -> term -> thm
nipkow@6102
    31
end;
nipkow@6102
    32
(*
nipkow@6102
    33
mk_Eq(~in) = `in == False'
nipkow@6102
    34
mk_Eq(in) = `in == True'
nipkow@6102
    35
where `in' is an (in)equality.
nipkow@6102
    36
webertj@23190
    37
neg_prop(t) = neg  if t is wrapped up in Trueprop and neg is the
webertj@23190
    38
  (logically) negated version of t (again wrapped up in Trueprop),
webertj@23190
    39
  where the negation of a negative term is the term itself (no
webertj@23190
    40
  double negation!); raises TERM ("neg_prop", [t]) if t is not of
webertj@23190
    41
  the form 'Trueprop $ _'
nipkow@6128
    42
nipkow@6128
    43
is_nat(parameter-types,t) =  t:nat
nipkow@6128
    44
mk_nat_thm(t) = "0 <= t"
nipkow@6102
    45
*)
nipkow@6102
    46
nipkow@5982
    47
signature LIN_ARITH_DATA =
nipkow@5982
    48
sig
wenzelm@24076
    49
  (*internal representation of linear (in-)equations:*)
wenzelm@26945
    50
  type decomp = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool
wenzelm@26945
    51
  val decomp: Proof.context -> term -> decomp option
wenzelm@24076
    52
  val domain_is_nat: term -> bool
wenzelm@24076
    53
wenzelm@24076
    54
  (*preprocessing, performed on a representation of subgoals as list of premises:*)
wenzelm@24076
    55
  val pre_decomp: Proof.context -> typ list * term list -> (typ list * term list) list
wenzelm@24076
    56
wenzelm@24076
    57
  (*preprocessing, performed on the goal -- must do the same as 'pre_decomp':*)
wenzelm@35230
    58
  val pre_tac: simpset -> int -> tactic
wenzelm@24076
    59
wenzelm@24076
    60
  (*the limit on the number of ~= allowed; because each ~= is split
wenzelm@24076
    61
    into two cases, this can lead to an explosion*)
wenzelm@24112
    62
  val fast_arith_neq_limit: int Config.T
nipkow@5982
    63
end;
nipkow@5982
    64
(*
nipkow@7551
    65
decomp(`x Rel y') should yield (p,i,Rel,q,j,d)
nipkow@5982
    66
   where Rel is one of "<", "~<", "<=", "~<=" and "=" and
webertj@20217
    67
         p (q, respectively) is the decomposition of the sum term x
webertj@20217
    68
         (y, respectively) into a list of summand * multiplicity
webertj@20217
    69
         pairs and a constant summand and d indicates if the domain
webertj@20217
    70
         is discrete.
webertj@20217
    71
webertj@20276
    72
domain_is_nat(`x Rel y') t should yield true iff x is of type "nat".
webertj@20276
    73
webertj@20217
    74
The relationship between pre_decomp and pre_tac is somewhat tricky.  The
webertj@20217
    75
internal representation of a subgoal and the corresponding theorem must
webertj@20217
    76
be modified by pre_decomp (pre_tac, resp.) in a corresponding way.  See
webertj@20217
    77
the comment for split_items below.  (This is even necessary for eta- and
webertj@20217
    78
beta-equivalent modifications, as some of the lin. arith. code is not
webertj@20217
    79
insensitive to them.)
nipkow@5982
    80
wenzelm@9420
    81
ss must reduce contradictory <= to False.
nipkow@5982
    82
   It should also cancel common summands to keep <= reduced;
nipkow@5982
    83
   otherwise <= can grow to massive proportions.
nipkow@5982
    84
*)
nipkow@5982
    85
nipkow@6062
    86
signature FAST_LIN_ARITH =
nipkow@6062
    87
sig
haftmann@31102
    88
  val cut_lin_arith_tac: simpset -> int -> tactic
haftmann@31102
    89
  val lin_arith_tac: Proof.context -> bool -> int -> tactic
haftmann@31102
    90
  val lin_arith_simproc: simpset -> term -> thm option
nipkow@15184
    91
  val map_data: ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
boehmes@31510
    92
                 lessD: thm list, neqE: thm list, simpset: Simplifier.simpset,
boehmes@31510
    93
                 number_of : serial * (theory -> typ -> int -> cterm)}
nipkow@15184
    94
                 -> {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
boehmes@31510
    95
                     lessD: thm list, neqE: thm list, simpset: Simplifier.simpset,
boehmes@31510
    96
                     number_of : serial * (theory -> typ -> int -> cterm)})
wenzelm@24076
    97
                -> Context.generic -> Context.generic
wenzelm@38762
    98
  val add_inj_thms: thm list -> Context.generic -> Context.generic
wenzelm@38762
    99
  val add_lessD: thm -> Context.generic -> Context.generic
wenzelm@38762
   100
  val add_simps: thm list -> Context.generic -> Context.generic
wenzelm@38762
   101
  val add_simprocs: simproc list -> Context.generic -> Context.generic
wenzelm@38762
   102
  val set_number_of: (theory -> typ -> int -> cterm) -> Context.generic -> Context.generic
wenzelm@32740
   103
  val trace: bool Unsynchronized.ref
nipkow@6062
   104
end;
nipkow@6062
   105
wenzelm@24076
   106
functor Fast_Lin_Arith
wenzelm@24076
   107
  (structure LA_Logic: LIN_ARITH_LOGIC and LA_Data: LIN_ARITH_DATA): FAST_LIN_ARITH =
nipkow@5982
   108
struct
nipkow@5982
   109
wenzelm@9420
   110
wenzelm@9420
   111
(** theory data **)
wenzelm@9420
   112
boehmes@31510
   113
fun no_number_of _ _ _ = raise CTERM ("number_of", [])
boehmes@31510
   114
wenzelm@33519
   115
structure Data = Generic_Data
wenzelm@22846
   116
(
wenzelm@24076
   117
  type T =
wenzelm@24076
   118
   {add_mono_thms: thm list,
wenzelm@24076
   119
    mult_mono_thms: thm list,
wenzelm@24076
   120
    inj_thms: thm list,
wenzelm@24076
   121
    lessD: thm list,
wenzelm@24076
   122
    neqE: thm list,
boehmes@31510
   123
    simpset: Simplifier.simpset,
boehmes@31510
   124
    number_of : serial * (theory -> typ -> int -> cterm)};
wenzelm@9420
   125
nipkow@10691
   126
  val empty = {add_mono_thms = [], mult_mono_thms = [], inj_thms = [],
boehmes@31510
   127
               lessD = [], neqE = [], simpset = Simplifier.empty_ss,
haftmann@31638
   128
               number_of = (serial (), no_number_of) } : T;
wenzelm@16458
   129
  val extend = I;
wenzelm@33519
   130
  fun merge
wenzelm@16458
   131
    ({add_mono_thms= add_mono_thms1, mult_mono_thms= mult_mono_thms1, inj_thms= inj_thms1,
boehmes@31510
   132
      lessD = lessD1, neqE=neqE1, simpset = simpset1,
boehmes@31510
   133
      number_of = (number_of1 as (s1, _))},
wenzelm@16458
   134
     {add_mono_thms= add_mono_thms2, mult_mono_thms= mult_mono_thms2, inj_thms= inj_thms2,
boehmes@31510
   135
      lessD = lessD2, neqE=neqE2, simpset = simpset2,
boehmes@31510
   136
      number_of = (number_of2 as (s2, _))}) =
wenzelm@24039
   137
    {add_mono_thms = Thm.merge_thms (add_mono_thms1, add_mono_thms2),
wenzelm@24039
   138
     mult_mono_thms = Thm.merge_thms (mult_mono_thms1, mult_mono_thms2),
wenzelm@24039
   139
     inj_thms = Thm.merge_thms (inj_thms1, inj_thms2),
wenzelm@24039
   140
