src/HOL/Nat_Numeral.thy
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```     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_semiring
```
```    19 begin
```
```    20
```
```    21 definition
```
```    22   nat_number_of_def [code_unfold, code del]: "number_of v = nat (number_of v)"
```
```    23
```
```    24 instance proof
```
```    25   fix n show "number_of (int n) = (of_nat n :: nat)"
```
```    26     unfolding nat_number_of_def number_of_eq by simp
```
```    27 qed
```
```    28
```
```    29 end
```
```    30
```
```    31 lemma [code_post]:
```
```    32   "nat (number_of v) = number_of v"
```
```    33   unfolding nat_number_of_def ..
```
```    34
```
```    35
```
```    36 subsection {* Special case: squares and cubes *}
```
```    37
```
```    38 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
```
```    39   by (simp add: nat_number_of_def)
```
```    40
```
```    41 lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
```
```    42   by (simp add: nat_number_of_def)
```
```    43
```
```    44 context power
```
```    45 begin
```
```    46
```
```    47 abbreviation (xsymbols)
```
```    48   power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
```
```    49   "x\<twosuperior> \<equiv> x ^ 2"
```
```    50
```
```    51 notation (latex output)
```
```    52   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    53
```
```    54 notation (HTML output)
```
```    55   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    56
```
```    57 end
```
```    58
```
```    59 context monoid_mult
```
```    60 begin
```
```    61
```
```    62 lemma power2_eq_square: "a\<twosuperior> = a * a"
```
```    63   by (simp add: numeral_2_eq_2)
```
```    64
```
```    65 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    66   by (simp add: numeral_3_eq_3 mult_assoc)
```
```    67
```
```    68 lemma power_even_eq:
```
```    69   "a ^ (2*n) = (a ^ n) ^ 2"
```
```    70   by (subst mult_commute) (simp add: power_mult)
```
```    71
```
```    72 lemma power_odd_eq:
```
```    73   "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
```
```    74   by (simp add: power_even_eq)
```
```    75
```
```    76 end
```
```    77
```
```    78 context semiring_1
```
```    79 begin
```
```    80
```
```    81 lemma zero_power2 [simp]: "0\<twosuperior> = 0"
```
```    82   by (simp add: power2_eq_square)
```
```    83
```
```    84 lemma one_power2 [simp]: "1\<twosuperior> = 1"
```
```    85   by (simp add: power2_eq_square)
```
```    86
```
```    87 end
```
```    88
```
```    89 context ring_1
```
```    90 begin
```
```    91
```
```    92 lemma power2_minus [simp]:
```
```    93   "(- a)\<twosuperior> = a\<twosuperior>"
```
```    94   by (simp add: power2_eq_square)
```
```    95
```
```    96 text{*
```
```    97   We cannot prove general results about the numeral @{term "-1"},
```
```    98   so we have to use @{term "- 1"} instead.
```
```    99 *}
```
```   100
```
```   101 lemma power_minus1_even [simp]:
```
```   102   "(- 1) ^ (2*n) = 1"
```
```   103 proof (induct n)
```
```   104   case 0 show ?case by simp
```
```   105 next
```
```   106   case (Suc n) then show ?case by (simp add: power_add)
```
```   107 qed
```
```   108
```
```   109 lemma power_minus1_odd:
```
```   110   "(- 1) ^ Suc (2*n) = - 1"
```
```   111   by simp
```
```   112
```
```   113 lemma power_minus_even [simp]:
```
```   114   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   115   by (simp add: power_minus [of a])
```
```   116
```
```   117 end
```
```   118
```
```   119 context ring_1_no_zero_divisors
```
```   120 begin
```
```   121
```
```   122 lemma zero_eq_power2 [simp]:
```
```   123   "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
```
```   124   unfolding power2_eq_square by simp
```
```   125
```
```   126 lemma power2_eq_1_iff:
```
```   127   "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   128   unfolding power2_eq_square by (rule square_eq_1_iff)
```
```   129
```
```   130 end
```
```   131
```
```   132 context idom
```
```   133 begin
```
```   134
```
```   135 lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y"
```
```   136   unfolding power2_eq_square by (rule square_eq_iff)
```
```   137
```
```   138 end
```
```   139
```
```   140 context linordered_ring
```
```   141 begin
```
```   142
```
```   143 lemma sum_squares_ge_zero:
```
```   144   "0 \<le> x * x + y * y"
```
```   145   by (intro add_nonneg_nonneg zero_le_square)
```
```   146
```
```   147 lemma not_sum_squares_lt_zero:
```
```   148   "\<not> x * x + y * y < 0"
```
```   149   by (simp add: not_less sum_squares_ge_zero)
```
```   150
```
```   151 end
```
```   152
```
