src/HOL/Integ/NatSimprocs.thy
 author haftmann Mon Jan 30 08:20:56 2006 +0100 (2006-01-30) changeset 18851 9502ce541f01 parent 18648 22f96cd085d5 child 20485 3078fd2eec7b permissions -rw-r--r--
```     1 (*  Title:      HOL/NatSimprocs.thy
```
```     2     ID:         \$Id\$
```
```     3     Copyright   2003 TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {*Simprocs for the Naturals*}
```
```     7
```
```     8 theory NatSimprocs
```
```     9 imports NatBin
```
```    10 uses "int_factor_simprocs.ML" "nat_simprocs.ML"
```
```    11 begin
```
```    12
```
```    13 setup nat_simprocs_setup
```
```    14
```
```    15 subsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
```
```    16
```
```    17 text{*Where K above is a literal*}
```
```    18
```
```    19 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
```
```    20 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
```
```    21
```
```    22 text {*Now just instantiating @{text n} to @{text "number_of v"} does
```
```    23   the right simplification, but with some redundant inequality
```
```    24   tests.*}
```
```    25 lemma neg_number_of_bin_pred_iff_0:
```
```    26      "neg (number_of (bin_pred v)::int) = (number_of v = (0::nat))"
```
```    27 apply (subgoal_tac "neg (number_of (bin_pred v)) = (number_of v < Suc 0) ")
```
```    28 apply (simp only: less_Suc_eq_le le_0_eq)
```
```    29 apply (subst less_number_of_Suc, simp)
```
```    30 done
```
```    31
```
```    32 text{*No longer required as a simprule because of the @{text inverse_fold}
```
```    33    simproc*}
```
```    34 lemma Suc_diff_number_of:
```
```    35      "neg (number_of (bin_minus v)::int) ==>
```
```    36       Suc m - (number_of v) = m - (number_of (bin_pred v))"
```
```    37 apply (subst Suc_diff_eq_diff_pred)
```
```    38 apply simp
```
```    39 apply (simp del: nat_numeral_1_eq_1)
```
```    40 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
```
```    41                         neg_number_of_bin_pred_iff_0)
```
```    42 done
```
```    43
```
```    44 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
```
```    45 by (simp add: numerals split add: nat_diff_split)
```
```    46
```
```    47
```
```    48 subsection{*For @{term nat_case} and @{term nat_rec}*}
```
```    49
```
```    50 lemma nat_case_number_of [simp]:
```
```    51      "nat_case a f (number_of v) =
```
```    52         (let pv = number_of (bin_pred v) in
```
```    53          if neg pv then a else f (nat pv))"
```
```    54 by (simp split add: nat.split add: Let_def neg_number_of_bin_pred_iff_0)
```
```    55
```
```    56 lemma nat_case_add_eq_if [simp]:
```
```    57      "nat_case a f ((number_of v) + n) =
```
```    58        (let pv = number_of (bin_pred v) in
```
```    59          if neg pv then nat_case a f n else f (nat pv + n))"
```
```    60 apply (subst add_eq_if)
```
```    61 apply (simp split add: nat.split
```
```    62             del: nat_numeral_1_eq_1
```
```    63 	    add: numeral_1_eq_Suc_0 [symmetric] Let_def
```
```    64                  neg_imp_number_of_eq_0 neg_number_of_bin_pred_iff_0)
```
```    65 done
```
```    66
```
```    67 lemma nat_rec_number_of [simp]:
```
```    68      "nat_rec a f (number_of v) =
```
```    69         (let pv = number_of (bin_pred v) in
```
```    70          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
```
```    71 apply (case_tac " (number_of v) ::nat")
```
```    72 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_bin_pred_iff_0)
```
```    73 apply (simp split add: split_if_asm)
```
```    74 done
```
```    75
```
```    76 lemma nat_rec_add_eq_if [simp]:
```
```    77      "nat_rec a f (number_of v + n) =
```
```    78         (let pv = number_of (bin_pred v) in
```
```    79          if neg pv then nat_rec a f n
```
```    80                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
```
```    81 apply (subst add_eq_if)
```
```    82 apply (simp split add: nat.split
```
```    83             del: nat_numeral_1_eq_1
```
```    84             add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0
```
```    85                  neg_number_of_bin_pred_iff_0)
```
```    86 done
```
```    87
```
```    88
```
```    89 subsection{*Various Other Lemmas*}
```
```    90
```
```    91 subsubsection{*Evens and Odds, for Mutilated Chess Board*}
```
```    92
```
```    93 text{*Lemmas for specialist use, NOT as default simprules*}
```
```    94 lemma nat_mult_2: "2 * z = (z+z::nat)"
```
```    95 proof -
```
```    96   have "2*z = (1 + 1)*z" by simp
```
```    97   also have "... = z+z" by (simp add: left_distrib)
```
```    98   finally show ?thesis .
