src/HOL/Nat_Numeral.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 36964 a354605f03dc child 40077 c8a9eaaa2f59 permissions -rw-r--r--
declare lex_prod_def [code del]
```     1 (*  Title:      HOL/Nat_Numeral.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1999  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Binary numerals for the natural numbers *}
```
```     7
```
```     8 theory Nat_Numeral
```
```     9 imports Int
```
```    10 begin
```
```    11
```
```    12 subsection {* Numerals for natural numbers *}
```
```    13
```
```    14 text {*
```
```    15   Arithmetic for naturals is reduced to that for the non-negative integers.
```
```    16 *}
```
```    17
```
```    18 instantiation nat :: number
```
```    19 begin
```
```    20
```
```    21 definition
```
```    22   nat_number_of_def [code_unfold, code del]: "number_of v = nat (number_of v)"
```
```    23
```
```    24 instance ..
```
```    25
```
```    26 end
```
```    27
```
```    28 lemma [code_post]:
```
```    29   "nat (number_of v) = number_of v"
```
```    30   unfolding nat_number_of_def ..
```
```    31
```
```    32
```
```    33 subsection {* Special case: squares and cubes *}
```
```    34
```
```    35 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
```
```    36   by (simp add: nat_number_of_def)
```
```    37
```
```    38 lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
```
```    39   by (simp add: nat_number_of_def)
```
```    40
```
```    41 context power
```
```    42 begin
```
```    43
```
```    44 abbreviation (xsymbols)
```
```    45   power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
```
```    46   "x\<twosuperior> \<equiv> x ^ 2"
```
```    47
```
```    48 notation (latex output)
```
```    49   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    50
```
```    51 notation (HTML output)
```
```    52   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    53
```
```    54 end
```
```    55
```
```    56 context monoid_mult
```
```    57 begin
```
```    58
```
```    59 lemma power2_eq_square: "a\<twosuperior> = a * a"
```
```    60   by (simp add: numeral_2_eq_2)
```
```    61
```
```    62 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    63   by (simp add: numeral_3_eq_3 mult_assoc)
```
```    64
```
```    65 lemma power_even_eq:
```
```    66   "a ^ (2*n) = (a ^ n) ^ 2"
```
```    67   by (subst mult_commute) (simp add: power_mult)
```
```    68
```
```    69 lemma power_odd_eq:
```
```    70   "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
```
```    71   by (simp add: power_even_eq)
```
```    72
```
```    73 end
```
```    74
```
```    75 context semiring_1
```
```    76 begin
```
```    77
```
```    78 lemma zero_power2 [simp]: "0\<twosuperior> = 0"
```
```    79   by (simp add: power2_eq_square)
```
```    80
```
```    81 lemma one_power2 [simp]: "1\<twosuperior> = 1"
```
```    82   by (simp add: power2_eq_square)
```
```    83
```
```    84 end
```
```    85
```
```    86 context ring_1
```
```    87 begin
```
```    88
```
```    89 lemma power2_minus [simp]:
```
```    90   "(- a)\<twosuperior> = a\<twosuperior>"
```
```    91   by (simp add: power2_eq_square)
```
```    92
```
```    93 text{*
```
```    94   We cannot prove general results about the numeral @{term "-1"},
```
```    95   so we have to use @{term "- 1"} instead.