```   153 context linordered_ring_strict
```
```   154 begin
```
```   155
```
```   156 lemma sum_squares_eq_zero_iff:
```
```   157   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   158   by (simp add: add_nonneg_eq_0_iff)
```
```   159
```
```   160 lemma sum_squares_le_zero_iff:
```
```   161   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   162   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   163
```
```   164 lemma sum_squares_gt_zero_iff:
```
```   165   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   166   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   167
```
```   168 end
```
```   169
```
```   170 context linordered_semidom
```
```   171 begin
```
```   172
```
```   173 lemma power2_le_imp_le:
```
```   174   "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   175   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   176
```
```   177 lemma power2_less_imp_less:
```
```   178   "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   179   by (rule power_less_imp_less_base)
```
```   180
```
```   181 lemma power2_eq_imp_eq:
```
```   182   "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   183   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   184
```
```   185 end
```
```   186
```
```   187 context linordered_idom
```
```   188 begin
```
```   189
```
```   190 lemma zero_le_power2 [simp]:
```
```   191   "0 \<le> a\<twosuperior>"
```
```   192   by (simp add: power2_eq_square)
```
```   193
```
```   194 lemma zero_less_power2 [simp]:
```
```   195   "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
```
```   196   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   197
```
```   198 lemma power2_less_0 [simp]:
```
```   199   "\<not> a\<twosuperior> < 0"
```
```   200   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   201
```
```   202 lemma abs_power2 [simp]:
```
```   203   "abs (a\<twosuperior>) = a\<twosuperior>"
```
```   204   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   205
```
```   206 lemma power2_abs [simp]:
```
```   207   "(abs a)\<twosuperior> = a\<twosuperior>"
```
```   208   by (simp add: power2_eq_square abs_mult_self)
```
```   209
```
```   210 lemma odd_power_less_zero:
```
```   211   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   212 proof (induct n)
```
```   213   case 0
```
```   214   then show ?case by simp
```
```   215 next
```
```   216   case (Suc n)
```
```   217   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   218     by (simp add: mult_ac power_add power2_eq_square)
```
```   219   thus ?case
```
```   220     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   221 qed
```
```   222
```
```   223 lemma odd_0_le_power_imp_0_le:
```
```   224   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   225   using odd_power_less_zero [of a n]
```
```   226     by (force simp add: linorder_not_less [symmetric])
```
```   227
```
```   228 lemma zero_le_even_power'[simp]:
```
```   229   "0 \<le> a ^ (2*n)"
```
```   230 proof (induct n)
```
```   231   case 0
```
```   232     show ?case by simp
```
```   233 next
```
```   234   case (Suc n)
```
```   235     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   236       by (simp add: mult_ac power_add power2_eq_square)
```
```   237     thus ?case
```
```   238       by (simp add: Suc zero_le_mult_iff)
```
```   239 qed
```
```   240
```
```   241 lemma sum_power2_ge_zero:
```
```   242   "0 \<le> x\<twosuperior> + y\<twosuperior>"
```
```   243   unfolding power2_eq_square by (rule sum_squares_ge_zero)
```
```   244
```
```   245 lemma not_sum_power2_lt_zero:
```
```   246   "\<not> x\<twosuperior> + y\<twosuperior> < 0"
```
```   247   unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
```
```   248
```
```   249 lemma sum_power2_eq_zero_iff:
```
```   250   "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   251   unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
```
```   252
```
```   253 lemma sum_power2_le_zero_iff:
```
```   254   "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   255   unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
```
```   256
```
```   257 lemma sum_power2_gt_zero_iff:
```
```   258   "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   259   unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
```
```   260
```
```   261 end
```
```   262
```
```   263 lemma power2_sum:
```
```   264   fixes x y :: "'a::number_semiring"
```
```   265   shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
```
```   266   by (simp add: algebra_simps power2_eq_square semiring_mult_2_right)
```
```   267
```
```   268 lemma power2_diff:
```
```   269   fixes x y :: "'a::number_ring"
```
