src/ZF/arith_data.ML
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 13487 1291c6375c29
child 15531 08c8dad8e399
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
     1 (*  Title:      ZF/arith_data.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   2000  University of Cambridge
     5 
     6 Arithmetic simplification: cancellation of common terms
     7 *)
     8 
     9 signature ARITH_DATA =
    10 sig
    11   (*the main outcome*)
    12   val nat_cancel: simproc list
    13   (*tools for use in similar applications*)
    14   val gen_trans_tac: thm -> thm option -> tactic
    15   val prove_conv: string -> tactic list -> Sign.sg ->
    16                   thm list -> string list -> term * term -> thm option
    17   val simplify_meta_eq: thm list -> thm -> thm
    18   (*debugging*)
    19   structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
    20   structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
    21   structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
    22 end;
    23 
    24 
    25 structure ArithData: ARITH_DATA =
    26 struct
    27 
    28 val iT = Ind_Syntax.iT;
    29 
    30 val zero = Const("0", iT);
    31 val succ = Const("succ", iT --> iT);
    32 fun mk_succ t = succ $ t;
    33 val one = mk_succ zero;
    34 
    35 val mk_plus = FOLogic.mk_binop "Arith.add";
    36 
    37 (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
    38 fun mk_sum []        = zero
    39   | mk_sum [t,u]     = mk_plus (t, u)
    40   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    41 
    42 (*this version ALWAYS includes a trailing zero*)
    43 fun long_mk_sum []        = zero
    44   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    45 
    46 val dest_plus = FOLogic.dest_bin "Arith.add" iT;
    47 
    48 (* dest_sum *)
    49 
    50 fun dest_sum (Const("0",_)) = []
    51   | dest_sum (Const("succ",_) $ t) = one :: dest_sum t
    52   | dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
    53   | dest_sum tm = [tm];
    54 
    55 (*Apply the given rewrite (if present) just once*)
    56 fun gen_trans_tac th2 None      = all_tac
    57   | gen_trans_tac th2 (Some th) = ALLGOALS (rtac (th RS th2));
    58 
    59 (*Use <-> or = depending on the type of t*)
    60 fun mk_eq_iff(t,u) =
    61   if fastype_of t = iT then FOLogic.mk_eq(t,u)
    62                        else FOLogic.mk_iff(t,u);
    63 
    64 (*We remove equality assumptions because they confuse the simplifier and
    65   because only type-checking assumptions are necessary.*)
    66 fun is_eq_thm th =
    67     can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));
    68 
    69 fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
    70 
    71 fun prove_conv name tacs sg hyps xs (t,u) =
    72   if t aconv u then None
    73   else
    74   let val hyps' = filter (not o is_eq_thm) hyps
    75       val goal = Logic.list_implies (map (#prop o Thm.rep_thm) hyps',
    76         FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
    77   in Some (hyps' MRS Tactic.prove sg xs [] goal (K (EVERY tacs)))
    78       handle ERROR_MESSAGE msg =>
    79         (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); None)
    80   end;
    81 
    82 fun prep_simproc (name, pats, proc) =
    83   Simplifier.simproc (Theory.sign_of (the_context ())) name pats proc;
    84 
    85 
    86 (*** Use CancelNumerals simproc without binary numerals,
    87      just for cancellation ***)
    88 
    89 val mk_times = FOLogic.mk_binop "Arith.mult";
    90 
    91 fun mk_prod [] = one
    92   | mk_prod [t] = t
    93   | mk_prod (t :: ts) = if t = one then mk_prod ts
    94                         else mk_times (t, mk_prod ts);
    95 
    96 val dest_times = FOLogic.dest_bin "Arith.mult" iT;
    97 
    98 fun dest_prod t =
    99       let val (t,u) = dest_times t
   100       in  dest_prod t @ dest_prod u  end
   101       handle TERM _ => [t];
   102 
   103 (*Dummy version: the only arguments are 0 and 1*)
   104 fun mk_coeff (0, t) = zero
   105   | mk_coeff (1, t) = t
   106   | mk_coeff _       = raise TERM("mk_coeff", []);
   107 
   108 (*Dummy version: the "coefficient" is always 1.
