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 IntDiv
```
```    10 uses ("Tools/nat_simprocs.ML")
```
```    11 begin
```
```    12
```
```    13 text {*
```
```    14   Arithmetic for naturals is reduced to that for the non-negative integers.
```
```    15 *}
```
```    16
```
```    17 instantiation nat :: number
```
```    18 begin
```
```    19
```
```    20 definition
```
```    21   nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
```
```    22
```
```    23 instance ..
```
```    24
```
```    25 end
```
```    26
```
```    27 lemma [code post]:
```
```    28   "nat (number_of v) = number_of v"
```
```    29   unfolding nat_number_of_def ..
```
```    30
```
```    31 context recpower
```
```    32 begin
```
```    33
```
```    34 abbreviation (xsymbols)
```
```    35   power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
```
```    36   "x\<twosuperior> \<equiv> x ^ 2"
```
```    37
```
```    38 notation (latex output)
```
```    39   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    40
```
```    41 notation (HTML output)
```
```    42   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    43
```
```    44 end
```
```    45
```
```    46
```
```    47 subsection {* Predicate for negative binary numbers *}
```
```    48
```
```    49 definition neg  :: "int \<Rightarrow> bool" where
```
```    50   "neg Z \<longleftrightarrow> Z < 0"
```
```    51
```
```    52 lemma not_neg_int [simp]: "~ neg (of_nat n)"
```
```    53 by (simp add: neg_def)
```
```    54
```
```    55 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
```
```    56 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
```
```    57
```
```    58 lemmas neg_eq_less_0 = neg_def
```
```    59
```
```    60 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```    61 by (simp add: neg_def linorder_not_less)
```
```    62
```
```    63 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```    64
```
```    65 lemma not_neg_0: "~ neg 0"
```
```    66 by (simp add: One_int_def neg_def)
```
```    67
```
```    68 lemma not_neg_1: "~ neg 1"
```
```    69 by (simp add: neg_def linorder_not_less zero_le_one)
```
```    70
```
```    71 lemma neg_nat: "neg z ==> nat z = 0"
```
```    72 by (simp add: neg_def order_less_imp_le)
```
```    73
```
```    74 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
```
```    75 by (simp add: linorder_not_less neg_def)
```
```    76
```
```    77 text {*
```
```    78   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```    79   @{term Numeral0} IS @{term "number_of Pls"}
```
```    80 *}
```
```    81
```
```    82 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
```
```    83   by (simp add: neg_def)
```
```    84
```
```    85 lemma neg_number_of_Min: "neg (number_of Int.Min)"
```
```    86   by (simp add: neg_def)
```
```    87
```
```    88 lemma neg_number_of_Bit0:
```
```    89   "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
```
```    90   by (simp add: neg_def)
```
```    91
```
```    92 lemma neg_number_of_Bit1:
```
```    93   "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
```
```    94   by (simp add: neg_def)
```
```    95
```
```    96 lemmas neg_simps [simp] =
```
```    97   not_neg_0 not_neg_1
```
```    98   not_neg_number_of_Pls neg_number_of_Min
```
```    99   neg_number_of_Bit0 neg_number_of_Bit1
```
```   100
```
```   101
```
```   102 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
```
```   103
```
```   104 declare nat_0 [simp] nat_1 [simp]
```
```   105
```
```   106 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
```
```   107 by (simp add: nat_number_of_def)
```
```   108
```
```   109 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
```
```   110 by (simp add: nat_number_of_def)
```
```   111
```
```   112 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
```
```   113 by (simp add: nat_1 nat_number_of_def)
```
```   114
```
```   115 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
```
```   116 by (simp add: nat_numeral_1_eq_1)
```
```   117
```
```   118 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
```
```   119 apply (unfold nat_number_of_def)
```
```   120 apply (rule nat_2)
```
```   121 done
```
```   122
```
```   123
```
```   124 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
```
```   125
```
```   126 lemma int_nat_number_of [simp]:
```
```   127      "int (number_of v) =
```
```   128          (if neg (number_of v :: int) then 0
```
```   129           else (number_of v :: int))"
```
```   130   unfolding nat_number_of_def number_of_is_id neg_def
```
```   131   by simp
```
```   132
```
```   133
```
```   134 subsubsection{*Successor *}
```
```   135
```
```   136 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   137 apply (rule sym)
```
```   138 apply (simp add: nat_eq_iff int_Suc)
```
```   139 done
```
```   140
```
```   141 lemma Suc_nat_number_of_add:
```
```   142      "Suc (number_of v + n) =
```
```   143         (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
```
```   144   unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
```
```   145   by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
