src/HOL/NatBin.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24093 5d0ecd0c8f3c child 25481 aa16cd919dcc permissions -rw-r--r--
Name.uu, Name.aT;
```     1 (*  Title:      HOL/NatBin.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1999  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Binary arithmetic for the natural numbers *}
```
```     8
```
```     9 theory NatBin
```
```    10 imports IntDiv
```
```    11 begin
```
```    12
```
```    13 text {*
```
```    14   Arithmetic for naturals is reduced to that for the non-negative integers.
```
```    15 *}
```
```    16
```
```    17 instance nat :: number
```
```    18   nat_number_of_def [code inline]: "number_of v == nat (number_of (v\<Colon>int))" ..
```
```    19
```
```    20 abbreviation (xsymbols)
```
```    21   square :: "'a::power => 'a"  ("(_\<twosuperior>)" [1000] 999) where
```
```    22   "x\<twosuperior> == x^2"
```
```    23
```
```    24 notation (latex output)
```
```    25   square  ("(_\<twosuperior>)" [1000] 999)
```
```    26
```
```    27 notation (HTML output)
```
```    28   square  ("(_\<twosuperior>)" [1000] 999)
```
```    29
```
```    30
```
```    31 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
```
```    32
```
```    33 declare nat_0 [simp] nat_1 [simp]
```
```    34
```
```    35 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
```
```    36 by (simp add: nat_number_of_def)
```
```    37
```
```    38 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
```
```    39 by (simp add: nat_number_of_def)
```
```    40
```
```    41 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
```
```    42 by (simp add: nat_1 nat_number_of_def)
```
```    43
```
```    44 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
```
```    45 by (simp add: nat_numeral_1_eq_1)
```
```    46
```
```    47 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
```
```    48 apply (unfold nat_number_of_def)
```
```    49 apply (rule nat_2)
```
```    50 done
```
```    51
```
```    52
```
```    53 text{*Distributive laws for type @{text nat}.  The others are in theory
```
```    54    @{text IntArith}, but these require div and mod to be defined for type
```
```    55    "int".  They also need some of the lemmas proved above.*}
```
```    56
```
```    57 lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'"
```
```    58 apply (case_tac "0 <= z'")
```
```    59 apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV)
```
```    60 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
```
```    61 apply (auto elim!: nonneg_eq_int)
```
```    62 apply (rename_tac m m')
```
```    63 apply (subgoal_tac "0 <= int m div int m'")
```
```    64  prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff)
```
```    65 apply (rule of_nat_eq_iff [where 'a=int, THEN iffD1], simp)
```
```    66 apply (rule_tac r = "int (m mod m') " in quorem_div)
```
```    67  prefer 2 apply force
```
```    68 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
```
```    69                  of_nat_add [symmetric] of_nat_mult [symmetric]
```
```    70             del: of_nat_add of_nat_mult)
```
```    71 done
```
```    72
```
```    73 (*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```    74 lemma nat_mod_distrib:
```
```    75      "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"
```
```    76 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
```
```    77 apply (auto elim!: nonneg_eq_int)
```
```    78 apply (rename_tac m m')
```
```    79 apply (subgoal_tac "0 <= int m mod int m'")
```
```    80  prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign)
```
```    81 apply (rule int_int_eq [THEN iffD1], simp)
```
```    82 apply (rule_tac q = "int (m div m') " in quorem_mod)
```
```    83  prefer 2 apply force
```
```    84 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
```
```    85                  of_nat_add [symmetric] of_nat_mult [symmetric]
```
```    86             del: of_nat_add of_nat_mult)
```
```    87 done
```
```    88
```
```    89 text{*Suggested by Matthias Daum*}
```
```    90 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```    91 apply (subgoal_tac "nat x div nat k < nat x")
```
```    92  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
```
```    93 apply (rule Divides.div_less_dividend, simp_all)
```
```    94 done
```
```    95
```
```    96 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
```
```    97
```
```    98 (*"neg" is used in rewrite rules for binary comparisons*)
```
```    99 lemma int_nat_number_of [simp]:
```
```   100      "int (number_of v) =
```
```   101          (if neg (number_of v :: int) then 0
```
```   102           else (number_of v :: int))"
```
```   103 by (simp del: nat_number_of
```
```   104 	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)
```
```   105
```
```   106
```
```   107 subsubsection{*Successor *}
```
```   108
```
```   109 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   110 apply (rule sym)
```
```   111 apply (simp add: nat_eq_iff int_Suc)
```
```   112 done
```
```   113
```