```   270   shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
```
```   271   by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
```
```   272
```
```   273
```
```   274 subsection {* Predicate for negative binary numbers *}
```
```   275
```
```   276 definition neg  :: "int \<Rightarrow> bool" where
```
```   277   "neg Z \<longleftrightarrow> Z < 0"
```
```   278
```
```   279 lemma not_neg_int [simp]: "~ neg (of_nat n)"
```
```   280 by (simp add: neg_def)
```
```   281
```
```   282 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
```
```   283 by (simp add: neg_def del: of_nat_Suc)
```
```   284
```
```   285 lemmas neg_eq_less_0 = neg_def
```
```   286
```
```   287 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   288 by (simp add: neg_def linorder_not_less)
```
```   289
```
```   290 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   291
```
```   292 lemma not_neg_0: "~ neg 0"
```
```   293 by (simp add: One_int_def neg_def)
```
```   294
```
```   295 lemma not_neg_1: "~ neg 1"
```
```   296 by (simp add: neg_def linorder_not_less)
```
```   297
```
```   298 lemma neg_nat: "neg z ==> nat z = 0"
```
```   299 by (simp add: neg_def order_less_imp_le)
```
```   300
```
```   301 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
```
```   302 by (simp add: linorder_not_less neg_def)
```
```   303
```
```   304 text {*
```
```   305   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```   306   @{term Numeral0} IS @{term "number_of Pls"}
```
```   307 *}
```
```   308
```
```   309 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
```
```   310   by (simp add: neg_def)
```
```   311
```
```   312 lemma neg_number_of_Min: "neg (number_of Int.Min)"
```
```   313   by (simp add: neg_def)
```
```   314
```
```   315 lemma neg_number_of_Bit0:
```
```   316   "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
```
```   317   by (simp add: neg_def)
```
```   318
```
```   319 lemma neg_number_of_Bit1:
```
```   320   "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
```
```   321   by (simp add: neg_def)
```
```   322
```
```   323 lemmas neg_simps [simp] =
```
```   324   not_neg_0 not_neg_1
```
```   325   not_neg_number_of_Pls neg_number_of_Min
```
```   326   neg_number_of_Bit0 neg_number_of_Bit1
```
```   327
```
```   328
```
```   329 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
```
```   330
```
```   331 declare nat_1 [simp]
```
```   332
```
```   333 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
```
```   334 by (simp add: nat_number_of_def)
```
```   335
```
```   336 lemma nat_numeral_0_eq_0 [simp, code_post]: "Numeral0 = (0::nat)"
```
```   337   by (rule semiring_numeral_0_eq_0)
```
```   338
```
```   339 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
```
```   340   by (rule semiring_numeral_1_eq_1)
```
```   341
```
```   342 lemma Numeral1_eq1_nat:
```
```   343   "(1::nat) = Numeral1"
```
```   344   by simp
```
```   345
```
```   346 lemma numeral_1_eq_Suc_0 [code_post]: "Numeral1 = Suc 0"
```
```   347 by (simp only: nat_numeral_1_eq_1 One_nat_def)
```
```   348
```
```   349
```
```   350 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
```
```   351
```
```   352 lemma int_nat_number_of [simp]:
```
```   353      "int (number_of v) =
```
```   354          (if neg (number_of v :: int) then 0
```
```   355           else (number_of v :: int))"
```
```   356   unfolding nat_number_of_def number_of_is_id neg_def
```
```   357   by simp (* FIXME: redundant with of_nat_number_of_eq *)
```
```   358
```
```   359 lemma nonneg_int_cases:
```
```   360   fixes k :: int assumes "0 \<le> k" obtains n where "k = of_nat n"
```
```   361   using assms by (cases k, simp, simp)
```
```   362
```
```   363 subsubsection{*Successor *}
```
```   364
```
```   365 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   366 apply (rule sym)
```
```   367 apply (simp add: nat_eq_iff)
```
```   368 done
```
```   369
```
```   370 lemma Suc_nat_number_of_add:
```
```   371      "Suc (number_of v + n) =
```
```   372         (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
```
```   373   unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
```
```   374   by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
```
```   375
```
```   376 lemma Suc_nat_number_of [simp]:
```
```   377      "Suc (number_of v) =
```
```   378         (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
```
```   379 apply (cut_tac n = 0 in Suc_nat_number_of_add)
```
```   380 apply (simp cong del: if_weak_cong)
```
```   381 done
```
```   382
```
```   383
```
```   384 subsubsection{*Addition *}
```
```   385
```
```   386 lemma add_nat_number_of [simp]:
```