     lessD = Thm.merge_thms (lessD1, lessD2),
wenzelm@24039
   141
     neqE = Thm.merge_thms (neqE1, neqE2),
boehmes@31510
   142
     simpset = Simplifier.merge_ss (simpset1, simpset2),
wenzelm@38762
   143
     (* FIXME depends on accidental load order !?! *)  (* FIXME *)
boehmes@31510
   144
     number_of = if s1 > s2 then number_of1 else number_of2};
wenzelm@22846
   145
);
wenzelm@9420
   146
wenzelm@9420
   147
val map_data = Data.map;
wenzelm@24076
   148
val get_data = Data.get o Context.Proof;
wenzelm@9420
   149
wenzelm@38762
   150
fun map_inj_thms f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
wenzelm@38762
   151
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = f inj_thms,
wenzelm@38762
   152
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = number_of};
wenzelm@38762
   153
wenzelm@38762
   154
fun map_lessD f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
wenzelm@38762
   155
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
wenzelm@38762
   156
    lessD = f lessD, neqE = neqE, simpset = simpset, number_of = number_of};
wenzelm@38762
   157
wenzelm@38762
   158
fun map_simpset f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
wenzelm@38762
   159
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
wenzelm@38762
   160
    lessD = lessD, neqE = neqE, simpset = f simpset, number_of = number_of};
wenzelm@38762
   161
wenzelm@38762
   162
fun map_number_of f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
wenzelm@38762
   163
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
wenzelm@38762
   164
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = f number_of};
wenzelm@38762
   165
wenzelm@38762
   166
fun add_inj_thms thms = map_data (map_inj_thms (append thms));
wenzelm@38762
   167
fun add_lessD thm = map_data (map_lessD (fn thms => thms @ [thm]));
wenzelm@38762
   168
fun add_simps thms = map_data (map_simpset (fn simpset => simpset addsimps thms));
wenzelm@38762
   169
fun add_simprocs procs = map_data (map_simpset (fn simpset => simpset addsimprocs procs));
wenzelm@38762
   170
wenzelm@38762
   171
fun set_number_of f = map_data (map_number_of (K (serial (), f)));
wenzelm@9420
   172
wenzelm@9420
   173
nipkow@5982
   174
(*** A fast decision procedure ***)
nipkow@5982
   175
(*** Code ported from HOL Light ***)
nipkow@6056
   176
(* possible optimizations:
nipkow@6056
   177
   use (var,coeff) rep or vector rep  tp save space;
nipkow@6056
   178
   treat non-negative atoms separately rather than adding 0 <= atom
nipkow@6056
   179
*)
nipkow@5982
   180
wenzelm@32740
   181
val trace = Unsynchronized.ref false;
paulson@9073
   182
nipkow@5982
   183
datatype lineq_type = Eq | Le | Lt;
nipkow@5982
   184
nipkow@6056
   185
datatype injust = Asm of int
nipkow@6056
   186
                | Nat of int (* index of atom *)
nipkow@6128
   187
                | LessD of injust
nipkow@6128
   188
                | NotLessD of injust
nipkow@6128
   189
                | NotLeD of injust
nipkow@7551
   190
                | NotLeDD of injust
wenzelm@24630
   191
                | Multiplied of int * injust
nipkow@5982
   192
                | Added of injust * injust;
nipkow@5982
   193
wenzelm@24630
   194
datatype lineq = Lineq of int * lineq_type * int list * injust;
nipkow@5982
   195
nipkow@13498
   196
(* ------------------------------------------------------------------------- *)
nipkow@13498
   197
(* Finding a (counter) example from the trace of a failed elimination        *)
nipkow@13498
   198
(* ------------------------------------------------------------------------- *)
nipkow@13498
   199
(* Examples are represented as rational numbers,                             *)
nipkow@13498
   200
(* Dont blame John Harrison for this code - it is entirely mine. TN          *)
nipkow@13498
   201
nipkow@13498
   202
exception NoEx;
nipkow@13498
   203
nipkow@14372
   204
(* Coding: (i,true,cs) means i <= cs and (i,false,cs) means i < cs.
nipkow@14372
   205
   In general, true means the bound is included, false means it is excluded.
nipkow@14372
   206
   Need to know if it is a lower or upper bound for unambiguous interpretation!
nipkow@14372
   207
*)
nipkow@14372
   208
haftmann@22950
   209
fun elim_eqns (Lineq (i, Le, cs, _)) = [(i, true, cs)]
haftmann@22950
   210
  | elim_eqns (Lineq (i, Eq, cs, _)) = [(i, true, cs),(~i, true, map ~ cs)]
haftmann@22950
   211
  | elim_eqns (Lineq (i, Lt, cs, _)) = [(i, false, cs)];
nipkow@13498
   212
nipkow@13498
   213
(* PRE: ex[v] must be 0! *)
wenzelm@24630
   214
fun eval ex v (a, le, cs) =
haftmann@22950
   215
  let
haftmann@22950
   216
    val rs = map Rat.rat_of_int cs;
haftmann@22950
   217
    val rsum = fold2 (Rat.add oo Rat.mult) rs ex Rat.zero;
haftmann@23063
   218
  in (Rat.mult (Rat.add (Rat.rat_of_int a) (Rat.neg rsum)) (Rat.inv (nth rs v)), le) end;
haftmann@23063
   219
(* If nth rs v < 0, le should be negated.
nipkow@14372
   220
   Instead this swap is taken into account in ratrelmin2.
nipkow@14372
   221
*)
nipkow@13498
   222
haftmann@22950
   223
fun ratrelmin2 (x as (r, ler), y as (s, les)) =
haftmann@23520
   224
  case Rat.ord (r, s)
haftmann@22950
   225
   of EQUAL => (r, (not ler) andalso (not les))
haftmann@22950
   226
    | LESS => x
haftmann@22950
   227
    | GREATER => y;
haftmann@22950
   228
haftmann@22950
   229
fun ratrelmax2 (x as (r, ler), y as (s, les)) =
haftmann@23520
   230
  case Rat.ord (r, s)
haftmann@22950
   231
   of EQUAL => (r, ler andalso les)
haftmann@22950
   232
    | LESS => y
haftmann@22950
   233
    | GREATER => x;
nipkow@13498
   234
nipkow@14372
   235
val ratrelmin = foldr1 ratrelmin2;
nipkow@14372
   236
val ratrelmax = foldr1 ratrelmax2;
nipkow@13498
   237
haftmann@22950
   238
fun ratexact up (r, exact) =
nipkow@14372
   239
  if exact then r else
haftmann@22950
   240
  let
haftmann@38052
   241
    val (_, q) = Rat.quotient_of_rat r;
haftmann@22950
   242
    val nth = Rat.inv (Rat.rat_of_int q);
haftmann@22950
   243
  in Rat.add r (if up then nth else Rat.neg nth) end;
nipkow@14372
   244
haftmann@22950
   245
fun ratmiddle (r, s) = Rat.mult (Rat.add r s) (Rat.inv Rat.two);
nipkow@14372
   246
webertj@20217
   247
fun choose2 d ((lb, exactl), (ub, exactu)) =
haftmann@23520
   248
  let val ord = Rat.sign lb in
haftmann@22950
   249
  if (ord = LESS orelse exactl) andalso (ord = GREATER orelse exactu)
haftmann@22950
   250
    then Rat.zero
haftmann@22950
   251
    else if not d then
haftmann@22950
   252
      if ord = GREATER
webertj@20217
   253
        then if exactl then lb else ratmiddle (lb, ub)
haftmann@22950
   254
        else if exactu then ub else ratmiddle (lb, ub)
haftmann@22950
   255