```   114 lemma Suc_nat_number_of_add:
```
```   115      "Suc (number_of v + n) =
```
```   116         (if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)"
```
```   117 by (simp del: nat_number_of
```
```   118          add: nat_number_of_def neg_nat
```
```   119               Suc_nat_eq_nat_zadd1 number_of_succ)
```
```   120
```
```   121 lemma Suc_nat_number_of [simp]:
```
```   122      "Suc (number_of v) =
```
```   123         (if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))"
```
```   124 apply (cut_tac n = 0 in Suc_nat_number_of_add)
```
```   125 apply (simp cong del: if_weak_cong)
```
```   126 done
```
```   127
```
```   128
```
```   129 subsubsection{*Addition *}
```
```   130
```
```   131 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   132 lemma add_nat_number_of [simp]:
```
```   133      "(number_of v :: nat) + number_of v' =
```
```   134          (if neg (number_of v :: int) then number_of v'
```
```   135           else if neg (number_of v' :: int) then number_of v
```
```   136           else number_of (v + v'))"
```
```   137 by (force dest!: neg_nat
```
```   138           simp del: nat_number_of
```
```   139           simp add: nat_number_of_def nat_add_distrib [symmetric])
```
```   140
```
```   141
```
```   142 subsubsection{*Subtraction *}
```
```   143
```
```   144 lemma diff_nat_eq_if:
```
```   145      "nat z - nat z' =
```
```   146         (if neg z' then nat z
```
```   147          else let d = z-z' in
```
```   148               if neg d then 0 else nat d)"
```
```   149 apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
```
```   150 done
```
```   151
```
```   152 lemma diff_nat_number_of [simp]:
```
```   153      "(number_of v :: nat) - number_of v' =
```
```   154         (if neg (number_of v' :: int) then number_of v
```
```   155          else let d = number_of (v + uminus v') in
```
```   156               if neg d then 0 else nat d)"
```
```   157 by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def)
```
```   158
```
```   159
```
```   160
```
```   161 subsubsection{*Multiplication *}
```
```   162
```
```   163 lemma mult_nat_number_of [simp]:
```
```   164      "(number_of v :: nat) * number_of v' =
```
```   165        (if neg (number_of v :: int) then 0 else number_of (v * v'))"
```
```   166 by (force dest!: neg_nat
```
```   167           simp del: nat_number_of
```
```   168           simp add: nat_number_of_def nat_mult_distrib [symmetric])
```
```   169
```
```   170
```
```   171
```
```   172 subsubsection{*Quotient *}
```
```   173
```
```   174 lemma div_nat_number_of [simp]:
```
```   175      "(number_of v :: nat)  div  number_of v' =
```
```   176           (if neg (number_of v :: int) then 0
```
```   177            else nat (number_of v div number_of v'))"
```
```   178 by (force dest!: neg_nat
```
```   179           simp del: nat_number_of
```
```   180           simp add: nat_number_of_def nat_div_distrib [symmetric])
```
```   181
```
```   182 lemma one_div_nat_number_of [simp]:
```
```   183      "(Suc 0)  div  number_of v' = (nat (1 div number_of v'))"
```
```   184 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   185
```
```   186
```
```   187 subsubsection{*Remainder *}
```
```   188
```
```   189 lemma mod_nat_number_of [simp]:
```
```   190      "(number_of v :: nat)  mod  number_of v' =
```
```   191         (if neg (number_of v :: int) then 0
```
```   192          else if neg (number_of v' :: int) then number_of v
```
```   193          else nat (number_of v mod number_of v'))"
```
```   194 by (force dest!: neg_nat
```
```   195           simp del: nat_number_of
```
```   196           simp add: nat_number_of_def nat_mod_distrib [symmetric])
```
```   197
```
```   198 lemma one_mod_nat_number_of [simp]:
```
```   199      "(Suc 0)  mod  number_of v' =
```
```   200         (if neg (number_of v' :: int) then Suc 0
```
```   201          else nat (1 mod number_of v'))"
```
```   202 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   203
```
```   204
```
```   205 subsubsection{* Divisibility *}
```
```   206
```
```   207 lemmas dvd_eq_mod_eq_0_number_of =
```
```   208   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
```
```   209
```
```   210 declare dvd_eq_mod_eq_0_number_of [simp]
```
```   211
```
```   212 ML
```
```   213 {*
```
```   214 val nat_number_of_def = thm"nat_number_of_def";
```
```   215
```
```   216 val nat_number_of = thm"nat_number_of";
```
```   217 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
```
```   218 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
```
```   219 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
```
```   220 val numeral_2_eq_2 = thm"numeral_2_eq_2";
```
```   221 val nat_div_distrib = thm"nat_div_distrib";
```
```   222 val nat_mod_distrib = thm"nat_mod_distrib";
```
```   223 val int_nat_number_of = thm"int_nat_number_of";
```
```   224 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
```
```   225 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
```
```   226 val Suc_nat_number_of = thm"Suc_nat_number_of";
