src/ZF/arith_data.ML
author wenzelm
Sat Aug 05 14:52:57 2006 +0200 (2006-08-05)
changeset 20342 4392003fcbfa
parent 20113 90a8d14f3610
child 24630 351a308ab58d
permissions -rw-r--r--
tuned;
     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 -> Proof.context -> thm list -> term * term -> thm option
    16   val simplify_meta_eq: thm list -> simpset -> thm -> thm
    17   (*debugging*)
    18   structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
    19   structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
    20   structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
    21 end;
    22 
    23 
    24 structure ArithData: ARITH_DATA =
    25 struct
    26 
    27 val iT = Ind_Syntax.iT;
    28 
    29 val zero = Const("0", iT);
    30 val succ = Const("succ", iT --> iT);
    31 fun mk_succ t = succ $ t;
    32 val one = mk_succ zero;
    33 
    34 val mk_plus = FOLogic.mk_binop "Arith.add";
    35 
    36 (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
    37 fun mk_sum []        = zero
    38   | mk_sum [t,u]     = mk_plus (t, u)
    39   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    40 
    41 (*this version ALWAYS includes a trailing zero*)
    42 fun long_mk_sum []        = zero
    43   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    44 
    45 val dest_plus = FOLogic.dest_bin "Arith.add" iT;
    46 
    47 (* dest_sum *)
    48 
    49 fun dest_sum (Const("0",_)) = []
    50   | dest_sum (Const("succ",_) $ t) = one :: dest_sum t
    51   | dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
    52   | dest_sum tm = [tm];
    53 
    54 (*Apply the given rewrite (if present) just once*)
    55 fun gen_trans_tac th2 NONE      = all_tac
    56   | gen_trans_tac th2 (SOME th) = ALLGOALS (rtac (th RS th2));
    57 
    58 (*Use <-> or = depending on the type of t*)
    59 fun mk_eq_iff(t,u) =
    60   if fastype_of t = iT then FOLogic.mk_eq(t,u)
    61                        else FOLogic.mk_iff(t,u);
    62 
    63 (*We remove equality assumptions because they confuse the simplifier and
    64   because only type-checking assumptions are necessary.*)
    65 fun is_eq_thm th =
    66     can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));
    67 
    68 fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
    69 
    70 fun prove_conv name tacs ctxt prems (t,u) =
    71   if t aconv u then NONE
    72   else
    73   let val prems' = List.filter (not o is_eq_thm) prems
    74       val goal = Logic.list_implies (map (#prop o Thm.rep_thm) prems',
    75         FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
    76   in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs)))
    77       handle ERROR msg =>
    78         (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)
    79   end;
    80 
    81 fun prep_simproc (name, pats, proc) =
    82   Simplifier.simproc (the_context ()) name pats proc;
    83 
    84 
    85 (*** Use CancelNumerals simproc without binary numerals,
    86      just for cancellation ***)
    87 
    88 val mk_times = FOLogic.mk_binop "Arith.mult";
    89 
    90 fun mk_prod [] = one
    91   | mk_prod [t] = t
    92   | mk_prod (t :: ts) = if t = one then mk_prod ts
    93                         else mk_times (t, mk_prod ts);
    94 
    95 val dest_times = FOLogic.dest_bin "Arith.mult" iT;
    96 
    97 fun dest_prod t =
    98       let val (t,u) = dest_times t
    99       in  dest_prod t @ dest_prod u  end
   100       handle TERM _ => [t];
   101 
   102 (*Dummy version: the only arguments are 0 and 1*)
   103 fun mk_coeff (0: IntInf.int, t) = zero
   104   | mk_coeff (1, t) = t
   105   | mk_coeff _       = raise TERM("mk_coeff", []);
   106 
   107 (*Dummy version: the "coefficient" is always 1.
