src/HOL/Arith_Tools.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30079 293b896b9c25 permissions -rw-r--r--
simplified method setup;
```     1 (*  Title:      HOL/Arith_Tools.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Author:     Amine Chaieb, TU Muenchen
```
```     5 *)
```
```     6
```
```     7 header {* Setup of arithmetic tools *}
```
```     8
```
```     9 theory Arith_Tools
```
```    10 imports NatBin
```
```    11 uses
```
```    12   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
```
```    13   "~~/src/Provers/Arith/extract_common_term.ML"
```
```    14   "Tools/int_factor_simprocs.ML"
```
```    15   "Tools/nat_simprocs.ML"
```
```    16   "Tools/Qelim/qelim.ML"
```
```    17 begin
```
```    18
```
```    19 subsection {* Simprocs for the Naturals *}
```
```    20
```
```    21 declaration {* K nat_simprocs_setup *}
```
```    22
```
```    23 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
```
```    24
```
```    25 text{*Where K above is a literal*}
```
```    26
```
```    27 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
```
```    28 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
```
```    29
```
```    30 text {*Now just instantiating @{text n} to @{text "number_of v"} does
```
```    31   the right simplification, but with some redundant inequality
```
```    32   tests.*}
```
```    33 lemma neg_number_of_pred_iff_0:
```
```    34   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
```
```    35 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
```
```    36 apply (simp only: less_Suc_eq_le le_0_eq)
```
```    37 apply (subst less_number_of_Suc, simp)
```
```    38 done
```
```    39
```
```    40 text{*No longer required as a simprule because of the @{text inverse_fold}
```
```    41    simproc*}
```
```    42 lemma Suc_diff_number_of:
```
```    43      "Int.Pls < v ==>
```
```    44       Suc m - (number_of v) = m - (number_of (Int.pred v))"
```
```    45 apply (subst Suc_diff_eq_diff_pred)
```
```    46 apply simp
```
```    47 apply (simp del: nat_numeral_1_eq_1)
```
```    48 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
```
```    49                         neg_number_of_pred_iff_0)
```
```    50 done
```
```    51
```
```    52 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
```
```    53 by (simp add: numerals split add: nat_diff_split)
```
```    54
```
```    55
```
```    56 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
```
```    57
```
```    58 lemma nat_case_number_of [simp]:
```
```    59      "nat_case a f (number_of v) =
```
```    60         (let pv = number_of (Int.pred v) in
```
```    61          if neg pv then a else f (nat pv))"
```
```    62 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
```
```    63
```
```    64 lemma nat_case_add_eq_if [simp]:
```
```    65      "nat_case a f ((number_of v) + n) =
```
```    66        (let pv = number_of (Int.pred v) in
```
```    67          if neg pv then nat_case a f n else f (nat pv + n))"
```
```    68 apply (subst add_eq_if)
```
```    69 apply (simp split add: nat.split
```
```    70             del: nat_numeral_1_eq_1
```
```    71             add: nat_numeral_1_eq_1 [symmetric]
```
```    72                  numeral_1_eq_Suc_0 [symmetric]
```
```    73                  neg_number_of_pred_iff_0)
```
```    74 done
```
```    75
```
```    76 lemma nat_rec_number_of [simp]:
```
```    77      "nat_rec a f (number_of v) =
```
```    78         (let pv = number_of (Int.pred v) in
```
```    79          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
```
```    80 apply (case_tac " (number_of v) ::nat")
```
```    81 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
```
```    82 apply (simp split add: split_if_asm)
```
```    83 done
```
```    84
```
```    85 lemma nat_rec_add_eq_if [simp]:
```
```    86      "nat_rec a f (number_of v + n) =
```
```    87         (let pv = number_of (Int.pred v) in
```
```    88          if neg pv then nat_rec a f n
```
```    89                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
```
```    90 apply (subst add_eq_if)
```
```    91 apply (simp split add: nat.split
```
```    92             del: nat_numeral_1_eq_1
```
```    93             add: nat_numeral_1_eq_1 [symmetric]
```
```    94                  numeral_1_eq_Suc_0 [symmetric]
```
```    95                  neg_number_of_pred_iff_0)
```
```    96 done
```
```    97
```
```    98
```
```    99 subsubsection{*Various Other Lemmas*}
```
```   100
```
```   101 text {*Evens and Odds, for Mutilated Chess Board*}
```
```   102
```
```   103 text{*Lemmas for specialist use, NOT as default simprules*}
```
```   104 lemma nat_mult_2: "2 * z = (z+z::nat)"
```
```   105 proof -
```
```   106   have "2*z = (1 + 1)*z" by simp
```
```   107   also have "... = z+z" by (simp add: left_distrib)
```
```   108   finally show ?thesis .