```   387      "(number_of v :: nat) + number_of v' =
```
```   388          (if v < Int.Pls then number_of v'
```
```   389           else if v' < Int.Pls then number_of v
```
```   390           else number_of (v + v'))"
```
```   391   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   392   by (simp add: nat_add_distrib)
```
```   393
```
```   394 lemma nat_number_of_add_1 [simp]:
```
```   395   "number_of v + (1::nat) =
```
```   396     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   397   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   398   by (simp add: nat_add_distrib)
```
```   399
```
```   400 lemma nat_1_add_number_of [simp]:
```
```   401   "(1::nat) + number_of v =
```
```   402     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   403   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   404   by (simp add: nat_add_distrib)
```
```   405
```
```   406 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
```
```   407   by (rule semiring_one_add_one_is_two)
```
```   408
```
```   409 text {* TODO: replace simp rules above with these generic ones: *}
```
```   410
```
```   411 lemma semiring_add_number_of:
```
```   412   "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
```
```   413     (number_of v :: 'a::number_semiring) + number_of v' = number_of (v + v')"
```
```   414   unfolding Int.Pls_def
```
```   415   by (elim nonneg_int_cases,
```
```   416     simp only: number_of_int of_nat_add [symmetric])
```
```   417
```
```   418 lemma semiring_number_of_add_1:
```
```   419   "Int.Pls \<le> v \<Longrightarrow>
```
```   420     number_of v + (1::'a::number_semiring) = number_of (Int.succ v)"
```
```   421   unfolding Int.Pls_def Int.succ_def
```
```   422   by (elim nonneg_int_cases,
```
```   423     simp only: number_of_int add_commute [where b=1] of_nat_Suc [symmetric])
```
```   424
```
```   425 lemma semiring_1_add_number_of:
```
```   426   "Int.Pls \<le> v \<Longrightarrow>
```
```   427     (1::'a::number_semiring) + number_of v = number_of (Int.succ v)"
```
```   428   unfolding Int.Pls_def Int.succ_def
```
```   429   by (elim nonneg_int_cases,
```
```   430     simp only: number_of_int add_commute [where b=1] of_nat_Suc [symmetric])
```
```   431
```
```   432
```
```   433 subsubsection{*Subtraction *}
```
```   434
```
```   435 lemma diff_nat_eq_if:
```
```   436      "nat z - nat z' =
```
```   437         (if neg z' then nat z
```
```   438          else let d = z-z' in
```
```   439               if neg d then 0 else nat d)"
```
```   440 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
```
```   441
```
```   442
```
```   443 lemma diff_nat_number_of [simp]:
```
```   444      "(number_of v :: nat) - number_of v' =
```
```   445         (if v' < Int.Pls then number_of v
```
```   446          else let d = number_of (v + uminus v') in
```
```   447               if neg d then 0 else nat d)"
```
```   448   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
```
```   449   by auto
```
```   450
```
```   451 lemma nat_number_of_diff_1 [simp]:
```
```   452   "number_of v - (1::nat) =
```
```   453     (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
```
```   454   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   455   by auto
```
```   456
```
```   457
```
```   458 subsubsection{*Multiplication *}
```
```   459
```
```   460 lemma mult_nat_number_of [simp]:
```
```   461      "(number_of v :: nat) * number_of v' =
```
```   462        (if v < Int.Pls then 0 else number_of (v * v'))"
```
```   463   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   464   by (simp add: nat_mult_distrib)
```
```   465
```
```   466 (* TODO: replace mult_nat_number_of with this next rule *)
```
```   467 lemma semiring_mult_number_of:
```
```   468   "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
```
```   469     (number_of v :: 'a::number_semiring) * number_of v' = number_of (v * v')"
```
```   470   unfolding Int.Pls_def
```
```   471   by (elim nonneg_int_cases,
```
```   472     simp only: number_of_int of_nat_mult [symmetric])
```
```   473
```
```   474
```
```   475 subsection{*Comparisons*}
```
```   476
```
```   477 subsubsection{*Equals (=) *}
```
```   478
```
```   479 lemma eq_nat_number_of [simp]:
```
```   480      "((number_of v :: nat) = number_of v') =
```
```   481       (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
```
```   482        else if neg (number_of v' :: int) then (number_of v :: int) = 0
```
```   483        else v = v')"
```
```   484   unfolding nat_number_of_def number_of_is_id neg_def
```
```   485   by auto
```
```   486
```
```   487
```
```   488 subsubsection{*Less-than (<) *}
```
```   489
```