```
```   227 val add_nat_number_of = thm"add_nat_number_of";
```
```   228 val diff_nat_eq_if = thm"diff_nat_eq_if";
```
```   229 val diff_nat_number_of = thm"diff_nat_number_of";
```
```   230 val mult_nat_number_of = thm"mult_nat_number_of";
```
```   231 val div_nat_number_of = thm"div_nat_number_of";
```
```   232 val mod_nat_number_of = thm"mod_nat_number_of";
```
```   233 *}
```
```   234
```
```   235
```
```   236 subsection{*Comparisons*}
```
```   237
```
```   238 subsubsection{*Equals (=) *}
```
```   239
```
```   240 lemma eq_nat_nat_iff:
```
```   241      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
```
```   242 by (auto elim!: nonneg_eq_int)
```
```   243
```
```   244 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   245 lemma eq_nat_number_of [simp]:
```
```   246      "((number_of v :: nat) = number_of v') =
```
```   247       (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))
```
```   248        else if neg (number_of v' :: int) then iszero (number_of v :: int)
```
```   249        else iszero (number_of (v + uminus v') :: int))"
```
```   250 apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
```
```   251                   eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
```
```   252             split add: split_if cong add: imp_cong)
```
```   253 apply (simp only: nat_eq_iff nat_eq_iff2)
```
```   254 apply (simp add: not_neg_eq_ge_0 [symmetric])
```
```   255 done
```
```   256
```
```   257
```
```   258 subsubsection{*Less-than (<) *}
```
```   259
```
```   260 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   261 lemma less_nat_number_of [simp]:
```
```   262      "((number_of v :: nat) < number_of v') =
```
```   263          (if neg (number_of v :: int) then neg (number_of (uminus v') :: int)
```
```   264           else neg (number_of (v + uminus v') :: int))"
```
```   265 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
```
```   266                 nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
```
```   267          cong add: imp_cong, simp add: Pls_def)
```
```   268
```
```   269
```
```   270 (*Maps #n to n for n = 0, 1, 2*)
```
```   271 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
```
```   272
```
```   273
```
```   274 subsection{*Powers with Numeric Exponents*}
```
```   275
```
```   276 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
```
```   277 We cannot prove general results about the numeral @{term "-1"}, so we have to
```
```   278 use @{term "- 1"} instead.*}
```
```   279
```
```   280 lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
```
```   281   by (simp add: numeral_2_eq_2 Power.power_Suc)
```
```   282
```
```   283 lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
```
```   284   by (simp add: power2_eq_square)
```
```   285
```
```   286 lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
```
```   287   by (simp add: power2_eq_square)
```
```   288
```
```   289 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
```
```   290   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
```
```   291   apply (erule ssubst)
```
```   292   apply (simp add: power_Suc mult_ac)
```
```   293   apply (unfold nat_number_of_def)
```
```   294   apply (subst nat_eq_iff)
```
```   295   apply simp
```
```   296 done
```
```   297
```
```   298 text{*Squares of literal numerals will be evaluated.*}
```
```   299 lemmas power2_eq_square_number_of =
```
```   300     power2_eq_square [of "number_of w", standard]
```
```   301 declare power2_eq_square_number_of [simp]
```
```   302
```
```   303
```
```   304 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   305   by (simp add: power2_eq_square)
```
```   306
```
```   307 lemma zero_less_power2[simp]:
```
```   308      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
```
```   309   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   310
```
```   311 lemma power2_less_0[simp]:
```
```   312   fixes a :: "'a::{ordered_idom,recpower}"
```
```   313   shows "~ (a\<twosuperior> < 0)"
```
```   314 by (force simp add: power2_eq_square mult_less_0_iff)
```
```   315
```
```   316 lemma zero_eq_power2[simp]:
```
```   317      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
```
```   318   by (force simp add: power2_eq_square mult_eq_0_iff)
```
```   319
```
```   320 lemma abs_power2[simp]:
```
```   321      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   322   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   323
```
```   324 lemma power2_abs[simp]:
```
```   325      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   326   by (simp add: power2_eq_square abs_mult_self)
```
```   327
```
```   328 lemma power2_minus[simp]:
```
```   329      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
```
```   330   by (simp add: power2_eq_square)
```
```   331
```
```   332 lemma power2_le_imp_le:
```
```   333   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   334   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
```