```
```    96 *}
```
```    97
```
```    98 lemma power_minus1_even [simp]:
```
```    99   "(- 1) ^ (2*n) = 1"
```
```   100 proof (induct n)
```
```   101   case 0 show ?case by simp
```
```   102 next
```
```   103   case (Suc n) then show ?case by (simp add: power_add)
```
```   104 qed
```
```   105
```
```   106 lemma power_minus1_odd:
```
```   107   "(- 1) ^ Suc (2*n) = - 1"
```
```   108   by simp
```
```   109
```
```   110 lemma power_minus_even [simp]:
```
```   111   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   112   by (simp add: power_minus [of a])
```
```   113
```
```   114 end
```
```   115
```
```   116 context ring_1_no_zero_divisors
```
```   117 begin
```
```   118
```
```   119 lemma zero_eq_power2 [simp]:
```
```   120   "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
```
```   121   unfolding power2_eq_square by simp
```
```   122
```
```   123 lemma power2_eq_1_iff:
```
```   124   "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   125   unfolding power2_eq_square by (rule square_eq_1_iff)
```
```   126
```
```   127 end
```
```   128
```
```   129 context linordered_ring
```
```   130 begin
```
```   131
```
```   132 lemma sum_squares_ge_zero:
```
```   133   "0 \<le> x * x + y * y"
```
```   134   by (intro add_nonneg_nonneg zero_le_square)
```
```   135
```
```   136 lemma not_sum_squares_lt_zero:
```
```   137   "\<not> x * x + y * y < 0"
```
```   138   by (simp add: not_less sum_squares_ge_zero)
```
```   139
```
```   140 end
```
```   141
```
```   142 context linordered_ring_strict
```
```   143 begin
```
```   144
```
```   145 lemma sum_squares_eq_zero_iff:
```
```   146   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   147   by (simp add: add_nonneg_eq_0_iff)
```
```   148
```
```   149 lemma sum_squares_le_zero_iff:
```
```   150   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   151   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   152
```
```   153 lemma sum_squares_gt_zero_iff:
```
```   154   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   155   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   156
```
```   157 end
```
```   158
```
```   159 context linordered_semidom
```
```   160 begin
```
```   161
```
```   162 lemma power2_le_imp_le:
```
```   163   "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   164   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   165
```
```   166 lemma power2_less_imp_less:
```
```   167   "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   168   by (rule power_less_imp_less_base)
```
```   169
```
```   170 lemma power2_eq_imp_eq:
```
```   171   "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   172   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   173
```
```   174 end
```
```   175
```
```   176 context linordered_idom
```
```   177 begin
```
```   178
```
```   179 lemma zero_le_power2 [simp]:
```
```   180   "0 \<le> a\<twosuperior>"
```
```   181   by (simp add: power2_eq_square)
```
```   182
```
```   183 lemma zero_less_power2 [simp]:
```
```   184   "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
```
```   185   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   186
```
```   187 lemma power2_less_0 [simp]:
```
```   188   "\<not> a\<twosuperior> < 0"
```
```   189   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   190
```
```   191 lemma abs_power2 [simp]:
```
```   192   "abs (a\<twosuperior>) = a\<twosuperior>"
```
```   193   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   194
```
```   195 lemma power2_abs [simp]:
```
```   196   "(abs a)\<twosuperior> = a\<twosuperior>"
```
```   197   by (simp add: power2_eq_square abs_mult_self)
```
```   198
```
```   199 lemma odd_power_less_zero:
```
```   200   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   201 proof (induct n)
```
```   202   case 0
```
```   203   then show ?case by simp
```
```   204 next
```
```   205   case (Suc n)
```
```   206   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   207     by (simp add: mult_ac power_add power2_eq_square)
```
```   208   thus ?case
```
```   209     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   210 qed
```
```   211
```
```   212 lemma odd_0_le_power_imp_0_le:
```
```   213   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   214   using odd_power_less_zero [of a n]
```
```   215     by (force simp add: linorder_not_less [symmetric])
```
```   216
```
```   217 lemma zero_le_even_power'[simp]:
```
```   218   "0 \<le> a ^ (2*n)"
```
```   219 proof (induct n)
```
```   220   case 0
```
```   221     show ?case by simp
```
```   222 next
```
```   223   case (Suc n)
```
```   224     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   225       by (simp add: mult_ac power_add power2_eq_square)
```
```   226     thus ?case
```
```   227       by (simp add: Suc zero_le_mult_iff)
```
```   228 qed
```
```   229
```
```   230 lemma sum_power2_ge_zero:
```
```   231   "0 \<le> x\<twosuperior> + y\<twosuperior>"
```
```   232   unfolding power2_eq_square by (rule sum_squares_ge_zero)
```
```   233
```
```   234 lemma not_sum_power2_lt_zero:
```