   109   In the result, the factors are sorted terms*)
   110 fun dest_coeff t = (1, mk_prod (sort Term.term_ord (dest_prod t)));
   111 
   112 (*Find first coefficient-term THAT MATCHES u*)
   113 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   114   | find_first_coeff past u (t::terms) =
   115         let val (n,u') = dest_coeff t
   116         in  if u aconv u' then (n, rev past @ terms)
   117                           else find_first_coeff (t::past) u terms
   118         end
   119         handle TERM _ => find_first_coeff (t::past) u terms;
   120 
   121 
   122 (*Simplify #1*n and n*#1 to n*)
   123 val add_0s = [add_0_natify, add_0_right_natify];
   124 val add_succs = [add_succ, add_succ_right];
   125 val mult_1s = [mult_1_natify, mult_1_right_natify];
   126 val tc_rules = [natify_in_nat, add_type, diff_type, mult_type];
   127 val natifys = [natify_0, natify_ident, add_natify1, add_natify2,
   128                diff_natify1, diff_natify2];
   129 
   130 (*Final simplification: cancel + and **)
   131 fun simplify_meta_eq rules =
   132     mk_meta_eq o
   133     simplify (FOL_ss addeqcongs[eq_cong2,iff_cong2]
   134                      delsimps iff_simps (*these could erase the whole rule!*)
   135                      addsimps rules);
   136 
   137 val final_rules = add_0s @ mult_1s @ [mult_0, mult_0_right];
   138 
   139 structure CancelNumeralsCommon =
   140   struct
   141   val mk_sum            = (fn T:typ => mk_sum)
   142   val dest_sum          = dest_sum
   143   val mk_coeff          = mk_coeff
   144   val dest_coeff        = dest_coeff
   145   val find_first_coeff  = find_first_coeff []
   146   val norm_tac_ss1 = ZF_ss addsimps add_0s@add_succs@mult_1s@add_ac
   147   val norm_tac_ss2 = ZF_ss addsimps add_0s@mult_1s@
   148                                     add_ac@mult_ac@tc_rules@natifys
   149   val norm_tac = ALLGOALS (asm_simp_tac norm_tac_ss1)
   150                  THEN ALLGOALS (asm_simp_tac norm_tac_ss2)
   151   val numeral_simp_tac_ss = ZF_ss addsimps add_0s@tc_rules@natifys
   152   val numeral_simp_tac  = ALLGOALS (asm_simp_tac numeral_simp_tac_ss)
   153   val simplify_meta_eq  = simplify_meta_eq final_rules
   154   end;
   155 
   156 (** The functor argumnets are declared as separate structures
   157     so that they can be exported to ease debugging. **)
   158 
   159 structure EqCancelNumeralsData =
   160   struct
   161   open CancelNumeralsCommon
   162   val prove_conv = prove_conv "nateq_cancel_numerals"
   163   val mk_bal   = FOLogic.mk_eq
   164   val dest_bal = FOLogic.dest_eq
   165   val bal_add1 = eq_add_iff RS iff_trans
   166   val bal_add2 = eq_add_iff RS iff_trans
   167   val trans_tac = gen_trans_tac iff_trans
   168   end;
   169 
   170 structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
   171 
   172 structure LessCancelNumeralsData =
   173   struct
   174   open CancelNumeralsCommon
   175   val prove_conv = prove_conv "natless_cancel_numerals"
   176   val mk_bal   = FOLogic.mk_binrel "Ordinal.lt"
   177   val dest_bal = FOLogic.dest_bin "Ordinal.lt" iT
   178   val bal_add1 = less_add_iff RS iff_trans
   179   val bal_add2 = less_add_iff RS iff_trans
   180   val trans_tac = gen_trans_tac iff_trans
   181   end;
   182 
   183 structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
   184 
   185 structure DiffCancelNumeralsData =
   186   struct
   187   open CancelNumeralsCommon
   188   val prove_conv = prove_conv "natdiff_cancel_numerals"
   189   val mk_bal   = FOLogic.mk_binop "Arith.diff"
   190   val dest_bal = FOLogic.dest_bin "Arith.diff" iT
   191   val bal_add1 = diff_add_eq RS trans
   192   val bal_add2 = diff_add_eq RS trans
   193   val trans_tac = gen_trans_tac trans
   194   end;
   195 
   196 structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
   197 
   198 
   199 val nat_cancel =
   200   map prep_simproc
   201    [("nateq_cancel_numerals",
   202      ["l #+ m = n", "l = m #+ n",
   203       "l #* m = n", "l = m #* n",
   204       "succ(m) = n", "m = succ(n)"],
   205      EqCancelNumerals.proc),
   206     ("natless_cancel_numerals",
   207      ["l #+ m < n", "l < m #+ n",
   208       "l #* m < n", "l < m #* n",
   209       "succ(m) < n", "m < succ(n)"],
   210      LessCancelNumerals.proc),
   211     ("natdiff_cancel_numerals",
   212      ["(l #+ m) #- n", "l #- (m #+ n)",
   213       "(l #* m) #- n", "l #- (m #* n)",
   214       "succ(m) #- n", "m #- succ(n)"],
   215      DiffCancelNumerals.proc)];
   216 
   217 end;
   218 
   219 Addsimprocs ArithData.nat_cancel;
   220 
   221 
   222 (*examples:
   223 print_depth 22;
   224 set timing;
   225 set trace_simp;
   226 fun test s = (Goal s; by (Asm_simp_tac 1));
   227 
   228 test "x #+ y = x #+ z";
   229 test "y #+ x = x #+ z";
   230 test "x #+ y #+ z = x #+ z";
   231 test "y #+ (z #+ x) = z #+ x";
   232 test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
   233 test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
   234 
   235 test "x #+ succ(y) = x #+ z";
   236 test "x #+ succ(y) = succ(z #+ x)";
   237 test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
   238 
   239 test "(x #+ y) #- (x #+ z) = w";
   240 test "(y #+ x) #- (x #+ z) = dd";
   241 test "(x #+ y #+ z) #- (x #+ z) = dd";
   242 test "(y #+ (z #+ x)) #- (z #+ x) = dd";
   243 test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
   244 test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
   245 
   246 (*BAD occurrence of natify*)
   247 test "(x #+ succ(y)) #- (x #+ z) = dd";
   248 
   249 test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
   250 
   251 test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
   252 test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
   253 
   254 (*use of typing information*)
   255 test "x : nat ==> x #+ y = x";
   256 test "x : nat --> x #+ y = x";
   257 test "x : nat ==> x #+ y < x";
   258 test "x : nat ==> x < y#+x";
   259 test "x : nat ==> x le succ(x)";
   260 
   261 (*fails: no typing information isn't visible*)
   262 test "x #+ y = x";
   263 
   264 test "x #+ y < x #+ z";
   265 test "y #+ x < x #+ z";
   266 test "x #+ y #+ z < x #+ z";
   267 test "y #+ z #+ x < x #+ z";
   268 test "y #+ (z #+ x) < z #+ x";
   269 test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
   270 test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
   271 
   272 test "x #+ succ(y) < x #+ z";
   273 test "x #+ succ(y) < succ(z #+ x)";
   274 test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
   275 
   276 test "x #+ succ(y) le succ(z #+ x)";
   277 *)