src/Provers/Arith/fast_lin_arith.ML
author wenzelm
Fri Apr 10 11:31:10 2015 +0200 (2015-04-10)
changeset 59996 4dca48557921
parent 59656 ddc5411c1cb9
child 60348 e66830e878e3
permissions -rw-r--r--
tuned signature;
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@46709
     4
A generic linear arithmetic package.
nipkow@6102
     5
wenzelm@24076
     6
Only take premises and conclusions into account that are already
wenzelm@24076
     7
(negated) (in)equations. lin_arith_simproc tries to prove or disprove
wenzelm@24076
     8
the term.
nipkow@5982
     9
*)
nipkow@5982
    10
nipkow@5982
    11
(*** Data needed for setting up the linear arithmetic package ***)
nipkow@5982
    12
nipkow@6102
    13
signature LIN_ARITH_LOGIC =
nipkow@6102
    14
sig
webertj@20276
    15
  val conjI       : thm  (* P ==> Q ==> P & Q *)
webertj@20276
    16
  val ccontr      : thm  (* (~ P ==> False) ==> P *)
webertj@20276
    17
  val notI        : thm  (* (P ==> False) ==> ~ P *)
webertj@20276
    18
  val not_lessD   : thm  (* ~(m < n) ==> n <= m *)
webertj@20276
    19
  val not_leD     : thm  (* ~(m <= n) ==> n < m *)
webertj@20276
    20
  val sym         : thm  (* x = y ==> y = x *)
boehmes@31510
    21
  val trueI       : thm  (* True *)
webertj@20276
    22
  val mk_Eq       : thm -> thm
webertj@20276
    23
  val atomize     : thm -> thm list
webertj@20276
    24
  val mk_Trueprop : term -> term
webertj@20276
    25
  val neg_prop    : term -> term
webertj@20276
    26
  val is_False    : thm -> bool
webertj@20276
    27
  val is_nat      : typ list * term -> bool
webertj@20276
    28
  val mk_nat_thm  : theory -> term -> thm
nipkow@6102
    29
end;
nipkow@6102
    30
(*
nipkow@6102
    31
mk_Eq(~in) = `in == False'
nipkow@6102
    32
mk_Eq(in) = `in == True'
nipkow@6102
    33
where `in' is an (in)equality.
nipkow@6102
    34
webertj@23190
    35
neg_prop(t) = neg  if t is wrapped up in Trueprop and neg is the
webertj@23190
    36
  (logically) negated version of t (again wrapped up in Trueprop),
webertj@23190
    37
  where the negation of a negative term is the term itself (no
webertj@23190
    38
  double negation!); raises TERM ("neg_prop", [t]) if t is not of
webertj@23190
    39
  the form 'Trueprop $ _'
nipkow@6128
    40
nipkow@6128
    41
is_nat(parameter-types,t) =  t:nat
nipkow@6128
    42
mk_nat_thm(t) = "0 <= t"
nipkow@6102
    43
*)
nipkow@6102
    44
nipkow@5982
    45
signature LIN_ARITH_DATA =
nipkow@5982
    46
sig
wenzelm@24076
    47
  (*internal representation of linear (in-)equations:*)
wenzelm@26945
    48
  type decomp = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool
wenzelm@26945
    49
  val decomp: Proof.context -> term -> decomp option
wenzelm@24076
    50
  val domain_is_nat: term -> bool
wenzelm@24076
    51
wenzelm@24076
    52
  (*preprocessing, performed on a representation of subgoals as list of premises:*)
wenzelm@24076
    53
  val pre_decomp: Proof.context -> typ list * term list -> (typ list * term list) list
wenzelm@24076
    54
wenzelm@24076
    55
  (*preprocessing, performed on the goal -- must do the same as 'pre_decomp':*)
wenzelm@51717
    56
  val pre_tac: Proof.context -> int -> tactic
wenzelm@24076
    57
wenzelm@24076
    58
  (*the limit on the number of ~= allowed; because each ~= is split
wenzelm@24076
    59
    into two cases, this can lead to an explosion*)
wenzelm@44654
    60
  val neq_limit: int Config.T
boehmes@43607
    61
wenzelm@44654
    62
  val trace: bool 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@51717
    81
Simpset 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
wenzelm@51717
    88
  val prems_lin_arith_tac: Proof.context -> int -> tactic
wenzelm@59656
    89
  val lin_arith_tac: Proof.context -> int -> tactic
wenzelm@51717
    90
  val lin_arith_simproc: Proof.context -> term -> thm option
wenzelm@38763
    91
  val map_data:
wenzelm@38763
    92
    ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
wenzelm@51717
    93
      lessD: thm list, neqE: thm list, simpset: simpset,
wenzelm@59996
    94
      number_of: (Proof.context -> typ -> int -> cterm) option} ->
wenzelm@38763
    95
     {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
wenzelm@51717
    96
      lessD: thm list, neqE: thm list, simpset: simpset,
wenzelm@59996
    97
      number_of: (Proof.context -> typ -> int -> cterm) option}) ->
wenzelm@38763
    98
      Context.generic -> Context.generic
wenzelm@38762
    99
  val add_inj_thms: thm list -> Context.generic -> Context.generic
wenzelm@38762
   100
  val add_lessD: thm -> Context.generic -> Context.generic
wenzelm@38762
   101
  val add_simps: thm list -> Context.generic -> Context.generic
wenzelm@38762
   102
  val add_simprocs: simproc list -> Context.generic -> Context.generic
wenzelm@59996
   103
  val set_number_of: (Proof.context -> typ -> int -> cterm) -> Context.generic -> Context.generic
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
wenzelm@33519
   113
structure Data = Generic_Data
wenzelm@22846
   114
(
wenzelm@24076
   115
  type T =
wenzelm@24076
   116
   {add_mono_thms: thm list,
wenzelm@24076
   117
    mult_mono_thms: thm list,
wenzelm@24076
   118
    inj_thms: thm list,
wenzelm@24076
   119
    lessD: thm list,
wenzelm@24076
   120
    neqE: thm list,
wenzelm@51717
   121
    simpset: simpset,
wenzelm@59996
   122
    number_of: (Proof.context -> typ -> int -> cterm) option};
wenzelm@9420
   123
wenzelm@38763
   124
  val empty : T =
wenzelm@38763
   125
   {add_mono_thms = [], mult_mono_thms = [], inj_thms = [],
wenzelm@51717
   126
    lessD = [], neqE = [], simpset = empty_ss,
wenzelm@38763
   127
    number_of = NONE};
wenzelm@16458
   128
  val extend = I;
wenzelm@33519
   129
  fun merge
wenzelm@38763
   130
    ({add_mono_thms = add_mono_thms1, mult_mono_thms = mult_mono_thms1, inj_thms = inj_thms1,
wenzelm@38763
   131
      lessD = lessD1, neqE = neqE1, simpset = simpset1, number_of = number_of1},
wenzelm@38763
   132
     {add_mono_thms = add_mono_thms2, mult_mono_thms = mult_mono_thms2, inj_thms = inj_thms2,
wenzelm@38763
   133
      lessD = lessD2, neqE = neqE2, simpset = simpset2, number_of = number_of2}) : T =
wenzelm@24039
   134
    {add_mono_thms = Thm.merge_thms (add_mono_thms1, add_mono_thms2),
wenzelm@24039
   135
     mult_mono_thms = Thm.merge_thms (mult_mono_thms1, mult_mono_thms2),
wenzelm@24039
   136
     inj_thms = Thm.merge_thms (inj_thms1, inj_thms2),
wenzelm@24039
   137
     lessD = Thm.merge_thms (lessD1, lessD2),
wenzelm@24039
   138
     neqE = Thm.merge_thms (neqE1, neqE2),
wenzelm@51717
   139
     simpset = merge_ss (simpset1, simpset2),
wenzelm@41493
   140
     number_of = merge_options (number_of1, number_of2)};
