src/ZF/int_arith.ML
author wenzelm
Tue Sep 01 22:32:58 2015 +0200 (2015-09-01)
changeset 61076 bdc1e2f0a86a
parent 59748 a1c35e6fe735
child 61144 5e94dfead1c2
permissions -rw-r--r--
eliminated \<Colon>;
     1 (*  Title:      ZF/int_arith.ML
     2     Author:     Larry Paulson
     3 
     4 Simprocs for linear arithmetic.
     5 *)
     6 
     7 signature INT_NUMERAL_SIMPROCS =
     8 sig
     9   val cancel_numerals: simproc list
    10   val combine_numerals: simproc
    11   val combine_numerals_prod: simproc
    12 end
    13 
    14 structure Int_Numeral_Simprocs: INT_NUMERAL_SIMPROCS =
    15 struct
    16 
    17 (* abstract syntax operations *)
    18 
    19 fun mk_bit 0 = @{term "0"}
    20   | mk_bit 1 = @{term "succ(0)"}
    21   | mk_bit _ = raise TERM ("mk_bit", []);
    22 
    23 fun dest_bit @{term "0"} = 0
    24   | dest_bit @{term "succ(0)"} = 1
    25   | dest_bit t = raise TERM ("dest_bit", [t]);
    26 
    27 fun mk_bin i =
    28   let
    29     fun term_of [] = @{term Pls}
    30       | term_of [~1] = @{term Min}
    31       | term_of (b :: bs) = @{term Bit} $ term_of bs $ mk_bit b;
    32   in term_of (Numeral_Syntax.make_binary i) end;
    33 
    34 fun dest_bin tm =
    35   let
    36     fun bin_of @{term Pls} = []
    37       | bin_of @{term Min} = [~1]
    38       | bin_of (@{term Bit} $ bs $ b) = dest_bit b :: bin_of bs
    39       | bin_of _ = raise TERM ("dest_bin", [tm]);
    40   in Numeral_Syntax.dest_binary (bin_of tm) end;
    41 
    42 
    43 (*Utilities*)
    44 
    45 fun mk_numeral i = @{const integ_of} $ mk_bin i;
    46 
    47 fun dest_numeral (Const(@{const_name integ_of}, _) $ w) = dest_bin w
    48   | dest_numeral t = raise TERM ("dest_numeral", [t]);
    49 
    50 fun find_first_numeral past (t::terms) =
    51         ((dest_numeral t, rev past @ terms)
    52          handle TERM _ => find_first_numeral (t::past) terms)
    53   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
    54 
    55 val zero = mk_numeral 0;
    56 val mk_plus = FOLogic.mk_binop @{const_name "zadd"};
    57 
    58 (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
    59 fun mk_sum []        = zero
    60   | mk_sum [t,u]     = mk_plus (t, u)
    61   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    62 
    63 (*this version ALWAYS includes a trailing zero*)
    64 fun long_mk_sum []        = zero
    65   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    66 
    67 (*decompose additions AND subtractions as a sum*)
    68 fun dest_summing (pos, Const (@{const_name "zadd"}, _) $ t $ u, ts) =
    69         dest_summing (pos, t, dest_summing (pos, u, ts))
    70   | dest_summing (pos, Const (@{const_name "zdiff"}, _) $ t $ u, ts) =
    71         dest_summing (pos, t, dest_summing (not pos, u, ts))
    72   | dest_summing (pos, t, ts) =
    73         if pos then t::ts else @{const zminus} $ t :: ts;
    74 
    75 fun dest_sum t = dest_summing (true, t, []);
    76 
    77 val one = mk_numeral 1;
    78 val mk_times = FOLogic.mk_binop @{const_name "zmult"};
    79 
    80 fun mk_prod [] = one
    81   | mk_prod [t] = t
    82   | mk_prod (t :: ts) = if t = one then mk_prod ts
    83                         else mk_times (t, mk_prod ts);
    84 
    85 val dest_times = FOLogic.dest_bin @{const_name "zmult"} @{typ i};
    86 
    87 fun dest_prod t =
    88       let val (t,u) = dest_times t
    89       in  dest_prod t @ dest_prod u  end
    90       handle TERM _ => [t];
    91 
    92 (*DON'T do the obvious simplifications; that would create special cases*)