```   490 lemma less_nat_number_of [simp]:
```
```   491   "(number_of v :: nat) < number_of v' \<longleftrightarrow>
```
```   492     (if v < v' then Int.Pls < v' else False)"
```
```   493   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   494   by auto
```
```   495
```
```   496
```
```   497 subsubsection{*Less-than-or-equal *}
```
```   498
```
```   499 lemma le_nat_number_of [simp]:
```
```   500   "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
```
```   501     (if v \<le> v' then True else v \<le> Int.Pls)"
```
```   502   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   503   by auto
```
```   504
```
```   505 (*Maps #n to n for n = 0, 1, 2*)
```
```   506 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
```
```   507
```
```   508
```
```   509 subsection{*Powers with Numeric Exponents*}
```
```   510
```
```   511 text{*Squares of literal numerals will be evaluated.*}
```
```   512 lemmas power2_eq_square_number_of [simp] =
```
```   513   power2_eq_square [of "number_of w"] for w
```
```   514
```
```   515
```
```   516 text{*Simprules for comparisons where common factors can be cancelled.*}
```
```   517 lemmas zero_compare_simps =
```
```   518     add_strict_increasing add_strict_increasing2 add_increasing
```
```   519     zero_le_mult_iff zero_le_divide_iff
```
```   520     zero_less_mult_iff zero_less_divide_iff
```
```   521     mult_le_0_iff divide_le_0_iff
```
```   522     mult_less_0_iff divide_less_0_iff
```
```   523     zero_le_power2 power2_less_0
```
```   524
```
```   525 subsubsection{*Nat *}
```
```   526
```
```   527 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
```
```   528 by simp
```
```   529
```
```   530 (*Expresses a natural number constant as the Suc of another one.
```
```   531   NOT suitable for rewriting because n recurs in the condition.*)
```
```   532 lemmas expand_Suc = Suc_pred' [of "number_of v"] for v
```
```   533
```
```   534 subsubsection{*Arith *}
```
```   535
```
```   536 lemma Suc_eq_plus1: "Suc n = n + 1"
```
```   537   unfolding One_nat_def by simp
```
```   538
```
```   539 lemma Suc_eq_plus1_left: "Suc n = 1 + n"
```
```   540   unfolding One_nat_def by simp
```
```   541
```
```   542 (* These two can be useful when m = number_of... *)
```
```   543
```
```   544 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
```
```   545   unfolding One_nat_def by (cases m) simp_all
```
```   546
```
```   547 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
```
```   548   unfolding One_nat_def by (cases m) simp_all
```
```   549
```
```   550 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   551   unfolding One_nat_def by (cases m) simp_all
```
```   552
```
```   553
```
```   554 subsection{*Comparisons involving (0::nat) *}
```
```   555
```
```   556 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
```
```   557
```
```   558 lemma eq_number_of_0 [simp]:
```
```   559   "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
```
```   560   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   561   by auto
```
```   562
```
```   563 lemma eq_0_number_of [simp]:
```
```   564   "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
```
```   565 by (rule trans [OF eq_sym_conv eq_number_of_0])
```
```   566
```
```   567 lemma less_0_number_of [simp]:
```
```   568    "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
```
```   569   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   570   by simp
```
```   571
```
```   572 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
```
```   573 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
```
```   574
```
```   575
```
```   576
```
```   577 subsection{*Comparisons involving  @{term Suc} *}
```
```   578
```
```   579 lemma eq_number_of_Suc [simp]:
```
```   580      "(number_of v = Suc n) =
```
```   581         (let pv = number_of (Int.pred v) in
```
```   582          if neg pv then False else nat pv = n)"
```
```   583 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   584                   number_of_pred nat_number_of_def
```
```   585             split add: split_if)
```
```   586 apply (rule_tac x = "number_of v" in spec)
```
```   587 apply (auto simp add: nat_eq_iff)
```
```   588 done
```
```   589
```
```   590 lemma Suc_eq_number_of [simp]:
```
```   591      "(Suc n = number_of v) =
```
```   592         (let pv = number_of (Int.pred v) in
```
```   593          if neg pv then False else nat pv = n)"
```
```   594 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
```
```   595
```
```   596 lemma less_number_of_Suc [simp]:
```
```   597      "(number_of v < Suc n) =
```
```   598         (let pv = number_of (Int.pred v) in
```