wenzelm@22846
   141
);
wenzelm@9420
   142
wenzelm@9420
   143
val map_data = Data.map;
wenzelm@24076
   144
val get_data = Data.get o Context.Proof;
wenzelm@9420
   145
wenzelm@38762
   146
fun map_inj_thms f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
wenzelm@38762
   147
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = f inj_thms,
wenzelm@38762
   148
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = number_of};
wenzelm@38762
   149
wenzelm@38762
   150
fun map_lessD 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 = inj_thms,
wenzelm@38762
   152
    lessD = f lessD, neqE = neqE, simpset = simpset, number_of = number_of};
wenzelm@38762
   153
wenzelm@51717
   154
fun map_simpset f context =
wenzelm@51717
   155
  map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =>
wenzelm@51717
   156
    {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
wenzelm@51717
   157
      lessD = lessD, neqE = neqE, simpset = simpset_map (Context.proof_of context) f simpset,
wenzelm@51717
   158
      number_of = number_of}) context;
wenzelm@38762
   159
wenzelm@38762
   160
fun add_inj_thms thms = map_data (map_inj_thms (append thms));
wenzelm@38762
   161
fun add_lessD thm = map_data (map_lessD (fn thms => thms @ [thm]));
wenzelm@51717
   162
fun add_simps thms = map_simpset (fn ctxt => ctxt addsimps thms);
wenzelm@51717
   163
fun add_simprocs procs = map_simpset (fn ctxt => ctxt addsimprocs procs);
wenzelm@38762
   164
wenzelm@38763
   165
fun set_number_of f =
wenzelm@38763
   166
  map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, ...} =>
wenzelm@38763
   167
   {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
wenzelm@38763
   168
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = SOME f});
wenzelm@38763
   169
wenzelm@38763
   170
fun number_of ctxt =
wenzelm@38763
   171
  (case Data.get (Context.Proof ctxt) of
wenzelm@59996
   172
    {number_of = SOME f, ...} => f ctxt
wenzelm@38763
   173
  | _ => fn _ => fn _ => raise CTERM ("number_of", []));
wenzelm@38763
   174
wenzelm@9420
   175
wenzelm@9420
   176
nipkow@5982
   177
(*** A fast decision procedure ***)
nipkow@5982
   178
(*** Code ported from HOL Light ***)
nipkow@6056
   179
(* possible optimizations:
nipkow@6056
   180
   use (var,coeff) rep or vector rep  tp save space;
nipkow@6056
   181
   treat non-negative atoms separately rather than adding 0 <= atom
nipkow@6056
   182
*)
nipkow@5982
   183
nipkow@5982
   184
datatype lineq_type = Eq | Le | Lt;
nipkow@5982
   185
nipkow@6056
   186
datatype injust = Asm of int
nipkow@6056
   187
                | Nat of int (* index of atom *)
nipkow@6128
   188
                | LessD of injust
nipkow@6128
   189
                | NotLessD of injust
nipkow@6128
   190
                | NotLeD of injust
nipkow@7551
   191
                | NotLeDD of injust
wenzelm@24630
   192
                | Multiplied of int * injust
nipkow@5982
   193
                | Added of injust * injust;
nipkow@5982
   194
wenzelm@24630
   195
datatype lineq = Lineq of int * lineq_type * int list * injust;
nipkow@5982
   196
nipkow@13498
   197
(* ------------------------------------------------------------------------- *)
nipkow@13498
   198
(* Finding a (counter) example from the trace of a failed elimination        *)
nipkow@13498
   199
(* ------------------------------------------------------------------------- *)
nipkow@13498
   200
(* Examples are represented as rational numbers,                             *)
nipkow@13498
   201
(* Dont blame John Harrison for this code - it is entirely mine. TN          *)
nipkow@13498
   202
nipkow@13498
   203
exception NoEx;
nipkow@13498
   204
nipkow@14372
   205
(* Coding: (i,true,cs) means i <= cs and (i,false,cs) means i < cs.
nipkow@14372
   206
   In general, true means the bound is included, false means it is excluded.
nipkow@14372
   207
   Need to know if it is a lower or upper bound for unambiguous interpretation!
nipkow@14372
   208
*)
nipkow@14372
   209
haftmann@22950
   210
fun elim_eqns (Lineq (i, Le, cs, _)) = [(i, true, cs)]
haftmann@22950
   211
  | elim_eqns (Lineq (i, Eq, cs, _)) = [(i, true, cs),(~i, true, map ~ cs)]
haftmann@22950
   212
  | elim_eqns (Lineq (i, Lt, cs, _)) = [(i, false, cs)];
nipkow@13498
   213
nipkow@13498
   214
(* PRE: ex[v] must be 0! *)
wenzelm@24630
   215
fun eval ex v (a, le, cs) =
haftmann@22950
   216
  let
haftmann@22950
   217
    val rs = map Rat.rat_of_int cs;
haftmann@22950
   218
    val rsum = fold2 (Rat.add oo Rat.mult) rs ex Rat.zero;
haftmann@23063
   219
  in (Rat.mult (Rat.add (Rat.rat_of_int a) (Rat.neg rsum)) (Rat.inv (nth rs v)), le) end;
haftmann@23063
   220
(* If nth rs v < 0, le should be negated.
nipkow@14372
   221
   Instead this swap is taken into account in ratrelmin2.
nipkow@14372
   222
*)
nipkow@13498
   223
haftmann@22950
   224
fun ratrelmin2 (x as (r, ler), y as (s, les)) =
haftmann@23520
   225
  case Rat.ord (r, s)
haftmann@22950
   226
   of EQUAL => (r, (not ler) andalso (not les))
haftmann@22950
   227
    | LESS => x
haftmann@22950
   228
    | GREATER => y;
haftmann@22950
   229
haftmann@22950
   230
fun ratrelmax2 (x as (r, ler), y as (s, les)) =
haftmann@23520
   231
  case Rat.ord (r, s)
haftmann@22950
   232
   of EQUAL => (r, ler andalso les)
haftmann@22950
   233
    | LESS => y
haftmann@22950
   234
    | GREATER => x;
nipkow@13498
   235
nipkow@14372
   236
val ratrelmin = foldr1 ratrelmin2;
nipkow@14372
   237
val ratrelmax = foldr1 ratrelmax2;
nipkow@13498
   238
haftmann@22950
   239
fun ratexact up (r, exact) =
nipkow@14372
   240
  if exact then r else
haftmann@22950
   241
  let
haftmann@38052
   242
    val (_, q) = Rat.quotient_of_rat r;
haftmann@22950
   243
    val nth = Rat.inv (Rat.rat_of_int q);
haftmann@22950
   244
  in Rat.add r (if up then nth else Rat.neg nth) end;
nipkow@14372
   245
haftmann@22950
   246
fun ratmiddle (r, s) = Rat.mult (Rat.add r s) (Rat.inv Rat.two);
nipkow@14372
   247
webertj@20217
   248
fun choose2 d ((lb, exactl), (ub, exactu)) =
haftmann@23520
   249
  let val ord = Rat.sign lb in
haftmann@22950
   250
  if (ord = LESS orelse exactl) andalso (ord = GREATER orelse exactu)
haftmann@22950
   251
    then Rat.zero
haftmann@22950
   252
    else if not d then
haftmann@22950
   253
      if ord = GREATER
webertj@20217
   254
        then if exactl then lb else ratmiddle (lb, ub)
haftmann@22950
   255
        else if exactu then ub else ratmiddle (lb, ub)
haftmann@22950
   256
      else (*discrete domain, both bounds must be exact*)
haftmann@23025
   257
      if ord = GREATER
haftmann@22950
   258
        then let val lb' = Rat.roundup lb in
haftmann@22950
   259
          if Rat.le lb' ub then lb' else raise NoEx end