```   235   "\<not> x\<twosuperior> + y\<twosuperior> < 0"
```
```   236   unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
```
```   237
```
```   238 lemma sum_power2_eq_zero_iff:
```
```   239   "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   240   unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
```
```   241
```
```   242 lemma sum_power2_le_zero_iff:
```
```   243   "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   244   unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
```
```   245
```
```   246 lemma sum_power2_gt_zero_iff:
```
```   247   "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   248   unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
```
```   249
```
```   250 end
```
```   251
```
```   252 lemma power2_sum:
```
```   253   fixes x y :: "'a::number_ring"
```
```   254   shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
```
```   255   by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
```
```   256
```
```   257 lemma power2_diff:
```
```   258   fixes x y :: "'a::number_ring"
```
```   259   shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
```
```   260   by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
```
```   261
```
```   262
```
```   263 subsection {* Predicate for negative binary numbers *}
```
```   264
```
```   265 definition neg  :: "int \<Rightarrow> bool" where
```
```   266   "neg Z \<longleftrightarrow> Z < 0"
```
```   267
```
```   268 lemma not_neg_int [simp]: "~ neg (of_nat n)"
```
```   269 by (simp add: neg_def)
```
```   270
```
```   271 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
```
```   272 by (simp add: neg_def del: of_nat_Suc)
```
```   273
```
```   274 lemmas neg_eq_less_0 = neg_def
```
```   275
```
```   276 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   277 by (simp add: neg_def linorder_not_less)
```
```   278
```
```   279 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   280
```
```   281 lemma not_neg_0: "~ neg 0"
```
```   282 by (simp add: One_int_def neg_def)
```
```   283
```
```   284 lemma not_neg_1: "~ neg 1"
```
```   285 by (simp add: neg_def linorder_not_less)
```
```   286
```
```   287 lemma neg_nat: "neg z ==> nat z = 0"
```
```   288 by (simp add: neg_def order_less_imp_le)
```
```   289
```
```   290 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
```
```   291 by (simp add: linorder_not_less neg_def)
```
```   292
```
```   293 text {*
```
```   294   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```   295   @{term Numeral0} IS @{term "number_of Pls"}
```
```   296 *}
```
```   297
```
```   298 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
```
```   299   by (simp add: neg_def)
```
```   300
```
```   301 lemma neg_number_of_Min: "neg (number_of Int.Min)"
```
```   302   by (simp add: neg_def)
```
```   303
```
```   304 lemma neg_number_of_Bit0:
```
```   305   "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
```
```   306   by (simp add: neg_def)
```
```   307
```
```   308 lemma neg_number_of_Bit1:
```
```   309   "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
```
```   310   by (simp add: neg_def)
```
```   311
```
```   312 lemmas neg_simps [simp] =
```
```   313   not_neg_0 not_neg_1
```
```   314   not_neg_number_of_Pls neg_number_of_Min
```
```   315   neg_number_of_Bit0 neg_number_of_Bit1
```
```   316
```
```   317
```
```   318 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
```
```   319
```
```   320 declare nat_1 [simp]
```
```   321
```
```   322 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
```
```   323 by (simp add: nat_number_of_def)
```
```   324
```
```   325 lemma nat_numeral_0_eq_0 [simp, code_post]: "Numeral0 = (0::nat)"
```
```   326 by (simp add: nat_number_of_def)
```
```   327
```
```   328 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
```
```   329 by (simp add: nat_number_of_def)
```
```   330
```
```   331 lemma Numeral1_eq1_nat:
```
```   332   "(1::nat) = Numeral1"
```
```   333   by simp
```
```   334
```
```   335 lemma numeral_1_eq_Suc_0 [code_post]: "Numeral1 = Suc 0"
```
```   336 by (simp only: nat_numeral_1_eq_1 One_nat_def)
```
```   337
```
```   338
```
```   339 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
```
```   340
```
```   341 lemma int_nat_number_of [simp]:
```
```   342      "int (number_of v) =
```
```   343          (if neg (number_of v :: int) then 0
```
```   344           else (number_of v :: int))"
```
```   345   unfolding nat_number_of_def number_of_is_id neg_def
```
```   346   by simp
```
```   347
```
```   348
```
```   349 subsubsection{*Successor *}
```
```   350
```
```   351 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   352 apply (rule sym)
```
```   353 apply (simp add: nat_eq_iff int_Suc)
```
```   354 done
```
```   355
```
```   356 lemma Suc_nat_number_of_add:
```
```   357      "Suc (number_of v + n) =
```
```   358         (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
```
```   359   unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
```
```   360   by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
```
```   361
```
```   362 lemma Suc_nat_number_of [simp]:
```
```   363      "Suc (number_of v) =