   108   In the result, the factors are sorted terms*)
   109 fun dest_coeff t = (1 : IntInf.int, mk_prod (sort Term.term_ord (dest_prod t)));
   110 
   111 (*Find first coefficient-term THAT MATCHES u*)
   112 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   113   | find_first_coeff past u (t::terms) =
   114         let val (n,u') = dest_coeff t
   115         in  if u aconv u' then (n, rev past @ terms)
   116                           else find_first_coeff (t::past) u terms
   117         end
   118         handle TERM _ => find_first_coeff (t::past) u terms;
   119 
   120 
   121 (*Simplify #1*n and n*#1 to n*)
   122 val add_0s = [add_0_natify, add_0_right_natify];
   123 val add_succs = [add_succ, add_succ_right];
   124 val mult_1s = [mult_1_natify, mult_1_right_natify];
   125 val tc_rules = [natify_in_nat, add_type, diff_type, mult_type];
   126 val natifys = [natify_0, natify_ident, add_natify1, add_natify2,
   127                diff_natify1, diff_natify2];
   128 
   129 (*Final simplification: cancel + and **)
   130 fun simplify_meta_eq rules =
   131   let val ss0 =
   132     FOL_ss addeqcongs [eq_cong2, iff_cong2]
   133       delsimps iff_simps (*these could erase the whole rule!*)
   134       addsimps rules
   135   in fn ss => mk_meta_eq o simplify (Simplifier.inherit_context ss ss0) end;
   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 
   147   val norm_ss1 = ZF_ss addsimps add_0s @ add_succs @ mult_1s @ add_ac
   148   val norm_ss2 = ZF_ss addsimps add_0s @ mult_1s @ add_ac @ mult_ac @ tc_rules @ natifys
   149   fun norm_tac ss =
   150     ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
   151     THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
   152   val numeral_simp_ss = ZF_ss addsimps add_0s @ tc_rules @ natifys
   153   fun numeral_simp_tac ss =
   154     ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   155   val simplify_meta_eq  = simplify_meta_eq final_rules
   156   end;
   157 
   158 (** The functor argumnets are declared as separate structures
   159     so that they can be exported to ease debugging. **)
   160 
   161 structure EqCancelNumeralsData =
   162   struct
   163   open CancelNumeralsCommon
   164   val prove_conv = prove_conv "nateq_cancel_numerals"
   165   val mk_bal   = FOLogic.mk_eq
   166   val dest_bal = FOLogic.dest_eq
   167   val bal_add1 = eq_add_iff RS iff_trans
   168   val bal_add2 = eq_add_iff RS iff_trans
   169   fun trans_tac _ = gen_trans_tac iff_trans
   170   end;
   171 
   172 structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
   173 
   174 structure LessCancelNumeralsData =
   175   struct
   176   open CancelNumeralsCommon
   177   val prove_conv = prove_conv "natless_cancel_numerals"
   178   val mk_bal   = FOLogic.mk_binrel "Ordinal.lt"
   179   val dest_bal = FOLogic.dest_bin "Ordinal.lt" iT
   180   val bal_add1 = less_add_iff RS iff_trans
   181   val bal_add2 = less_add_iff RS iff_trans
   182   fun trans_tac _ = gen_trans_tac iff_trans
   183   end;
   184 
   185 structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
   186 
   187 structure DiffCancelNumeralsData =
   188   struct
   189   open CancelNumeralsCommon
   190   val prove_conv = prove_conv "natdiff_cancel_numerals"
   191   val mk_bal   = FOLogic.mk_binop "Arith.diff"
   192   val dest_bal = FOLogic.dest_bin "Arith.diff" iT
   193   val bal_add1 = diff_add_eq RS trans
   194   val bal_add2 = diff_add_eq RS trans
   195   fun trans_tac _ = gen_trans_tac trans
   196   end;
   197 
   198 structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
   199 
   200 
   201 val nat_cancel =
   202   map prep_simproc
   203    [("nateq_cancel_numerals",
   204      ["l #+ m = n", "l = m #+ n",
   205       "l #* m = n", "l = m #* n",
   206       "succ(m) = n", "m = succ(n)"],
   207      (K EqCancelNumerals.proc)),
   208     ("natless_cancel_numerals",
   209      ["l #+ m < n", "l < m #+ n",
   210       "l #* m < n", "l < m #* n",
   211       "succ(m) < n", "m < succ(n)"],
   212      (K LessCancelNumerals.proc)),
   213     ("natdiff_cancel_numerals",
   214      ["(l #+ m) #- n", "l #- (m #+ n)",
   215       "(l #* m) #- n", "l #- (m #* n)",
   216       "succ(m) #- n", "m #- succ(n)"],
   217      (K DiffCancelNumerals.proc))];
   218 
   219 end;
   220 
   221 Addsimprocs ArithData.nat_cancel;
   222 
   223 
   224 (*examples:
   225 print_depth 22;
   226 set timing;
   227 set trace_simp;
   228 fun test s = (Goal s; by (Asm_simp_tac 1));
   229 
   230 test "x #+ y = x #+ z";
   231 test "y #+ x = x #+ z";
   232 test "x #+ y #+ z = x #+ z";
   233 test "y #+ (z #+ x) = z #+ x";
   234 test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
   235 test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
   236 
   237 test "x #+ succ(y) = x #+ z";
   238 test "x #+ succ(y) = succ(z #+ x)";
   239 test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
   240 
   241 test "(x #+ y) #- (x #+ z) = w";
   242 test "(y #+ x) #- (x #+ z) = dd";
   243 test "(x #+ y #+ z) #- (x #+ z) = dd";
   244 test "(y #+ (z #+ x)) #- (z #+ x) = dd";
   245 test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
   246 test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
   247 
   248 (*BAD occurrence of natify*)
   249 test "(x #+ succ(y)) #- (x #+ z) = dd";
   250 
   251 test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
   252 
   253 test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
   254 test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
   255 
   256 (*use of typing information*)
   257 test "x : nat ==> x #+ y = x";
   258 test "x : nat --> x #+ y = x";
   259 test "x : nat ==> x #+ y < x";
   260 test "x : nat ==> x < y#+x";
   261 test "x : nat ==> x le succ(x)";
   262 
   263 (*fails: no typing information isn't visible*)
   264 test "x #+ y = x";
   265 
   266 test "x #+ y < x #+ z";
   267 test "y #+ x < x #+ z";
   268 test "x #+ y #+ z < x #+ z";
   269 test "y #+ z #+ x < x #+ z";
   270 test "y #+ (z #+ x) < z #+ x";
   271 test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
   272 test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
   273 
   274 test "x #+ succ(y) < x #+ z";
   275 test "x #+ succ(y) < succ(z #+ x)";
   276 test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
   277 
   278 test "x #+ succ(y) le succ(z #+ x)";
   279 *)