src/HOL/Integ/presburger.ML
author skalberg
Fri Mar 04 15:07:34 2005 +0100 (2005-03-04)
changeset 15574 b1d1b5bfc464
parent 15570 8d8c70b41bab
child 15620 8ccdc8bc66a2
permissions -rw-r--r--
Removed practically all references to Library.foldr.
berghofe@13876
     1
(*  Title:      HOL/Integ/presburger.ML
berghofe@13876
     2
    ID:         $Id$
berghofe@13876
     3
    Author:     Amine Chaieb and Stefan Berghofer, TU Muenchen
berghofe@13876
     4
berghofe@13876
     5
Tactic for solving arithmetical Goals in Presburger Arithmetic
berghofe@13876
     6
*)
berghofe@13876
     7
chaieb@14811
     8
(* This version of presburger deals with occurences of functional symbols in the subgoal and abstract over them to try to prove the more general formula. It then resolves with the subgoal. To enable this feature call the procedure with the parameter abs
chaieb@14758
     9
*)
chaieb@14758
    10
berghofe@13876
    11
signature PRESBURGER = 
berghofe@13876
    12
sig
chaieb@14758
    13
 val presburger_tac : bool -> bool -> int -> tactic
chaieb@14758
    14
 val presburger_method : bool -> bool -> int -> Proof.method
berghofe@13876
    15
 val setup : (theory -> theory) list
berghofe@13876
    16
 val trace : bool ref
berghofe@13876
    17
end;
berghofe@13876
    18
berghofe@13876
    19
structure Presburger: PRESBURGER =
berghofe@13876
    20
struct
berghofe@13876
    21
berghofe@13876
    22
val trace = ref false;
berghofe@13876
    23
fun trace_msg s = if !trace then tracing s else ();
berghofe@13876
    24
berghofe@13876
    25
(*-----------------------------------------------------------------*)
berghofe@13876
    26
(*cooper_pp: provefunction for the one-exstance quantifier elimination*)
berghofe@13876
    27
(* Here still only one problem : The proof for the arithmetical transformations done on the dvd atomic formulae*)
berghofe@13876
    28
(*-----------------------------------------------------------------*)
berghofe@13876
    29
chaieb@14941
    30
chaieb@14941
    31
(* Invoking the oracle *)
chaieb@14941
    32
chaieb@14941
    33
fun pres_oracle sg t = 
berghofe@15240
    34
  invoke_oracle (theory "Presburger") "presburger_oracle" 
chaieb@14941
    35
     (sg, CooperDec.COOPER_ORACLE t) ;
chaieb@14941
    36
berghofe@14801
    37
val presburger_ss = simpset_of (theory "Presburger");
berghofe@14801
    38
chaieb@14758
    39
fun cooper_pp sg (fm as e$Abs(xn,xT,p)) = 
berghofe@13876
    40
  let val (xn1,p1) = variant_abs (xn,xT,p)
chaieb@14758
    41
  in (CooperProof.cooper_prv sg (Free (xn1, xT)) p1) end;
berghofe@13876
    42
berghofe@13876
    43
fun mnnf_pp sg fm = CooperProof.proof_of_cnnf sg fm
berghofe@13876
    44
  (CooperProof.proof_of_evalc sg);
berghofe@13876
    45
chaieb@14758
    46
fun tmproof_of_int_qelim sg fm =
chaieb@14758
    47
  Qelim.tproof_of_mlift_qelim sg CooperDec.is_arith_rel
berghofe@13876
    48
    (CooperProof.proof_of_linform sg) (mnnf_pp sg) (cooper_pp sg) fm;
berghofe@13876
    49
chaieb@14758
    50
berghofe@13876
    51
(* Theorems to be used in this tactic*)
berghofe@13876
    52
berghofe@13876
    53
val zdvd_int = thm "zdvd_int";
berghofe@13876
    54
val zdiff_int_split = thm "zdiff_int_split";