```   335 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   336
```
```   337 lemma power2_less_imp_less:
```
```   338   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   339   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
```
```   340 by (rule power_less_imp_less_base)
```
```   341
```
```   342 lemma power2_eq_imp_eq:
```
```   343   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   344   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
```
```   345 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
```
```   346
```
```   347 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
```
```   348 apply (induct "n")
```
```   349 apply (auto simp add: power_Suc power_add)
```
```   350 done
```
```   351
```
```   352 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
```
```   353 by (subst mult_commute) (simp add: power_mult)
```
```   354
```
```   355 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
```
```   356 by (simp add: power_even_eq)
```
```   357
```
```   358 lemma power_minus_even [simp]:
```
```   359      "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
```
```   360 by (simp add: power_minus1_even power_minus [of a])
```
```   361
```
```   362 lemma zero_le_even_power'[simp]:
```
```   363      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
```
```   364 proof (induct "n")
```
```   365   case 0
```
```   366     show ?case by (simp add: zero_le_one)
```
```   367 next
```
```   368   case (Suc n)
```
```   369     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   370       by (simp add: mult_ac power_add power2_eq_square)
```
```   371     thus ?case
```
```   372       by (simp add: prems zero_le_mult_iff)
```
```   373 qed
```
```   374
```
```   375 lemma odd_power_less_zero:
```
```   376      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
```
```   377 proof (induct "n")
```
```   378   case 0
```
```   379   then show ?case by (simp add: Power.power_Suc)
```
```   380 next
```
```   381   case (Suc n)
```
```   382   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   383     by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
```
```   384   thus ?case
```
```   385     by (simp add: prems mult_less_0_iff mult_neg_neg)
```
```   386 qed
```
```   387
```
```   388 lemma odd_0_le_power_imp_0_le:
```
```   389      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
```
```   390 apply (insert odd_power_less_zero [of a n])
```
```   391 apply (force simp add: linorder_not_less [symmetric])
```
```   392 done
```
```   393
```
```   394 text{*Simprules for comparisons where common factors can be cancelled.*}
```
```   395 lemmas zero_compare_simps =
```
```   396     add_strict_increasing add_strict_increasing2 add_increasing
```
```   397     zero_le_mult_iff zero_le_divide_iff
```
```   398     zero_less_mult_iff zero_less_divide_iff
```
```   399     mult_le_0_iff divide_le_0_iff
```
```   400     mult_less_0_iff divide_less_0_iff
```
```   401     zero_le_power2 power2_less_0
```
```   402
```
```   403 subsubsection{*Nat *}
```
```   404
```
```   405 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
```
```   406 by (simp add: numerals)
```
```   407
```
```   408 (*Expresses a natural number constant as the Suc of another one.
```
```   409   NOT suitable for rewriting because n recurs in the condition.*)
```
```   410 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
```
```   411
```
```   412 subsubsection{*Arith *}
```
```   413
```
```   414 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
```
```   415 by (simp add: numerals)
```
```   416
```
```   417 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
```
```   418 by (simp add: numerals)
```
```   419
```
```   420 (* These two can be useful when m = number_of... *)
```
```   421
```
```   422 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
```
```   423 apply (case_tac "m")
```
```   424 apply (simp_all add: numerals)
```
```   425 done
```
```   426
```
```   427 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
```
```   428 apply (case_tac "m")
```
```   429 apply (simp_all add: numerals)
```
```   430 done
```
```   431
```
```   432 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   433 apply (case_tac "m")
```
```   434 apply (simp_all add: numerals)
```
```   435 done
```
```   436
```
```   437
```
```   438 subsection{*Comparisons involving (0::nat) *}
```
```   439
```
```   440 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
```
```   441
```
```   442 lemma eq_number_of_0 [simp]:
```
```   443      "(number_of v = (0::nat)) =
```
```   444       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
```
```   445 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
```
```   446
```
```   447 lemma eq_0_number_of [simp]:
```
```   448      "((0::nat) = number_of v) =
```
```   449       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
```
```   450 by (rule trans [OF eq_sym_conv eq_number_of_0])
```
```   451
```
```   452 lemma less_0_number_of [simp]:
```
```   453      "((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