haftmann@22950
   260
        else let val ub' = Rat.rounddown ub in
haftmann@22950
   261
          if Rat.le lb ub' then ub' else raise NoEx end
haftmann@22950
   262
  end;
nipkow@13498
   263
haftmann@22950
   264
fun findex1 discr (v, lineqs) ex =
haftmann@22950
   265
  let
haftmann@23063
   266
    val nz = filter (fn (Lineq (_, _, cs, _)) => nth cs v <> 0) lineqs;
haftmann@22950
   267
    val ineqs = maps elim_eqns nz;
haftmann@23063
   268
    val (ge, le) = List.partition (fn (_,_,cs) => nth cs v > 0) ineqs
haftmann@22950
   269
    val lb = ratrelmax (map (eval ex v) ge)
haftmann@22950
   270
    val ub = ratrelmin (map (eval ex v) le)
haftmann@21109
   271
  in nth_map v (K (choose2 (nth discr v) (lb, ub))) ex end;
nipkow@13498
   272
nipkow@13498
   273
fun elim1 v x =
haftmann@23063
   274
  map (fn (a,le,bs) => (Rat.add a (Rat.neg (Rat.mult (nth bs v) x)), le,
haftmann@21109
   275
                        nth_map v (K Rat.zero) bs));
nipkow@13498
   276
haftmann@23520
   277
fun single_var v (_, _, cs) = case filter_out (curry (op =) EQUAL o Rat.sign) cs
haftmann@23063
   278
 of [x] => x =/ nth cs v
haftmann@23063
   279
  | _ => false;
nipkow@13498
   280
nipkow@13498
   281
(* The base case:
nipkow@13498
   282
   all variables occur only with positive or only with negative coefficients *)
nipkow@13498
   283
fun pick_vars discr (ineqs,ex) =
haftmann@23520
   284
  let val nz = filter_out (fn (_,_,cs) => forall (curry (op =) EQUAL o Rat.sign) cs) ineqs
nipkow@14372
   285
  in case nz of [] => ex
nipkow@14372
   286
     | (_,_,cs) :: _ =>
haftmann@23520
   287
       let val v = find_index (not o curry (op =) EQUAL o Rat.sign) cs
haftmann@22950
   288
           val d = nth discr v;
haftmann@23520
   289
           val pos = not (Rat.sign (nth cs v) = LESS);
haftmann@22950
   290
           val sv = filter (single_var v) nz;
nipkow@14372
   291
           val minmax =
haftmann@17951
   292
             if pos then if d then Rat.roundup o fst o ratrelmax
nipkow@14372
   293
                         else ratexact true o ratrelmax
haftmann@17951
   294
                    else if d then Rat.rounddown o fst o ratrelmin
nipkow@14372
   295
                         else ratexact false o ratrelmin
haftmann@23063
   296
           val bnds = map (fn (a,le,bs) => (Rat.mult a (Rat.inv (nth bs v)), le)) sv
haftmann@17951
   297
           val x = minmax((Rat.zero,if pos then true else false)::bnds)
nipkow@14372
   298
           val ineqs' = elim1 v x nz
haftmann@21109
   299
           val ex' = nth_map v (K x) ex
nipkow@14372
   300
       in pick_vars discr (ineqs',ex') end
nipkow@13498
   301
  end;
nipkow@13498
   302
nipkow@13498
   303
fun findex0 discr n lineqs =
haftmann@22950
   304
  let val ineqs = maps elim_eqns lineqs
haftmann@22950
   305
      val rineqs = map (fn (a,le,cs) => (Rat.rat_of_int a, le, map Rat.rat_of_int cs))
nipkow@14372
   306
                       ineqs
haftmann@17951
   307
  in pick_vars discr (rineqs,replicate n Rat.zero) end;
nipkow@13498
   308
nipkow@13498
   309
(* ------------------------------------------------------------------------- *)
webertj@23197
   310
(* End of counterexample finder. The actual decision procedure starts here.  *)
nipkow@13498
   311
(* ------------------------------------------------------------------------- *)
nipkow@13498
   312
nipkow@5982
   313
(* ------------------------------------------------------------------------- *)
nipkow@5982
   314
(* Calculate new (in)equality type after addition.                           *)
nipkow@5982
   315
(* ------------------------------------------------------------------------- *)
nipkow@5982
   316
nipkow@5982
   317
fun find_add_type(Eq,x) = x
nipkow@5982
   318
  | find_add_type(x,Eq) = x
nipkow@5982
   319
  | find_add_type(_,Lt) = Lt
nipkow@5982
   320
  | find_add_type(Lt,_) = Lt
nipkow@5982
   321
  | find_add_type(Le,Le) = Le;
nipkow@5982
   322
nipkow@5982
   323
(* ------------------------------------------------------------------------- *)
nipkow@5982
   324
(* Multiply out an (in)equation.                                             *)
nipkow@5982
   325
(* ------------------------------------------------------------------------- *)
nipkow@5982
   326
nipkow@5982
   327
fun multiply_ineq n (i as Lineq(k,ty,l,just)) =
nipkow@5982
   328
  if n = 1 then i
wenzelm@40316
   329
  else if n = 0 andalso ty = Lt then raise Fail "multiply_ineq"
wenzelm@40316
   330
  else if n < 0 andalso (ty=Le orelse ty=Lt) then raise Fail "multiply_ineq"
wenzelm@33002
   331
  else Lineq (n * k, ty, map (Integer.mult n) l, Multiplied (n, just));
nipkow@5982
   332
nipkow@5982
   333
(* ------------------------------------------------------------------------- *)
nipkow@5982
   334
(* Add together (in)equations.                                               *)
nipkow@5982
   335
(* ------------------------------------------------------------------------- *)
nipkow@5982
   336
haftmann@38052
   337
fun add_ineq (Lineq (k1,ty1,l1,just1)) (Lineq (k2,ty2,l2,just2)) =
wenzelm@33002
   338
  let val l = map2 Integer.add l1 l2
nipkow@5982
   339
  in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end;
nipkow@5982
   340
nipkow@5982
   341
(* ------------------------------------------------------------------------- *)
nipkow@5982
   342
(* Elimination of variable between a single pair of (in)equations.           *)
nipkow@5982
   343
(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve.       *)
nipkow@5982
   344
(* ------------------------------------------------------------------------- *)
nipkow@5982
   345
haftmann@49387
   346
fun elim_var v (i1 as Lineq(_,ty1,l1,_)) (i2 as Lineq(_,ty2,l2,_)) =
haftmann@23063
   347
  let val c1 = nth l1 v and c2 = nth l2 v
haftmann@23261
   348
      val m = Integer.lcm (abs c1) (abs c2)
nipkow@5982
   349
      val m1 = m div (abs c1) and m2 = m div (abs c2)
nipkow@5982
   350
      val (n1,n2) =
nipkow@5982
   351
        if (c1 >= 0) = (c2 >= 0)
nipkow@5982
   352
        then if ty1 = Eq then (~m1,m2)
nipkow@5982
   353
             else if ty2 = Eq then (m1,~m2)
wenzelm@40316
   354
                  else raise Fail "elim_var"
nipkow@5982
   355
        else (m1,m2)
nipkow@5982
   356
      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
nipkow@5982
   357
                    then (~n1,~n2) else (n1,n2)
boehmes@31510
   358
  in add_ineq (multiply_ineq p1 i1) (multiply_ineq p2 i2) end;
nipkow@5982
   359
nipkow@5982
   360
(* ------------------------------------------------------------------------- *)
nipkow@5982
   361
(* The main refutation-finding code.                                         *)
nipkow@5982
   362