```
```    99 qed
```
```   100
```
```   101 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
```
```   102 by (subst mult_commute, rule nat_mult_2)
```
```   103
```
```   104 text{*Case analysis on @{term "n<2"}*}
```
```   105 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
```
```   106 by arith
```
```   107
```
```   108 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
```
```   109 by arith
```
```   110
```
```   111 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```   112 by (simp add: nat_mult_2 [symmetric])
```
```   113
```
```   114 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
```
```   115 apply (subgoal_tac "m mod 2 < 2")
```
```   116 apply (erule less_2_cases [THEN disjE])
```
```   117 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
```
```   118 done
```
```   119
```
```   120 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
```
```   121 apply (subgoal_tac "m mod 2 < 2")
```
```   122 apply (force simp del: mod_less_divisor, simp)
```
```   123 done
```
```   124
```
```   125 subsubsection{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
```
```   126
```
```   127 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```   128 by simp
```
```   129
```
```   130 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```   131 by simp
```
```   132
```
```   133 text{*Can be used to eliminate long strings of Sucs, but not by default*}
```
```   134 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```   135 by simp
```
```   136
```
```   137
```
```   138 text{*These lemmas collapse some needless occurrences of Suc:
```
```   139     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```   140     We already have some rules to simplify operands smaller than 3.*}
```
```   141
```
```   142 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```   143 by (simp add: Suc3_eq_add_3)
```
```   144
```
```   145 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```   146 by (simp add: Suc3_eq_add_3)
```
```   147
```
```   148 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```   149 by (simp add: Suc3_eq_add_3)
```
```   150
```
```   151 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```   152 by (simp add: Suc3_eq_add_3)
```
```   153
```
```   154 lemmas Suc_div_eq_add3_div_number_of =
```
```   155     Suc_div_eq_add3_div [of _ "number_of v", standard]
```
```   156 declare Suc_div_eq_add3_div_number_of [simp]
```
```   157
```
```   158 lemmas Suc_mod_eq_add3_mod_number_of =
```
```   159     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
```
```   160 declare Suc_mod_eq_add3_mod_number_of [simp]
```
```   161
```
```   162
```
```   163
```
```   164 subsection{*Special Simplification for Constants*}
```
```   165
```
```   166 text{*These belong here, late in the development of HOL, to prevent their
```
```   167 interfering with proofs of abstract properties of instances of the function
```
```   168 @{term number_of}*}
```
```   169
```
```   170 text{*These distributive laws move literals inside sums and differences.*}
```
```   171 lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
```
```   172 declare left_distrib_number_of [simp]
```
```   173
```
```   174 lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
```
```   175 declare right_distrib_number_of [simp]
```
```   176
```
```   177
```
```   178 lemmas left_diff_distrib_number_of =
```
```   179     left_diff_distrib [of _ _ "number_of v", standard]
```
```   180 declare left_diff_distrib_number_of [simp]
```
```   181
```
```   182 lemmas right_diff_distrib_number_of =
```
```   183     right_diff_distrib [of "number_of v", standard]
```
```   184 declare right_diff_distrib_number_of [simp]
```
```   185
```
```   186
```
```   187 text{*These are actually for fields, like real: but where else to put them?*}
```
```   188 lemmas zero_less_divide_iff_number_of =
```
```   189     zero_less_divide_iff [of "number_of w", standard]
```
```   190 declare zero_less_divide_iff_number_of [simp]
```
```   191
```
```   192 lemmas divide_less_0_iff_number_of =
```
```   193     divide_less_0_iff [of "number_of w", standard]
```
```   194 declare divide_less_0_iff_number_of [simp]
```
```   195
```
```   196 lemmas zero_le_divide_iff_number_of =
```
```   197     zero_le_divide_iff [of "number_of w", standard]
```
```   198 declare zero_le_divide_iff_number_of [simp]
```
```   199
```
```   200 lemmas divide_le_0_iff_number_of =
```
```   201     divide_le_0_iff [of "number_of w", standard]
```
```   202 declare divide_le_0_iff_number_of [simp]
```
```   203
```
```   204
```
```   205 (****
```
```   206 IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
```
```   207 then these special-case declarations may be useful.
