src/HOL/Integ/NatSimprocs.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16417 9bc16273c2d4 child 16775 c1b87ef4a1c3 permissions -rw-r--r--
Constant "If" is now local
```     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 add: );
```
```    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 declare Suc_div_eq_add3_div [of _ "number_of v", standard, simp]
```
```   155 declare Suc_mod_eq_add3_mod [of _ "number_of v", standard, simp]
```
```   156
```
```   157
```
```   158 subsection{*Special Simplification for Constants*}
```
```   159
```
```   160 text{*These belong here, late in the development of HOL, to prevent their
```
```   161 interfering with proofs of abstract properties of instances of the function
```
```   162 @{term number_of}*}
```
```   163
```
```   164 text{*These distributive laws move literals inside sums and differences.*}
```
```   165 declare left_distrib [of _ _ "number_of v", standard, simp]
```
```   166 declare right_distrib [of "number_of v", standard, simp]
```
```   167
```
```   168 declare left_diff_distrib [of _ _ "number_of v", standard, simp]
```
```   169 declare right_diff_distrib [of "number_of v", standard, simp]
```
```   170
```
```   171 text{*These are actually for fields, like real: but where else to put them?*}
```
```   172 declare zero_less_divide_iff [of "number_of w", standard, simp]
```
```   173 declare divide_less_0_iff [of "number_of w", standard, simp]
```
```   174 declare zero_le_divide_iff [of "number_of w", standard, simp]
```
```   175 declare divide_le_0_iff [of "number_of w", standard, simp]
```
```   176
```
```   177 (****
```
```   178 IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
```
```   179 then these special-case declarations may be useful.
```
```   180
```
```   181 text{*These simprules move numerals into numerators and denominators.*}
```
```   182 lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
```
```   183 by (simp add: times_divide_eq)
```
```   184
```
```   185 lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
```
```   186 by (simp add: times_divide_eq)
```
```   187
```
```   188 declare times_divide_eq_right [of "number_of w", standard, simp]
```
```   189 declare times_divide_eq_right [of _ _ "number_of w", standard, simp]
```
```   190 declare times_divide_eq_left [of _ "number_of w", standard, simp]
```
```   191 declare times_divide_eq_left [of _ _ "number_of w", standard, simp]
```
```   192 ****)
```
```   193
```
```   194 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```   195   strange, but then other simprocs simplify the quotient.*}
```
```   196
```
```   197 declare inverse_eq_divide [of "number_of w", standard, simp]
```
```   198
```
```   199 text{*These laws simplify inequalities, moving unary minus from a term
```
```   200 into the literal.*}
```
```   201 declare less_minus_iff [of "number_of v", standard, simp]
```
```   202 declare le_minus_iff [of "number_of v", standard, simp]
```
```   203 declare equation_minus_iff [of "number_of v", standard, simp]
```
```   204
```
```   205 declare minus_less_iff [of _ "number_of v", standard, simp]
```
```   206 declare minus_le_iff [of _ "number_of v", standard, simp]
```
```   207 declare minus_equation_iff [of _ "number_of v", standard, simp]
```
```   208
```
```   209 text{*These simplify inequalities where one side is the constant 1.*}
```
```   210 declare less_minus_iff [of 1, simplified, simp]
```
```   211 declare le_minus_iff [of 1, simplified, simp]
```
```   212 declare equation_minus_iff [of 1, simplified, simp]
```
```   213
```
```   214 declare minus_less_iff [of _ 1, simplified, simp]
```
```   215 declare minus_le_iff [of _ 1, simplified, simp]
```
```   216 declare minus_equation_iff [of _ 1, simplified, simp]
```
```   217
```
```   218 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```   219
```
```   220 declare mult_less_cancel_left [of "number_of v", standard, simp]
```
```   221 declare mult_less_cancel_right [of _ "number_of v", standard, simp]
```
```   222 declare mult_le_cancel_left [of "number_of v", standard, simp]
```
```   223 declare mult_le_cancel_right [of _ "number_of v", standard, simp]
```
```   224
```
```   225 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```   226
```
```   227 declare le_divide_eq [of _ _ "number_of w", standard, simp]
```
```   228 declare divide_le_eq [of _ "number_of w", standard, simp]
```
```   229 declare less_divide_eq [of _ _ "number_of w", standard, simp]
```
```   230 declare divide_less_eq [of _ "number_of w", standard, simp]
```
```   231 declare eq_divide_eq [of _ _ "number_of w", standard, simp]
```
```   232 declare divide_eq_eq [of _ "number_of w", standard, simp]
```
```   233
```
```   234
```
```   235 subsection{*Optional Simplification Rules Involving Constants*}
```
```   236
```
```   237 text{*Simplify quotients that are compared with a literal constant.*}
```
```   238
```