(* ------------------------------------------------------------------------- *)
nipkow@5982
   363
nipkow@5982
   364
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
nipkow@5982
   365
haftmann@38052
   366
fun is_contradictory (Lineq(k,ty,_,_)) =
nipkow@5982
   367
  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
nipkow@5982
   368
wenzelm@24630
   369
fun calc_blowup l =
wenzelm@33317
   370
  let val (p,n) = List.partition (curry (op <) 0) (filter (curry (op <>) 0) l)
wenzelm@24630
   371
  in length p * length n end;
nipkow@5982
   372
nipkow@5982
   373
(* ------------------------------------------------------------------------- *)
nipkow@5982
   374
(* Main elimination code:                                                    *)
nipkow@5982
   375
(*                                                                           *)
nipkow@5982
   376
(* (1) Looks for immediate solutions (false assertions with no variables).   *)
nipkow@5982
   377
(*                                                                           *)
nipkow@5982
   378
(* (2) If there are any equations, picks a variable with the lowest absolute *)
nipkow@5982
   379
(* coefficient in any of them, and uses it to eliminate.                     *)
nipkow@5982
   380
(*                                                                           *)
nipkow@5982
   381
(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
nipkow@5982
   382
(* blowup (number of consequences generated) and eliminates it.              *)
nipkow@5982
   383
(* ------------------------------------------------------------------------- *)
nipkow@5982
   384
nipkow@5982
   385
fun extract_first p =
boehmes@31510
   386
  let
boehmes@31510
   387
    fun extract xs (y::ys) = if p y then (y, xs @ ys) else extract (y::xs) ys
wenzelm@47060
   388
      | extract xs [] = raise List.Empty
nipkow@5982
   389
  in extract [] end;
nipkow@5982
   390
wenzelm@44654
   391
fun print_ineqs ctxt ineqs =
wenzelm@44654
   392
  if Config.get ctxt LA_Data.trace then
wenzelm@12262
   393
     tracing(cat_lines(""::map (fn Lineq(c,t,l,_) =>
wenzelm@24630
   394
       string_of_int c ^
paulson@9073
   395
       (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
wenzelm@24630
   396
       commas(map string_of_int l)) ineqs))
paulson@9073
   397
  else ();
nipkow@6056
   398
nipkow@13498
   399
type history = (int * lineq list) list;
nipkow@13498
   400
datatype result = Success of injust | Failure of history;
nipkow@13498
   401
wenzelm@44654
   402
fun elim ctxt (ineqs, hist) =
wenzelm@44654
   403
  let val _ = print_ineqs ctxt ineqs
webertj@20217
   404
      val (triv, nontriv) = List.partition is_trivial ineqs in
webertj@20217
   405
  if not (null triv)
wenzelm@59584
   406
  then case find_first is_contradictory triv of
wenzelm@44654
   407
         NONE => elim ctxt (nontriv, hist)
skalberg@15531
   408
       | SOME(Lineq(_,_,_,j)) => Success j
nipkow@5982
   409
  else
webertj@20217
   410
  if null nontriv then Failure hist
nipkow@13498
   411
  else
webertj@20217
   412
  let val (eqs, noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
webertj@20217
   413
  if not (null eqs) then
boehmes@31510
   414
     let val c =
haftmann@33042
   415
           fold (fn Lineq(_,_,l,_) => fn cs => union (op =) l cs) eqs []
boehmes@31510
   416
           |> filter (fn i => i <> 0)
wenzelm@59058
   417
           |> sort (int_ord o apply2 abs)
boehmes@31510
   418
           |> hd
boehmes@31510
   419
         val (eq as Lineq(_,_,ceq,_),othereqs) =
haftmann@36692
   420
               extract_first (fn Lineq(_,_,l,_) => member (op =) l c) eqs
haftmann@31986
   421
         val v = find_index (fn v => v = c) ceq
haftmann@23063
   422
         val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0)
nipkow@5982
   423
                                     (othereqs @ noneqs)
nipkow@5982
   424
         val others = map (elim_var v eq) roth @ ioth
wenzelm@44654
   425
     in elim ctxt (others,(v,nontriv)::hist) end
nipkow@5982
   426
  else
nipkow@5982
   427
  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
haftmann@23063
   428
      val numlist = 0 upto (length (hd lists) - 1)
haftmann@23063
   429
      val coeffs = map (fn i => map (fn xs => nth xs i) lists) numlist
nipkow@5982
   430
      val blows = map calc_blowup coeffs
nipkow@5982
   431
      val iblows = blows ~~ numlist
haftmann@23063
   432
      val nziblows = filter_out (fn (i, _) => i = 0) iblows
nipkow@13498
   433
  in if null nziblows then Failure((~1,nontriv)::hist)
nipkow@13498
   434
     else
nipkow@5982
   435
     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
haftmann@23063
   436
         val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) ineqs
haftmann@23063
   437
         val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => nth l v > 0) yes
wenzelm@44654
   438
     in elim ctxt (no @ map_product (elim_var v) pos neg, (v,nontriv)::hist) end
nipkow@5982
   439
  end
nipkow@5982
   440
  end
nipkow@5982
   441
  end;
nipkow@5982
   442
nipkow@5982
   443
(* ------------------------------------------------------------------------- *)
nipkow@5982
   444
(* Translate back a proof.                                                   *)
nipkow@5982
   445
(* ------------------------------------------------------------------------- *)
nipkow@5982
   446
wenzelm@44654
   447
fun trace_thm ctxt msgs th =
wenzelm@44654
   448
 (if Config.get ctxt LA_Data.trace
wenzelm@44654
   449
  then tracing (cat_lines (msgs @ [Display.string_of_thm ctxt th]))
wenzelm@44654
   450
  else (); th);
paulson@9073
   451
wenzelm@44654
   452
fun trace_term ctxt msgs t =
wenzelm@44654
   453
 (if Config.get ctxt LA_Data.trace
wenzelm@44654
   454
  then tracing (cat_lines (msgs @ [Syntax.string_of_term ctxt t]))
wenzelm@44654
   455
  else (); t);
wenzelm@24076
   456
wenzelm@44654
   457
fun trace_msg ctxt msg =
wenzelm@44654
   458
  if Config.get ctxt LA_Data.trace then tracing msg else ();
paulson@9073
   459
wenzelm@52131
   460
val union_term = union Envir.aeconv;
wenzelm@52131
   461
val union_bterm = union (fn ((b:bool, t), (b', t')) => b = b' andalso Envir.aeconv (t, t'));
berghofe@26835
   462
boehmes@31510
   463
fun add_atoms (lhs, _, _, rhs, _, _) =
boehmes@31510
   464
  union_term (map fst lhs) o union_term (map fst rhs);
nipkow@6056
   465
boehmes@31510
   466
fun atoms_of ds = fold add_atoms ds [];
boehmes@31510
   467
boehmes@31510
   468
(*
nipkow@6056
   469
Simplification may detect a contradiction 'prematurely' due to type
nipkow@6056
   470
information: n+1 <= 0 is simplified to False and does not need to be crossed
nipkow@6056
   471
with 0 <= n.