      else (*discrete domain, both bounds must be exact*)
haftmann@23025
   256
      if ord = GREATER
haftmann@22950
   257
        then let val lb' = Rat.roundup lb in
haftmann@22950
   258
          if Rat.le lb' ub then lb' else raise NoEx end
haftmann@22950
   259
        else let val ub' = Rat.rounddown ub in
haftmann@22950
   260
          if Rat.le lb ub' then ub' else raise NoEx end
haftmann@22950
   261
  end;
nipkow@13498
   262
haftmann@22950
   263
fun findex1 discr (v, lineqs) ex =
haftmann@22950
   264
  let
haftmann@23063
   265
    val nz = filter (fn (Lineq (_, _, cs, _)) => nth cs v <> 0) lineqs;
haftmann@22950
   266
    val ineqs = maps elim_eqns nz;
haftmann@23063
   267
    val (ge, le) = List.partition (fn (_,_,cs) => nth cs v > 0) ineqs
haftmann@22950
   268
    val lb = ratrelmax (map (eval ex v) ge)
haftmann@22950
   269
    val ub = ratrelmin (map (eval ex v) le)
haftmann@21109
   270
  in nth_map v (K (choose2 (nth discr v) (lb, ub))) ex end;
nipkow@13498
   271
nipkow@13498
   272
fun elim1 v x =
haftmann@23063
   273
  map (fn (a,le,bs) => (Rat.add a (Rat.neg (Rat.mult (nth bs v) x)), le,
haftmann@21109
   274
                        nth_map v (K Rat.zero) bs));
nipkow@13498
   275
haftmann@23520
   276
fun single_var v (_, _, cs) = case filter_out (curry (op =) EQUAL o Rat.sign) cs
haftmann@23063
   277
 of [x] => x =/ nth cs v
haftmann@23063
   278
  | _ => false;
nipkow@13498
   279
nipkow@13498
   280
(* The base case:
nipkow@13498
   281
   all variables occur only with positive or only with negative coefficients *)
nipkow@13498
   282
fun pick_vars discr (ineqs,ex) =
haftmann@23520
   283
  let val nz = filter_out (fn (_,_,cs) => forall (curry (op =) EQUAL o Rat.sign) cs) ineqs
nipkow@14372
   284
  in case nz of [] => ex
nipkow@14372
   285
     | (_,_,cs) :: _ =>
haftmann@23520
   286
       let val v = find_index (not o curry (op =) EQUAL o Rat.sign) cs
haftmann@22950
   287
           val d = nth discr v;
haftmann@23520
   288
           val pos = not (Rat.sign (nth cs v) = LESS);
haftmann@22950
   289
           val sv = filter (single_var v) nz;
nipkow@14372
   290
           val minmax =
haftmann@17951
   291
             if pos then if d then Rat.roundup o fst o ratrelmax
nipkow@14372
   292
                         else ratexact true o ratrelmax
haftmann@17951
   293
                    else if d then Rat.rounddown o fst o ratrelmin
nipkow@14372
   294
                         else ratexact false o ratrelmin
haftmann@23063
   295
           val bnds = map (fn (a,le,bs) => (Rat.mult a (Rat.inv (nth bs v)), le)) sv
haftmann@17951
   296
           val x = minmax((Rat.zero,if pos then true else false)::bnds)
nipkow@14372
   297
           val ineqs' = elim1 v x nz
haftmann@21109
   298
           val ex' = nth_map v (K x) ex
nipkow@14372
   299
       in pick_vars discr (ineqs',ex') end
nipkow@13498
   300
  end;
nipkow@13498
   301
nipkow@13498
   302
fun findex0 discr n lineqs =
haftmann@22950
   303
  let val ineqs = maps elim_eqns lineqs
haftmann@22950
   304
      val rineqs = map (fn (a,le,cs) => (Rat.rat_of_int a, le, map Rat.rat_of_int cs))
nipkow@14372
   305
                       ineqs
haftmann@17951
   306
  in pick_vars discr (rineqs,replicate n Rat.zero) end;
nipkow@13498
   307
nipkow@13498
   308
(* ------------------------------------------------------------------------- *)
webertj@23197
   309
(* End of counterexample finder. The actual decision procedure starts here.  *)
nipkow@13498
   310
(* ------------------------------------------------------------------------- *)
nipkow@13498
   311
nipkow@5982
   312
(* ------------------------------------------------------------------------- *)
nipkow@5982
   313
(* Calculate new (in)equality type after addition.                           *)
nipkow@5982
   314
(* ------------------------------------------------------------------------- *)
nipkow@5982
   315
nipkow@5982
   316
fun find_add_type(Eq,x) = x
nipkow@5982
   317
  | find_add_type(x,Eq) = x
nipkow@5982
   318
  | find_add_type(_,Lt) = Lt
nipkow@5982
   319
  | find_add_type(Lt,_) = Lt
nipkow@5982
   320
  | find_add_type(Le,Le) = Le;
nipkow@5982
   321
nipkow@5982
   322
(* ------------------------------------------------------------------------- *)
nipkow@5982
   323
(* Multiply out an (in)equation.                                             *)
nipkow@5982
   324
(* ------------------------------------------------------------------------- *)
nipkow@5982
   325
nipkow@5982
   326
fun multiply_ineq n (i as Lineq(k,ty,l,just)) =
nipkow@5982
   327
  if n = 1 then i
nipkow@5982
   328
  else if n = 0 andalso ty = Lt then sys_error "multiply_ineq"
nipkow@5982
   329
  else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq"
wenzelm@33002
   330
  else Lineq (n * k, ty, map (Integer.mult n) l, Multiplied (n, just));
nipkow@5982
   331
nipkow@5982
   332
(* ------------------------------------------------------------------------- *)
nipkow@5982
   333
(* Add together (in)equations.                                               *)
nipkow@5982
   334
(* ------------------------------------------------------------------------- *)
nipkow@5982
   335
haftmann@38052
   336
fun add_ineq (Lineq (k1,ty1,l1,just1)) (Lineq (k2,ty2,l2,just2)) =
wenzelm@33002
   337
  let val l = map2 Integer.add l1 l2
nipkow@5982
   338
  in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end;
nipkow@5982
   339
nipkow@5982
   340
(* ------------------------------------------------------------------------- *)
nipkow@5982
   341
(* Elimination of variable between a single pair of (in)equations.           *)
nipkow@5982
   342
(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve.       *)
nipkow@5982
   343
(* ------------------------------------------------------------------------- *)
nipkow@5982
   344
nipkow@5982
   345
fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) =
haftmann@23063
   346
  let val c1 = nth l1 v and c2 = nth l2 v
haftmann@23261
   347
      val m = Integer.lcm (abs c1) (abs c2)
nipkow@5982
   348
      val m1 = m div (abs c1) and m2 = m div (abs c2)
nipkow@5982
   349
      val (n1,n2) =
nipkow@5982
   350
        if (c1 >= 0) = (c2 >= 0)
nipkow@5982
   351
        then if ty1 = Eq then (~m1,m2)
nipkow@5982
   352
             else if ty2 = Eq then (m1,~m2)
nipkow@5982
   353
                  else sys_error "elim_var"
nipkow@5982
   354
        else (m1,m2)
nipkow@5982
   355
      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
nipkow@5982
   356
                    then (~n1,~n2) else (n1,n2)
boehmes@31510
   357
  in add_ineq (multiply_ineq p1 i1) (multiply_ineq p2 i2) end;
nipkow@5982
   358
nipkow@5982