berghofe@13876
    55
val all_nat = thm "all_nat";
berghofe@13876
    56
val ex_nat = thm "ex_nat";
berghofe@13876
    57
val number_of1 = thm "number_of1";
berghofe@13876
    58
val number_of2 = thm "number_of2";
berghofe@13876
    59
val split_zdiv = thm "split_zdiv";
berghofe@13876
    60
val split_zmod = thm "split_zmod";
berghofe@13876
    61
val mod_div_equality' = thm "mod_div_equality'";
berghofe@13876
    62
val split_div' = thm "split_div'";
berghofe@13876
    63
val Suc_plus1 = thm "Suc_plus1";
berghofe@13876
    64
val imp_le_cong = thm "imp_le_cong";
berghofe@13876
    65
val conj_le_cong = thm "conj_le_cong";
berghofe@13876
    66
berghofe@13876
    67
(* extract all the constants in a term*)
berghofe@13876
    68
fun add_term_typed_consts (Const (c, T), cs) = (c,T) ins cs
berghofe@13876
    69
  | add_term_typed_consts (t $ u, cs) =
berghofe@13876
    70
      add_term_typed_consts (t, add_term_typed_consts (u, cs))
berghofe@13876
    71
  | add_term_typed_consts (Abs (_, _, t), cs) = add_term_typed_consts (t, cs)
berghofe@13876
    72
  | add_term_typed_consts (_, cs) = cs;
berghofe@13876
    73
berghofe@13876
    74
fun term_typed_consts t = add_term_typed_consts(t,[]);
berghofe@13876
    75
skalberg@15531
    76
(* SOME Types*)
berghofe@13876
    77
val bT = HOLogic.boolT;
berghofe@13876
    78
val iT = HOLogic.intT;
berghofe@13876
    79
val binT = HOLogic.binT;
berghofe@13876
    80
val nT = HOLogic.natT;
berghofe@13876
    81
berghofe@13876
    82
(* Allowed Consts in formulae for presburger tactic*)
berghofe@13876
    83
berghofe@13876
    84
val allowed_consts =
berghofe@13876
    85
  [("All", (iT --> bT) --> bT),
berghofe@13876
    86
   ("Ex", (iT --> bT) --> bT),
berghofe@13876
    87
   ("All", (nT --> bT) --> bT),
berghofe@13876
    88
   ("Ex", (nT --> bT) --> bT),
berghofe@13876
    89
berghofe@13876
    90
   ("op &", bT --> bT --> bT),
berghofe@13876
    91
   ("op |", bT --> bT --> bT),
berghofe@13876
    92
   ("op -->", bT --> bT --> bT),
berghofe@13876
    93
   ("op =", bT --> bT --> bT),
berghofe@13876
    94
   ("Not", bT --> bT),
berghofe@13876
    95
berghofe@13876
    96
   ("op <=", iT --> iT --> bT),
berghofe@13876
    97
   ("op =", iT --> iT --> bT),
berghofe@13876
    98
   ("op <", iT --> iT --> bT),
berghofe@13876
    99
   ("Divides.op dvd", iT --> iT --> bT),
berghofe@13876
   100
   ("Divides.op div", iT --> iT --> iT),
berghofe@13876
   101
   ("Divides.op mod", iT --> iT --> iT),
berghofe@13876
   102
   ("op +", iT --> iT --> iT),
berghofe@13876
   103
   ("op -", iT --> iT --> iT),
berghofe@13876
   104
   ("op *", iT --> iT --> iT), 
berghofe@13876
   105
   ("HOL.abs", iT --> iT),
berghofe@13876
   106
   ("uminus", iT --> iT),
berghofe@14801
   107
   ("HOL.max", iT --> iT --> iT),
berghofe@14801
   108
   ("HOL.min", iT --> iT --> iT),
berghofe@13876
   109
berghofe@13876
   110
   ("op <=", nT --> nT --> bT),
berghofe@13876
   111
   ("op =", nT --> nT --> bT),
berghofe@13876
   112
   ("op <", nT --> nT --> bT),
berghofe@13876
   113
   ("Divides.op dvd", nT --> nT --> bT),
berghofe@13876
   114
   ("Divides.op div", nT --> nT --> nT),
berghofe@13876
   115
   ("Divides.op mod", nT --> nT --> nT),