```
```   208
```
```   209 text{*These simprules move numerals into numerators and denominators.*}
```
```   210 lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
```
```   211 by (simp add: times_divide_eq)
```
```   212
```
```   213 lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
```
```   214 by (simp add: times_divide_eq)
```
```   215
```
```   216 lemmas times_divide_eq_right_number_of =
```
```   217     times_divide_eq_right [of "number_of w", standard]
```
```   218 declare times_divide_eq_right_number_of [simp]
```
```   219
```
```   220 lemmas times_divide_eq_right_number_of =
```
```   221     times_divide_eq_right [of _ _ "number_of w", standard]
```
```   222 declare times_divide_eq_right_number_of [simp]
```
```   223
```
```   224 lemmas times_divide_eq_left_number_of =
```
```   225     times_divide_eq_left [of _ "number_of w", standard]
```
```   226 declare times_divide_eq_left_number_of [simp]
```
```   227
```
```   228 lemmas times_divide_eq_left_number_of =
```
```   229     times_divide_eq_left [of _ _ "number_of w", standard]
```
```   230 declare times_divide_eq_left_number_of [simp]
```
```   231
```
```   232 ****)
```
```   233
```
```   234 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```   235   strange, but then other simprocs simplify the quotient.*}
```
```   236
```
```   237 lemmas inverse_eq_divide_number_of =
```
```   238     inverse_eq_divide [of "number_of w", standard]
```
```   239 declare inverse_eq_divide_number_of [simp]
```
```   240
```
```   241
```
```   242 text{*These laws simplify inequalities, moving unary minus from a term
```
```   243 into the literal.*}
```
```   244 lemmas less_minus_iff_number_of =
```
```   245     less_minus_iff [of "number_of v", standard]
```
```   246 declare less_minus_iff_number_of [simp]
```
```   247
```
```   248 lemmas le_minus_iff_number_of =
```
```   249     le_minus_iff [of "number_of v", standard]
```
```   250 declare le_minus_iff_number_of [simp]
```
```   251
```
```   252 lemmas equation_minus_iff_number_of =
```
```   253     equation_minus_iff [of "number_of v", standard]
```
```   254 declare equation_minus_iff_number_of [simp]
```
```   255
```
```   256
```
```   257 lemmas minus_less_iff_number_of =
```
```   258     minus_less_iff [of _ "number_of v", standard]
```
```   259 declare minus_less_iff_number_of [simp]
```
```   260
```
```   261 lemmas minus_le_iff_number_of =
```
```   262     minus_le_iff [of _ "number_of v", standard]
```
```   263 declare minus_le_iff_number_of [simp]
```
```   264
```
```   265 lemmas minus_equation_iff_number_of =
```
```   266     minus_equation_iff [of _ "number_of v", standard]
```
```   267 declare minus_equation_iff_number_of [simp]
```
```   268
```
```   269
```
```   270 text{*These simplify inequalities where one side is the constant 1.*}
```
```   271 lemmas less_minus_iff_1 = less_minus_iff [of 1, simplified]
```
```   272 declare less_minus_iff_1 [simp]
```
```   273
```
```   274 lemmas le_minus_iff_1 = le_minus_iff [of 1, simplified]
```
```   275 declare le_minus_iff_1 [simp]
```
```   276
```
```   277 lemmas equation_minus_iff_1 = equation_minus_iff [of 1, simplified]
```
```   278 declare equation_minus_iff_1 [simp]
```
```   279
```
```   280 lemmas minus_less_iff_1 = minus_less_iff [of _ 1, simplified]
```
```   281 declare minus_less_iff_1 [simp]
```
```   282
```
```   283 lemmas minus_le_iff_1 = minus_le_iff [of _ 1, simplified]
```
```   284 declare minus_le_iff_1 [simp]
```
```   285
```
```   286 lemmas minus_equation_iff_1 = minus_equation_iff [of _ 1, simplified]
```
```   287 declare minus_equation_iff_1 [simp]
```
```   288
```
```   289
```
```   290 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```   291
```
```   292 lemmas mult_less_cancel_left_number_of =
```