    93 fun mk_coeff (k, t) = mk_times (mk_numeral k, t);
    94 
    95 (*Express t as a product of (possibly) a numeral with other sorted terms*)
    96 fun dest_coeff sign (Const (@{const_name "zminus"}, _) $ t) = dest_coeff (~sign) t
    97   | dest_coeff sign t =
    98     let val ts = sort Term_Ord.term_ord (dest_prod t)
    99         val (n, ts') = find_first_numeral [] ts
   100                           handle TERM _ => (1, ts)
   101     in (sign*n, mk_prod ts') end;
   102 
   103 (*Find first coefficient-term THAT MATCHES u*)
   104 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
   105   | find_first_coeff past u (t::terms) =
   106         let val (n,u') = dest_coeff 1 t
   107         in  if u aconv u' then (n, rev past @ terms)
   108                           else find_first_coeff (t::past) u terms
   109         end
   110         handle TERM _ => find_first_coeff (t::past) u terms;
   111 
   112 
   113 (*Simplify #1*n and n*#1 to n*)
   114 val add_0s = [@{thm zadd_0_intify}, @{thm zadd_0_right_intify}];
   115 
   116 val mult_1s = [@{thm zmult_1_intify}, @{thm zmult_1_right_intify},
   117                @{thm zmult_minus1}, @{thm zmult_minus1_right}];
   118 
   119 val tc_rules = [@{thm integ_of_type}, @{thm intify_in_int},
   120                 @{thm int_of_type}, @{thm zadd_type}, @{thm zdiff_type}, @{thm zmult_type}] @ 
   121                @{thms bin.intros};
   122 val intifys = [@{thm intify_ident}, @{thm zadd_intify1}, @{thm zadd_intify2},
   123                @{thm zdiff_intify1}, @{thm zdiff_intify2}, @{thm zmult_intify1}, @{thm zmult_intify2},
   124                @{thm zless_intify1}, @{thm zless_intify2}, @{thm zle_intify1}, @{thm zle_intify2}];
   125 
   126 (*To perform binary arithmetic*)
   127 val bin_simps = [@{thm add_integ_of_left}] @ @{thms bin_arith_simps} @ @{thms bin_rel_simps};
   128 
   129 (*To evaluate binary negations of coefficients*)
   130 val zminus_simps = @{thms NCons_simps} @
   131                    [@{thm integ_of_minus} RS @{thm sym},
   132                     @{thm bin_minus_1}, @{thm bin_minus_0}, @{thm bin_minus_Pls}, @{thm bin_minus_Min},
   133                     @{thm bin_pred_1}, @{thm bin_pred_0}, @{thm bin_pred_Pls}, @{thm bin_pred_Min}];
   134 
   135 (*To let us treat subtraction as addition*)
   136 val diff_simps = [@{thm zdiff_def}, @{thm zminus_zadd_distrib}, @{thm zminus_zminus}];
   137 
   138 (*push the unary minus down*)
   139 val int_minus_mult_eq_1_to_2 = @{lemma "$- w $* z = w $* $- z" by simp};
   140 
   141 (*to extract again any uncancelled minuses*)
   142 val int_minus_from_mult_simps =
   143     [@{thm zminus_zminus}, @{thm zmult_zminus}, @{thm zmult_zminus_right}];
   144 
   145 (*combine unary minus with numeric literals, however nested within a product*)
   146 val int_mult_minus_simps =
   147     [@{thm zmult_assoc}, @{thm zmult_zminus} RS @{thm sym}, int_minus_mult_eq_1_to_2];
   148 
   149 fun prep_simproc thy (name, pats, proc) =
   150   Simplifier.simproc_global thy name pats proc;
   151 
   152 structure CancelNumeralsCommon =
   153   struct
   154   val mk_sum = (fn _ : typ => mk_sum)
   155   val dest_sum = dest_sum
   156   val mk_coeff = mk_coeff
   157   val dest_coeff = dest_coeff 1
   158   val find_first_coeff = find_first_coeff []
   159   fun trans_tac ctxt = ArithData.gen_trans_tac ctxt @{thm iff_trans}
   160 
   161   val norm_ss1 =
   162     simpset_of (put_simpset ZF_ss @{context}
   163       addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac})