```
```   146
```
```   147 lemma Suc_nat_number_of [simp]:
```
```   148      "Suc (number_of v) =
```
```   149         (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
```
```   150 apply (cut_tac n = 0 in Suc_nat_number_of_add)
```
```   151 apply (simp cong del: if_weak_cong)
```
```   152 done
```
```   153
```
```   154
```
```   155 subsubsection{*Addition *}
```
```   156
```
```   157 lemma add_nat_number_of [simp]:
```
```   158      "(number_of v :: nat) + number_of v' =
```
```   159          (if v < Int.Pls then number_of v'
```
```   160           else if v' < Int.Pls then number_of v
```
```   161           else number_of (v + v'))"
```
```   162   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   163   by (simp add: nat_add_distrib)
```
```   164
```
```   165 lemma nat_number_of_add_1 [simp]:
```
```   166   "number_of v + (1::nat) =
```
```   167     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   168   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   169   by (simp add: nat_add_distrib)
```
```   170
```
```   171 lemma nat_1_add_number_of [simp]:
```
```   172   "(1::nat) + number_of v =
```
```   173     (if v < Int.Pls then 1 else number_of (Int.succ v))"
```
```   174   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   175   by (simp add: nat_add_distrib)
```
```   176
```
```   177 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
```
```   178   by (rule int_int_eq [THEN iffD1]) simp
```
```   179
```
```   180
```
```   181 subsubsection{*Subtraction *}
```
```   182
```
```   183 lemma diff_nat_eq_if:
```
```   184      "nat z - nat z' =
```
```   185         (if neg z' then nat z
```
```   186          else let d = z-z' in
```
```   187               if neg d then 0 else nat d)"
```
```   188 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
```
```   189
```
```   190
```
```   191 lemma diff_nat_number_of [simp]:
```
```   192      "(number_of v :: nat) - number_of v' =
```
```   193         (if v' < Int.Pls then number_of v
```
```   194          else let d = number_of (v + uminus v') in
```
```   195               if neg d then 0 else nat d)"
```
```   196   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
```
```   197   by auto
```
```   198
```
```   199 lemma nat_number_of_diff_1 [simp]:
```
```   200   "number_of v - (1::nat) =
```
```   201     (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
```
```   202   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   203   by auto
```
```   204
```
```   205
```
```   206 subsubsection{*Multiplication *}
```
```   207
```
```   208 lemma mult_nat_number_of [simp]:
```
```   209      "(number_of v :: nat) * number_of v' =
```
```   210        (if v < Int.Pls then 0 else number_of (v * v'))"
```
```   211   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   212   by (simp add: nat_mult_distrib)
```
```   213
```
```   214
```
```   215 subsubsection{*Quotient *}
```
```   216
```
```   217 lemma div_nat_number_of [simp]:
```
```   218      "(number_of v :: nat)  div  number_of v' =
```
```   219           (if neg (number_of v :: int) then 0
```
```   220            else nat (number_of v div number_of v'))"
```
```   221   unfolding nat_number_of_def number_of_is_id neg_def
```
```   222   by (simp add: nat_div_distrib)
```
```   223
```
```   224 lemma one_div_nat_number_of [simp]:
```
```   225      "Suc 0 div number_of v' = nat (1 div number_of v')"
```
```   226 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   227
```
```   228
```
```   229 subsubsection{*Remainder *}
```
```   230
```
```   231 lemma mod_nat_number_of [simp]:
```
```   232      "(number_of v :: nat)  mod  number_of v' =
```
```   233         (if neg (number_of v :: int) then 0
```
```   234          else if neg (number_of v' :: int) then number_of v
```
```   235          else nat (number_of v mod number_of v'))"
```
```   236   unfolding nat_number_of_def number_of_is_id neg_def
```
```   237   by (simp add: nat_mod_distrib)
```
```   238
```
```   239 lemma one_mod_nat_number_of [simp]:
```
```   240      "Suc 0 mod number_of v' =
```
```   241         (if neg (number_of v' :: int) then Suc 0
```
```   242          else nat (1 mod number_of v'))"
```
```   243 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   244
```
```   245
```
```   246 subsubsection{* Divisibility *}
```
```   247
```
```   248 lemmas dvd_eq_mod_eq_0_number_of =
```
```   249   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
```
```   250
```
```   251 declare dvd_eq_mod_eq_0_number_of [simp]
```
```   252
```
```   253 ML
```
```   254 {*
```
```   255 val nat_number_of_def = thm"nat_number_of_def";
```
```   256
```
```   257 val nat_number_of = thm"nat_number_of";
```
```   258 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
```
```   259 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
```
```   260 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
```
```   261 val numeral_2_eq_2 = thm"numeral_2_eq_2";
```
```   262 val nat_div_distrib = thm"nat_div_distrib";
```
```   263 val nat_mod_distrib = thm"nat_mod_distrib";
```