```   599          if neg pv then True else nat pv < n)"
```
```   600 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   601                   number_of_pred nat_number_of_def
```
```   602             split add: split_if)
```
```   603 apply (rule_tac x = "number_of v" in spec)
```
```   604 apply (auto simp add: nat_less_iff)
```
```   605 done
```
```   606
```
```   607 lemma less_Suc_number_of [simp]:
```
```   608      "(Suc n < number_of v) =
```
```   609         (let pv = number_of (Int.pred v) in
```
```   610          if neg pv then False else n < nat pv)"
```
```   611 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   612                   number_of_pred nat_number_of_def
```
```   613             split add: split_if)
```
```   614 apply (rule_tac x = "number_of v" in spec)
```
```   615 apply (auto simp add: zless_nat_eq_int_zless)
```
```   616 done
```
```   617
```
```   618 lemma le_number_of_Suc [simp]:
```
```   619      "(number_of v <= Suc n) =
```
```   620         (let pv = number_of (Int.pred v) in
```
```   621          if neg pv then True else nat pv <= n)"
```
```   622 by (simp add: Let_def linorder_not_less [symmetric])
```
```   623
```
```   624 lemma le_Suc_number_of [simp]:
```
```   625      "(Suc n <= number_of v) =
```
```   626         (let pv = number_of (Int.pred v) in
```
```   627          if neg pv then False else n <= nat pv)"
```
```   628 by (simp add: Let_def linorder_not_less [symmetric])
```
```   629
```
```   630
```
```   631 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
```
```   632 by auto
```
```   633
```
```   634
```
```   635
```
```   636 subsection{*Max and Min Combined with @{term Suc} *}
```
```   637
```
```   638 lemma max_number_of_Suc [simp]:
```
```   639      "max (Suc n) (number_of v) =
```
```   640         (let pv = number_of (Int.pred v) in
```
```   641          if neg pv then Suc n else Suc(max n (nat pv)))"
```
```   642 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   643             split add: split_if nat.split)
```
```   644 apply (rule_tac x = "number_of v" in spec)
```
```   645 apply auto
```
```   646 done
```
```   647
```
```   648 lemma max_Suc_number_of [simp]:
```
```   649      "max (number_of v) (Suc n) =
```
```   650         (let pv = number_of (Int.pred v) in
```
```   651          if neg pv then Suc n else Suc(max (nat pv) n))"
```
```   652 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   653             split add: split_if nat.split)
```
```   654 apply (rule_tac x = "number_of v" in spec)
```
```   655 apply auto
```
```   656 done
```
```   657
```
```   658 lemma min_number_of_Suc [simp]:
```
```   659      "min (Suc n) (number_of v) =
```
```   660         (let pv = number_of (Int.pred v) in
```
```   661          if neg pv then 0 else Suc(min n (nat pv)))"
```
```   662 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   663             split add: split_if nat.split)
```
```   664 apply (rule_tac x = "number_of v" in spec)
```
```   665 apply auto
```
```   666 done
```
```   667
```
```   668 lemma min_Suc_number_of [simp]:
```
```   669      "min (number_of v) (Suc n) =
```
```   670         (let pv = number_of (Int.pred v) in
```
```   671          if neg pv then 0 else Suc(min (nat pv) n))"
```
```   672 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   673             split add: split_if nat.split)
```
```   674 apply (rule_tac x = "number_of v" in spec)
```
```   675 apply auto
```
```   676 done
```
```   677
```
```   678 subsection{*Literal arithmetic involving powers*}
```
```   679
```
```   680 lemma power_nat_number_of:
```
```   681      "(number_of v :: nat) ^ n =
```
```   682        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
```
```   683 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
```
```   684          split add: split_if cong: imp_cong)
```
```   685
```
```   686
```
```   687 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w"] for w
```
```   688 declare power_nat_number_of_number_of [simp]
```
```   689
```
```   690
```
```   691
```
```   692 text{*For arbitrary rings*}
```
```   693
```
```   694 lemma power_number_of_even:
```
```   695   fixes z :: "'a::monoid_mult"
```
```   696   shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
```
```   697 by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
```
```   698   nat_add_distrib power_add simp del: nat_number_of)
```
```   699
```
```   700 lemma power_number_of_odd:
```
```   701   fixes z :: "'a::monoid_mult"
```
```   702   shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
```
```   703      then (let w = z ^ (number_of w) in z * w * w) else 1)"
```
```   704 unfolding Let_def Bit1_def nat_number_of_def number_of_is_id