```
```   364         (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
```
```   365 apply (cut_tac n = 0 in Suc_nat_number_of_add)
```
```   366 apply (simp cong del: if_weak_cong)
```
```   367 done
```
```   368
```
```   369
```
```   370 subsubsection{*Addition *}
```
```   371
```
```   372 lemma add_nat_number_of [simp]:
```
```   373      "(number_of v :: nat) + number_of v' =
```
```   374          (if v < Int.Pls then number_of v'
```
```   375           else if v' < Int.Pls then number_of v
```
```   376           else number_of (v + v'))"
```
```   377   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   378   by (simp add: nat_add_distrib)
```
```   379
```
```   380 lemma nat_number_of_add_1 [simp]:
```
```   381   "number_of v + (1::nat) =
```
```   382     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   383   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   384   by (simp add: nat_add_distrib)
```
```   385
```
```   386 lemma nat_1_add_number_of [simp]:
```
```   387   "(1::nat) + number_of v =
```
```   388     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   389   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   390   by (simp add: nat_add_distrib)
```
```   391
```
```   392 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
```
```   393   by (rule int_int_eq [THEN iffD1]) simp
```
```   394
```
```   395
```
```   396 subsubsection{*Subtraction *}
```
```   397
```
```   398 lemma diff_nat_eq_if:
```
```   399      "nat z - nat z' =
```
```   400         (if neg z' then nat z
```
```   401          else let d = z-z' in
```
```   402               if neg d then 0 else nat d)"
```
```   403 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
```
```   404
```
```   405
```
```   406 lemma diff_nat_number_of [simp]:
```
```   407      "(number_of v :: nat) - number_of v' =
```
```   408         (if v' < Int.Pls then number_of v
```
```   409          else let d = number_of (v + uminus v') in
```
```   410               if neg d then 0 else nat d)"
```
```   411   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
```
```   412   by auto
```
```   413
```
```   414 lemma nat_number_of_diff_1 [simp]:
```
```   415   "number_of v - (1::nat) =
```
```   416     (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
```
```   417   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   418   by auto
```
```   419
```
```   420
```
```   421 subsubsection{*Multiplication *}
```
```   422
```
```   423 lemma mult_nat_number_of [simp]:
```
```   424      "(number_of v :: nat) * number_of v' =
```
```   425        (if v < Int.Pls then 0 else number_of (v * v'))"
```
```   426   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   427   by (simp add: nat_mult_distrib)
```
```   428
```
```   429
```
```   430 subsection{*Comparisons*}
```
```   431
```
```   432 subsubsection{*Equals (=) *}
```
```   433
```
```   434 lemma eq_nat_number_of [simp]:
```
```   435      "((number_of v :: nat) = number_of v') =
```
```   436       (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
```
```   437        else if neg (number_of v' :: int) then (number_of v :: int) = 0
```
```   438        else v = v')"
```
```   439   unfolding nat_number_of_def number_of_is_id neg_def
```
```   440   by auto
```
```   441
```
```   442
```
```   443 subsubsection{*Less-than (<) *}
```
```   444
```
```   445 lemma less_nat_number_of [simp]:
```
```   446   "(number_of v :: nat) < number_of v' \<longleftrightarrow>
```
```   447     (if v < v' then Int.Pls < v' else False)"
```
```   448   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   449   by auto
```
```   450
```
```   451
```
```   452 subsubsection{*Less-than-or-equal *}
```
```   453
```
```   454 lemma le_nat_number_of [simp]:
```
```   455   "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
```
```   456     (if v \<le> v' then True else v \<le> Int.Pls)"
```
```   457   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   458   by auto
```
```   459
```
```   460 (*Maps #n to n for n = 0, 1, 2*)
```
```   461 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
```
```   462
```
```   463
```
```   464 subsection{*Powers with Numeric Exponents*}
```
```   465
```
```   466 text{*Squares of literal numerals will be evaluated.*}
```
```   467 lemmas power2_eq_square_number_of [simp] =
```
```   468     power2_eq_square [of "number_of w", standard]
```
```   469
```
```   470
```
```   471 text{*Simprules for comparisons where common factors can be cancelled.*}
```
```   472 lemmas zero_compare_simps =
```
```   473     add_strict_increasing add_strict_increasing2 add_increasing
```
```   474     zero_le_mult_iff zero_le_divide_iff
```
```   475     zero_less_mult_iff zero_less_divide_iff
```
```   476     mult_le_0_iff divide_le_0_iff
```
```   477     mult_less_0_iff divide_less_0_iff
```
```   478     zero_le_power2 power2_less_0
```
```   479
```
```   480 subsubsection{*Nat *}
```
```   481
```
```   482 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
```
```   483 by simp
```
```   484
```
```   485 (*Expresses a natural number constant as the Suc of another one.