```   264 val int_nat_number_of = thm"int_nat_number_of";
```
```   265 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
```
```   266 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
```
```   267 val Suc_nat_number_of = thm"Suc_nat_number_of";
```
```   268 val add_nat_number_of = thm"add_nat_number_of";
```
```   269 val diff_nat_eq_if = thm"diff_nat_eq_if";
```
```   270 val diff_nat_number_of = thm"diff_nat_number_of";
```
```   271 val mult_nat_number_of = thm"mult_nat_number_of";
```
```   272 val div_nat_number_of = thm"div_nat_number_of";
```
```   273 val mod_nat_number_of = thm"mod_nat_number_of";
```
```   274 *}
```
```   275
```
```   276
```
```   277 subsection{*Comparisons*}
```
```   278
```
```   279 subsubsection{*Equals (=) *}
```
```   280
```
```   281 lemma eq_nat_nat_iff:
```
```   282      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
```
```   283 by (auto elim!: nonneg_eq_int)
```
```   284
```
```   285 lemma eq_nat_number_of [simp]:
```
```   286      "((number_of v :: nat) = number_of v') =
```
```   287       (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
```
```   288        else if neg (number_of v' :: int) then (number_of v :: int) = 0
```
```   289        else v = v')"
```
```   290   unfolding nat_number_of_def number_of_is_id neg_def
```
```   291   by auto
```
```   292
```
```   293
```
```   294 subsubsection{*Less-than (<) *}
```
```   295
```
```   296 lemma less_nat_number_of [simp]:
```
```   297   "(number_of v :: nat) < number_of v' \<longleftrightarrow>
```
```   298     (if v < v' then Int.Pls < v' else False)"
```
```   299   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   300   by auto
```
```   301
```
```   302
```
```   303 subsubsection{*Less-than-or-equal *}
```
```   304
```
```   305 lemma le_nat_number_of [simp]:
```
```   306   "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
```
```   307     (if v \<le> v' then True else v \<le> Int.Pls)"
```
```   308   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   309   by auto
```
```   310
```
```   311 (*Maps #n to n for n = 0, 1, 2*)
```
```   312 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
```
```   313
```
```   314
```
```   315 subsection{*Powers with Numeric Exponents*}
```
```   316
```
```   317 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
```
```   318 We cannot prove general results about the numeral @{term "-1"}, so we have to
```
```   319 use @{term "- 1"} instead.*}
```
```   320
```
```   321 lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
```
```   322   by (simp add: numeral_2_eq_2 Power.power_Suc)
```
```   323
```
```   324 lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
```
```   325   by (simp add: power2_eq_square)
```
```   326
```
```   327 lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
```
```   328   by (simp add: power2_eq_square)
```
```   329
```
```   330 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
```
```   331   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
```
```   332   apply (erule ssubst)
```
```   333   apply (simp add: power_Suc mult_ac)
```
```   334   apply (unfold nat_number_of_def)
```
```   335   apply (subst nat_eq_iff)
```
```   336   apply simp
```
```   337 done
```
```   338
```
```   339 text{*Squares of literal numerals will be evaluated.*}
```
```   340 lemmas power2_eq_square_number_of =
```
```   341     power2_eq_square [of "number_of w", standard]
```
```   342 declare power2_eq_square_number_of [simp]
```
```   343
```
```   344
```
```   345 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   346   by (simp add: power2_eq_square)
```
```   347
```
```   348 lemma zero_less_power2[simp]:
```
```   349      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
```
```   350   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   351
```
```   352 lemma power2_less_0[simp]:
```
```   353   fixes a :: "'a::{ordered_idom,recpower}"
```
```   354   shows "~ (a\<twosuperior> < 0)"
```
```   355 by (force simp add: power2_eq_square mult_less_0_iff)
```
```   356
```
```   357 lemma zero_eq_power2[simp]:
```
```   358      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
```
```   359   by (force simp add: power2_eq_square mult_eq_0_iff)
```
```   360
```
```   361 lemma abs_power2[simp]:
```
```   362      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   363   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   364
```
```   365 lemma power2_abs[simp]:
```
```   366      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   367   by (simp add: power2_eq_square abs_mult_self)
```
```   368
```
```   369 lemma power2_minus[simp]:
```
```   370      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
```
```   371   by (simp add: power2_eq_square)
```
```   372
```
```   373 lemma power2_le_imp_le:
```
```   374   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   375   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
```
```   376 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   377
```
```   378 lemma power2_less_imp_less:
```
```   379   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   380   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