```
```   109 qed
```
```   110
```
```   111 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
```
```   112 by (subst mult_commute, rule nat_mult_2)
```
```   113
```
```   114 text{*Case analysis on @{term "n<2"}*}
```
```   115 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
```
```   116 by arith
```
```   117
```
```   118 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
```
```   119 by arith
```
```   120
```
```   121 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```   122 by (simp add: nat_mult_2 [symmetric])
```
```   123
```
```   124 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
```
```   125 apply (subgoal_tac "m mod 2 < 2")
```
```   126 apply (erule less_2_cases [THEN disjE])
```
```   127 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
```
```   128 done
```
```   129
```
```   130 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
```
```   131 apply (subgoal_tac "m mod 2 < 2")
```
```   132 apply (force simp del: mod_less_divisor, simp)
```
```   133 done
```
```   134
```
```   135 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
```
```   136
```
```   137 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```   138 by simp
```
```   139
```
```   140 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```   141 by simp
```
```   142
```
```   143 text{*Can be used to eliminate long strings of Sucs, but not by default*}
```
```   144 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```   145 by simp
```
```   146
```
```   147
```
```   148 text{*These lemmas collapse some needless occurrences of Suc:
```
```   149     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```   150     We already have some rules to simplify operands smaller than 3.*}
```
```   151
```
```   152 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```   153 by (simp add: Suc3_eq_add_3)
```
```   154
```
```   155 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```   156 by (simp add: Suc3_eq_add_3)
```
```   157
```
```   158 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```   159 by (simp add: Suc3_eq_add_3)
```
```   160
```
```   161 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```   162 by (simp add: Suc3_eq_add_3)
```
```   163
```
```   164 lemmas Suc_div_eq_add3_div_number_of =
```
```   165     Suc_div_eq_add3_div [of _ "number_of v", standard]
```
```   166 declare Suc_div_eq_add3_div_number_of [simp]
```
```   167
```
```   168 lemmas Suc_mod_eq_add3_mod_number_of =
```
```   169     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
```
```   170 declare Suc_mod_eq_add3_mod_number_of [simp]
```
```   171
```
```   172
```
```   173 subsubsection{*Special Simplification for Constants*}
```
```   174
```
```   175 text{*These belong here, late in the development of HOL, to prevent their
```
```   176 interfering with proofs of abstract properties of instances of the function
```
```   177 @{term number_of}*}
```
```   178
```
```   179 text{*These distributive laws move literals inside sums and differences.*}
```
```   180 lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
```
```   181 declare left_distrib_number_of [simp]
```
```   182
```
```   183 lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
```
```   184 declare right_distrib_number_of [simp]
```
```   185
```
```   186
```
```   187 lemmas left_diff_distrib_number_of =
```
```   188     left_diff_distrib [of _ _ "number_of v", standard]
```
```   189 declare left_diff_distrib_number_of [simp]
```
```   190
```
```   191 lemmas right_diff_distrib_number_of =
```
```   192     right_diff_distrib [of "number_of v", standard]
```
```   193 declare right_diff_distrib_number_of [simp]
```
```   194
```
```   195
```
```   196 text{*These are actually for fields, like real: but where else to put them?*}
```
```   197 lemmas zero_less_divide_iff_number_of =
```
```   198     zero_less_divide_iff [of "number_of w", standard]
```
```   199 declare zero_less_divide_iff_number_of [simp,noatp]
```
```   200
```
```   201 lemmas divide_less_0_iff_number_of =
```
```   202     divide_less_0_iff [of "number_of w", standard]
```
```   203 declare divide_less_0_iff_number_of [simp,noatp]
```
```   204
```
```   205 lemmas zero_le_divide_iff_number_of =
```
```   206     zero_le_divide_iff [of "number_of w", standard]
```
```   207 declare zero_le_divide_iff_number_of [simp,noatp]
```
```   208
```
```   209 lemmas divide_le_0_iff_number_of =
```
```   210     divide_le_0_iff [of "number_of w", standard]
```
```   211 declare divide_le_0_iff_number_of [simp,noatp]
```
```   212
```
```   213
```
```   214 (****
```
```   215 IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
```
```   216 then these special-case declarations may be useful.