```
```   486   NOT suitable for rewriting because n recurs in the condition.*)
```
```   487 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
```
```   488
```
```   489 subsubsection{*Arith *}
```
```   490
```
```   491 lemma Suc_eq_plus1: "Suc n = n + 1"
```
```   492   unfolding One_nat_def by simp
```
```   493
```
```   494 lemma Suc_eq_plus1_left: "Suc n = 1 + n"
```
```   495   unfolding One_nat_def by simp
```
```   496
```
```   497 (* These two can be useful when m = number_of... *)
```
```   498
```
```   499 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
```
```   500   unfolding One_nat_def by (cases m) simp_all
```
```   501
```
```   502 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
```
```   503   unfolding One_nat_def by (cases m) simp_all
```
```   504
```
```   505 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   506   unfolding One_nat_def by (cases m) simp_all
```
```   507
```
```   508
```
```   509 subsection{*Comparisons involving (0::nat) *}
```
```   510
```
```   511 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
```
```   512
```
```   513 lemma eq_number_of_0 [simp]:
```
```   514   "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
```
```   515   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   516   by auto
```
```   517
```
```   518 lemma eq_0_number_of [simp]:
```
```   519   "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
```
```   520 by (rule trans [OF eq_sym_conv eq_number_of_0])
```
```   521
```
```   522 lemma less_0_number_of [simp]:
```
```   523    "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
```
```   524   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   525   by simp
```
```   526
```
```   527 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
```
```   528 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
```
```   529
```
```   530
```
```   531
```
```   532 subsection{*Comparisons involving  @{term Suc} *}
```
```   533
```
```   534 lemma eq_number_of_Suc [simp]:
```
```   535      "(number_of v = Suc n) =
```
```   536         (let pv = number_of (Int.pred v) in
```
```   537          if neg pv then False else nat pv = n)"
```
```   538 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   539                   number_of_pred nat_number_of_def
```
```   540             split add: split_if)
```
```   541 apply (rule_tac x = "number_of v" in spec)
```
```   542 apply (auto simp add: nat_eq_iff)
```
```   543 done
```
```   544
```
```   545 lemma Suc_eq_number_of [simp]:
```
```   546      "(Suc n = number_of v) =
```
```   547         (let pv = number_of (Int.pred v) in
```
```   548          if neg pv then False else nat pv = n)"
```
```   549 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
```
```   550
```
```   551 lemma less_number_of_Suc [simp]:
```
```   552      "(number_of v < Suc n) =
```
```   553         (let pv = number_of (Int.pred v) in
```
```   554          if neg pv then True else nat pv < n)"
```
```   555 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   556                   number_of_pred nat_number_of_def
```
```   557             split add: split_if)
```
```   558 apply (rule_tac x = "number_of v" in spec)
```
```   559 apply (auto simp add: nat_less_iff)
```
```   560 done
```
```   561
```
```   562 lemma less_Suc_number_of [simp]:
```
```   563      "(Suc n < number_of v) =
```
```   564         (let pv = number_of (Int.pred v) in
```
```   565          if neg pv then False else n < nat pv)"
```
```   566 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   567                   number_of_pred nat_number_of_def
```
```   568             split add: split_if)
```
```   569 apply (rule_tac x = "number_of v" in spec)
```
```   570 apply (auto simp add: zless_nat_eq_int_zless)
```
```   571 done
```
```   572
```
```   573 lemma le_number_of_Suc [simp]:
```
```   574      "(number_of v <= Suc n) =
```
```   575         (let pv = number_of (Int.pred v) in
```
```   576          if neg pv then True else nat pv <= n)"
```
```   577 by (simp add: Let_def linorder_not_less [symmetric])
```
```   578
```
```   579 lemma le_Suc_number_of [simp]:
```
```   580      "(Suc n <= number_of v) =
```
```   581         (let pv = number_of (Int.pred v) in
```