```
```   381 by (rule power_less_imp_less_base)
```
```   382
```
```   383 lemma power2_eq_imp_eq:
```
```   384   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   385   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
```
```   386 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
```
```   387
```
```   388 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
```
```   389 proof (induct n)
```
```   390   case 0 show ?case by simp
```
```   391 next
```
```   392   case (Suc n) then show ?case by (simp add: power_Suc power_add)
```
```   393 qed
```
```   394
```
```   395 lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
```
```   396   by (simp add: power_Suc)
```
```   397
```
```   398 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
```
```   399   by (subst mult_commute) (simp add: power_mult)
```
```   400
```
```   401 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
```
```   402   by (simp add: power_even_eq)
```
```   403
```
```   404 lemma power_minus_even [simp]:
```
```   405   "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
```
```   406   by (simp add: power_minus [of a])
```
```   407
```
```   408 lemma zero_le_even_power'[simp]:
```
```   409      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
```
```   410 proof (induct "n")
```
```   411   case 0
```
```   412     show ?case by (simp add: zero_le_one)
```
```   413 next
```
```   414   case (Suc n)
```
```   415     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   416       by (simp add: mult_ac power_add power2_eq_square)
```
```   417     thus ?case
```
```   418       by (simp add: prems zero_le_mult_iff)
```
```   419 qed
```
```   420
```
```   421 lemma odd_power_less_zero:
```
```   422      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
```
```   423 proof (induct "n")
```
```   424   case 0
```
```   425   then show ?case by simp
```
```   426 next
```
```   427   case (Suc n)
```
```   428   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   429     by (simp add: mult_ac power_add power2_eq_square)
```
```   430   thus ?case
```
```   431     by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
```
```   432 qed
```
```   433
```
```   434 lemma odd_0_le_power_imp_0_le:
```
```   435      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
```
```   436 apply (insert odd_power_less_zero [of a n])
```
```   437 apply (force simp add: linorder_not_less [symmetric])
```
```   438 done
```
```   439
```
```   440 text{*Simprules for comparisons where common factors can be cancelled.*}
```
```   441 lemmas zero_compare_simps =
```
```   442     add_strict_increasing add_strict_increasing2 add_increasing
```
```   443     zero_le_mult_iff zero_le_divide_iff
```
```   444     zero_less_mult_iff zero_less_divide_iff
```
```   445     mult_le_0_iff divide_le_0_iff
```
```   446     mult_less_0_iff divide_less_0_iff
```
```   447     zero_le_power2 power2_less_0
```
```   448
```
```   449 subsubsection{*Nat *}
```
```   450
```
```   451 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
```
```   452 by (simp add: numerals)
```
```   453
```
```   454 (*Expresses a natural number constant as the Suc of another one.
```
```   455   NOT suitable for rewriting because n recurs in the condition.*)
```
```   456 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
```
```   457
```
```   458 subsubsection{*Arith *}
```
```   459
```
```   460 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
```
```   461 by (simp add: numerals)
```
```   462
```
```   463 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
```
```   464 by (simp add: numerals)
```
```   465
```
```   466 (* These two can be useful when m = number_of... *)
```
```   467
```
```   468 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
```
```   469   unfolding One_nat_def by (cases m) simp_all
```
```   470
```
```   471 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
```
```   472   unfolding One_nat_def by (cases m) simp_all
```
```   473
```
```   474 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   475   unfolding One_nat_def by (cases m) simp_all
```
```   476
```
```   477
```
```   478 subsection{*Comparisons involving (0::nat) *}
```
```   479
```
```   480 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
```
```   481
```
```   482 lemma eq_number_of_0 [simp]:
```
```   483   "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
```
```   484   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   485   by auto
```
```   486
```
```   487 lemma eq_0_number_of [simp]:
```
```   488   "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
```
```   489 by (rule trans [OF eq_sym_conv eq_number_of_0])
```
```   490
```
```   491 lemma less_0_number_of [simp]:
```
```   492    "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
```
```   493   unfolding nat_number_of_def number_of_is_id numeral_simps
```
```   494   by simp
```
```   495
```
```   496 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
```
```   497 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