```   239 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
```
```   240 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
```
```   241 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
```
```   242 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
```
```   243 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
```
```   244 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
```
```   245
```
```   246 text{*Simplify quotients that are compared with the value 1.*}
```
```   247
```
```   248 lemma le_divide_eq_1:
```
```   249   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   250   shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
```
```   251 by (auto simp add: le_divide_eq)
```
```   252
```
```   253 lemma divide_le_eq_1:
```
```   254   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   255   shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
```
```   256 by (auto simp add: divide_le_eq)
```
```   257
```
```   258 lemma less_divide_eq_1:
```
```   259   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   260   shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
```
```   261 by (auto simp add: less_divide_eq)
```
```   262
```
```   263 lemma divide_less_eq_1:
```
```   264   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   265   shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
```
```   266 by (auto simp add: divide_less_eq)
```
```   267
```
```   268
```
```   269 text{*Not good as automatic simprules because they cause case splits.*}
```
```   270 lemmas divide_const_simps =
```
```   271   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
```
```   272   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
```
```   273   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```   274
```
```   275
```
```   276 subsection{*Conditional Simplification Rules: No Case Splits*}
```
```   277
```
```   278 lemma le_divide_eq_1_pos [simp]:
```
```   279   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   280   shows "0 < a \<Longrightarrow> (1 \<le> b / a) = (a \<le> b)"
```
```   281 by (auto simp add: le_divide_eq)
```
```   282
```
```   283 lemma le_divide_eq_1_neg [simp]:
```
```   284   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   285   shows "a < 0 \<Longrightarrow> (1 \<le> b / a) = (b \<le> a)"
```
```   286 by (auto simp add: le_divide_eq)
```
```   287
```
```   288 lemma divide_le_eq_1_pos [simp]:
```
```   289   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   290   shows "0 < a \<Longrightarrow> (b / a \<le> 1) = (b \<le> a)"
```
```   291 by (auto simp add: divide_le_eq)
```
```   292
```
```   293 lemma divide_le_eq_1_neg [simp]:
```
```   294   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   295   shows "a < 0 \<Longrightarrow> (b / a \<le> 1) = (a \<le> b)"
```
```   296 by (auto simp add: divide_le_eq)
```
```   297
```
```   298 lemma less_divide_eq_1_pos [simp]:
```
```   299   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   300   shows "0 < a \<Longrightarrow> (1 < b / a) = (a < b)"
```
```   301 by (auto simp add: less_divide_eq)
```
```   302
```
```   303 lemma less_divide_eq_1_neg [simp]:
```
```   304   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   305   shows "a < 0 \<Longrightarrow> (1 < b / a) = (b < a)"
```
```   306 by (auto simp add: less_divide_eq)
```
```   307
```
```   308 lemma divide_less_eq_1_pos [simp]:
```
```   309   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   310   shows "0 < a \<Longrightarrow> (b / a < 1) = (b < a)"
```
```   311 by (auto simp add: divide_less_eq)
```
```   312
```
```   313 lemma eq_divide_eq_1 [simp]:
```
```   314   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   315   shows "(1 = b / a) = ((a \<noteq> 0 & a = b))"
```
```   316 by (auto simp add: eq_divide_eq)
```
```   317
```
```   318 lemma divide_eq_eq_1 [simp]:
```
```   319   fixes a :: "'a :: {ordered_field,division_by_zero}"
```
```   320   shows "(b / a = 1) = ((a \<noteq> 0 & a = b))"
```
```   321 by (auto simp add: divide_eq_eq)
```
```   322
```
```   323
```
```   324 subsubsection{*Division By @{term "-1"}*}
```
```   325
```
```   326 lemma divide_minus1 [simp]:
```
```   327      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
```
```   328 by simp
```
```   329
```
```   330 lemma minus1_divide [simp]:
```
```   331      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
```
```   332 by (simp add: divide_inverse inverse_minus_eq)
```
```   333
```
```   334 lemma half_gt_zero_iff:
```
```   335      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
```
```   336 by auto
```
```   337
```
```   338 lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, simp]
```
```   339
```
```   340
```
```   341
```
```   342
```
```   343 ML
```
```   344 {*
```
```   345 val divide_minus1 = thm "divide_minus1";
```
```   346 val minus1_divide = thm "minus1_divide";
```
```   347 *}
```
```   348
```
```   349 end
```