src/ZF/arith_data.ML
author paulson
Wed Sep 06 11:48:51 2000 +0200 (2000-09-06)
changeset 9874 0aa0874ab66b
parent 9649 89155e48fa53
child 12134 7049eead7a50
permissions -rw-r--r--
bug fix for arithmetic simprocs (nat & int)
paulson@9548
     1
(*  Title:      ZF/arith_data.ML
paulson@9548
     2
    ID:         $Id$
paulson@9548
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@9548
     4
    Copyright   2000  University of Cambridge
paulson@9548
     5
paulson@9548
     6
Arithmetic simplification: cancellation of common terms
paulson@9548
     7
*)
paulson@9548
     8
paulson@9548
     9
signature ARITH_DATA =
paulson@9548
    10
sig
paulson@9570
    11
  (*the main outcome*)
paulson@9548
    12
  val nat_cancel: simproc list
paulson@9570
    13
  (*tools for use in similar applications*)
paulson@9570
    14
  val gen_trans_tac: thm -> thm option -> tactic
paulson@9570
    15
  val prove_conv: string -> tactic list -> Sign.sg -> 
paulson@9570
    16
                  thm list -> term * term -> thm option
paulson@9570
    17
  val simplify_meta_eq: thm list -> thm -> thm
paulson@9874
    18
  (*debugging*)
paulson@9874
    19
  structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
paulson@9874
    20
  structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
paulson@9874
    21
  structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
paulson@9548
    22
end;
paulson@9548
    23
paulson@9570
    24
paulson@9548
    25
structure ArithData: ARITH_DATA =
paulson@9548
    26
struct
paulson@9548
    27
paulson@9548
    28
val iT = Ind_Syntax.iT;
paulson@9548
    29
paulson@9548
    30
val zero = Const("0", iT);
paulson@9548
    31
val succ = Const("succ", iT --> iT);
paulson@9548
    32
fun mk_succ t = succ $ t;
paulson@9548
    33
val one = mk_succ zero;
paulson@9548
    34
paulson@9570
    35
val mk_plus = FOLogic.mk_binop "Arith.add";
paulson@9548
    36
paulson@9548
    37
(*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
paulson@9548
    38
fun mk_sum []        = zero
paulson@9548
    39
  | mk_sum [t,u]     = mk_plus (t, u)
paulson@9548
    40
  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
paulson@9548
    41
paulson@9548
    42
(*this version ALWAYS includes a trailing zero*)
paulson@9548
    43
fun long_mk_sum []        = zero
paulson@9548
    44
  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
paulson@9548
    45
paulson@9548
    46
val dest_plus = FOLogic.dest_bin "Arith.add" iT;
paulson@9548
    47
paulson@9548
    48
(* dest_sum *)
paulson@9548
    49
paulson@9548
    50
fun dest_sum (Const("0",_)) = []
paulson@9548
    51
  | dest_sum (Const("succ",_) $ t) = one :: dest_sum t
paulson@9548
    52
  | dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
paulson@9548
    53
  | dest_sum tm = [tm];
paulson@9548
    54
paulson@9548
    55
(*Apply the given rewrite (if present) just once*)
paulson@9548
    56
fun gen_trans_tac th2 None      = all_tac
paulson@9548
    57
  | gen_trans_tac th2 (Some th) = ALLGOALS (rtac (th RS th2));
paulson@9548
    58
paulson@9548
    59
(*Use <-> or = depending on the type of t*)
paulson@9548
    60
fun mk_eq_iff(t,u) =
paulson@9548
    61
  if fastype_of t = iT then FOLogic.mk_eq(t,u)
paulson@9548
    62
                       else FOLogic.mk_iff(t,u);
paulson@9548
    63
paulson@9874
    64
(*We remove equality assumptions because they confuse the simplifier and
paulson@9874
    65
  because only type-checking assumptions are necessary.*)
paulson@9874
    66
fun is_eq_thm th = 
paulson@9874
    67
    can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));
paulson@9649
    68