```
```   498
```
```   499
```
```   500
```
```   501 subsection{*Comparisons involving  @{term Suc} *}
```
```   502
```
```   503 lemma eq_number_of_Suc [simp]:
```
```   504      "(number_of v = Suc n) =
```
```   505         (let pv = number_of (Int.pred v) in
```
```   506          if neg pv then False else nat pv = n)"
```
```   507 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   508                   number_of_pred nat_number_of_def
```
```   509             split add: split_if)
```
```   510 apply (rule_tac x = "number_of v" in spec)
```
```   511 apply (auto simp add: nat_eq_iff)
```
```   512 done
```
```   513
```
```   514 lemma Suc_eq_number_of [simp]:
```
```   515      "(Suc n = number_of v) =
```
```   516         (let pv = number_of (Int.pred v) in
```
```   517          if neg pv then False else nat pv = n)"
```
```   518 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
```
```   519
```
```   520 lemma less_number_of_Suc [simp]:
```
```   521      "(number_of v < Suc n) =
```
```   522         (let pv = number_of (Int.pred v) in
```
```   523          if neg pv then True else nat pv < n)"
```
```   524 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   525                   number_of_pred nat_number_of_def
```
```   526             split add: split_if)
```
```   527 apply (rule_tac x = "number_of v" in spec)
```
```   528 apply (auto simp add: nat_less_iff)
```
```   529 done
```
```   530
```
```   531 lemma less_Suc_number_of [simp]:
```
```   532      "(Suc n < number_of v) =
```
```   533         (let pv = number_of (Int.pred v) in
```
```   534          if neg pv then False else n < nat pv)"
```
```   535 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   536                   number_of_pred nat_number_of_def
```
```   537             split add: split_if)
```
```   538 apply (rule_tac x = "number_of v" in spec)
```
```   539 apply (auto simp add: zless_nat_eq_int_zless)
```
```   540 done
```
```   541
```
```   542 lemma le_number_of_Suc [simp]:
```
```   543      "(number_of v <= Suc n) =
```
```   544         (let pv = number_of (Int.pred v) in
```
```   545          if neg pv then True else nat pv <= n)"
```
```   546 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
```
```   547
```
```   548 lemma le_Suc_number_of [simp]:
```
```   549      "(Suc n <= number_of v) =
```
```   550         (let pv = number_of (Int.pred v) in
```
```   551          if neg pv then False else n <= nat pv)"
```
```   552 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
```
```   553
```
```   554
```
```   555 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
```
```   556 by auto
```
```   557
```
```   558
```
```   559
```
```   560 subsection{*Max and Min Combined with @{term Suc} *}
```
```   561
```
```   562 lemma max_number_of_Suc [simp]:
```
```   563      "max (Suc n) (number_of v) =
```
```   564         (let pv = number_of (Int.pred v) in
```
```   565          if neg pv then Suc n else Suc(max n (nat pv)))"
```
```   566 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   567             split add: split_if nat.split)
```
```   568 apply (rule_tac x = "number_of v" in spec)
```
```   569 apply auto
```
```   570 done
```
```   571
```
```   572 lemma max_Suc_number_of [simp]:
```
```   573      "max (number_of v) (Suc n) =
```
```   574         (let pv = number_of (Int.pred v) in
```
```   575          if neg pv then Suc n else Suc(max (nat pv) n))"
```
```   576 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   577             split add: split_if nat.split)
```
```   578 apply (rule_tac x = "number_of v" in spec)
```
```   579 apply auto
```
```   580 done
```
```   581
```
```   582 lemma min_number_of_Suc [simp]:
```
```   583      "min (Suc n) (number_of v) =
```
```   584         (let pv = number_of (Int.pred v) in
```
```   585          if neg pv then 0 else Suc(min n (nat pv)))"
```
```   586 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   587             split add: split_if nat.split)
```
```   588 apply (rule_tac x = "number_of v" in spec)
```
```   589 apply auto
```
```   590 done
```
```   591
```
```   592 lemma min_Suc_number_of [simp]:
```
```   593      "min (number_of v) (Suc n) =
```
```   594         (let pv = number_of (Int.pred v) in
```
```   595          if neg pv then 0 else Suc(min (nat pv) n))"
```
```   596 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   597             split add: split_if nat.split)
```
```   598 apply (rule_tac x = "number_of v" in spec)
```
```   599 apply auto
```
```   600 done
```
```   601
```
```   602 subsection{*Literal arithmetic involving powers*}
```
```   603
```
```   604 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
```
```   605 apply (induct "n")
```
```   606 apply (simp_all (no_asm_simp) add: nat_mult_distrib)
```
```   607 done
```
```   608
```
```   609 lemma power_nat_number_of:
```
```   610      "(number_of v :: nat) ^ n =
```
```   611        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
```
```   612 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
```
```   613          split add: split_if cong: imp_cong)