```
```   705 apply (cases "0 <= w")
```
```   706 apply (simp only: mult_assoc nat_add_distrib power_add, simp)
```
```   707 apply (simp add: not_le mult_2 [symmetric] add_assoc)
```
```   708 done
```
```   709
```
```   710 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
```
```   711 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
```
```   712
```
```   713 lemmas power_number_of_even_number_of [simp] =
```
```   714     power_number_of_even [of "number_of v"] for v
```
```   715
```
```   716 lemmas power_number_of_odd_number_of [simp] =
```
```   717     power_number_of_odd [of "number_of v"] for v
```
```   718
```
```   719 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
```
```   720   by (simp add: nat_number_of_def)
```
```   721
```
```   722 lemma nat_number_of_Min [no_atp]: "number_of Int.Min = (0::nat)"
```
```   723   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
```
```   724   done
```
```   725
```
```   726 lemma nat_number_of_Bit0:
```
```   727     "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
```
```   728 by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
```
```   729   nat_add_distrib simp del: nat_number_of)
```
```   730
```
```   731 lemma nat_number_of_Bit1:
```
```   732   "number_of (Int.Bit1 w) =
```
```   733     (if neg (number_of w :: int) then 0
```
```   734      else let n = number_of w in Suc (n + n))"
```
```   735 unfolding Let_def Bit1_def nat_number_of_def number_of_is_id neg_def
```
```   736 apply (cases "w < 0")
```
```   737 apply (simp add: mult_2 [symmetric] add_assoc)
```
```   738 apply (simp only: nat_add_distrib, simp)
```
```   739 done
```
```   740
```
```   741 lemmas eval_nat_numeral =
```
```   742   nat_number_of_Bit0 nat_number_of_Bit1
```
```   743
```
```   744 lemmas nat_arith =
```
```   745   add_nat_number_of
```
```   746   diff_nat_number_of
```
```   747   mult_nat_number_of
```
```   748   eq_nat_number_of
```
```   749   less_nat_number_of
```
```   750
```
```   751 lemmas semiring_norm =
```
```   752   Let_def arith_simps nat_arith rel_simps neg_simps if_False
```
```   753   if_True add_0 add_Suc add_number_of_left mult_number_of_left
```
```   754   numeral_1_eq_1 [symmetric] Suc_eq_plus1
```
```   755   numeral_0_eq_0 [symmetric] numerals [symmetric]
```
```   756   not_iszero_Numeral1
```
```   757
```
```   758 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
```
```   759   by (fact Let_def)
```
```   760
```
```   761 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring})"
```
```   762   by (simp only: number_of_Min power_minus1_even)
```
```   763
```
```   764 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring})"
```
```   765   by (simp only: number_of_Min power_minus1_odd)
```
```   766
```
```   767 lemma nat_number_of_add_left:
```
```   768      "number_of v + (number_of v' + (k::nat)) =
```
```   769          (if neg (number_of v :: int) then number_of v' + k
```
```   770           else if neg (number_of v' :: int) then number_of v + k
```
```   771           else number_of (v + v') + k)"
```
```   772 by (auto simp add: neg_def)
```
```   773
```
```   774 lemma nat_number_of_mult_left:
```
```   775      "number_of v * (number_of v' * (k::nat)) =
```
```   776          (if v < Int.Pls then 0
```
```   777           else number_of (v * v') * k)"
```
```   778 by (auto simp add: not_less Pls_def nat_number_of_def number_of_is_id
```
```   779   nat_mult_distrib simp del: nat_number_of)
```
```   780
```
```   781
```
```   782 subsection{*Literal arithmetic and @{term of_nat}*}
```
```   783
```
```   784 lemma of_nat_double:
```
```   785      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
```
```   786 by (simp only: mult_2 nat_add_distrib of_nat_add)
```
```   787
```
```   788 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
```
```   789 by (simp only: nat_number_of_def)
```
```   790
```
```   791 lemma of_nat_number_of_lemma:
```
```   792      "of_nat (number_of v :: nat) =
```
```   793          (if 0 \<le> (number_of v :: int)
```
```   794           then (number_of v :: 'a :: number_semiring)
```
```   795           else 0)"
```
```   796   by (auto simp add: int_number_of_def nat_number_of_def number_of_int
```
```   797     elim!: nonneg_int_cases)
```
```   798
```
```   799 lemma of_nat_number_of_eq [simp]:
```
```   800      "of_nat (number_of v :: nat) =
```
```   801          (if neg (number_of v :: int) then 0
```
```   802           else (number_of v :: 'a :: number_semiring))"
```
```   803   by (simp only: of_nat_number_of_lemma neg_def, simp)
```
```   804
```
```   805
```
```   806 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