```   582          if neg pv then False else n <= nat pv)"
```
```   583 by (simp add: Let_def linorder_not_less [symmetric])
```
```   584
```
```   585
```
```   586 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
```
```   587 by auto
```
```   588
```
```   589
```
```   590
```
```   591 subsection{*Max and Min Combined with @{term Suc} *}
```
```   592
```
```   593 lemma max_number_of_Suc [simp]:
```
```   594      "max (Suc n) (number_of v) =
```
```   595         (let pv = number_of (Int.pred v) in
```
```   596          if neg pv then Suc n else Suc(max n (nat pv)))"
```
```   597 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   598             split add: split_if nat.split)
```
```   599 apply (rule_tac x = "number_of v" in spec)
```
```   600 apply auto
```
```   601 done
```
```   602
```
```   603 lemma max_Suc_number_of [simp]:
```
```   604      "max (number_of v) (Suc n) =
```
```   605         (let pv = number_of (Int.pred v) in
```
```   606          if neg pv then Suc n else Suc(max (nat pv) n))"
```
```   607 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   608             split add: split_if nat.split)
```
```   609 apply (rule_tac x = "number_of v" in spec)
```
```   610 apply auto
```
```   611 done
```
```   612
```
```   613 lemma min_number_of_Suc [simp]:
```
```   614      "min (Suc n) (number_of v) =
```
```   615         (let pv = number_of (Int.pred v) in
```
```   616          if neg pv then 0 else Suc(min n (nat pv)))"
```
```   617 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   618             split add: split_if nat.split)
```
```   619 apply (rule_tac x = "number_of v" in spec)
```
```   620 apply auto
```
```   621 done
```
```   622
```
```   623 lemma min_Suc_number_of [simp]:
```
```   624      "min (number_of v) (Suc n) =
```
```   625         (let pv = number_of (Int.pred v) in
```
```   626          if neg pv then 0 else Suc(min (nat pv) n))"
```
```   627 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   628             split add: split_if nat.split)
```
```   629 apply (rule_tac x = "number_of v" in spec)
```
```   630 apply auto
```
```   631 done
```
```   632
```
```   633 subsection{*Literal arithmetic involving powers*}
```
```   634
```
```   635 lemma power_nat_number_of:
```
```   636      "(number_of v :: nat) ^ n =
```
```   637        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
```
```   638 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
```
```   639          split add: split_if cong: imp_cong)
```
```   640
```
```   641
```
```   642 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
```
```   643 declare power_nat_number_of_number_of [simp]
```
```   644
```
```   645
```
```   646
```
```   647 text{*For arbitrary rings*}
```
```   648
```
```   649 lemma power_number_of_even:
```
```   650   fixes z :: "'a::number_ring"
```
```   651   shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
```
```   652 by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
```
```   653   nat_add_distrib power_add simp del: nat_number_of)
```
```   654
```
```   655 lemma power_number_of_odd:
```
```   656   fixes z :: "'a::number_ring"
```
```   657   shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
```
```   658      then (let w = z ^ (number_of w) in z * w * w) else 1)"
```
```   659 unfolding Let_def Bit1_def nat_number_of_def number_of_is_id
```
```   660 apply (cases "0 <= w")
```
```   661 apply (simp only: mult_assoc nat_add_distrib power_add, simp)
```
```   662 apply (simp add: not_le mult_2 [symmetric] add_assoc)
```
```   663 done
```
```   664
```
```   665 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
```
```   666 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
```
```   667
```
```   668 lemmas power_number_of_even_number_of [simp] =
```
```   669     power_number_of_even [of "number_of v", standard]
```
```   670
```
```   671 lemmas power_number_of_odd_number_of [simp] =
```
```   672     power_number_of_odd [of "number_of v", standard]
```
```   673
```
```   674 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
```
```   675   by (simp add: nat_number_of_def)
```
```   676
```
```   677 lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
```
```   678   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