nipkow@6056
   472
*)
nipkow@6056
   473
local
wenzelm@24076
   474
  exception FalseE of thm
nipkow@6056
   475
in
wenzelm@27020
   476
wenzelm@51717
   477
fun mkthm ctxt asms (just: injust) =
wenzelm@24076
   478
  let
wenzelm@42361
   479
    val thy = Proof_Context.theory_of ctxt;
wenzelm@38763
   480
    val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = get_data ctxt;
wenzelm@38763
   481
    val number_of = number_of ctxt;
wenzelm@51717
   482
    val simpset_ctxt = put_simpset simpset ctxt;
boehmes@31510
   483
    fun only_concl f thm =
boehmes@31510
   484
      if Thm.no_prems thm then f (Thm.concl_of thm) else NONE;
boehmes@31510
   485
    val atoms = atoms_of (map_filter (only_concl (LA_Data.decomp ctxt)) asms);
boehmes@31510
   486
boehmes@31510
   487
    fun use_first rules thm =
boehmes@31510
   488
      get_first (fn th => SOME (thm RS th) handle THM _ => NONE) rules
boehmes@31510
   489
boehmes@31510
   490
    fun add2 thm1 thm2 =
boehmes@31510
   491
      use_first add_mono_thms (thm1 RS (thm2 RS LA_Logic.conjI));
boehmes@31510
   492
    fun try_add thms thm = get_first (fn th => add2 th thm) thms;
nipkow@6056
   493
boehmes@31510
   494
    fun add_thms thm1 thm2 =
boehmes@31510
   495
      (case add2 thm1 thm2 of
boehmes@31510
   496
        NONE =>
boehmes@31510
   497
          (case try_add ([thm1] RL inj_thms) thm2 of
boehmes@31510
   498
            NONE =>
boehmes@31510
   499
              (the (try_add ([thm2] RL inj_thms) thm1)
wenzelm@51930
   500
                handle Option.Option =>
wenzelm@44654
   501
                  (trace_thm ctxt [] thm1; trace_thm ctxt [] thm2;
wenzelm@40316
   502
                   raise Fail "Linear arithmetic: failed to add thms"))
boehmes@31510
   503
          | SOME thm => thm)
boehmes@31510
   504
      | SOME thm => thm);
boehmes@31510
   505
boehmes@31510
   506
    fun mult_by_add n thm =
boehmes@31510
   507
      let fun mul i th = if i = 1 then th else mul (i - 1) (add_thms thm th)
boehmes@31510
   508
      in mul n thm end;
nipkow@10575
   509
wenzelm@51717
   510
    val rewr = Simplifier.rewrite simpset_ctxt;
boehmes@31510
   511
    val rewrite_concl = Conv.fconv_rule (Conv.concl_conv ~1 (Conv.arg_conv
boehmes@31510
   512
      (Conv.binop_conv rewr)));
boehmes@31510
   513
    fun discharge_prem thm = if Thm.nprems_of thm = 0 then thm else
boehmes@31510
   514
      let val cv = Conv.arg1_conv (Conv.arg_conv rewr)
boehmes@31510
   515
      in Thm.implies_elim (Conv.fconv_rule cv thm) LA_Logic.trueI end
webertj@20217
   516
boehmes@31510
   517
    fun mult n thm =
boehmes@31510
   518
      (case use_first mult_mono_thms thm of
boehmes@31510
   519
        NONE => mult_by_add n thm
boehmes@31510
   520
      | SOME mth =>
boehmes@31510
   521
          let
boehmes@31510
   522
            val cv = mth |> Thm.cprop_of |> Drule.strip_imp_concl
boehmes@31510
   523
              |> Thm.dest_arg |> Thm.dest_arg1 |> Thm.dest_arg1
wenzelm@59586
   524
            val T = Thm.typ_of_cterm cv
boehmes@31510
   525
          in
boehmes@31510
   526
            mth
wenzelm@38763
   527
            |> Thm.instantiate ([], [(cv, number_of T n)])
boehmes@31510
   528
            |> rewrite_concl
boehmes@31510
   529
            |> discharge_prem
boehmes@31510
   530
            handle CTERM _ => mult_by_add n thm
boehmes@31510
   531
                 | THM _ => mult_by_add n thm
boehmes@31510
   532
          end);
nipkow@10691
   533
boehmes@31510
   534
    fun mult_thm (n, thm) =
boehmes@31510
   535
      if n = ~1 then thm RS LA_Logic.sym
boehmes@31510
   536
      else if n < 0 then mult (~n) thm RS LA_Logic.sym
boehmes@31510
   537
      else mult n thm;
boehmes@31510
   538
boehmes@31510
   539
    fun simp thm =
wenzelm@51717
   540
      let val thm' = trace_thm ctxt ["Simplified:"] (full_simplify simpset_ctxt thm)
boehmes@31510
   541
      in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end;
nipkow@6056
   542
wenzelm@44654
   543
    fun mk (Asm i) = trace_thm ctxt ["Asm " ^ string_of_int i] (nth asms i)
wenzelm@44654
   544
      | mk (Nat i) = trace_thm ctxt ["Nat " ^ string_of_int i] (LA_Logic.mk_nat_thm thy (nth atoms i))
wenzelm@44654
   545
      | mk (LessD j) = trace_thm ctxt ["L"] (hd ([mk j] RL lessD))
wenzelm@44654
   546
      | mk (NotLeD j) = trace_thm ctxt ["NLe"] (mk j RS LA_Logic.not_leD)
wenzelm@44654
   547
      | mk (NotLeDD j) = trace_thm ctxt ["NLeD"] (hd ([mk j RS LA_Logic.not_leD] RL lessD))
wenzelm@44654
   548
      | mk (NotLessD j) = trace_thm ctxt ["NL"] (mk j RS LA_Logic.not_lessD)
wenzelm@44654
   549
      | mk (Added (j1, j2)) = simp (trace_thm ctxt ["+"] (add_thms (mk j1) (mk j2)))
wenzelm@32091
   550
      | mk (Multiplied (n, j)) =
wenzelm@44654
   551
          (trace_msg ctxt ("*" ^ string_of_int n); trace_thm ctxt ["*"] (mult_thm (n, mk j)))
nipkow@5982
   552
wenzelm@27020
   553
  in
wenzelm@27020
   554
    let
wenzelm@44654
   555
      val _ = trace_msg ctxt "mkthm";
wenzelm@44654
   556
      val thm = trace_thm ctxt ["Final thm:"] (mk just);
wenzelm@51717
   557
      val fls = simplify simpset_ctxt thm;
wenzelm@44654
   558
      val _ = trace_thm ctxt ["After simplification:"] fls;
wenzelm@27020
   559
      val _ =
wenzelm@27020
   560
        if LA_Logic.is_False fls then ()
wenzelm@27020
   561
        else
boehmes@35872
   562
         (tracing (cat_lines
boehmes@35872
   563
           (["Assumptions:"] @ map (Display.string_of_thm ctxt) asms @ [""] @
boehmes@35872
   564
            ["Proved:", Display.string_of_thm ctxt fls, ""]));
boehmes@35872
   565
          warning "Linear arithmetic should have refuted the assumptions.\n\
boehmes@35872
   566
            \Please inform Tobias Nipkow.")