```
```   807
```
```   808 text{*Where K above is a literal*}
```
```   809
```
```   810 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
```
```   811 by (simp split: nat_diff_split)
```
```   812
```
```   813 text {*Now just instantiating @{text n} to @{text "number_of v"} does
```
```   814   the right simplification, but with some redundant inequality
```
```   815   tests.*}
```
```   816 lemma neg_number_of_pred_iff_0:
```
```   817   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
```
```   818 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
```
```   819 apply (simp only: less_Suc_eq_le le_0_eq)
```
```   820 apply (subst less_number_of_Suc, simp)
```
```   821 done
```
```   822
```
```   823 text{*No longer required as a simprule because of the @{text inverse_fold}
```
```   824    simproc*}
```
```   825 lemma Suc_diff_number_of:
```
```   826      "Int.Pls < v ==>
```
```   827       Suc m - (number_of v) = m - (number_of (Int.pred v))"
```
```   828 apply (subst Suc_diff_eq_diff_pred)
```
```   829 apply simp
```
```   830 apply (simp del: nat_numeral_1_eq_1)
```
```   831 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
```
```   832                         neg_number_of_pred_iff_0)
```
```   833 done
```
```   834
```
```   835 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
```
```   836 by (simp split: nat_diff_split)
```
```   837
```
```   838
```
```   839 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
```
```   840
```
```   841 lemma nat_case_number_of [simp]:
```
```   842      "nat_case a f (number_of v) =
```
```   843         (let pv = number_of (Int.pred v) in
```
```   844          if neg pv then a else f (nat pv))"
```
```   845 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
```
```   846
```
```   847 lemma nat_case_add_eq_if [simp]:
```
```   848      "nat_case a f ((number_of v) + n) =
```
```   849        (let pv = number_of (Int.pred v) in
```
```   850          if neg pv then nat_case a f n else f (nat pv + n))"
```
```   851 apply (subst add_eq_if)
```
```   852 apply (simp split add: nat.split
```
```   853             del: nat_numeral_1_eq_1
```
```   854             add: nat_numeral_1_eq_1 [symmetric]
```
```   855                  numeral_1_eq_Suc_0 [symmetric]
```
```   856                  neg_number_of_pred_iff_0)
```
```   857 done
```
```   858
```
```   859 lemma nat_rec_number_of [simp]:
```
```   860      "nat_rec a f (number_of v) =
```
```   861         (let pv = number_of (Int.pred v) in
```
```   862          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
```
```   863 apply (case_tac " (number_of v) ::nat")
```
```   864 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
```
```   865 apply (simp split add: split_if_asm)
```
```   866 done
```
```   867
```
```   868 lemma nat_rec_add_eq_if [simp]:
```
```   869      "nat_rec a f (number_of v + n) =
```
```   870         (let pv = number_of (Int.pred v) in
```
```   871          if neg pv then nat_rec a f n
```
```   872                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
```
```   873 apply (subst add_eq_if)
```
```   874 apply (simp split add: nat.split
```
```   875             del: nat_numeral_1_eq_1
```
```   876             add: nat_numeral_1_eq_1 [symmetric]
```
```   877                  numeral_1_eq_Suc_0 [symmetric]
```
```   878                  neg_number_of_pred_iff_0)
```
```   879 done
```
```   880
```
```   881
```
```   882 subsubsection{*Various Other Lemmas*}
```
```   883
```
```   884 lemma card_UNIV_bool[simp]: "card (UNIV :: bool set) = 2"
```
```   885 by(simp add: UNIV_bool)
```
```   886
```
```   887 text {*Evens and Odds, for Mutilated Chess Board*}
```
```   888
```
```   889 text{*Lemmas for specialist use, NOT as default simprules*}
```
```   890 lemma nat_mult_2: "2 * z = (z+z::nat)"
```
```   891 by (rule semiring_mult_2)
```
```   892
```
```   893 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
```
```   894 by (rule semiring_mult_2_right)
```
```   895
```
```   896 text{*Case analysis on @{term "n<2"}*}
```
```   897 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
```
```   898 by (auto simp add: nat_1_add_1 [symmetric])
```
```   899
```
```   900 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
```
```   901
```
```   902 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```   903 by simp
```
```   904
```
```   905 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```   906 by simp
```
```   907
```
```   908 text{*Can be used to eliminate long strings of Sucs, but not by default*}
```
```   909 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```   910 by simp
```
```   911
```
```   912 end
```