   359
(* ------------------------------------------------------------------------- *)
nipkow@5982
   360
(* The main refutation-finding code.                                         *)
nipkow@5982
   361
(* ------------------------------------------------------------------------- *)
nipkow@5982
   362
nipkow@5982
   363
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
nipkow@5982
   364
haftmann@38052
   365
fun is_contradictory (Lineq(k,ty,_,_)) =
nipkow@5982
   366
  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
nipkow@5982
   367
wenzelm@24630
   368
fun calc_blowup l =
wenzelm@33317
   369
  let val (p,n) = List.partition (curry (op <) 0) (filter (curry (op <>) 0) l)
wenzelm@24630
   370
  in length p * length n end;
nipkow@5982
   371
nipkow@5982
   372
(* ------------------------------------------------------------------------- *)
nipkow@5982
   373
(* Main elimination code:                                                    *)
nipkow@5982
   374
(*                                                                           *)
nipkow@5982
   375
(* (1) Looks for immediate solutions (false assertions with no variables).   *)
nipkow@5982
   376
(*                                                                           *)
nipkow@5982
   377
(* (2) If there are any equations, picks a variable with the lowest absolute *)
nipkow@5982
   378
(* coefficient in any of them, and uses it to eliminate.                     *)
nipkow@5982
   379
(*                                                                           *)
nipkow@5982
   380
(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
nipkow@5982
   381
(* blowup (number of consequences generated) and eliminates it.              *)
nipkow@5982
   382
(* ------------------------------------------------------------------------- *)
nipkow@5982
   383
nipkow@5982
   384
fun extract_first p =
boehmes@31510
   385
  let
boehmes@31510
   386
    fun extract xs (y::ys) = if p y then (y, xs @ ys) else extract (y::xs) ys
boehmes@31510
   387
      | extract xs [] = raise Empty
nipkow@5982
   388
  in extract [] end;
nipkow@5982
   389
nipkow@6056
   390
fun print_ineqs ineqs =
paulson@9073
   391
  if !trace then
wenzelm@12262
   392
     tracing(cat_lines(""::map (fn Lineq(c,t,l,_) =>
wenzelm@24630
   393
       string_of_int c ^
paulson@9073
   394
       (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
wenzelm@24630
   395
       commas(map string_of_int l)) ineqs))
paulson@9073
   396
  else ();
nipkow@6056
   397
nipkow@13498
   398
type history = (int * lineq list) list;
nipkow@13498
   399
datatype result = Success of injust | Failure of history;
nipkow@13498
   400
webertj@20217
   401
fun elim (ineqs, hist) =
boehmes@31510
   402
  let val _ = print_ineqs ineqs
webertj@20217
   403
      val (triv, nontriv) = List.partition is_trivial ineqs in
webertj@20217
   404
  if not (null triv)
boehmes@31510
   405
  then case Library.find_first is_contradictory triv of
webertj@20217
   406
         NONE => elim (nontriv, hist)
skalberg@15531
   407
       | SOME(Lineq(_,_,_,j)) => Success j
nipkow@5982
   408
  else
webertj@20217
   409
  if null nontriv then Failure hist
nipkow@13498
   410
  else
webertj@20217
   411
  let val (eqs, noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
webertj@20217
   412
  if not (null eqs) then
boehmes@31510
   413
     let val c =
haftmann@33042
   414
           fold (fn Lineq(_,_,l,_) => fn cs => union (op =) l cs) eqs []
boehmes@31510
   415
           |> filter (fn i => i <> 0)
boehmes@31510
   416
           |> sort (int_ord o pairself abs)
boehmes@31510
   417
           |> hd
boehmes@31510
   418
         val (eq as Lineq(_,_,ceq,_),othereqs) =
haftmann@36692
   419
               extract_first (fn Lineq(_,_,l,_) => member (op =) l c) eqs
haftmann@31986
   420
         val v = find_index (fn v => v = c) ceq
haftmann@23063
   421
         val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0)
nipkow@5982
   422
                                     (othereqs @ noneqs)
nipkow@5982
   423
         val others = map (elim_var v eq) roth @ ioth
nipkow@13498
   424
     in elim(others,(v,nontriv)::hist) end
nipkow@5982
   425
  else
nipkow@5982
   426
  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
haftmann@23063
   427
      val numlist = 0 upto (length (hd lists) - 1)
haftmann@23063
   428
      val coeffs = map (fn i => map (fn xs => nth xs i) lists) numlist
nipkow@5982
   429
      val blows = map calc_blowup coeffs
nipkow@5982
   430
      val iblows = blows ~~ numlist
haftmann@23063
   431
      val nziblows = filter_out (fn (i, _) => i = 0) iblows
nipkow@13498
   432
  in if null nziblows then Failure((~1,nontriv)::hist)
nipkow@13498
   433
     else
nipkow@5982
   434
     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
haftmann@23063
   435
         val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) ineqs
haftmann@23063
   436
         val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => nth l v > 0) yes
boehmes@31510
   437
     in elim(no @ map_product (elim_var v) pos neg, (v,nontriv)::hist) end
nipkow@5982
   438
  end
nipkow@5982
   439
  end
nipkow@5982
   440
  end;
nipkow@5982
   441
nipkow@5982
   442
(* ------------------------------------------------------------------------- *)
nipkow@5982
   443
(* Translate back a proof.                                                   *)
nipkow@5982
   444
(* ------------------------------------------------------------------------- *)
nipkow@5982
   445
wenzelm@32091
   446
fun trace_thm ctxt msg th =
wenzelm@32091
   447
  (if !trace then (tracing msg; tracing (Display.string_of_thm ctxt th)) else (); th);
paulson@9073
   448
wenzelm@24076
   449
fun trace_term ctxt msg t =
wenzelm@24920
   450
  (if !trace then tracing (cat_lines [msg, Syntax.string_of_term ctxt t]) else (); t)
wenzelm@24076
   451
wenzelm@24076
   452
fun trace_msg msg =
wenzelm@24076
   453
  if !trace then tracing msg else ();
paulson@9073
   454
haftmann@33042
   455
val union_term = union Pattern.aeconv;
haftmann@33042
   456
val union_bterm = union (fn ((b:bool, t), (b', t')) => b = b' andalso Pattern.aeconv (t, t'));
berghofe@26835
   457
boehmes@31510
   458
fun add_atoms (lhs, _, _, rhs, _, _) =
boehmes@31510
   459
  union_term (map fst lhs) o union_term (map fst rhs);
nipkow@6056
   460
boehmes@31510
   461
fun atoms_of ds = fold add_atoms ds [];
boehmes@31510
   462
boehmes@31510
   463
(*
nipkow@6056
   464
Simplification may detect a contradiction 'prematurely' due to type
nipkow@6056
   465
information: n+1 <= 0 is simplified to False and does not need to be crossed
nipkow@6056
   466
with 0 <= n.