wenzelm@27020
   567
    in fls end
wenzelm@44654
   568
    handle FalseE thm => trace_thm ctxt ["False reached early:"] thm
wenzelm@27020
   569
  end;
wenzelm@27020
   570
nipkow@6056
   571
end;
nipkow@5982
   572
haftmann@23261
   573
fun coeff poly atom =
wenzelm@52131
   574
  AList.lookup Envir.aeconv poly atom |> the_default 0;
nipkow@10691
   575
nipkow@10691
   576
fun integ(rlhs,r,rel,rrhs,s,d) =
haftmann@17951
   577
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s
wenzelm@24630
   578
    val m = Integer.lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs))
wenzelm@22846
   579
    fun mult(t,r) =
haftmann@17951
   580
        let val (i,j) = Rat.quotient_of_rat r
paulson@15965
   581
        in (t,i * (m div j)) end
nipkow@12932
   582
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end
nipkow@10691
   583
haftmann@38052
   584
fun mklineq atoms =
webertj@20217
   585
  fn (item, k) =>
webertj@20217
   586
  let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item
nipkow@13498
   587
      val lhsa = map (coeff lhs) atoms
nipkow@13498
   588
      and rhsa = map (coeff rhs) atoms
haftmann@18330
   589
      val diff = map2 (curry (op -)) rhsa lhsa
nipkow@13498
   590
      val c = i-j
nipkow@13498
   591
      val just = Asm k
boehmes@31511
   592
      fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied(m,j))
nipkow@13498
   593
  in case rel of
nipkow@13498
   594
      "<="   => lineq(c,Le,diff,just)
nipkow@13498
   595
     | "~<=" => if discrete
nipkow@13498
   596
                then lineq(1-c,Le,map (op ~) diff,NotLeDD(just))
nipkow@13498
   597
                else lineq(~c,Lt,map (op ~) diff,NotLeD(just))
nipkow@13498
   598
     | "<"   => if discrete
nipkow@13498
   599
                then lineq(c+1,Le,diff,LessD(just))
nipkow@13498
   600
                else lineq(c,Lt,diff,just)
nipkow@13498
   601
     | "~<"  => lineq(~c,Le,map (op~) diff,NotLessD(just))
nipkow@13498
   602
     | "="   => lineq(c,Eq,diff,just)
wenzelm@40316
   603
     | _     => raise Fail ("mklineq" ^ rel)
nipkow@5982
   604
  end;
nipkow@5982
   605
nipkow@13498
   606
(* ------------------------------------------------------------------------- *)
nipkow@13498
   607
(* Print (counter) example                                                   *)
nipkow@13498
   608
(* ------------------------------------------------------------------------- *)
nipkow@13498
   609
nipkow@13498
   610
fun print_atom((a,d),r) =
haftmann@17951
   611
  let val (p,q) = Rat.quotient_of_rat r
wenzelm@24630
   612
      val s = if d then string_of_int p else
nipkow@13498
   613
              if p = 0 then "0"
wenzelm@24630
   614
              else string_of_int p ^ "/" ^ string_of_int q
nipkow@13498
   615
  in a ^ " = " ^ s end;
nipkow@13498
   616
boehmes@43607
   617
fun is_variable (Free _) = true
boehmes@43607
   618
  | is_variable (Var _) = true
boehmes@43607
   619
  | is_variable (Bound _) = true
boehmes@43607
   620
  | is_variable _ = false
nipkow@13498
   621
wenzelm@24076
   622
fun trace_ex ctxt params atoms discr n (hist: history) =
webertj@20217
   623
  case hist of
webertj@20217
   624
    [] => ()
webertj@20217
   625
  | (v, lineqs) :: hist' =>
wenzelm@24076
   626
      let
wenzelm@24076
   627
        val frees = map Free params
wenzelm@24920
   628
        fun show_term t = Syntax.string_of_term ctxt (subst_bounds (frees, t))
wenzelm@24076
   629
        val start =
wenzelm@24076
   630
          if v = ~1 then (hist', findex0 discr n lineqs)
haftmann@22950
   631
          else (hist, replicate n Rat.zero)
boehmes@43607
   632
        val produce_ex =
boehmes@43607
   633
          map print_atom #> commas #>
boehmes@43607
   634
          prefix "Counterexample (possibly spurious):\n"
boehmes@43607
   635
        val ex = (
boehmes@43607
   636
          uncurry (fold (findex1 discr)) start
boehmes@43607
   637
          |> map2 pair (atoms ~~ discr)
boehmes@43607
   638
          |> filter (fn ((t, _), _) => is_variable t)
boehmes@43607
   639
          |> map (apfst (apfst show_term))
boehmes@43607
   640
          |> (fn [] => NONE | sdss => SOME (produce_ex sdss)))
webertj@20217
   641
          handle NoEx => NONE
wenzelm@24076
   642
      in
wenzelm@24076
   643
        case ex of
wenzelm@44654
   644
          SOME s =>
wenzelm@44654
   645
            (warning "Linear arithmetic failed -- see trace for a (potentially spurious) counterexample.";
wenzelm@44654
   646
              tracing s)
haftmann@30687
   647
        | NONE => warning "Linear arithmetic failed"
wenzelm@24076
   648
      end;
nipkow@13498
   649
webertj@20217
   650
(* ------------------------------------------------------------------------- *)
webertj@20217
   651
webertj@20268
   652
fun mknat (pTs : typ list) (ixs : int list) (atom : term, i : int) : lineq option =
webertj@20217
   653
  if LA_Logic.is_nat (pTs, atom)
nipkow@6056
   654
  then let val l = map (fn j => if j=i then 1 else 0) ixs
webertj@20217
   655
       in SOME (Lineq (0, Le, l, Nat i)) end
webertj@20217
   656
  else NONE;
nipkow@6056
   657
nipkow@13186
   658
(* This code is tricky. It takes a list of premises in the order they occur
skalberg@15531
   659
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical
skalberg@15531
   660
ones as NONE. Going through the premises, each numeric one is converted into
nipkow@13186
   661
a Lineq. The tricky bit is to convert ~= which is split into two cases < and
nipkow@13498
   662
>. Thus split_items returns a list of equation systems. This may blow up if
nipkow@13186
   663
there are many ~=, but in practice it does not seem to happen. The really
nipkow@13186
   664
tricky bit is to arrange the order of the cases such that they coincide with
nipkow@13186
   665
the order in which the cases are in the end generated by the tactic that
nipkow@13186
   666
applies the generated refutation thms (see function 'refute_tac').
nipkow@13186
   667
nipkow@13186
   668
For variables n of type nat, a constraint 0 <= n is added.
nipkow@13186
   669
*)
webertj@20217
   670
webertj@20217
   671
(* FIXME: To optimize, the splitting of cases and the search for refutations *)
webertj@20276
   672
(*        could be intertwined: separate the first (fully split) case,       *)
webertj@20217
   673
(*        refute it, continue with splitting and refuting.  Terminate with   *)
webertj@20217
   674
(*        failure as soon as a case could not be refuted; i.e. delay further *)
webertj@20217
   675
(*        splitting until after a refutation for other cases has been found. *)
webertj@20217
   676
webertj@30406
   677
fun split_items ctxt do_pre split_neq (Ts, terms) : (typ list * (LA_Data.decomp * int) list) list =
webertj@20276
   678
let
webertj@20276
   679
  (* splits inequalities '~=' into '<' and '>'; this corresponds to *)
webertj@20276
   680
  (* 'REPEAT_DETERM (eresolve_tac neqE i)' at the theorem/tactic    *)
webertj@20276
   681
  (* level                                                          *)
webertj@20276
   682
  (* FIXME: this is currently sensitive to the order of theorems in *)
webertj@20276
   683
  (*        neqE:  The theorem for type "nat" must come first.  A   *)
webertj@20276
   684
  (*        better (i.e. less likely to break when neqE changes)    *)
webertj@20276
   685
  (*        implementation should *test* which theorem from neqE    *)
webertj@20276
   686
  (*        can be applied, and split the premise accordingly.      *)
wenzelm@26945
   687
  fun elim_neq (ineqs : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   688
               (LA_Data.decomp option * bool) list list =
webertj@20276
   689
  let
wenzelm@26945
   690
    fun elim_neq' nat_only ([] : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   691
                  (LA_Data.decomp option * bool) list list =
webertj@20276
   692
          [[]]
webertj@20276
   693
      | elim_neq' nat_only ((NONE, is_nat) :: ineqs) =
webertj@20276
   694
          map (cons (NONE, is_nat)) (elim_neq' nat_only ineqs)
webertj@20276
   695
      | elim_neq' nat_only ((ineq as (SOME (l, i, rel, r, j, d), is_nat)) :: ineqs) =
webertj@20276
   696
          if rel = "~=" andalso (not nat_only orelse is_nat) then
webertj@20276
   697
            (* [| ?l ~= ?r; ?l < ?r ==> ?R; ?r < ?l ==> ?R |] ==> ?R *)
webertj@20276
   698
            elim_neq' nat_only (ineqs @ [(SOME (l, i, "<", r, j, d), is_nat)]) @
webertj@20276
   699
            elim_neq' nat_only (ineqs @ [(SOME (r, j, "<", l, i, d), is_nat)])
webertj@20276
   700
          else
webertj@20276
   701
            map (cons ineq) (elim_neq' nat_only ineqs)
webertj@20276
   702
  in
webertj@20276
   703
    ineqs |> elim_neq' true
wenzelm@26945
   704
          |> maps (elim_neq' false)
webertj@20276
   705
  end
nipkow@13464
   706
webertj@30406
   707
  fun ignore_neq (NONE, bool) = (NONE, bool)
webertj@30406
   708
    | ignore_neq (ineq as SOME (_, _, rel, _, _, _), bool) =
webertj@30406
   709
      if rel = "~=" then (NONE, bool) else (ineq, bool)
webertj@30406
   710
webertj@20276
   711
  fun number_hyps _ []             = []
webertj@20276
   712
    | number_hyps n (NONE::xs)     = number_hyps (n+1) xs
webertj@20276
   713
    | number_hyps n ((SOME x)::xs) = (x, n) :: number_hyps (n+1) xs
webertj@20276
   714
webertj@20276
   715
  val result = (Ts, terms)
webertj@20276
   716
    |> (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   717
       (if do_pre then LA_Data.pre_decomp ctxt else Library.single)
wenzelm@44654
   718
    |> tap (fn subgoals => trace_msg ctxt ("Preprocessing yields " ^
webertj@23195
   719
         string_of_int (length subgoals) ^ " subgoal(s) total."))