berghofe@13876
   116
   ("op +", nT --> nT --> nT),
berghofe@13876
   117
   ("op -", nT --> nT --> nT),
berghofe@13876
   118
   ("op *", nT --> nT --> nT), 
berghofe@13876
   119
   ("Suc", nT --> nT),
berghofe@14801
   120
   ("HOL.max", nT --> nT --> nT),
berghofe@14801
   121
   ("HOL.min", nT --> nT --> nT),
berghofe@13876
   122
paulson@15013
   123
   ("Numeral.Bit", binT --> bT --> binT),
paulson@15013
   124
   ("Numeral.Pls", binT),
paulson@15013
   125
   ("Numeral.Min", binT),
berghofe@13876
   126
   ("Numeral.number_of", binT --> iT),
berghofe@13876
   127
   ("Numeral.number_of", binT --> nT),
berghofe@13876
   128
   ("0", nT),
berghofe@13876
   129
   ("0", iT),
berghofe@13876
   130
   ("1", nT),
berghofe@13876
   131
   ("1", iT),
berghofe@13876
   132
   ("False", bT),
berghofe@13876
   133
   ("True", bT)];
berghofe@13876
   134
berghofe@13876
   135
(* Preparation of the formula to be sent to the Integer quantifier *)
berghofe@13876
   136
(* elimination procedure                                           *)
berghofe@13876
   137
(* Transforms meta implications and meta quantifiers to object     *)
berghofe@13876
   138
(* implications and object quantifiers                             *)
berghofe@13876
   139
chaieb@14758
   140
chaieb@14758
   141
(*==================================*)
chaieb@14758
   142
(* Abstracting on subterms  ========*)
chaieb@14758
   143
(*==================================*)
chaieb@14758
   144
(* Returns occurences of terms that are function application of type int or nat*)
chaieb@14758
   145
chaieb@14758
   146
fun getfuncs fm = case strip_comb fm of
chaieb@14758
   147
    (Free (_, T), ts as _ :: _) =>
chaieb@14758
   148
      if body_type T mem [iT, nT] 
chaieb@14758
   149
         andalso not (ts = []) andalso forall (null o loose_bnos) ts 
chaieb@14758
   150
      then [fm]
skalberg@15570
   151
      else Library.foldl op union ([], map getfuncs ts)
chaieb@14758
   152
  | (Var (_, T), ts as _ :: _) =>
chaieb@14758
   153
      if body_type T mem [iT, nT] 
chaieb@14758
   154
         andalso not (ts = []) andalso forall (null o loose_bnos) ts then [fm]
skalberg@15570
   155
      else Library.foldl op union ([], map getfuncs ts)
chaieb@14758
   156
  | (Const (s, T), ts) =>
chaieb@14758
   157
      if (s, T) mem allowed_consts orelse not (body_type T mem [iT, nT])
skalberg@15570
   158
      then Library.foldl op union ([], map getfuncs ts)
chaieb@14758
   159
      else [fm]
chaieb@14758
   160
  | (Abs (s, T, t), _) => getfuncs t
chaieb@14758
   161
  | _ => [];
chaieb@14758
   162
chaieb@14758
   163
chaieb@14758
   164
fun abstract_pres sg fm = 
skalberg@15574
   165
  foldr (fn (t, u) =>
chaieb@14758
   166
      let val T = fastype_of t
chaieb@14758
   167
      in all T $ Abs ("x", T, abstract_over (t, u)) end)
skalberg@15574
   168
         fm (getfuncs fm);
chaieb@14758
   169
chaieb@14758
   170
chaieb@14758
   171
chaieb@14758
   172
(* hasfuncs_on_bounds dont care of the type of the functions applied!
chaieb@14758
   173
 It returns true if there is a subterm coresponding to the application of
chaieb@14758
   174
 a function on a bounded variable.