nipkow@6056
   467
*)
nipkow@6056
   468
local
wenzelm@24076
   469
  exception FalseE of thm
nipkow@6056
   470
in
wenzelm@27020
   471
wenzelm@24076
   472
fun mkthm ss asms (just: injust) =
wenzelm@24076
   473
  let
wenzelm@24076
   474
    val ctxt = Simplifier.the_context ss;
wenzelm@24076
   475
    val thy = ProofContext.theory_of ctxt;
boehmes@31510
   476
    val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset,
boehmes@31510
   477
      number_of = (_, num_of), ...} = get_data ctxt;
wenzelm@24076
   478
    val simpset' = Simplifier.inherit_context ss simpset;
boehmes@31510
   479
    fun only_concl f thm =
boehmes@31510
   480
      if Thm.no_prems thm then f (Thm.concl_of thm) else NONE;
boehmes@31510
   481
    val atoms = atoms_of (map_filter (only_concl (LA_Data.decomp ctxt)) asms);
boehmes@31510
   482
boehmes@31510
   483
    fun use_first rules thm =
boehmes@31510
   484
      get_first (fn th => SOME (thm RS th) handle THM _ => NONE) rules
boehmes@31510
   485
boehmes@31510
   486
    fun add2 thm1 thm2 =
boehmes@31510
   487
      use_first add_mono_thms (thm1 RS (thm2 RS LA_Logic.conjI));
boehmes@31510
   488
    fun try_add thms thm = get_first (fn th => add2 th thm) thms;
nipkow@6056
   489
boehmes@31510
   490
    fun add_thms thm1 thm2 =
boehmes@31510
   491
      (case add2 thm1 thm2 of
boehmes@31510
   492
        NONE =>
boehmes@31510
   493
          (case try_add ([thm1] RL inj_thms) thm2 of
boehmes@31510
   494
            NONE =>
boehmes@31510
   495
              (the (try_add ([thm2] RL inj_thms) thm1)
boehmes@31510
   496
                handle Option =>
wenzelm@32091
   497
                  (trace_thm ctxt "" thm1; trace_thm ctxt "" thm2;
boehmes@31510
   498
                   sys_error "Linear arithmetic: failed to add thms"))
boehmes@31510
   499
          | SOME thm => thm)
boehmes@31510
   500
      | SOME thm => thm);
boehmes@31510
   501
boehmes@31510
   502
    fun mult_by_add n thm =
boehmes@31510
   503
      let fun mul i th = if i = 1 then th else mul (i - 1) (add_thms thm th)
boehmes@31510
   504
      in mul n thm end;
nipkow@10575
   505
boehmes@31510
   506
    val rewr = Simplifier.rewrite simpset';
boehmes@31510
   507
    val rewrite_concl = Conv.fconv_rule (Conv.concl_conv ~1 (Conv.arg_conv
boehmes@31510
   508
      (Conv.binop_conv rewr)));
boehmes@31510
   509
    fun discharge_prem thm = if Thm.nprems_of thm = 0 then thm else
boehmes@31510
   510
      let val cv = Conv.arg1_conv (Conv.arg_conv rewr)
boehmes@31510
   511
      in Thm.implies_elim (Conv.fconv_rule cv thm) LA_Logic.trueI end
webertj@20217
   512
boehmes@31510
   513
    fun mult n thm =
boehmes@31510
   514
      (case use_first mult_mono_thms thm of
boehmes@31510
   515
        NONE => mult_by_add n thm
boehmes@31510
   516
      | SOME mth =>
boehmes@31510
   517
          let
boehmes@31510
   518
            val cv = mth |> Thm.cprop_of |> Drule.strip_imp_concl
boehmes@31510
   519
              |> Thm.dest_arg |> Thm.dest_arg1 |> Thm.dest_arg1
boehmes@31510
   520
            val T = #T (Thm.rep_cterm cv)
boehmes@31510
   521
          in
boehmes@31510
   522
            mth
boehmes@31510
   523
            |> Thm.instantiate ([], [(cv, num_of thy T n)])
boehmes@31510
   524
            |> rewrite_concl
boehmes@31510
   525
            |> discharge_prem
boehmes@31510
   526
            handle CTERM _ => mult_by_add n thm
boehmes@31510
   527
                 | THM _ => mult_by_add n thm
boehmes@31510
   528
          end);
nipkow@10691
   529
boehmes@31510
   530
    fun mult_thm (n, thm) =
boehmes@31510
   531
      if n = ~1 then thm RS LA_Logic.sym
boehmes@31510
   532
      else if n < 0 then mult (~n) thm RS LA_Logic.sym
boehmes@31510
   533
      else mult n thm;
boehmes@31510
   534
boehmes@31510
   535
    fun simp thm =
wenzelm@32091
   536
      let val thm' = trace_thm ctxt "Simplified:" (full_simplify simpset' thm)
boehmes@31510
   537
      in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end;
nipkow@6056
   538
wenzelm@32091
   539
    fun mk (Asm i) = trace_thm ctxt ("Asm " ^ string_of_int i) (nth asms i)
wenzelm@32091
   540
      | mk (Nat i) = trace_thm ctxt ("Nat " ^ string_of_int i) (LA_Logic.mk_nat_thm thy (nth atoms i))
wenzelm@32091
   541
      | mk (LessD j) = trace_thm ctxt "L" (hd ([mk j] RL lessD))
wenzelm@32091
   542
      | mk (NotLeD j) = trace_thm ctxt "NLe" (mk j RS LA_Logic.not_leD)
wenzelm@32091
   543
      | mk (NotLeDD j) = trace_thm ctxt "NLeD" (hd ([mk j RS LA_Logic.not_leD] RL lessD))
wenzelm@32091
   544
      | mk (NotLessD j) = trace_thm ctxt "NL" (mk j RS LA_Logic.not_lessD)
wenzelm@32091
   545
      | mk (Added (j1, j2)) = simp (trace_thm ctxt "+" (add_thms (mk j1) (mk j2)))
wenzelm@32091
   546
      | mk (Multiplied (n, j)) =
wenzelm@32091
   547
          (trace_msg ("*" ^ string_of_int n); trace_thm ctxt "*" (mult_thm (n, mk j)))
nipkow@5982
   548
wenzelm@27020
   549
  in
wenzelm@27020
   550
    let
wenzelm@27020
   551
      val _ = trace_msg "mkthm";
wenzelm@32091
   552
      val thm = trace_thm ctxt "Final thm:" (mk just);
wenzelm@27020
   553
      val fls = simplify simpset' thm;
wenzelm@32091
   554
      val _ = trace_thm ctxt "After simplification:" fls;
wenzelm@27020
   555
      val _ =
wenzelm@27020
   556
        if LA_Logic.is_False fls then ()
wenzelm@27020
   557
        else
boehmes@35872
   558
         (tracing (cat_lines
boehmes@35872
   559
           (["Assumptions:"] @ map (Display.string_of_thm ctxt) asms @ [""] @
boehmes@35872
   560
            ["Proved:", Display.string_of_thm ctxt fls, ""]));
boehmes@35872
   561
          warning "Linear arithmetic should have refuted the assumptions.\n\
boehmes@35872
   562
            \Please inform Tobias Nipkow.")
wenzelm@27020
   563
    in fls end
wenzelm@32091
   564
    handle FalseE thm => trace_thm ctxt "False reached early:" thm
wenzelm@27020
   565
  end;
wenzelm@27020
   566
nipkow@6056
   567
end;
nipkow@5982
   568
haftmann@23261
   569
fun coeff poly atom =
berghofe@26835
   570
  AList.lookup Pattern.aeconv poly atom |> the_default 0;
nipkow@10691
   571
nipkow@10691
   572
fun integ(rlhs,r,rel,rrhs,s,d) =
haftmann@17951
   573
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s
wenzelm@24630
   574
    val m = Integer.lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs))
wenzelm@22846
   575
    fun mult(t,r) =
haftmann@17951
   576
        let val (i,j) = Rat.quotient_of_rat r
paulson@15965
   577
        in (t,i * (m div j)) end
nipkow@12932
   578
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end
nipkow@10691
   579
haftmann@38052
   580
fun mklineq atoms =
webertj@20217
   581
  fn (item, k) =>
webertj@20217
   582
  let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item
nipkow@13498
   583
      val lhsa = map (coeff lhs) atoms
nipkow@13498
   584
      and rhsa = map (coeff rhs) atoms
haftmann@18330
   585
      val diff = map2 (curry (op -)) rhsa lhsa
nipkow@13498
   586
      val c = i-j
nipkow@13498
   587
      val just = Asm k
boehmes@31511
   588
      fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied(m,j))
nipkow@13498
   589
  in case rel of
nipkow@13498
   590
      "<="   => lineq(c,Le,diff,just)
nipkow@13498
   591
     | "~<=" => if discrete
nipkow@13498
   592
                then lineq(1-c,Le,map (op ~) diff,NotLeDD(just))
nipkow@13498
   593
                else lineq(~c,Lt,map (op ~) diff,NotLeD(just))
nipkow@13498
   594
     | "<"   => if discrete
