src/ZF/arith_data.ML
author wenzelm
Thu Oct 29 17:58:26 2009 +0100 (2009-10-29)
changeset 33317 b4534348b8fd
parent 32155 e2bf2f73b0c8
child 35408 b48ab741683b
permissions -rw-r--r--
standardized filter/filter_out;
     1 (*  Title:      ZF/arith_data.ML
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 
     4 Arithmetic simplification: cancellation of common terms
     5 *)
     6 
     7 signature ARITH_DATA =
     8 sig
     9   (*the main outcome*)
    10   val nat_cancel: simproc list
    11   (*tools for use in similar applications*)
    12   val gen_trans_tac: thm -> thm option -> tactic
    13   val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option
    14   val simplify_meta_eq: thm list -> simpset -> thm -> thm
    15   (*debugging*)
    16   structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
    17   structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
    18   structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
    19 end;
    20 
    21 
    22 structure ArithData: ARITH_DATA =
    23 struct
    24 
    25 val iT = Ind_Syntax.iT;
    26 
    27 val zero = Const("0", iT);
    28 val succ = Const("succ", iT --> iT);
    29 fun mk_succ t = succ $ t;
    30 val one = mk_succ zero;
    31 
    32 val mk_plus = FOLogic.mk_binop "Arith.add";
    33 
    34 (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
    35 fun mk_sum []        = zero
    36   | mk_sum [t,u]     = mk_plus (t, u)
    37   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    38 
    39 (*this version ALWAYS includes a trailing zero*)
    40 fun long_mk_sum []        = zero
    41   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    42 
    43 val dest_plus = FOLogic.dest_bin "Arith.add" iT;
    44 
    45 (* dest_sum *)
    46 
    47 fun dest_sum (Const("0",_)) = []
    48   | dest_sum (Const("succ",_) $ t) = one :: dest_sum t
    49   | dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
    50   | dest_sum tm = [tm];
    51 
    52 (*Apply the given rewrite (if present) just once*)
    53 fun gen_trans_tac th2 NONE      = all_tac
    54   | gen_trans_tac th2 (SOME th) = ALLGOALS (rtac (th RS th2));
    55 
    56 (*Use <-> or = depending on the type of t*)
    57 fun mk_eq_iff(t,u) =
    58   if fastype_of t = iT then FOLogic.mk_eq(t,u)
    59                        else FOLogic.mk_iff(t,u);
    60 
    61 (*We remove equality assumptions because they confuse the simplifier and
    62   because only type-checking assumptions are necessary.*)
    63 fun is_eq_thm th =
    64     can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));
    65 
    66 fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
    67 
    68 fun prove_conv name tacs ctxt prems (t,u) =
    69   if t aconv u then NONE
    70   else
    71   let val prems' = filter_out is_eq_thm prems
    72       val goal = Logic.list_implies (map (#prop o Thm.rep_thm) prems',
    73         FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
    74   in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs)))
    75       handle ERROR msg =>
    76         (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)
    77   end;
    78 
    79 fun prep_simproc thy (name, pats, proc) =
    80   Simplifier.simproc thy name pats proc;
    81 
    82 
    83 (*** Use CancelNumerals simproc without binary numerals,
    84      just for cancellation ***)
    85 
    86 val mk_times = FOLogic.mk_binop "Arith.mult";
    87 
    88 fun mk_prod [] = one
    89   | mk_prod [t] = t
    90   | mk_prod (t :: ts) = if t = one then mk_prod ts
    91                         else mk_times (t, mk_prod ts);
    92 
    93 val dest_times = FOLogic.dest_bin "Arith.mult" iT;
    94 
    95 fun dest_prod t =
    96       let val (t,u) = dest_times t
    97       in  dest_prod t @ dest_prod u  end
    98       handle TERM _ => [t];
    99 
   100 (*Dummy version: the only arguments are 0 and 1*)
   101 fun mk_coeff (0, t) = zero
   102   | mk_coeff (1, t) = t
   103   | mk_coeff _       = raise TERM("mk_coeff", []);
   104 
   105 (*Dummy version: the "coefficient" is always 1.