wenzelm@22846
   720
    |> (* produce the internal encoding of (in-)equalities *)
wenzelm@24076
   721
       map (apsnd (map (fn t => (LA_Data.decomp ctxt t, LA_Data.domain_is_nat t))))
webertj@20276
   722
    |> (* splitting of inequalities *)
webertj@30406
   723
       map (apsnd (if split_neq then elim_neq else
webertj@30406
   724
                     Library.single o map ignore_neq))
wenzelm@22846
   725
    |> maps (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals)
webertj@20276
   726
    |> (* numbering of hypotheses, ignoring irrelevant ones *)
webertj@20276
   727
       map (apsnd (number_hyps 0))
webertj@23195
   728
in
wenzelm@44654
   729
  trace_msg ctxt ("Splitting of inequalities yields " ^
webertj@23195
   730
    string_of_int (length result) ^ " subgoal(s) total.");
webertj@23195
   731
  result
webertj@23195
   732
end;
nipkow@13464
   733
wenzelm@33245
   734
fun add_datoms ((lhs,_,_,rhs,_,d) : LA_Data.decomp, _) (dats : (bool * term) list) =
berghofe@26835
   735
  union_bterm (map (pair d o fst) lhs) (union_bterm (map (pair d o fst) rhs) dats);
nipkow@13498
   736
wenzelm@59656
   737
fun refutes ctxt :
wenzelm@26945
   738
    (typ list * (LA_Data.decomp * int) list) list -> injust list -> injust list option =
wenzelm@26945
   739
  let
wenzelm@26945
   740
    fun refute ((Ts, initems : (LA_Data.decomp * int) list) :: initemss) (js: injust list) =
wenzelm@26945
   741
          let
boehmes@31510
   742
            val atoms = atoms_of (map fst initems)
wenzelm@26945
   743
            val n = length atoms
haftmann@38052
   744
            val mkleq = mklineq atoms
wenzelm@26945
   745
            val ixs = 0 upto (n - 1)
wenzelm@26945
   746
            val iatoms = atoms ~~ ixs
wenzelm@32952
   747
            val natlineqs = map_filter (mknat Ts ixs) iatoms
wenzelm@26945
   748
            val ineqs = map mkleq initems @ natlineqs
wenzelm@59656
   749
          in
wenzelm@59656
   750
            (case elim ctxt (ineqs, []) of
wenzelm@26945
   751
               Success j =>
wenzelm@44654
   752
                 (trace_msg ctxt ("Contradiction! (" ^ string_of_int (length js + 1) ^ ")");
wenzelm@26945
   753
                  refute initemss (js @ [j]))
wenzelm@59656
   754
             | Failure _ => NONE)
wenzelm@26945
   755
          end
wenzelm@26945
   756
      | refute [] js = SOME js
wenzelm@26945
   757
  in refute end;
nipkow@5982
   758
wenzelm@59656
   759
fun refute ctxt params do_pre split_neq terms : injust list option =
wenzelm@59656
   760
  refutes ctxt (split_items ctxt do_pre split_neq (map snd params, terms)) [];
webertj@20254
   761
haftmann@22950
   762
fun count P xs = length (filter P xs);
webertj@20254
   763
wenzelm@59656
   764
fun prove ctxt params do_pre Hs concl : bool * injust list option =
webertj@20254
   765
  let
wenzelm@44654
   766
    val _ = trace_msg ctxt "prove:"
webertj@20254
   767
    (* append the negated conclusion to 'Hs' -- this corresponds to     *)
webertj@20254
   768
    (* 'DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i)' at the *)
webertj@20254
   769
    (* theorem/tactic level                                             *)
webertj@20254
   770
    val Hs' = Hs @ [LA_Logic.neg_prop concl]
webertj@20254
   771
    fun is_neq NONE                 = false
webertj@20254
   772
      | is_neq (SOME (_,_,r,_,_,_)) = (r = "~=")
wenzelm@44654
   773
    val neq_limit = Config.get ctxt LA_Data.neq_limit
webertj@30406
   774
    val split_neq = count is_neq (map (LA_Data.decomp ctxt) Hs') <= neq_limit
webertj@20254
   775
  in
webertj@30406
   776
    if split_neq then ()
wenzelm@24076
   777
    else
wenzelm@44654
   778
      trace_msg ctxt ("neq_limit exceeded (current value is " ^
webertj@30406
   779
        string_of_int neq_limit ^ "), ignoring all inequalities");
wenzelm@59656
   780
    (split_neq, refute ctxt params do_pre split_neq Hs')
webertj@23190
   781
  end handle TERM ("neg_prop", _) =>
webertj@23190
   782
    (* since no meta-logic negation is available, we can only fail if   *)
webertj@23190
   783
    (* the conclusion is not of the form 'Trueprop $ _' (simply         *)
webertj@23190
   784
    (* dropping the conclusion doesn't work either, because even        *)
webertj@23190
   785
    (* 'False' does not imply arbitrary 'concl::prop')                  *)
wenzelm@44654
   786
    (trace_msg ctxt "prove failed (cannot negate conclusion).";
webertj@30406
   787
      (false, NONE));
webertj@20217
   788
wenzelm@51717
   789
fun refute_tac ctxt (i, split_neq, justs) =
nipkow@6074
   790
  fn state =>
wenzelm@24076
   791
    let
wenzelm@32091
   792
      val _ = trace_thm ctxt
wenzelm@44654
   793
        ["refute_tac (on subgoal " ^ string_of_int i ^ ", with " ^
wenzelm@44654
   794
          string_of_int (length justs) ^ " justification(s)):"] state
wenzelm@24076
   795
      val {neqE, ...} = get_data ctxt;
wenzelm@24076
   796
      fun just1 j =
wenzelm@24076
   797
        (* eliminate inequalities *)
webertj@30406
   798
        (if split_neq then
wenzelm@59498
   799
          REPEAT_DETERM (eresolve_tac ctxt neqE i)
webertj@30406
   800
        else
webertj@30406
   801
          all_tac) THEN
wenzelm@44654
   802
          PRIMITIVE (trace_thm ctxt ["State after neqE:"]) THEN
wenzelm@24076
   803
          (* use theorems generated from the actual justifications *)
wenzelm@59498
   804
          Subgoal.FOCUS (fn {prems, ...} => resolve_tac ctxt [mkthm ctxt prems j] 1) ctxt i
wenzelm@24076
   805
    in
wenzelm@24076
   806
      (* rewrite "[| A1; ...; An |] ==> B" to "[| A1; ...; An; ~B |] ==> False" *)
wenzelm@59498
   807
      DETERM (resolve_tac ctxt [LA_Logic.notI, LA_Logic.ccontr] i) THEN
wenzelm@24076
   808
      (* user-defined preprocessing of the subgoal *)
wenzelm@51717
   809
      DETERM (LA_Data.pre_tac ctxt i) THEN
wenzelm@44654
   810
      PRIMITIVE (trace_thm ctxt ["State after pre_tac:"]) THEN
wenzelm@24076
   811
      (* prove every resulting subgoal, using its justification *)
wenzelm@24076
   812
      EVERY (map just1 justs)
webertj@20217
   813
    end  state;
nipkow@6074
   814
nipkow@5982
   815
(*
nipkow@5982
   816
Fast but very incomplete decider. Only premises and conclusions
nipkow@5982
   817
that are already (negated) (in)equations are taken into account.