nipkow@13498
   595
                then lineq(c+1,Le,diff,LessD(just))
nipkow@13498
   596
                else lineq(c,Lt,diff,just)
nipkow@13498
   597
     | "~<"  => lineq(~c,Le,map (op~) diff,NotLessD(just))
nipkow@13498
   598
     | "="   => lineq(c,Eq,diff,just)
wenzelm@22846
   599
     | _     => sys_error("mklineq" ^ rel)
nipkow@5982
   600
  end;
nipkow@5982
   601
nipkow@13498
   602
(* ------------------------------------------------------------------------- *)
nipkow@13498
   603
(* Print (counter) example                                                   *)
nipkow@13498
   604
(* ------------------------------------------------------------------------- *)
nipkow@13498
   605
nipkow@13498
   606
fun print_atom((a,d),r) =
haftmann@17951
   607
  let val (p,q) = Rat.quotient_of_rat r
wenzelm@24630
   608
      val s = if d then string_of_int p else
nipkow@13498
   609
              if p = 0 then "0"
wenzelm@24630
   610
              else string_of_int p ^ "/" ^ string_of_int q
nipkow@13498
   611
  in a ^ " = " ^ s end;
nipkow@13498
   612
wenzelm@19049
   613
fun produce_ex sds =
haftmann@17496
   614
  curry (op ~~) sds
haftmann@17496
   615
  #> map print_atom
haftmann@17496
   616
  #> commas
webertj@23197
   617
  #> curry (op ^) "Counterexample (possibly spurious):\n";
nipkow@13498
   618
wenzelm@24076
   619
fun trace_ex ctxt params atoms discr n (hist: history) =
webertj@20217
   620
  case hist of
webertj@20217
   621
    [] => ()
webertj@20217
   622
  | (v, lineqs) :: hist' =>
wenzelm@24076
   623
      let
wenzelm@24076
   624
        val frees = map Free params
wenzelm@24920
   625
        fun show_term t = Syntax.string_of_term ctxt (subst_bounds (frees, t))
wenzelm@24076
   626
        val start =
wenzelm@24076
   627
          if v = ~1 then (hist', findex0 discr n lineqs)
haftmann@22950
   628
          else (hist, replicate n Rat.zero)
wenzelm@24076
   629
        val ex = SOME (produce_ex (map show_term atoms ~~ discr)
wenzelm@24076
   630
            (uncurry (fold (findex1 discr)) start))
webertj@20217
   631
          handle NoEx => NONE
wenzelm@24076
   632
      in
wenzelm@24076
   633
        case ex of
haftmann@30687
   634
          SOME s => (warning "Linear arithmetic failed - see trace for a counterexample."; tracing s)
haftmann@30687
   635
        | NONE => warning "Linear arithmetic failed"
wenzelm@24076
   636
      end;
nipkow@13498
   637
webertj@20217
   638
(* ------------------------------------------------------------------------- *)
webertj@20217
   639
webertj@20268
   640
fun mknat (pTs : typ list) (ixs : int list) (atom : term, i : int) : lineq option =
webertj@20217
   641
  if LA_Logic.is_nat (pTs, atom)
nipkow@6056
   642
  then let val l = map (fn j => if j=i then 1 else 0) ixs
webertj@20217
   643
       in SOME (Lineq (0, Le, l, Nat i)) end
webertj@20217
   644
  else NONE;
nipkow@6056
   645
nipkow@13186
   646
(* This code is tricky. It takes a list of premises in the order they occur
skalberg@15531
   647
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical
skalberg@15531
   648
ones as NONE. Going through the premises, each numeric one is converted into
nipkow@13186
   649
a Lineq. The tricky bit is to convert ~= which is split into two cases < and
nipkow@13498
   650
>. Thus split_items returns a list of equation systems. This may blow up if
nipkow@13186
   651
there are many ~=, but in practice it does not seem to happen. The really
nipkow@13186
   652
tricky bit is to arrange the order of the cases such that they coincide with
nipkow@13186
   653
the order in which the cases are in the end generated by the tactic that
nipkow@13186
   654
applies the generated refutation thms (see function 'refute_tac').
nipkow@13186
   655
nipkow@13186
   656
For variables n of type nat, a constraint 0 <= n is added.
nipkow@13186
   657
*)
webertj@20217
   658
webertj@20217
   659
(* FIXME: To optimize, the splitting of cases and the search for refutations *)
webertj@20276
   660
(*        could be intertwined: separate the first (fully split) case,       *)
webertj@20217
   661
(*        refute it, continue with splitting and refuting.  Terminate with   *)
webertj@20217
   662
(*        failure as soon as a case could not be refuted; i.e. delay further *)
webertj@20217
   663
(*        splitting until after a refutation for other cases has been found. *)
webertj@20217
   664
webertj@30406
   665
fun split_items ctxt do_pre split_neq (Ts, terms) : (typ list * (LA_Data.decomp * int) list) list =
webertj@20276
   666
let
webertj@20276
   667
  (* splits inequalities '~=' into '<' and '>'; this corresponds to *)
webertj@20276
   668
  (* 'REPEAT_DETERM (eresolve_tac neqE i)' at the theorem/tactic    *)
webertj@20276
   669
  (* level                                                          *)
webertj@20276
   670
  (* FIXME: this is currently sensitive to the order of theorems in *)
webertj@20276
   671
  (*        neqE:  The theorem for type "nat" must come first.  A   *)
webertj@20276
   672
  (*        better (i.e. less likely to break when neqE changes)    *)
webertj@20276
   673
  (*        implementation should *test* which theorem from neqE    *)
webertj@20276
   674
  (*        can be applied, and split the premise accordingly.      *)
wenzelm@26945
   675
  fun elim_neq (ineqs : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   676
               (LA_Data.decomp option * bool) list list =
webertj@20276
   677
  let
wenzelm@26945
   678
    fun elim_neq' nat_only ([] : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   679
                  (LA_Data.decomp option * bool) list list =
webertj@20276
   680
          [[]]
webertj@20276
   681
      | elim_neq' nat_only ((NONE, is_nat) :: ineqs) =
webertj@20276
   682
          map (cons (NONE, is_nat)) (elim_neq' nat_only ineqs)
webertj@20276
   683
      | elim_neq' nat_only ((ineq as (SOME (l, i, rel, r, j, d), is_nat)) :: ineqs) =
webertj@20276
   684
          if rel = "~=" andalso (not nat_only orelse is_nat) then
webertj@20276
   685
            (* [| ?l ~= ?r; ?l < ?r ==> ?R; ?r < ?l ==> ?R |] ==> ?R *)
webertj@20276
   686
            elim_neq' nat_only (ineqs @ [(SOME (l, i, "<", r, j, d), is_nat)]) @
webertj@20276
   687
            elim_neq' nat_only (ineqs @ [(SOME (r, j, "<", l, i, d), is_nat)])
webertj@20276
   688
          else
webertj@20276
   689
            map (cons ineq) (elim_neq' nat_only ineqs)
webertj@20276
   690
  in
webertj@20276
   691
    ineqs |> elim_neq' true
wenzelm@26945
   692
          |> maps (elim_neq' false)
webertj@20276
   693
  end
nipkow@13464
   694
webertj@30406
   695
  fun ignore_neq (NONE, bool) = (NONE, bool)
webertj@30406
   696
    | ignore_neq (ineq as SOME (_, _, rel, _, _, _), bool) =
webertj@30406
   697
      if rel = "~=" then (NONE, bool) else (ineq, bool)
webertj@30406
   698
webertj@20276
   699
  fun number_hyps _ []             = []
webertj@20276
   700
    | number_hyps n (NONE::xs)     = number_hyps (n+1) xs
webertj@20276
   701
    | number_hyps n ((SOME x)::xs) = (x, n) :: number_hyps (n+1) xs
webertj@20276
   702
webertj@20276
   703
  val result = (Ts, terms)
webertj@20276
   704
    |> (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   705
       (if do_pre then LA_Data.pre_decomp ctxt else Library.single)
webertj@23195
   706
    |> tap (fn subgoals => trace_msg ("Preprocessing yields " ^
webertj@23195
   707
         string_of_int (length subgoals) ^ " subgoal(s) total."))