chaieb@14758
   175
chaieb@14758
   176
 Function applications are allowed only for well predefined functions a 
chaieb@14758
   177
 consts*)
chaieb@14758
   178
chaieb@14758
   179
fun has_free_funcs fm  = case strip_comb fm of
chaieb@14758
   180
    (Free (_, T), ts as _ :: _) => 
chaieb@14758
   181
      if (body_type T mem [iT,nT]) andalso (not (T mem [iT,nT]))
chaieb@14758
   182
      then true
chaieb@14758
   183
      else exists (fn x => x) (map has_free_funcs ts)
chaieb@14758
   184
  | (Var (_, T), ts as _ :: _) =>
chaieb@14758
   185
      if (body_type T mem [iT,nT]) andalso not (T mem [iT,nT])
chaieb@14758
   186
      then true
chaieb@14758
   187
      else exists (fn x => x) (map has_free_funcs ts)
chaieb@14758
   188
  | (Const (s, T), ts) =>  exists (fn x => x) (map has_free_funcs ts)
chaieb@14758
   189
  | (Abs (s, T, t), _) => has_free_funcs t
chaieb@14758
   190
  |_ => false;
chaieb@14758
   191
chaieb@14758
   192
chaieb@14758
   193
(*returns true if the formula is relevant for presburger arithmetic tactic
chaieb@14758
   194
The constants occuring in term t should be a subset of the allowed_consts
chaieb@14758
   195
 There also should be no occurences of application of functions on bounded 
chaieb@14758
   196
 variables. Whenever this function will be used, it will be ensured that t 
chaieb@14758
   197
 will not contain subterms with function symbols that could have been 
chaieb@14758
   198
 abstracted over.*)
chaieb@14758
   199
 
chaieb@14758
   200
fun relevant ps t = (term_typed_consts t) subset allowed_consts andalso 
skalberg@15570
   201
  map (fn i => snd (List.nth (ps, i))) (loose_bnos t) @
chaieb@14758
   202
  map (snd o dest_Free) (term_frees t) @ map (snd o dest_Var) (term_vars t)
chaieb@14758
   203
  subset [iT, nT]
chaieb@14758
   204
  andalso not (has_free_funcs t);
chaieb@14758
   205
chaieb@14758
   206
chaieb@14758
   207
fun prepare_for_presburger sg q fm = 
berghofe@13876
   208
  let
berghofe@13876
   209
    val ps = Logic.strip_params fm
berghofe@13876
   210
    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
berghofe@13876
   211
    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
chaieb@14758
   212
    val _ = if relevant (rev ps) c then () 
chaieb@14758
   213
               else  (trace_msg ("Conclusion is not a presburger term:\n" ^
chaieb@14758
   214
             Sign.string_of_term sg c); raise CooperDec.COOPER)
berghofe@13876
   215
    fun mk_all ((s, T), (P,n)) =
berghofe@13876
   216
      if 0 mem loose_bnos P then
berghofe@13876
   217
        (HOLogic.all_const T $ Abs (s, T, P), n)
berghofe@13876
   218
      else (incr_boundvars ~1 P, n-1)
berghofe@13876
   219
    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
skalberg@15570
   220
    val (rhs,irhs) = List.partition (relevant (rev ps)) hs
berghofe@13876
   221
    val np = length ps
skalberg@15574
   222
    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
skalberg@15574
   223
      (foldr HOLogic.mk_imp c rhs, np) ps
skalberg@15570
   224
    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
berghofe@13876
   225
      (term_frees fm' @ term_vars fm');
skalberg@15574
   226
    val fm2 = foldr mk_all2 fm' vs
berghofe@13876
   227
  in (fm2, np + length vs, length rhs) end;
berghofe@13876
   228
berghofe@13876
   229
(*Object quantifier to meta --*)
berghofe@13876
   230
fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
berghofe@13876
   231
berghofe@13876
   232
(* object implication to meta---*)
berghofe@13876
   233
fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
berghofe@13876
   234
berghofe@13876
   235
(* the presburger tactic*)
chaieb@14758
   236
chaieb@14758
   237
(* Parameters : q = flag for quantify ofer free variables ; 
chaieb@14758
   238
                a = flag for abstracting over function occurences
chaieb@14758
   239
                i = subgoal  *)
chaieb@14758
   240
chaieb@14758
   241
fun presburger_tac q a i = ObjectLogic.atomize_tac i THEN (fn st =>
berghofe@13876
   242
  let
chaieb@14758
   243
    val g = BasisLibrary.List.nth (prems_of st, i - 1)
chaieb@14758
   244
    val sg = sign_of_thm st
chaieb@14758
   245
    (* The Abstraction step *)
chaieb@14758
   246
    val g' = if a then abstract_pres sg g else g
berghofe@13876
   247
    (* Transform the term*)
chaieb@14758
   248
    val (t,np,nh) = prepare_for_presburger sg q g'