paulson@9548
    69
fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
paulson@9548
    70
paulson@9548
    71
fun prove_conv name tacs sg hyps (t,u) =
paulson@9548
    72
  if t aconv u then None
paulson@9548
    73
  else
paulson@9874
    74
  let val hyps' = filter (not o is_eq_thm) hyps
paulson@9649
    75
      val ct = add_chyps hyps'
paulson@9548
    76
                  (cterm_of sg (FOLogic.mk_Trueprop (mk_eq_iff(t, u))))
paulson@9548
    77
  in Some
paulson@9649
    78
      (hyps' MRS 
paulson@9649
    79
       (prove_goalw_cterm [] ct 
paulson@9548
    80
	(fn prems => cut_facts_tac prems 1 :: tacs)))
paulson@9548
    81
      handle ERROR => 
paulson@9548
    82
	(warning 
paulson@9548
    83
	 ("Cancellation failed: no typing information? (" ^ name ^ ")"); 
paulson@9548
    84
	 None)
paulson@9548
    85
  end;
paulson@9548
    86
paulson@9548
    87
fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
paulson@9548
    88
fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context ()))
paulson@9548
    89
                      (s, TypeInfer.anyT ["logic"]);
paulson@9548
    90
val prep_pats = map prep_pat;
paulson@9548
    91
paulson@9548
    92
paulson@9548
    93
(*** Use CancelNumerals simproc without binary numerals, 
paulson@9548
    94
     just for cancellation ***)
paulson@9548
    95
paulson@9570
    96
val mk_times = FOLogic.mk_binop "Arith.mult";
paulson@9548
    97
paulson@9548
    98
fun mk_prod [] = one
paulson@9548
    99
  | mk_prod [t] = t
paulson@9548
   100
  | mk_prod (t :: ts) = if t = one then mk_prod ts
paulson@9548
   101
                        else mk_times (t, mk_prod ts);
paulson@9548
   102
paulson@9548
   103
val dest_times = FOLogic.dest_bin "Arith.mult" iT;
paulson@9548
   104
paulson@9548
   105
fun dest_prod t =
paulson@9548
   106
      let val (t,u) = dest_times t
paulson@9548
   107
      in  dest_prod t @ dest_prod u  end
paulson@9548
   108
      handle TERM _ => [t];
paulson@9548
   109
paulson@9548
   110
(*Dummy version: the only arguments are 0 and 1*)
paulson@9548
   111
fun mk_coeff (0, t) = zero
paulson@9548
   112
  | mk_coeff (1, t) = t
paulson@9548
   113
  | mk_coeff _       = raise TERM("mk_coeff", []);
paulson@9548
   114
paulson@9548
   115
(*Dummy version: the "coefficient" is always 1.
paulson@9548
   116
  In the result, the factors are sorted terms*)
paulson@9548
   117
fun dest_coeff t = (1, mk_prod (sort Term.term_ord (dest_prod t)));
paulson@9548
   118
paulson@9548
   119
(*Find first coefficient-term THAT MATCHES u*)
paulson@9548
   120
fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
paulson@9548
   121
  | find_first_coeff past u (t::terms) =
paulson@9548
   122
        let val (n,u') = dest_coeff t
paulson@9548
   123
        in  if u aconv u' then (n, rev past @ terms)
paulson@9548
   124
                          else find_first_coeff (t::past) u terms
paulson@9548
   125
        end
paulson@9548
   126
        handle TERM _ => find_first_coeff (t::past) u terms;
paulson@9548
   127
paulson@9548
   128
paulson@9548
   129
(*Simplify #1*n and n*#1 to n*)
paulson@9548
   130
val add_0s = [add_0_natify, add_0_right_natify];
paulson@9548
   131
val add_succs = [add_succ, add_succ_right];
paulson@9548
   132
val mult_1s = [mult_1_natify, mult_1_right_natify];
paulson@9548
   133
val tc_rules = [natify_in_nat, add_type, diff_type, mult_type];
paulson@9548
   134
val natifys = [natify_0, natify_ident, add_natify1, add_natify2,
paulson@9570
   135
               diff_natify1, diff_natify2];
paulson@9548
   136
paulson@9548
   137
(*Final simplification: cancel + and **)
paulson@9548