```
```   679   done
```
```   680
```
```   681 lemma nat_number_of_Bit0:
```
```   682     "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
```
```   683 by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
```
```   684   nat_add_distrib simp del: nat_number_of)
```
```   685
```
```   686 lemma nat_number_of_Bit1:
```
```   687   "number_of (Int.Bit1 w) =
```
```   688     (if neg (number_of w :: int) then 0
```
```   689      else let n = number_of w in Suc (n + n))"
```
```   690 unfolding Let_def Bit1_def nat_number_of_def number_of_is_id neg_def
```
```   691 apply (cases "w < 0")
```
```   692 apply (simp add: mult_2 [symmetric] add_assoc)
```
```   693 apply (simp only: nat_add_distrib, simp)
```
```   694 done
```
```   695
```
```   696 lemmas nat_number =
```
```   697   nat_number_of_Pls nat_number_of_Min
```
```   698   nat_number_of_Bit0 nat_number_of_Bit1
```
```   699
```
```   700 lemmas nat_number' =
```
```   701   nat_number_of_Bit0 nat_number_of_Bit1
```
```   702
```
```   703 lemmas nat_arith =
```
```   704   add_nat_number_of
```
```   705   diff_nat_number_of
```
```   706   mult_nat_number_of
```
```   707   eq_nat_number_of
```
```   708   less_nat_number_of
```
```   709
```
```   710 lemmas semiring_norm =
```
```   711   Let_def arith_simps nat_arith rel_simps neg_simps if_False
```
```   712   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   713   numeral_1_eq_1 [symmetric] Suc_eq_plus1
```
```   714   numeral_0_eq_0 [symmetric] numerals [symmetric]
```
```   715   not_iszero_Numeral1
```
```   716
```
```   717 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
```
```   718   by (fact Let_def)
```
```   719
```
```   720 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring})"
```
```   721   by (simp only: number_of_Min power_minus1_even)
```
```   722
```
```   723 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring})"
```
```   724   by (simp only: number_of_Min power_minus1_odd)
```
```   725
```
```   726 lemma nat_number_of_add_left:
```
```   727      "number_of v + (number_of v' + (k::nat)) =
```
```   728          (if neg (number_of v :: int) then number_of v' + k
```
```   729           else if neg (number_of v' :: int) then number_of v + k
```
```   730           else number_of (v + v') + k)"
```
```   731 by (auto simp add: neg_def)
```
```   732
```
```   733 lemma nat_number_of_mult_left:
```
```   734      "number_of v * (number_of v' * (k::nat)) =
```
```   735          (if v < Int.Pls then 0
```
```   736           else number_of (v * v') * k)"
```
```   737 by (auto simp add: not_less Pls_def nat_number_of_def number_of_is_id
```
```   738   nat_mult_distrib simp del: nat_number_of)
```
```   739
```
```   740
```
```   741 subsection{*Literal arithmetic and @{term of_nat}*}
```
```   742
```
```   743 lemma of_nat_double:
```
```   744      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
```
```   745 by (simp only: mult_2 nat_add_distrib of_nat_add)
```
```   746
```
```   747 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
```
```   748 by (simp only: nat_number_of_def)
```
```   749
```
```   750 lemma of_nat_number_of_lemma:
```
```   751      "of_nat (number_of v :: nat) =
```
```   752          (if 0 \<le> (number_of v :: int)
```
```   753           then (number_of v :: 'a :: number_ring)
```
```   754           else 0)"
```
```   755 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat)
```
```   756
```
```   757 lemma of_nat_number_of_eq [simp]:
```
```   758      "of_nat (number_of v :: nat) =
```
```   759          (if neg (number_of v :: int) then 0
```
```   760           else (number_of v :: 'a :: number_ring))"
```
```   761 by (simp only: of_nat_number_of_lemma neg_def, simp)
```
```   762
```
```   763
```
```   764 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
```
```   765
```
```   766 text{*Where K above is a literal*}
```
```   767
```
```   768 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
```
```   769 by (simp split: nat_diff_split)
```
```   770
```
```   771 text {*Now just instantiating @{text n} to @{text "number_of v"} does
```
```   772   the right simplification, but with some redundant inequality
```
```   773   tests.*}
```
```   774 lemma neg_number_of_pred_iff_0:
```