```
```   614
```
```   615
```
```   616 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
```
```   617 declare power_nat_number_of_number_of [simp]
```
```   618
```
```   619
```
```   620
```
```   621 text{*For arbitrary rings*}
```
```   622
```
```   623 lemma power_number_of_even:
```
```   624   fixes z :: "'a::{number_ring,recpower}"
```
```   625   shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
```
```   626 unfolding Let_def nat_number_of_def number_of_Bit0
```
```   627 apply (rule_tac x = "number_of w" in spec, clarify)
```
```   628 apply (case_tac " (0::int) <= x")
```
```   629 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
```
```   630 done
```
```   631
```
```   632 lemma power_number_of_odd:
```
```   633   fixes z :: "'a::{number_ring,recpower}"
```
```   634   shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
```
```   635      then (let w = z ^ (number_of w) in z * w * w) else 1)"
```
```   636 unfolding Let_def nat_number_of_def number_of_Bit1
```
```   637 apply (rule_tac x = "number_of w" in spec, auto)
```
```   638 apply (simp only: nat_add_distrib nat_mult_distrib)
```
```   639 apply simp
```
```   640 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
```
```   641 done
```
```   642
```
```   643 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
```
```   644 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
```
```   645
```
```   646 lemmas power_number_of_even_number_of [simp] =
```
```   647     power_number_of_even [of "number_of v", standard]
```
```   648
```
```   649 lemmas power_number_of_odd_number_of [simp] =
```
```   650     power_number_of_odd [of "number_of v", standard]
```
```   651
```
```   652
```
```   653
```
```   654 ML
```
```   655 {*
```
```   656 val numeral_ss = @{simpset} addsimps @{thms numerals};
```
```   657
```
```   658 val nat_bin_arith_setup =
```
```   659  Lin_Arith.map_data
```
```   660    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
```
```   661      {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
```
```   662       inj_thms = inj_thms,
```
```   663       lessD = lessD, neqE = neqE,
```
```   664       simpset = simpset addsimps @{thms neg_simps} @
```
```   665         [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
```
```   666 *}
```
```   667
```
```   668 declaration {* K nat_bin_arith_setup *}
```
```   669
```
```   670 (* Enable arith to deal with div/mod k where k is a numeral: *)
```
```   671 declare split_div[of _ _ "number_of k", standard, arith_split]
```
```   672 declare split_mod[of _ _ "number_of k", standard, arith_split]
```
```   673
```
```   674 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
```
```   675   by (simp add: number_of_Pls 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   unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
```
```   684   by auto
```
```   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 nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
```
```   691   by auto
```
```   692
```
```   693 lemmas nat_number =
```
```   694   nat_number_of_Pls nat_number_of_Min
```
```   695   nat_number_of_Bit0 nat_number_of_Bit1
```
```   696
```
```   697 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
```
```   698   by (simp add: Let_def)
```
```   699
```
```   700 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
```
```   701 by (simp add: power_mult power_Suc);
```
```   702
```
```   703 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
```
```   704 by (simp add: power_mult power_Suc);
```
```   705
```
```   706
```
```   707 subsection{*Literal arithmetic and @{term of_nat}*}
```
```   708
```
```   709 lemma of_nat_double:
```
```   710      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
```
```   711 by (simp only: mult_2 nat_add_distrib of_nat_add)
```
```   712
```
```   713 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
```
```   714 by (simp only: nat_number_of_def)
```
```   715
```
```   716 lemma of_nat_number_of_lemma:
```
```   717      "of_nat (number_of v :: nat) =
```
```   718          (if 0 \<le> (number_of v :: int)
```
```   719           then (number_of v :: 'a :: number_ring)
```
```   720           else 0)"
```
```   721 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
```
```   722
```
```   723 lemma of_nat_number_of_eq [simp]:
```
```   724      "of_nat (number_of v :: nat) =
```
```   725          (if neg (number_of v :: int) then 0
```
```   726           else (number_of v :: 'a :: number_ring))"
```
```   727 by (simp only: of_nat_number_of_lemma neg_def, simp)
```
```   728
```
```   729
```
```   730 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
```
```   731
```
```   732 lemma nat_number_of_add_left:
```
```   733      "number_of v + (number_of v' + (k::nat)) =
```
```   734          (if neg (number_of v :: int) then number_of v' + k
```
```   735           else if neg (number_of v' :: int) then number_of v + k
```
```   736           else number_of (v + v') + k)"
```
```   737   unfolding nat_number_of_def number_of_is_id neg_def
```
```   738   by auto
```
```   739
```