wenzelm@22846
   708
    |> (* produce the internal encoding of (in-)equalities *)
wenzelm@24076
   709
       map (apsnd (map (fn t => (LA_Data.decomp ctxt t, LA_Data.domain_is_nat t))))
webertj@20276
   710
    |> (* splitting of inequalities *)
webertj@30406
   711
       map (apsnd (if split_neq then elim_neq else
webertj@30406
   712
                     Library.single o map ignore_neq))
wenzelm@22846
   713
    |> maps (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals)
webertj@20276
   714
    |> (* numbering of hypotheses, ignoring irrelevant ones *)
webertj@20276
   715
       map (apsnd (number_hyps 0))
webertj@23195
   716
in
webertj@23195
   717
  trace_msg ("Splitting of inequalities yields " ^
webertj@23195
   718
    string_of_int (length result) ^ " subgoal(s) total.");
webertj@23195
   719
  result
webertj@23195
   720
end;
nipkow@13464
   721
wenzelm@33245
   722
fun add_datoms ((lhs,_,_,rhs,_,d) : LA_Data.decomp, _) (dats : (bool * term) list) =
berghofe@26835
   723
  union_bterm (map (pair d o fst) lhs) (union_bterm (map (pair d o fst) rhs) dats);
nipkow@13498
   724
wenzelm@26945
   725
fun discr (initems : (LA_Data.decomp * int) list) : bool list =
wenzelm@33245
   726
  map fst (fold add_datoms initems []);
webertj@20217
   727
wenzelm@24076
   728
fun refutes ctxt params show_ex :
wenzelm@26945
   729
    (typ list * (LA_Data.decomp * int) list) list -> injust list -> injust list option =
wenzelm@26945
   730
  let
wenzelm@26945
   731
    fun refute ((Ts, initems : (LA_Data.decomp * int) list) :: initemss) (js: injust list) =
wenzelm@26945
   732
          let
boehmes@31510
   733
            val atoms = atoms_of (map fst initems)
wenzelm@26945
   734
            val n = length atoms
haftmann@38052
   735
            val mkleq = mklineq atoms
wenzelm@26945
   736
            val ixs = 0 upto (n - 1)
wenzelm@26945
   737
            val iatoms = atoms ~~ ixs
wenzelm@32952
   738
            val natlineqs = map_filter (mknat Ts ixs) iatoms
wenzelm@26945
   739
            val ineqs = map mkleq initems @ natlineqs
wenzelm@26945
   740
          in case elim (ineqs, []) of
wenzelm@26945
   741
               Success j =>
wenzelm@26945
   742
                 (trace_msg ("Contradiction! (" ^ string_of_int (length js + 1) ^ ")");
wenzelm@26945
   743
                  refute initemss (js @ [j]))
wenzelm@26945
   744
             | Failure hist =>
wenzelm@26945
   745
                 (if not show_ex then ()
wenzelm@26945
   746
                  else
wenzelm@26945
   747
                    let
wenzelm@26945
   748
                      val (param_names, ctxt') = ctxt |> Variable.variant_fixes (map fst params)
wenzelm@26945
   749
                      val (more_names, ctxt'') = ctxt' |> Variable.variant_fixes
wenzelm@26945
   750
                        (Name.invents (Variable.names_of ctxt') Name.uu (length Ts - length params))
wenzelm@26945
   751
                      val params' = (more_names @ param_names) ~~ Ts
wenzelm@26945
   752
                    in
wenzelm@26945
   753
                      trace_ex ctxt'' params' atoms (discr initems) n hist
wenzelm@26945
   754
                    end; NONE)
wenzelm@26945
   755
          end
wenzelm@26945
   756
      | refute [] js = SOME js
wenzelm@26945
   757
  in refute end;
nipkow@5982
   758
webertj@30406
   759
fun refute ctxt params show_ex do_pre split_neq terms : injust list option =
webertj@30406
   760
  refutes ctxt params show_ex (split_items ctxt do_pre split_neq
webertj@30406
   761
    (map snd params, terms)) [];
webertj@20254
   762
haftmann@22950
   763
fun count P xs = length (filter P xs);
webertj@20254
   764
webertj@30406
   765
fun prove ctxt params show_ex do_pre Hs concl : bool * injust list option =
webertj@20254
   766
  let
webertj@23190
   767
    val _ = trace_msg "prove:"
webertj@20254
   768
    (* append the negated conclusion to 'Hs' -- this corresponds to     *)
webertj@20254
   769
    (* 'DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i)' at the *)
webertj@20254
   770
    (* theorem/tactic level                                             *)
webertj@20254
   771
    val Hs' = Hs @ [LA_Logic.neg_prop concl]
webertj@20254
   772
    fun is_neq NONE                 = false
webertj@20254
   773
      | is_neq (SOME (_,_,r,_,_,_)) = (r = "~=")
webertj@30406
   774
    val neq_limit = Config.get ctxt LA_Data.fast_arith_neq_limit
webertj@30406
   775
    val split_neq = count is_neq (map (LA_Data.decomp ctxt) Hs') <= neq_limit
webertj@20254
   776
  in
webertj@30406
   777
    if split_neq then ()
wenzelm@24076
   778
    else
webertj@30406
   779
      trace_msg ("fast_arith_neq_limit exceeded (current value is " ^
webertj@30406
   780
        string_of_int neq_limit ^ "), ignoring all inequalities");
webertj@30406
   781
    (split_neq, refute ctxt params show_ex do_pre split_neq Hs')
webertj@23190
   782
  end handle TERM ("neg_prop", _) =>
webertj@23190
   783
    (* since no meta-logic negation is available, we can only fail if   *)
webertj@23190
   784
    (* the conclusion is not of the form 'Trueprop $ _' (simply         *)
webertj@23190
   785
    (* dropping the conclusion doesn't work either, because even        *)
webertj@23190
   786
    (* 'False' does not imply arbitrary 'concl::prop')                  *)
webertj@30406
   787
    (trace_msg "prove failed (cannot negate conclusion).";
webertj@30406
   788
      (false, NONE));
webertj@20217
   789
webertj@30406
   790
fun refute_tac ss (i, split_neq, justs) =
nipkow@6074
   791
  fn state =>
wenzelm@24076
   792
    let
wenzelm@24076
   793
      val ctxt = Simplifier.the_context ss;
wenzelm@32091
   794
      val _ = trace_thm ctxt
wenzelm@32091
   795
        ("refute_tac (on subgoal " ^ string_of_int i ^ ", with " ^
wenzelm@32091
   796
          string_of_int (length justs) ^ " justification(s)):") state
wenzelm@24076
   797
      val {neqE, ...} = get_data ctxt;
wenzelm@24076
   798
      fun just1 j =
wenzelm@24076
   799
        (* eliminate inequalities *)
webertj@30406
   800
        (if split_neq then
webertj@30406
   801
          REPEAT_DETERM (eresolve_tac neqE i)
webertj@30406
   802
        else
webertj@30406
   803
          all_tac) THEN
wenzelm@32091
   804
          PRIMITIVE (trace_thm ctxt "State after neqE:") THEN
wenzelm@24076
   805
          (* use theorems generated from the actual justifications *)
wenzelm@32283
   806
          Subgoal.FOCUS (fn {prems, ...} => rtac (mkthm ss prems j) 1) ctxt i
wenzelm@24076
   807
    in
wenzelm@24076
   808
      (* rewrite "[| A1; ...; An |] ==> B" to "[| A1; ...; An; ~B |] ==> False" *)
wenzelm@24076
   809
      DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i) THEN
wenzelm@24076
   810
      (* user-defined preprocessing of the subgoal *)
wenzelm@35230
   811
      DETERM (LA_Data.pre_tac ss i) THEN
wenzelm@32091
   812
      PRIMITIVE (trace_thm ctxt "State after pre_tac:") THEN
wenzelm@24076
   813
      (* prove every resulting subgoal, using its justification *)
wenzelm@24076
   814
      EVERY (map just1 justs)
webertj@20217
   815
    end  state;
nipkow@6074
   816
nipkow@5982
   817
(*
nipkow@5982
   818
Fast but very incomplete decider. Only premises and conclusions
nipkow@5982
   819
that are already (negated) (in)equations are taken into account.