```
```   217
```
```   218 text{*These simprules move numerals into numerators and denominators.*}
```
```   219 lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
```
```   220 by (simp add: times_divide_eq)
```
```   221
```
```   222 lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
```
```   223 by (simp add: times_divide_eq)
```
```   224
```
```   225 lemmas times_divide_eq_right_number_of =
```
```   226     times_divide_eq_right [of "number_of w", standard]
```
```   227 declare times_divide_eq_right_number_of [simp]
```
```   228
```
```   229 lemmas times_divide_eq_right_number_of =
```
```   230     times_divide_eq_right [of _ _ "number_of w", standard]
```
```   231 declare times_divide_eq_right_number_of [simp]
```
```   232
```
```   233 lemmas times_divide_eq_left_number_of =
```
```   234     times_divide_eq_left [of _ "number_of w", standard]
```
```   235 declare times_divide_eq_left_number_of [simp]
```
```   236
```
```   237 lemmas times_divide_eq_left_number_of =
```
```   238     times_divide_eq_left [of _ _ "number_of w", standard]
```
```   239 declare times_divide_eq_left_number_of [simp]
```
```   240
```
```   241 ****)
```
```   242
```
```   243 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```   244   strange, but then other simprocs simplify the quotient.*}
```
```   245
```
```   246 lemmas inverse_eq_divide_number_of =
```
```   247     inverse_eq_divide [of "number_of w", standard]
```
```   248 declare inverse_eq_divide_number_of [simp]
```
```   249
```
```   250
```
```   251 text {*These laws simplify inequalities, moving unary minus from a term
```
```   252 into the literal.*}
```
```   253 lemmas less_minus_iff_number_of =
```
```   254     less_minus_iff [of "number_of v", standard]
```
```   255 declare less_minus_iff_number_of [simp,noatp]
```
```   256
```
```   257 lemmas le_minus_iff_number_of =
```
```   258     le_minus_iff [of "number_of v", standard]
```
```   259 declare le_minus_iff_number_of [simp,noatp]
```
```   260
```
```   261 lemmas equation_minus_iff_number_of =
```
```   262     equation_minus_iff [of "number_of v", standard]
```
```   263 declare equation_minus_iff_number_of [simp,noatp]
```
```   264
```
```   265
```
```   266 lemmas minus_less_iff_number_of =
```
```   267     minus_less_iff [of _ "number_of v", standard]
```
```   268 declare minus_less_iff_number_of [simp,noatp]
```
```   269
```
```   270 lemmas minus_le_iff_number_of =
```
```   271     minus_le_iff [of _ "number_of v", standard]
```
```   272 declare minus_le_iff_number_of [simp,noatp]
```
```   273
```
```   274 lemmas minus_equation_iff_number_of =
```
```   275     minus_equation_iff [of _ "number_of v", standard]
```
```   276 declare minus_equation_iff_number_of [simp,noatp]
```
```   277
```
```   278
```
```   279 text{*To Simplify Inequalities Where One Side is the Constant 1*}
```
```   280
```
```   281 lemma less_minus_iff_1 [simp,noatp]:
```
```   282   fixes b::"'b::{ordered_idom,number_ring}"
```
```   283   shows "(1 < - b) = (b < -1)"
```
```   284 by auto
```
```   285
```
```   286 lemma le_minus_iff_1 [simp,noatp]:
```
```   287   fixes b::"'b::{ordered_idom,number_ring}"
```
```   288   shows "(1 \<le> - b) = (b \<le> -1)"
```
```   289 by auto
```
```   290
```
```   291 lemma equation_minus_iff_1 [simp,noatp]:
```
```   292   fixes b::"'b::number_ring"
```
```   293   shows "(1 = - b) = (b = -1)"
```
```   294 by (subst equation_minus_iff, auto)
```
```   295
```
```   296 lemma minus_less_iff_1 [simp,noatp]:
```
```   297   fixes a::"'b::{ordered_idom,number_ring}"
```
```   298   shows "(- a < 1) = (-1 < a)"
```
```   299 by auto
```
```   300
```
```   301 lemma minus_le_iff_1 [simp,noatp]:
```
```   302   fixes a::"'b::{ordered_idom,number_ring}"
```
```   303   shows "(- a \<le> 1) = (-1 \<le> a)"