```   740 lemma nat_number_of_mult_left:
```
```   741      "number_of v * (number_of v' * (k::nat)) =
```
```   742          (if v < Int.Pls then 0
```
```   743           else number_of (v * v') * k)"
```
```   744 by simp
```
```   745
```
```   746
```
```   747 subsubsection{*For @{text combine_numerals}*}
```
```   748
```
```   749 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
```
```   750 by (simp add: add_mult_distrib)
```
```   751
```
```   752
```
```   753 subsubsection{*For @{text cancel_numerals}*}
```
```   754
```
```   755 lemma nat_diff_add_eq1:
```
```   756      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
```
```   757 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   758
```
```   759 lemma nat_diff_add_eq2:
```
```   760      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
```
```   761 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   762
```
```   763 lemma nat_eq_add_iff1:
```
```   764      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
```
```   765 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   766
```
```   767 lemma nat_eq_add_iff2:
```
```   768      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
```
```   769 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   770
```
```   771 lemma nat_less_add_iff1:
```
```   772      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
```
```   773 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   774
```
```   775 lemma nat_less_add_iff2:
```
```   776      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
```
```   777 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   778
```
```   779 lemma nat_le_add_iff1:
```
```   780      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
```
```   781 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   782
```
```   783 lemma nat_le_add_iff2:
```
```   784      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
```
```   785 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   786
```
```   787
```
```   788 subsubsection{*For @{text cancel_numeral_factors} *}
```
```   789
```
```   790 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
```
```   791 by auto
```
```   792
```
```   793 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
```
```   794 by auto
```
```   795
```
```   796 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
```
```   797 by auto
```
```   798
```
```   799 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
```
```   800 by auto
```
```   801
```
```   802 lemma nat_mult_dvd_cancel_disj[simp]:
```
```   803   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
```
```   804 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
```
```   805
```
```   806 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
```
```   807 by(auto)
```
```   808
```
```   809
```
```   810 subsubsection{*For @{text cancel_factor} *}
```
```   811
```
```   812 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
```
```   813 by auto
```
```   814
```
```   815 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
```
```   816 by auto
```
```   817
```
```   818 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
```
```   819 by auto
```
```   820
```
```   821 lemma nat_mult_div_cancel_disj[simp]:
```
```   822      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
```
```   823 by (simp add: nat_mult_div_cancel1)
```
```   824
```
```   825
```
```   826 subsection {* Simprocs for the Naturals *}
```
```   827
```
```   828 use "Tools/nat_simprocs.ML"
```
```   829 declaration {* K nat_simprocs_setup *}
```
```   830
```
```   831 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
```
```   832
```
```   833 text{*Where K above is a literal*}
```
```   834
```
```   835 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
```
```   836 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
```
```   837
```
```   838 text {*Now just instantiating @{text n} to @{text "number_of v"} does
```
```   839   the right simplification, but with some redundant inequality
```
```   840   tests.*}
```
```   841 lemma neg_number_of_pred_iff_0:
```
```   842   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
```
```   843 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
```
```   844 apply (simp only: less_Suc_eq_le le_0_eq)
```
```   845 apply (subst less_number_of_Suc, simp)
```
```   846 done
```
```   847
```
```   848 text{*No longer required as a simprule because of the @{text inverse_fold}
```
```   849    simproc*}
```
```   850 lemma Suc_diff_number_of:
```
```   851      "Int.Pls < v ==>
```
```   852       Suc m - (number_of v) = m - (number_of (Int.pred v))"
```
```   853 apply (subst Suc_diff_eq_diff_pred)
```
```   854 apply simp
```
```   855 apply (simp del: nat_numeral_1_eq_1)
```
```   856 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
```
```   857                         neg_number_of_pred_iff_0)
```
```   858 done
```
```   859
```
```   860 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
```
```   861 by (simp add: numerals split add: nat_diff_split)
```
```   862
```
```   863
```