```   775   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
```
```   776 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
```
```   777 apply (simp only: less_Suc_eq_le le_0_eq)
```
```   778 apply (subst less_number_of_Suc, simp)
```
```   779 done
```
```   780
```
```   781 text{*No longer required as a simprule because of the @{text inverse_fold}
```
```   782    simproc*}
```
```   783 lemma Suc_diff_number_of:
```
```   784      "Int.Pls < v ==>
```
```   785       Suc m - (number_of v) = m - (number_of (Int.pred v))"
```
```   786 apply (subst Suc_diff_eq_diff_pred)
```
```   787 apply simp
```
```   788 apply (simp del: nat_numeral_1_eq_1)
```
```   789 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
```
```   790                         neg_number_of_pred_iff_0)
```
```   791 done
```
```   792
```
```   793 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
```
```   794 by (simp split: nat_diff_split)
```
```   795
```
```   796
```
```   797 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
```
```   798
```
```   799 lemma nat_case_number_of [simp]:
```
```   800      "nat_case a f (number_of v) =
```
```   801         (let pv = number_of (Int.pred v) in
```
```   802          if neg pv then a else f (nat pv))"
```
```   803 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
```
```   804
```
```   805 lemma nat_case_add_eq_if [simp]:
```
```   806      "nat_case a f ((number_of v) + n) =
```
```   807        (let pv = number_of (Int.pred v) in
```
```   808          if neg pv then nat_case a f n else f (nat pv + n))"
```
```   809 apply (subst add_eq_if)
```
```   810 apply (simp split add: nat.split
```
```   811             del: nat_numeral_1_eq_1
```
```   812             add: nat_numeral_1_eq_1 [symmetric]
```
```   813                  numeral_1_eq_Suc_0 [symmetric]
```
```   814                  neg_number_of_pred_iff_0)
```
```   815 done
```
```   816
```
```   817 lemma nat_rec_number_of [simp]:
```
```   818      "nat_rec a f (number_of v) =
```
```   819         (let pv = number_of (Int.pred v) in
```
```   820          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
```
```   821 apply (case_tac " (number_of v) ::nat")
```
```   822 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
```
```   823 apply (simp split add: split_if_asm)
```
```   824 done
```
```   825
```
```   826 lemma nat_rec_add_eq_if [simp]:
```
```   827      "nat_rec a f (number_of v + n) =
```
```   828         (let pv = number_of (Int.pred v) in
```
```   829          if neg pv then nat_rec a f n
```
```   830                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
```
```   831 apply (subst add_eq_if)
```
```   832 apply (simp split add: nat.split
```
```   833             del: nat_numeral_1_eq_1
```
```   834             add: nat_numeral_1_eq_1 [symmetric]
```
```   835                  numeral_1_eq_Suc_0 [symmetric]
```
```   836                  neg_number_of_pred_iff_0)
```
```   837 done
```
```   838
```
```   839
```
```   840 subsubsection{*Various Other Lemmas*}
```
```   841
```
```   842 lemma card_UNIV_bool[simp]: "card (UNIV :: bool set) = 2"
```
```   843 by(simp add: UNIV_bool)
```
```   844
```
```   845 text {*Evens and Odds, for Mutilated Chess Board*}
```
```   846
```
```   847 text{*Lemmas for specialist use, NOT as default simprules*}
```
```   848 lemma nat_mult_2: "2 * z = (z+z::nat)"
```
```   849 unfolding nat_1_add_1 [symmetric] left_distrib by simp
```
```   850
```
```   851 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
```
```   852 by (subst mult_commute, rule nat_mult_2)
```
```   853
```
```   854 text{*Case analysis on @{term "n<2"}*}
```
```   855 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
```
```   856 by (auto simp add: nat_1_add_1 [symmetric])
```
```   857
```
```   858 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
```
```   859
```
```   860 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```   861 by simp
```
```   862
```
```   863 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```   864 by simp
```
```   865
```
```   866 text{*Can be used to eliminate long strings of Sucs, but not by default*}
```
```   867 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```   868 by simp
```
```   869
```
```   870 end
```