```
```   304 by auto
```
```   305
```
```   306 lemma minus_equation_iff_1 [simp,noatp]:
```
```   307   fixes a::"'b::number_ring"
```
```   308   shows "(- a = 1) = (a = -1)"
```
```   309 by (subst minus_equation_iff, auto)
```
```   310
```
```   311
```
```   312 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```   313
```
```   314 lemmas mult_less_cancel_left_number_of =
```
```   315     mult_less_cancel_left [of "number_of v", standard]
```
```   316 declare mult_less_cancel_left_number_of [simp,noatp]
```
```   317
```
```   318 lemmas mult_less_cancel_right_number_of =
```
```   319     mult_less_cancel_right [of _ "number_of v", standard]
```
```   320 declare mult_less_cancel_right_number_of [simp,noatp]
```
```   321
```
```   322 lemmas mult_le_cancel_left_number_of =
```
```   323     mult_le_cancel_left [of "number_of v", standard]
```
```   324 declare mult_le_cancel_left_number_of [simp,noatp]
```
```   325
```
```   326 lemmas mult_le_cancel_right_number_of =
```
```   327     mult_le_cancel_right [of _ "number_of v", standard]
```
```   328 declare mult_le_cancel_right_number_of [simp,noatp]
```
```   329
```
```   330
```
```   331 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```   332
```
```   333 lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
```
```   334 lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
```
```   335 lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
```
```   336 lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
```
```   337 lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
```
```   338 lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
```
```   339
```
```   340
```
```   341 subsubsection{*Optional Simplification Rules Involving Constants*}
```
```   342
```
```   343 text{*Simplify quotients that are compared with a literal constant.*}
```
```   344
```
```   345 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
```
```   346 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
```
```   347 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
```
```   348 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
```
```   349 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
```
```   350 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
```
```   351
```
```   352
```
```   353 text{*Not good as automatic simprules because they cause case splits.*}
```
```   354 lemmas divide_const_simps =
```
```   355   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
```
```   356   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
```
```   357   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```   358
```
```   359 text{*Division By @{text "-1"}*}
```
```   360
```
```   361 lemma divide_minus1 [simp]:
```
```   362      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
```
```   363 by simp
```
```   364
```
```   365 lemma minus1_divide [simp]:
```
```   366      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
```
```   367 by (simp add: divide_inverse inverse_minus_eq)
```
```   368
```
```   369 lemma half_gt_zero_iff:
```
```   370      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
```
```   371 by auto
```
```   372
```
```   373 lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
```
```   374 declare half_gt_zero [simp]
```
```   375
```
```   376 (* The following lemma should appear in Divides.thy, but there the proof
```
```   377    doesn't work. *)
```
```   378
```
```   379 lemma nat_dvd_not_less:
```
```   380   "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
```
```   381   by (unfold dvd_def) auto
```
```   382
```
```   383 ML {*
```
```   384 val divide_minus1 = @{thm divide_minus1};
```
```   385 val minus1_divide = @{thm minus1_divide};
```
```   386 *}
```
```   387
```
```   388 end
```