   164   val norm_ss2 =
   165     simpset_of (put_simpset ZF_ss @{context}
   166       addsimps bin_simps @ int_mult_minus_simps @ intifys)
   167   val norm_ss3 =
   168     simpset_of (put_simpset ZF_ss @{context}
   169       addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys)
   170   fun norm_tac ctxt =
   171     ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt))
   172     THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt))
   173     THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss3 ctxt))
   174 
   175   val numeral_simp_ss =
   176     simpset_of (put_simpset ZF_ss @{context}
   177       addsimps add_0s @ bin_simps @ tc_rules @ intifys)
   178   fun numeral_simp_tac ctxt =
   179     ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
   180     THEN ALLGOALS (asm_simp_tac ctxt)
   181   val simplify_meta_eq  = ArithData.simplify_meta_eq (add_0s @ mult_1s)
   182   end;
   183 
   184 
   185 structure EqCancelNumerals = CancelNumeralsFun
   186  (open CancelNumeralsCommon
   187   val prove_conv = ArithData.prove_conv "inteq_cancel_numerals"
   188   val mk_bal   = FOLogic.mk_eq
   189   val dest_bal = FOLogic.dest_eq
   190   val bal_add1 = @{thm eq_add_iff1} RS @{thm iff_trans}
   191   val bal_add2 = @{thm eq_add_iff2} RS @{thm iff_trans}
   192 );
   193 
   194 structure LessCancelNumerals = CancelNumeralsFun
   195  (open CancelNumeralsCommon
   196   val prove_conv = ArithData.prove_conv "intless_cancel_numerals"
   197   val mk_bal   = FOLogic.mk_binrel @{const_name "zless"}
   198   val dest_bal = FOLogic.dest_bin @{const_name "zless"} @{typ i}
   199   val bal_add1 = @{thm less_add_iff1} RS @{thm iff_trans}
   200   val bal_add2 = @{thm less_add_iff2} RS @{thm iff_trans}
   201 );
   202 
   203 structure LeCancelNumerals = CancelNumeralsFun
   204  (open CancelNumeralsCommon
   205   val prove_conv = ArithData.prove_conv "intle_cancel_numerals"
   206   val mk_bal   = FOLogic.mk_binrel @{const_name "zle"}
   207   val dest_bal = FOLogic.dest_bin @{const_name "zle"} @{typ i}
   208   val bal_add1 = @{thm le_add_iff1} RS @{thm iff_trans}
   209   val bal_add2 = @{thm le_add_iff2} RS @{thm iff_trans}
   210 );
   211 
   212 val cancel_numerals =
   213   map (prep_simproc @{theory})
   214    [("inteq_cancel_numerals",
   215      ["l $+ m = n", "l = m $+ n",
   216       "l $- m = n", "l = m $- n",
   217       "l $* m = n", "l = m $* n"],
   218      EqCancelNumerals.proc),
   219     ("intless_cancel_numerals",
   220      ["l $+ m $< n", "l $< m $+ n",
   221       "l $- m $< n", "l $< m $- n",
   222       "l $* m $< n", "l $< m $* n"],
   223      LessCancelNumerals.proc),
   224     ("intle_cancel_numerals",
   225      ["l $+ m $<= n", "l $<= m $+ n",
   226       "l $- m $<= n", "l $<= m $- n",
   227       "l $* m $<= n", "l $<= m $* n"],
   228      LeCancelNumerals.proc)];
   229 
   230 
   231 (*version without the hyps argument*)
   232 fun prove_conv_nohyps name tacs sg = ArithData.prove_conv name tacs sg [];
   233 
   234 structure CombineNumeralsData =
   235   struct
   236   type coeff = int
   237   val iszero = (fn x => x = 0)
   238   val add = op + 
   239   val mk_sum = (fn _ : typ => long_mk_sum) (*to work for #2*x $+ #3*x *)
   240   val dest_sum = dest_sum
   241   val mk_coeff = mk_coeff
   242   val dest_coeff = dest_coeff 1
   243   val left_distrib = @{thm left_zadd_zmult_distrib} RS @{thm trans}
   244   val prove_conv = prove_conv_nohyps "int_combine_numerals"