   138
fun simplify_meta_eq rules =
paulson@9548
   139
    mk_meta_eq o
paulson@9548
   140
    simplify (FOL_ss addeqcongs[eq_cong2,iff_cong2] 
paulson@9548
   141
                     delsimps iff_simps (*these could erase the whole rule!*)
paulson@9570
   142
		     addsimps rules);
paulson@9548
   143
paulson@9548
   144
val final_rules = add_0s @ mult_1s @ [mult_0, mult_0_right];
paulson@9548
   145
paulson@9548
   146
structure CancelNumeralsCommon =
paulson@9548
   147
  struct
paulson@9548
   148
  val mk_sum            = mk_sum
paulson@9548
   149
  val dest_sum          = dest_sum
paulson@9548
   150
  val mk_coeff          = mk_coeff
paulson@9548
   151
  val dest_coeff        = dest_coeff
paulson@9548
   152
  val find_first_coeff  = find_first_coeff []
paulson@9548
   153
  val norm_tac_ss1 = ZF_ss addsimps add_0s@add_succs@mult_1s@add_ac
paulson@9548
   154
  val norm_tac_ss2 = ZF_ss addsimps add_ac@mult_ac@tc_rules@natifys
paulson@9548
   155
  val norm_tac = ALLGOALS (asm_simp_tac norm_tac_ss1)
paulson@9548
   156
                 THEN ALLGOALS (asm_simp_tac norm_tac_ss2)
paulson@9548
   157
  val numeral_simp_tac_ss = ZF_ss addsimps add_0s@tc_rules@natifys
paulson@9548
   158
  val numeral_simp_tac  = ALLGOALS (asm_simp_tac numeral_simp_tac_ss)
paulson@9548
   159
  val simplify_meta_eq  = simplify_meta_eq final_rules
paulson@9548
   160
  end;
paulson@9548
   161
paulson@9874
   162
(** The functor argumnets are declared as separate structures
paulson@9874
   163
    so that they can be exported to ease debugging. **)
paulson@9548
   164
paulson@9874
   165
structure EqCancelNumeralsData = 
paulson@9874
   166
  struct
paulson@9874
   167
  open CancelNumeralsCommon
paulson@9548
   168
  val prove_conv = prove_conv "nateq_cancel_numerals"
paulson@9548
   169
  val mk_bal   = FOLogic.mk_eq
paulson@9649
   170
  val dest_bal = FOLogic.dest_eq
paulson@9548
   171
  val bal_add1 = eq_add_iff RS iff_trans
paulson@9548
   172
  val bal_add2 = eq_add_iff RS iff_trans
paulson@9548
   173
  val trans_tac = gen_trans_tac iff_trans
paulson@9874
   174
  end;
paulson@9874
   175
paulson@9874
   176
structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
paulson@9548
   177
paulson@9874
   178
structure LessCancelNumeralsData = 
paulson@9874
   179
  struct
paulson@9874
   180
  open CancelNumeralsCommon
paulson@9548
   181
  val prove_conv = prove_conv "natless_cancel_numerals"
paulson@9570
   182
  val mk_bal   = FOLogic.mk_binrel "Ordinal.op <"
paulson@9548
   183
  val dest_bal = FOLogic.dest_bin "Ordinal.op <" iT
paulson@9548
   184
  val bal_add1 = less_add_iff RS iff_trans
paulson@9548
   185
  val bal_add2 = less_add_iff RS iff_trans
paulson@9548
   186
  val trans_tac = gen_trans_tac iff_trans
paulson@9874
   187
  end;
paulson@9874
   188
paulson@9874
   189
structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
paulson@9548
   190
paulson@9874
   191
structure DiffCancelNumeralsData = 
paulson@9874
   192
  struct
paulson@9874
   193
  open CancelNumeralsCommon
paulson@9548
   194
  val prove_conv = prove_conv "natdiff_cancel_numerals"
paulson@9570
   195
  val mk_bal   = FOLogic.mk_binop "Arith.diff"
paulson@9548
   196
  val dest_bal = FOLogic.dest_bin "Arith.diff" iT
paulson@9548
   197
  val bal_add1 = diff_add_eq RS trans
paulson@9548
   198
  val bal_add2 = diff_add_eq RS trans
paulson@9548
   199
  val trans_tac = gen_trans_tac trans
paulson@9874
   200
  end;
paulson@9874
   201
paulson@9874
   202
structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
paulson@9548
   203
paulson@9548
   204
paulson@9548
   205