nipkow@5982
   818
*)
wenzelm@59656
   819
fun simpset_lin_arith_tac ctxt = SUBGOAL (fn (A, i) =>
wenzelm@24076
   820
  let
wenzelm@24076
   821
    val params = rev (Logic.strip_params A)
wenzelm@24076
   822
    val Hs = Logic.strip_assums_hyp A
wenzelm@24076
   823
    val concl = Logic.strip_assums_concl A
wenzelm@44654
   824
    val _ = trace_term ctxt ["Trying to refute subgoal " ^ string_of_int i] A
wenzelm@24076
   825
  in
wenzelm@59656
   826
    case prove ctxt params true Hs concl of
wenzelm@44654
   827
      (_, NONE) => (trace_msg ctxt "Refutation failed."; no_tac)
wenzelm@44654
   828
    | (split_neq, SOME js) => (trace_msg ctxt "Refutation succeeded.";
wenzelm@51717
   829
                               refute_tac ctxt (i, split_neq, js))
wenzelm@24076
   830
  end);
nipkow@5982
   831
wenzelm@51717
   832
fun prems_lin_arith_tac ctxt =
wenzelm@51717
   833
  Method.insert_tac (Simplifier.prems_of ctxt) THEN'
wenzelm@59656
   834
  simpset_lin_arith_tac ctxt;
wenzelm@17613
   835
wenzelm@24076
   836
fun lin_arith_tac ctxt =
wenzelm@51717
   837
  simpset_lin_arith_tac (empty_simpset ctxt);
wenzelm@24076
   838
wenzelm@24076
   839
nipkow@5982
   840
nipkow@13186
   841
(** Forward proof from theorems **)
nipkow@13186
   842
webertj@20433
   843
(* More tricky code. Needs to arrange the proofs of the multiple cases (due
webertj@20433
   844
to splits of ~= premises) such that it coincides with the order of the cases
webertj@20433
   845
generated by function split_items. *)
webertj@20433
   846
webertj@20433
   847
datatype splittree = Tip of thm list
webertj@20433
   848
                   | Spl of thm * cterm * splittree * cterm * splittree;
webertj@20433
   849
webertj@20433
   850
(* "(ct1 ==> ?R) ==> (ct2 ==> ?R) ==> ?R" is taken to (ct1, ct2) *)
webertj@20433
   851
webertj@20433
   852
fun extract (imp : cterm) : cterm * cterm =
webertj@20433
   853
let val (Il, r)    = Thm.dest_comb imp
webertj@20433
   854
    val (_, imp1)  = Thm.dest_comb Il
webertj@20433
   855
    val (Ict1, _)  = Thm.dest_comb imp1
webertj@20433
   856
    val (_, ct1)   = Thm.dest_comb Ict1
webertj@20433
   857
    val (Ir, _)    = Thm.dest_comb r
webertj@20433
   858
    val (_, Ict2r) = Thm.dest_comb Ir
webertj@20433
   859
    val (Ict2, _)  = Thm.dest_comb Ict2r
webertj@20433
   860
    val (_, ct2)   = Thm.dest_comb Ict2
webertj@20433
   861
in (ct1, ct2) end;
webertj@20433
   862
wenzelm@24076
   863
fun splitasms ctxt (asms : thm list) : splittree =
wenzelm@24076
   864
let val {neqE, ...} = get_data ctxt
hoelzl@35693
   865
    fun elim_neq [] (asms', []) = Tip (rev asms')
hoelzl@35693
   866
      | elim_neq [] (asms', asms) = Tip (rev asms' @ asms)
haftmann@49387
   867
      | elim_neq (_ :: neqs) (asms', []) = elim_neq neqs ([],rev asms')
hoelzl@35693
   868
      | elim_neq (neqs as (neq :: _)) (asms', asm::asms) =
hoelzl@35693
   869
      (case get_first (fn th => SOME (asm COMP th) handle THM _ => NONE) [neq] of
webertj@20433
   870
        SOME spl =>
wenzelm@59582
   871
          let val (ct1, ct2) = extract (Thm.cprop_of spl)
wenzelm@36945
   872
              val thm1 = Thm.assume ct1
wenzelm@36945
   873
              val thm2 = Thm.assume ct2
hoelzl@35693
   874
          in Spl (spl, ct1, elim_neq neqs (asms', asms@[thm1]),
hoelzl@35693
   875
            ct2, elim_neq neqs (asms', asms@[thm2]))
webertj@20433
   876
          end
hoelzl@35693
   877
      | NONE => elim_neq neqs (asm::asms', asms))
hoelzl@35693
   878
in elim_neq neqE ([], asms) end;
webertj@20433
   879
wenzelm@51717
   880
fun fwdproof ctxt (Tip asms : splittree) (j::js : injust list) = (mkthm ctxt asms j, js)
wenzelm@51717
   881
  | fwdproof ctxt (Spl (thm, ct1, tree1, ct2, tree2)) js =
wenzelm@24076
   882
      let
wenzelm@51717
   883
        val (thm1, js1) = fwdproof ctxt tree1 js
wenzelm@51717
   884
        val (thm2, js2) = fwdproof ctxt tree2 js1
wenzelm@36945
   885
        val thm1' = Thm.implies_intr ct1 thm1
wenzelm@36945
   886
        val thm2' = Thm.implies_intr ct2 thm2
wenzelm@24076
   887
      in (thm2' COMP (thm1' COMP thm), js2) end;
wenzelm@24076
   888
      (* FIXME needs handle THM _ => NONE ? *)
webertj@20433
   889
wenzelm@51717
   890
fun prover ctxt thms Tconcl (js : injust list) split_neq pos : thm option =
wenzelm@24076
   891
  let
wenzelm@24076
   892
    val nTconcl = LA_Logic.neg_prop Tconcl
wenzelm@59642
   893
    val cnTconcl = Thm.cterm_of ctxt nTconcl
wenzelm@36945
   894
    val nTconclthm = Thm.assume cnTconcl
webertj@30406
   895
    val tree = (if split_neq then splitasms ctxt else Tip) (thms @ [nTconclthm])
wenzelm@51717
   896
    val (Falsethm, _) = fwdproof ctxt tree js
wenzelm@24076
   897
    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
wenzelm@36945
   898
    val concl = Thm.implies_intr cnTconcl Falsethm COMP contr
wenzelm@44654
   899
  in SOME (trace_thm ctxt ["Proved by lin. arith. prover:"] (LA_Logic.mk_Eq concl)) end
wenzelm@24076
   900
  (*in case concl contains ?-var, which makes assume fail:*)   (* FIXME Variable.import_terms *)
wenzelm@24076
   901
  handle THM _ => NONE;
nipkow@13186
   902
nipkow@13186
   903
(* PRE: concl is not negated!
nipkow@13186
   904
   This assumption is OK because
wenzelm@24076
   905
   1. lin_arith_simproc tries both to prove and disprove concl and
wenzelm@24076
   906
   2. lin_arith_simproc is applied by the Simplifier which
nipkow@13186
   907
      dives into terms and will thus try the non-negated concl anyway.
nipkow@13186
   908
*)
wenzelm@51717
   909
fun lin_arith_simproc ctxt concl =
wenzelm@24076
   910
  let
wenzelm@51717
   911
    val thms = maps LA_Logic.atomize (Simplifier.prems_of ctxt)
wenzelm@24076
   912
    val Hs = map Thm.prop_of thms
nipkow@6102
   913
    val Tconcl = LA_Logic.mk_Trueprop concl
wenzelm@24076
   914
  in
wenzelm@59656
   915
    case prove ctxt [] false Hs Tconcl of (* concl provable? *)
wenzelm@51717
   916
      (split_neq, SOME js) => prover ctxt thms Tconcl js split_neq true
webertj@30406
   917
    | (_, NONE) =>
wenzelm@24076
   918
        let val nTconcl = LA_Logic.neg_prop Tconcl in
wenzelm@59656
   919
          case prove ctxt [] false Hs nTconcl of (* ~concl provable? *)
wenzelm@51717
   920
            (split_neq, SOME js) => prover ctxt thms nTconcl js split_neq false
webertj@30406
   921
          | (_, NONE) => NONE
wenzelm@24076
   922
        end
wenzelm@24076
   923
  end;
nipkow@6074
   924
nipkow@6074
   925
end;