   245   fun trans_tac ctxt = ArithData.gen_trans_tac ctxt @{thm trans}
   246 
   247   val norm_ss1 =
   248     simpset_of (put_simpset ZF_ss @{context}
   249       addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac} @ intifys)
   250   val norm_ss2 =
   251     simpset_of (put_simpset ZF_ss @{context}
   252       addsimps bin_simps @ int_mult_minus_simps @ intifys)
   253   val norm_ss3 =
   254     simpset_of (put_simpset ZF_ss @{context}
   255       addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys)
   256   fun norm_tac ctxt =
   257     ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt))
   258     THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt))
   259     THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss3 ctxt))
   260 
   261   val numeral_simp_ss =
   262     simpset_of (put_simpset ZF_ss @{context} addsimps add_0s @ bin_simps @ tc_rules @ intifys)
   263   fun numeral_simp_tac ctxt =
   264     ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
   265   val simplify_meta_eq  = ArithData.simplify_meta_eq (add_0s @ mult_1s)
   266   end;
   267 
   268 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   269 
   270 val combine_numerals =
   271   prep_simproc @{theory}
   272     ("int_combine_numerals", ["i $+ j", "i $- j"], CombineNumerals.proc);
   273 
   274 
   275 
   276 (** Constant folding for integer multiplication **)
   277 
   278 (*The trick is to regard products as sums, e.g. #3 $* x $* #4 as
   279   the "sum" of #3, x, #4; the literals are then multiplied*)
   280 
   281 
   282 structure CombineNumeralsProdData =
   283 struct
   284   type coeff = int
   285   val iszero = (fn x => x = 0)
   286   val add = op *
   287   val mk_sum = (fn _ : typ => mk_prod)
   288   val dest_sum = dest_prod
   289   fun mk_coeff(k,t) =
   290     if t = one then mk_numeral k
   291     else raise TERM("mk_coeff", [])
   292   fun dest_coeff t = (dest_numeral t, one)  (*We ONLY want pure numerals.*)
   293   val left_distrib = @{thm zmult_assoc} RS @{thm sym} RS @{thm trans}
   294   val prove_conv = prove_conv_nohyps "int_combine_numerals_prod"
   295   fun trans_tac ctxt = ArithData.gen_trans_tac ctxt @{thm trans}
   296 
   297   val norm_ss1 =
   298     simpset_of (put_simpset ZF_ss @{context} addsimps mult_1s @ diff_simps @ zminus_simps)
   299   val norm_ss2 =
   300     simpset_of (put_simpset ZF_ss @{context} addsimps [@{thm zmult_zminus_right} RS @{thm sym}] @
   301     bin_simps @ @{thms zmult_ac} @ tc_rules @ intifys)
   302   fun norm_tac ctxt =
   303     ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt))
   304     THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt))
   305 
   306   val numeral_simp_ss =
   307     simpset_of (put_simpset ZF_ss @{context} addsimps bin_simps @ tc_rules @ intifys)
   308   fun numeral_simp_tac ctxt =
   309     ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
   310   val simplify_meta_eq  = ArithData.simplify_meta_eq (mult_1s);
   311 end;
   312 
   313 
   314 structure CombineNumeralsProd = CombineNumeralsFun(CombineNumeralsProdData);
   315 
   316 val combine_numerals_prod =
   317   prep_simproc @{theory}
   318     ("int_combine_numerals_prod", ["i $* j"], CombineNumeralsProd.proc);
   319 
   320 end;
   321 
   322 val _ =
   323   Theory.setup (Simplifier.map_theory_simpset (fn ctxt =>
   324     ctxt addsimprocs
   325       (Int_Numeral_Simprocs.cancel_numerals @
   326        [Int_Numeral_Simprocs.combine_numerals,
   327         Int_Numeral_Simprocs.combine_numerals_prod])));