val nat_cancel =
paulson@9548
   206
      map prep_simproc
paulson@9548
   207
       [("nateq_cancel_numerals",
paulson@9548
   208
	 prep_pats ["l #+ m = n", "l = m #+ n",
paulson@9548
   209
		    "l #* m = n", "l = m #* n",
paulson@9548
   210
		    "succ(m) = n", "m = succ(n)"],
paulson@9548
   211
	 EqCancelNumerals.proc),
paulson@9548
   212
	("natless_cancel_numerals",
paulson@9548
   213
	 prep_pats ["l #+ m < n", "l < m #+ n",
paulson@9548
   214
		    "l #* m < n", "l < m #* n",
paulson@9548
   215
		    "succ(m) < n", "m < succ(n)"],
paulson@9548
   216
	 LessCancelNumerals.proc),
paulson@9548
   217
	("natdiff_cancel_numerals",
paulson@9548
   218
	 prep_pats ["(l #+ m) #- n", "l #- (m #+ n)",
paulson@9548
   219
		    "(l #* m) #- n", "l #- (m #* n)",
paulson@9548
   220
		    "succ(m) #- n", "m #- succ(n)"],
paulson@9548
   221
	 DiffCancelNumerals.proc)];
paulson@9548
   222
paulson@9548
   223
end;
paulson@9548
   224
paulson@9548
   225
(*examples:
paulson@9548
   226
print_depth 22;
paulson@9548
   227
set timing;
paulson@9548
   228
set trace_simp;
paulson@9548
   229
fun test s = (Goal s; by (Asm_simp_tac 1));
paulson@9548
   230
paulson@9548
   231
test "x #+ y = x #+ z";
paulson@9548
   232
test "y #+ x = x #+ z";
paulson@9548
   233
test "x #+ y #+ z = x #+ z";
paulson@9548
   234
test "y #+ (z #+ x) = z #+ x";
paulson@9548
   235
test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
paulson@9548
   236
test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
paulson@9548
   237
paulson@9548
   238
test "x #+ succ(y) = x #+ z";
paulson@9548
   239
test "x #+ succ(y) = succ(z #+ x)";
paulson@9548
   240
test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
paulson@9548
   241
paulson@9548
   242
test "(x #+ y) #- (x #+ z) = w";
paulson@9548
   243
test "(y #+ x) #- (x #+ z) = dd";
paulson@9548
   244
test "(x #+ y #+ z) #- (x #+ z) = dd";
paulson@9548
   245
test "(y #+ (z #+ x)) #- (z #+ x) = dd";
paulson@9548
   246
test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
paulson@9548
   247
test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
paulson@9548
   248
paulson@9548
   249
(*BAD occurrence of natify*)
paulson@9548
   250
test "(x #+ succ(y)) #- (x #+ z) = dd";
paulson@9548
   251
paulson@9548
   252
test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
paulson@9548
   253
paulson@9548
   254
test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
paulson@9548
   255
test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
paulson@9548
   256
paulson@9548
   257
(*use of typing information*)
paulson@9548
   258
test "x : nat ==> x #+ y = x";
paulson@9548
   259
test "x : nat --> x #+ y = x";
paulson@9548
   260
test "x : nat ==> x #+ y < x";
paulson@9548
   261
test "x : nat ==> x < y#+x";
paulson@9548
   262
paulson@9548
   263
(*fails: no typing information isn't visible*)
paulson@9548
   264
test "x #+ y = x";
paulson@9548
   265
paulson@9548
   266
test "x #+ y < x #+ z";
paulson@9548
   267
test "y #+ x < x #+ z";
paulson@9548
   268
test "x #+ y #+ z < x #+ z";
paulson@9548
   269
test "y #+ z #+ x < x #+ z";
paulson@9548
   270
test "y #+ (z #+ x) < z #+ x";
paulson@9548
   271
test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
paulson@9548
   272
test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
paulson@9548
   273
paulson@9548
   274
test "x #+ succ(y) < x #+ z";
paulson@9548
   275
test "x #+ succ(y) < succ(z #+ x)";
paulson@9548
   276
test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
paulson@9548
   277
paulson@9548
   278
test "x #+ succ(y) le succ(z #+ x)";
paulson@9548
   279
*)