```   293     mult_less_cancel_left [of "number_of v", standard]
```
```   294 declare mult_less_cancel_left_number_of [simp]
```
```   295
```
```   296 lemmas mult_less_cancel_right_number_of =
```
```   297     mult_less_cancel_right [of _ "number_of v", standard]
```
```   298 declare mult_less_cancel_right_number_of [simp]
```
```   299
```
```   300 lemmas mult_le_cancel_left_number_of =
```
```   301     mult_le_cancel_left [of "number_of v", standard]
```
```   302 declare mult_le_cancel_left_number_of [simp]
```
```   303
```
```   304 lemmas mult_le_cancel_right_number_of =
```
```   305     mult_le_cancel_right [of _ "number_of v", standard]
```
```   306 declare mult_le_cancel_right_number_of [simp]
```
```   307
```
```   308
```
```   309 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```   310
```
```   311 lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard]
```
```   312 declare le_divide_eq_number_of [simp]
```
```   313
```
```   314 lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard]
```
```   315 declare divide_le_eq_number_of [simp]
```
```   316
```
```   317 lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard]
```
```   318 declare less_divide_eq_number_of [simp]
```
```   319
```
```   320 lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard]
```
```   321 declare divide_less_eq_number_of [simp]
```
```   322
```
```   323 lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard]
```
```   324 declare eq_divide_eq_number_of [simp]
```
```   325
```
```   326 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard]
```
```   327 declare divide_eq_eq_number_of [simp]
```
```   328
```
```   329
```
```   330
```
```   331 subsection{*Optional Simplification Rules Involving Constants*}
```
```   332
```
```   333 text{*Simplify quotients that are compared with a literal constant.*}
```
```   334
```
```   335 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
```
```   336 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
```
```   337 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
```
```   338 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
```
```   339 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
```
```   340 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
```
```   341
```
```   342
```
```   343 text{*Not good as automatic simprules because they cause case splits.*}
```
```   344 lemmas divide_const_simps =
```
```   345   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
```
```   346   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
```
```   347   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```   348
```
```   349 subsubsection{*Division By @{text "-1"}*}
```
```   350
```
```   351 lemma divide_minus1 [simp]:
```
```   352      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
```
```   353 by simp
```
```   354
```
```   355 lemma minus1_divide [simp]:
```
```   356      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
```
```   357 by (simp add: divide_inverse inverse_minus_eq)
```
```   358
```
```   359 lemma half_gt_zero_iff:
```
```   360      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
```
```   361 by auto
```
```   362
```
```   363 lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
```
```   364 declare half_gt_zero [simp]
```
```   365
```
```   366 (* The following lemma should appear in Divides.thy, but there the proof
```
```   367    doesn't work. *)
```
```   368
```
```   369 lemma nat_dvd_not_less:
```
```   370   "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
```
```   371   by (unfold dvd_def) auto
```
```   372
```
```   373 ML
```
```   374 {*
```
```   375 val divide_minus1 = thm "divide_minus1";
```
```   376 val minus1_divide = thm "minus1_divide";
```
```   377 *}
```
```   378
```
```   379 end
```