```   864 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
```
```   865
```
```   866 lemma nat_case_number_of [simp]:
```
```   867      "nat_case a f (number_of v) =
```
```   868         (let pv = number_of (Int.pred v) in
```
```   869          if neg pv then a else f (nat pv))"
```
```   870 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
```
```   871
```
```   872 lemma nat_case_add_eq_if [simp]:
```
```   873      "nat_case a f ((number_of v) + n) =
```
```   874        (let pv = number_of (Int.pred v) in
```
```   875          if neg pv then nat_case a f n else f (nat pv + n))"
```
```   876 apply (subst add_eq_if)
```
```   877 apply (simp split add: nat.split
```
```   878             del: nat_numeral_1_eq_1
```
```   879             add: nat_numeral_1_eq_1 [symmetric]
```
```   880                  numeral_1_eq_Suc_0 [symmetric]
```
```   881                  neg_number_of_pred_iff_0)
```
```   882 done
```
```   883
```
```   884 lemma nat_rec_number_of [simp]:
```
```   885      "nat_rec a f (number_of v) =
```
```   886         (let pv = number_of (Int.pred v) in
```
```   887          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
```
```   888 apply (case_tac " (number_of v) ::nat")
```
```   889 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
```
```   890 apply (simp split add: split_if_asm)
```
```   891 done
```
```   892
```
```   893 lemma nat_rec_add_eq_if [simp]:
```
```   894      "nat_rec a f (number_of v + n) =
```
```   895         (let pv = number_of (Int.pred v) in
```
```   896          if neg pv then nat_rec a f n
```
```   897                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
```
```   898 apply (subst add_eq_if)
```
```   899 apply (simp split add: nat.split
```
```   900             del: nat_numeral_1_eq_1
```
```   901             add: nat_numeral_1_eq_1 [symmetric]
```
```   902                  numeral_1_eq_Suc_0 [symmetric]
```
```   903                  neg_number_of_pred_iff_0)
```
```   904 done
```
```   905
```
```   906
```
```   907 subsubsection{*Various Other Lemmas*}
```
```   908
```
```   909 text {*Evens and Odds, for Mutilated Chess Board*}
```
```   910
```
```   911 text{*Lemmas for specialist use, NOT as default simprules*}
```
```   912 lemma nat_mult_2: "2 * z = (z+z::nat)"
```
```   913 proof -
```
```   914   have "2*z = (1 + 1)*z" by simp
```
```   915   also have "... = z+z" by (simp add: left_distrib)
```
```   916   finally show ?thesis .
```
```   917 qed
```
```   918
```
```   919 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
```
```   920 by (subst mult_commute, rule nat_mult_2)
```
```   921
```
```   922 text{*Case analysis on @{term "n<2"}*}
```
```   923 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
```
```   924 by arith
```
```   925
```
```   926 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
```
```   927 by arith
```
```   928
```
```   929 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
```
```   930 by (simp add: nat_mult_2 [symmetric])
```
```   931
```
```   932 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
```
```   933 apply (subgoal_tac "m mod 2 < 2")
```
```   934 apply (erule less_2_cases [THEN disjE])
```
```   935 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
```
```   936 done
```
```   937
```
```   938 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
```
```   939 apply (subgoal_tac "m mod 2 < 2")
```
```   940 apply (force simp del: mod_less_divisor, simp)
```
```   941 done
```
```   942
```
```   943 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
```
```   944
```
```   945 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```   946 by simp
```
```   947
```
```   948 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```   949 by simp
```
```   950
```
```   951 text{*Can be used to eliminate long strings of Sucs, but not by default*}
```
```   952 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```   953 by simp
```
```   954
```
```   955
```
```   956 text{*These lemmas collapse some needless occurrences of Suc:
```
```   957     at least three Sucs, since two and fewer are rewritten back to Suc again!
```
```   958     We already have some rules to simplify operands smaller than 3.*}
```
```   959
```
```   960 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
```
```   961 by (simp add: Suc3_eq_add_3)
```
```   962
```
```   963 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
```
```   964 by (simp add: Suc3_eq_add_3)
```
```   965
```
```   966 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
```
```   967 by (simp add: Suc3_eq_add_3)
```
```   968
```
```   969 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
```
```   970 by (simp add: Suc3_eq_add_3)
```
```   971
```
```   972 lemmas Suc_div_eq_add3_div_number_of =
```
```   973     Suc_div_eq_add3_div [of _ "number_of v", standard]
```
```   974 declare Suc_div_eq_add3_div_number_of [simp]
```
```   975
```
```   976 lemmas Suc_mod_eq_add3_mod_number_of =
```
```   977     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
```
```   978 declare Suc_mod_eq_add3_mod_number_of [simp]
```
```   979
```
`   980 end`