   106   In the result, the factors are sorted terms*)
   107 fun dest_coeff t = (1, mk_prod (sort TermOrd.term_ord (dest_prod t)));
   108 
   109 (*Find first coefficient-term THAT MATCHES u*)
   110 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   111   | find_first_coeff past u (t::terms) =
   112         let val (n,u') = dest_coeff t
   113         in  if u aconv u' then (n, rev past @ terms)
   114                           else find_first_coeff (t::past) u terms
   115         end
   116         handle TERM _ => find_first_coeff (t::past) u terms;
   117 
   118 
   119 (*Simplify #1*n and n*#1 to n*)
   120 val add_0s = [@{thm add_0_natify}, @{thm add_0_right_natify}];
   121 val add_succs = [@{thm add_succ}, @{thm add_succ_right}];
   122 val mult_1s = [@{thm mult_1_natify}, @{thm mult_1_right_natify}];
   123 val tc_rules = [@{thm natify_in_nat}, @{thm add_type}, @{thm diff_type}, @{thm mult_type}];
   124 val natifys = [@{thm natify_0}, @{thm natify_ident}, @{thm add_natify1}, @{thm add_natify2},
   125                @{thm diff_natify1}, @{thm diff_natify2}];
   126 
   127 (*Final simplification: cancel + and **)
   128 fun simplify_meta_eq rules =
   129   let val ss0 =
   130     FOL_ss addeqcongs [@{thm eq_cong2}, @{thm iff_cong2}]
   131       delsimps @{thms iff_simps} (*these could erase the whole rule!*)
   132       addsimps rules
   133   in fn ss => mk_meta_eq o simplify (Simplifier.inherit_context ss ss0) end;
   134 
   135 val final_rules = add_0s @ mult_1s @ [@{thm mult_0}, @{thm mult_0_right}];
   136 
   137 structure CancelNumeralsCommon =
   138   struct
   139   val mk_sum            = (fn T:typ => mk_sum)
   140   val dest_sum          = dest_sum
   141   val mk_coeff          = mk_coeff
   142   val dest_coeff        = dest_coeff
   143   val find_first_coeff  = find_first_coeff []
   144 
   145   val norm_ss1 = ZF_ss addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac}
   146   val norm_ss2 = ZF_ss addsimps add_0s @ mult_1s @ @{thms add_ac} @
   147     @{thms mult_ac} @ tc_rules @ natifys
   148   fun norm_tac ss =
   149     ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
   150     THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
   151   val numeral_simp_ss = ZF_ss addsimps add_0s @ tc_rules @ natifys
   152   fun numeral_simp_tac ss =
   153     ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
   154   val simplify_meta_eq  = simplify_meta_eq final_rules
   155   end;
   156 
   157 (** The functor argumnets are declared as separate structures
   158     so that they can be exported to ease debugging. **)
   159 
   160 structure EqCancelNumeralsData =
   161   struct
   162   open CancelNumeralsCommon
   163   val prove_conv = prove_conv "nateq_cancel_numerals"
   164   val mk_bal   = FOLogic.mk_eq
   165   val dest_bal = FOLogic.dest_eq
   166   val bal_add1 = @{thm eq_add_iff} RS iff_trans
   167   val bal_add2 = @{thm eq_add_iff} RS iff_trans
   168   fun trans_tac _ = gen_trans_tac iff_trans
   169   end;
   170 
   171 structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
   172 
   173 structure LessCancelNumeralsData =
   174   struct
   175   open CancelNumeralsCommon
   176   val prove_conv = prove_conv "natless_cancel_numerals"
   177   val mk_bal   = FOLogic.mk_binrel "Ordinal.lt"
   178   val dest_bal = FOLogic.dest_bin "Ordinal.lt" iT
   179   val bal_add1 = @{thm less_add_iff} RS iff_trans
   180   val bal_add2 = @{thm less_add_iff} RS iff_trans
   181   fun trans_tac _ = gen_trans_tac iff_trans
   182   end;
   183 
   184 structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
   185 
   186 structure DiffCancelNumeralsData =
   187   struct
   188   open CancelNumeralsCommon
   189   val prove_conv = prove_conv "natdiff_cancel_numerals"
   190   val mk_bal   = FOLogic.mk_binop "Arith.diff"
   191   val dest_bal = FOLogic.dest_bin "Arith.diff" iT
   192   val bal_add1 = @{thm diff_add_eq} RS trans
   193   val bal_add2 = @{thm diff_add_eq} RS trans
   194   fun trans_tac _ = gen_trans_tac trans
   195   end;
   196 
   197 structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
   198 
   199 
   200 val nat_cancel =
   201   map (prep_simproc @{theory})
   202    [("nateq_cancel_numerals",
   203      ["l #+ m = n", "l = m #+ n",
   204       "l #* m = n", "l = m #* n",
   205       "succ(m) = n", "m = succ(n)"],
   206      (K EqCancelNumerals.proc)),
   207     ("natless_cancel_numerals",
   208      ["l #+ m < n", "l < m #+ n",
   209       "l #* m < n", "l < m #* n",
   210       "succ(m) < n", "m < succ(n)"],
   211      (K LessCancelNumerals.proc)),
   212     ("natdiff_cancel_numerals",
   213      ["(l #+ m) #- n", "l #- (m #+ n)",
   214       "(l #* m) #- n", "l #- (m #* n)",
   215       "succ(m) #- n", "m #- succ(n)"],
   216      (K DiffCancelNumerals.proc))];
   217 
   218 end;
   219 
   220 Addsimprocs ArithData.nat_cancel;
   221 
   222 
   223 (*examples:
   224 print_depth 22;
   225 set timing;
   226 set trace_simp;
   227 fun test s = (Goal s; by (Asm_simp_tac 1));
   228 
   229 test "x #+ y = x #+ z";
   230 test "y #+ x = x #+ z";
   231 test "x #+ y #+ z = x #+ z";
   232 test "y #+ (z #+ x) = z #+ x";
   233 test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
   234 test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
   235 
   236 test "x #+ succ(y) = x #+ z";
   237 test "x #+ succ(y) = succ(z #+ x)";
   238 test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
   239 
   240 test "(x #+ y) #- (x #+ z) = w";
   241 test "(y #+ x) #- (x #+ z) = dd";
   242 test "(x #+ y #+ z) #- (x #+ z) = dd";
   243 test "(y #+ (z #+ x)) #- (z #+ x) = dd";
   244 test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
   245 test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
   246 
   247 (*BAD occurrence of natify*)
   248 test "(x #+ succ(y)) #- (x #+ z) = dd";
   249 
   250 test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
   251 
   252 test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
   253 test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
   254 
   255 (*use of typing information*)
   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 < y#+x";
   260 test "x : nat ==> x le succ(x)";
   261 
   262 (*fails: no typing information isn't visible*)
   263 test "x #+ y = x";
   264 
   265 test "x #+ y < x #+ z";
   266 test "y #+ x < x #+ z";
   267 test "x #+ y #+ z < x #+ z";
   268 test "y #+ z #+ x < x #+ z";
   269 test "y #+ (z #+ x) < z #+ x";
   270 test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
   271 test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
   272 
   273 test "x #+ succ(y) < x #+ z";
   274 test "x #+ succ(y) < succ(z #+ x)";
   275 test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
   276 
   277 test "x #+ succ(y) le succ(z #+ x)";
   278 *)