skalberg@15531
   249
    (* SOME simpsets for dealing with mod div abs and nat*)
berghofe@13876
   250
    val simpset0 = HOL_basic_ss
berghofe@13876
   251
      addsimps [mod_div_equality', Suc_plus1]
berghofe@13997
   252
      addsplits [split_zdiv, split_zmod, split_div', split_min, split_max]
berghofe@13876
   253
    (* Simp rules for changing (n::int) to int n *)
berghofe@13876
   254
    val simpset1 = HOL_basic_ss
berghofe@13876
   255
      addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)
berghofe@13876
   256
        [int_int_eq, zle_int, zless_int, zadd_int, zmult_int]
berghofe@13876
   257
      addsplits [zdiff_int_split]
berghofe@13876
   258
    (*simp rules for elimination of int n*)
berghofe@13876
   259
berghofe@13876
   260
    val simpset2 = HOL_basic_ss
berghofe@13876
   261
      addsimps [nat_0_le, all_nat, ex_nat, number_of1, number_of2, int_0, int_1]
berghofe@13876
   262
      addcongs [conj_le_cong, imp_le_cong]
berghofe@13876
   263
    (* simp rules for elimination of abs *)
paulson@14353
   264
    val simpset3 = HOL_basic_ss addsplits [abs_split]
berghofe@13876
   265
    val ct = cterm_of sg (HOLogic.mk_Trueprop t)
berghofe@13876
   266
    (* Theorem for the nat --> int transformation *)
berghofe@13876
   267
    val pre_thm = Seq.hd (EVERY
chaieb@14758
   268
      [simp_tac simpset0 1,
chaieb@14758
   269
       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),
berghofe@14801
   270
       TRY (simp_tac simpset3 1), TRY (simp_tac presburger_ss 1)]
berghofe@13876
   271
      (trivial ct))
chaieb@14758
   272
    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
berghofe@13876
   273
    (* The result of the quantifier elimination *)
berghofe@13876
   274
    val (th, tac) = case (prop_of pre_thm) of
berghofe@13876
   275
        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
chaieb@14920
   276
    let val pth = 
chaieb@14920
   277
          (* If quick_and_dirty then run without proof generation as oracle*)
chaieb@14920
   278
             if !quick_and_dirty 
chaieb@14941
   279
             then pres_oracle sg (Pattern.eta_long [] t1)
chaieb@14941
   280
(*
chaieb@14941
   281
assume (cterm_of sg 
chaieb@14920
   282
	       (HOLogic.mk_Trueprop(HOLogic.mk_eq(t1,CooperDec.integer_qelim (Pattern.eta_long [] t1)))))
chaieb@14941
   283
*)
chaieb@14920
   284
	     else tmproof_of_int_qelim sg (Pattern.eta_long [] t1)
chaieb@14920
   285
    in 
berghofe@13876
   286
          (trace_msg ("calling procedure with term:\n" ^
berghofe@13876
   287
             Sign.string_of_term sg t1);
chaieb@14920
   288
           ((pth RS iffD2) RS pre_thm,
berghofe@13876
   289
            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)))
chaieb@14920
   290
    end
berghofe@13876
   291
      | _ => (pre_thm, assm_tac i)
chaieb@14758
   292
  in (rtac (((mp_step nh) o (spec_step np)) th) i 
chaieb@14758
   293
      THEN tac) st
berghofe@14130
   294
  end handle Subscript => no_tac st | CooperDec.COOPER => no_tac st);
berghofe@13876
   295
berghofe@13876
   296
fun presburger_args meth =
chaieb@14758
   297
 let val parse_flag = 
chaieb@14758
   298
         Args.$$$ "no_quantify" >> K (apfst (K false))
chaieb@14811
   299
      || Args.$$$ "abs" >> K (apsnd (K true));
chaieb@14758
   300
 in
chaieb@14758
   301
   Method.simple_args 
wenzelm@14882
   302
  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
skalberg@15570
   303
    curry (Library.foldl op |>) (true, false))
chaieb@14758
   304
    (fn (q,a) => fn _ => meth q a 1)
chaieb@14758
   305
  end;
berghofe@13876
   306
chaieb@14758
   307
fun presburger_method q a i = Method.METHOD (fn facts =>
chaieb@14758
   308
  Method.insert_tac facts 1 THEN presburger_tac q a i)
berghofe@13876
   309
berghofe@13876
   310
val setup =
berghofe@13876
   311
  [Method.add_method ("presburger",
berghofe@13876
   312
     presburger_args presburger_method,
berghofe@13876
   313
     "decision procedure for Presburger arithmetic"),
berghofe@13876
   314
   ArithTheoryData.map (fn {splits, inj_consts, discrete, presburger} =>
berghofe@13876
   315
     {splits = splits, inj_consts = inj_consts, discrete = discrete,
skalberg@15531
   316
      presburger = SOME (presburger_tac true false)})];
berghofe@13876
   317
berghofe@13876
   318
end;
berghofe@13876
   319
chaieb@14920
   320
val presburger_tac = Presburger.presburger_tac true false;