nipkow@5982
   820
*)
wenzelm@24076
   821
fun simpset_lin_arith_tac ss show_ex = SUBGOAL (fn (A, i) =>
wenzelm@24076
   822
  let
wenzelm@24076
   823
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   824
    val params = rev (Logic.strip_params A)
wenzelm@24076
   825
    val Hs = Logic.strip_assums_hyp A
wenzelm@24076
   826
    val concl = Logic.strip_assums_concl A
wenzelm@24076
   827
    val _ = trace_term ctxt ("Trying to refute subgoal " ^ string_of_int i) A
wenzelm@24076
   828
  in
wenzelm@24076
   829
    case prove ctxt params show_ex true Hs concl of
webertj@30406
   830
      (_, NONE) => (trace_msg "Refutation failed."; no_tac)
webertj@30406
   831
    | (split_neq, SOME js) => (trace_msg "Refutation succeeded.";
webertj@30406
   832
                               refute_tac ss (i, split_neq, js))
wenzelm@24076
   833
  end);
nipkow@5982
   834
wenzelm@24076
   835
fun cut_lin_arith_tac ss =
wenzelm@24076
   836
  cut_facts_tac (Simplifier.prems_of_ss ss) THEN'
wenzelm@24076
   837
  simpset_lin_arith_tac ss false;
wenzelm@17613
   838
wenzelm@24076
   839
fun lin_arith_tac ctxt =
wenzelm@24076
   840
  simpset_lin_arith_tac (Simplifier.context ctxt Simplifier.empty_ss);
wenzelm@24076
   841
wenzelm@24076
   842
nipkow@5982
   843
nipkow@13186
   844
(** Forward proof from theorems **)
nipkow@13186
   845
webertj@20433
   846
(* More tricky code. Needs to arrange the proofs of the multiple cases (due
webertj@20433
   847
to splits of ~= premises) such that it coincides with the order of the cases
webertj@20433
   848
generated by function split_items. *)
webertj@20433
   849
webertj@20433
   850
datatype splittree = Tip of thm list
webertj@20433
   851
                   | Spl of thm * cterm * splittree * cterm * splittree;
webertj@20433
   852
webertj@20433
   853
(* "(ct1 ==> ?R) ==> (ct2 ==> ?R) ==> ?R" is taken to (ct1, ct2) *)
webertj@20433
   854
webertj@20433
   855
fun extract (imp : cterm) : cterm * cterm =
webertj@20433
   856
let val (Il, r)    = Thm.dest_comb imp
webertj@20433
   857
    val (_, imp1)  = Thm.dest_comb Il
webertj@20433
   858
    val (Ict1, _)  = Thm.dest_comb imp1
webertj@20433
   859
    val (_, ct1)   = Thm.dest_comb Ict1
webertj@20433
   860
    val (Ir, _)    = Thm.dest_comb r
webertj@20433
   861
    val (_, Ict2r) = Thm.dest_comb Ir
webertj@20433
   862
    val (Ict2, _)  = Thm.dest_comb Ict2r
webertj@20433
   863
    val (_, ct2)   = Thm.dest_comb Ict2
webertj@20433
   864
in (ct1, ct2) end;
webertj@20433
   865
wenzelm@24076
   866
fun splitasms ctxt (asms : thm list) : splittree =
wenzelm@24076
   867
let val {neqE, ...} = get_data ctxt
hoelzl@35693
   868
    fun elim_neq [] (asms', []) = Tip (rev asms')
hoelzl@35693
   869
      | elim_neq [] (asms', asms) = Tip (rev asms' @ asms)
hoelzl@35693
   870
      | elim_neq (neq :: neqs) (asms', []) = elim_neq neqs ([],rev asms')
hoelzl@35693
   871
      | elim_neq (neqs as (neq :: _)) (asms', asm::asms) =
hoelzl@35693
   872
      (case get_first (fn th => SOME (asm COMP th) handle THM _ => NONE) [neq] of
webertj@20433
   873
        SOME spl =>
webertj@20433
   874
          let val (ct1, ct2) = extract (cprop_of spl)
wenzelm@36945
   875
              val thm1 = Thm.assume ct1
wenzelm@36945
   876
              val thm2 = Thm.assume ct2
hoelzl@35693
   877
          in Spl (spl, ct1, elim_neq neqs (asms', asms@[thm1]),
hoelzl@35693
   878
            ct2, elim_neq neqs (asms', asms@[thm2]))
webertj@20433
   879
          end
hoelzl@35693
   880
      | NONE => elim_neq neqs (asm::asms', asms))
hoelzl@35693
   881
in elim_neq neqE ([], asms) end;
webertj@20433
   882
wenzelm@24076
   883
fun fwdproof ss (Tip asms : splittree) (j::js : injust list) = (mkthm ss asms j, js)
wenzelm@24076
   884
  | fwdproof ss (Spl (thm, ct1, tree1, ct2, tree2)) js =
wenzelm@24076
   885
      let
wenzelm@24076
   886
        val (thm1, js1) = fwdproof ss tree1 js
wenzelm@24076
   887
        val (thm2, js2) = fwdproof ss tree2 js1
wenzelm@36945
   888
        val thm1' = Thm.implies_intr ct1 thm1
wenzelm@36945
   889
        val thm2' = Thm.implies_intr ct2 thm2
wenzelm@24076
   890
      in (thm2' COMP (thm1' COMP thm), js2) end;
wenzelm@24076
   891
      (* FIXME needs handle THM _ => NONE ? *)
webertj@20433
   892
webertj@30406
   893
fun prover ss thms Tconcl (js : injust list) split_neq pos : thm option =
wenzelm@24076
   894
  let
wenzelm@24076
   895
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   896
    val thy = ProofContext.theory_of ctxt
wenzelm@24076
   897
    val nTconcl = LA_Logic.neg_prop Tconcl
wenzelm@24076
   898
    val cnTconcl = cterm_of thy nTconcl
wenzelm@36945
   899
    val nTconclthm = Thm.assume cnTconcl
webertj@30406
   900
    val tree = (if split_neq then splitasms ctxt else Tip) (thms @ [nTconclthm])
wenzelm@24076
   901
    val (Falsethm, _) = fwdproof ss tree js
wenzelm@24076
   902
    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
wenzelm@36945
   903
    val concl = Thm.implies_intr cnTconcl Falsethm COMP contr
wenzelm@32091
   904
  in SOME (trace_thm ctxt "Proved by lin. arith. prover:" (LA_Logic.mk_Eq concl)) end
wenzelm@24076
   905
  (*in case concl contains ?-var, which makes assume fail:*)   (* FIXME Variable.import_terms *)
wenzelm@24076
   906
  handle THM _ => NONE;
nipkow@13186
   907
nipkow@13186
   908
(* PRE: concl is not negated!
nipkow@13186
   909
   This assumption is OK because
wenzelm@24076
   910
   1. lin_arith_simproc tries both to prove and disprove concl and
wenzelm@24076
   911
   2. lin_arith_simproc is applied by the Simplifier which
nipkow@13186
   912
      dives into terms and will thus try the non-negated concl anyway.
nipkow@13186
   913
*)
wenzelm@24076
   914
fun lin_arith_simproc ss concl =
wenzelm@24076
   915
  let
wenzelm@24076
   916
    val ctxt = Simplifier.the_context ss
wenzelm@26945
   917
    val thms = maps LA_Logic.atomize (Simplifier.prems_of_ss ss)
wenzelm@24076
   918
    val Hs = map Thm.prop_of thms
nipkow@6102
   919
    val Tconcl = LA_Logic.mk_Trueprop concl
wenzelm@24076
   920
  in
wenzelm@24076
   921
    case prove ctxt [] false false Hs Tconcl of (* concl provable? *)
webertj@30406
   922
      (split_neq, SOME js) => prover ss thms Tconcl js split_neq true
webertj@30406
   923
    | (_, NONE) =>
wenzelm@24076
   924
        let val nTconcl = LA_Logic.neg_prop Tconcl in
wenzelm@24076
   925
          case prove ctxt [] false false Hs nTconcl of (* ~concl provable? *)
webertj@30406
   926
            (split_neq, SOME js) => prover ss thms nTconcl js split_neq false
webertj@30406
   927
          | (_, NONE) => NONE
wenzelm@24076
   928
        end
wenzelm@24076
   929
  end;
nipkow@6074
   930
nipkow@6074
   931
end;