```
```   454 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
```
```   455
```
```   456
```
```   457 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
```
```   458 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
```
```   459
```
```   460
```
```   461
```
```   462 subsection{*Comparisons involving  @{term Suc} *}
```
```   463
```
```   464 lemma eq_number_of_Suc [simp]:
```
```   465      "(number_of v = Suc n) =
```
```   466         (let pv = number_of (Numeral.pred v) in
```
```   467          if neg pv then False else nat pv = n)"
```
```   468 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   469                   number_of_pred nat_number_of_def
```
```   470             split add: split_if)
```
```   471 apply (rule_tac x = "number_of v" in spec)
```
```   472 apply (auto simp add: nat_eq_iff)
```
```   473 done
```
```   474
```
```   475 lemma Suc_eq_number_of [simp]:
```
```   476      "(Suc n = number_of v) =
```
```   477         (let pv = number_of (Numeral.pred v) in
```
```   478          if neg pv then False else nat pv = n)"
```
```   479 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
```
```   480
```
```   481 lemma less_number_of_Suc [simp]:
```
```   482      "(number_of v < Suc n) =
```
```   483         (let pv = number_of (Numeral.pred v) in
```
```   484          if neg pv then True else nat pv < n)"
```
```   485 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   486                   number_of_pred nat_number_of_def
```
```   487             split add: split_if)
```
```   488 apply (rule_tac x = "number_of v" in spec)
```
```   489 apply (auto simp add: nat_less_iff)
```
```   490 done
```
```   491
```
```   492 lemma less_Suc_number_of [simp]:
```
```   493      "(Suc n < number_of v) =
```
```   494         (let pv = number_of (Numeral.pred v) in
```
```   495          if neg pv then False else n < nat pv)"
```
```   496 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   497                   number_of_pred nat_number_of_def
```
```   498             split add: split_if)
```
```   499 apply (rule_tac x = "number_of v" in spec)
```
```   500 apply (auto simp add: zless_nat_eq_int_zless)
```
```   501 done
```
```   502
```
```   503 lemma le_number_of_Suc [simp]:
```
```   504      "(number_of v <= Suc n) =
```
```   505         (let pv = number_of (Numeral.pred v) in
```
```   506          if neg pv then True else nat pv <= n)"
```
```   507 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
```
```   508
```
```   509 lemma le_Suc_number_of [simp]:
```
```   510      "(Suc n <= number_of v) =
```
```   511         (let pv = number_of (Numeral.pred v) in
```
```   512          if neg pv then False else n <= nat pv)"
```
```   513 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
```
```   514
```
```   515
```
```   516 lemma lemma1: "(m+m = n+n) = (m = (n::int))"
```
```   517 by auto
```
```   518
```
```   519 lemma lemma2: "m+m ~= (1::int) + (n + n)"
```
```   520 apply auto
```
```   521 apply (drule_tac f = "%x. x mod 2" in arg_cong)
```
```   522 apply (simp add: zmod_zadd1_eq)
```
```   523 done
```
```   524
```
```   525 lemma eq_number_of_BIT_BIT:
```
```   526      "((number_of (v BIT x) ::int) = number_of (w BIT y)) =
```
```   527       (x=y & (((number_of v) ::int) = number_of w))"
```
```   528 apply (simp only: number_of_BIT lemma1 lemma2 eq_commute
```
```   529                OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0_left
```
```   530             split add: bit.split)
```
```   531 apply simp
```
```   532 done
```
```   533
```
```   534 lemma eq_number_of_BIT_Pls:
```
```   535      "((number_of (v BIT x) ::int) = Numeral0) =
```
```   536       (x=bit.B0 & (((number_of v) ::int) = Numeral0))"
```
```   537 apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute
```
```   538             split add: bit.split cong: imp_cong)
```
```   539 apply (rule_tac x = "number_of v" in spec, safe)
```
```   540 apply (simp_all (no_asm_use))
```
```   541 apply (drule_tac f = "%x. x mod 2" in arg_cong)
```
```   542 apply (simp add: zmod_zadd1_eq)
```
```   543 done
```
```   544
```
```   545 lemma eq_number_of_BIT_Min:
```
```   546      "((number_of (v BIT x) ::int) = number_of Numeral.Min) =
```
```   547       (x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))"
```
```   548 apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute
```
```   549             split add: bit.split cong: imp_cong)
```
```   550 apply (rule_tac x = "number_of v" in spec, auto)
```
```   551 apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)
```
```   552 done
```
```   553
```
```   554 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min"
```
```   555 by auto
```
```   556
```
```   557
```
```   558
```
```   559 subsection{*Max and Min Combined with @{term Suc} *}
```
```   560
```
```   561 lemma max_number_of_Suc [simp]:
```
```   562      "max (Suc n) (number_of v) =
```
```   563         (let pv = number_of (Numeral.pred v) in
```
```   564          if neg pv then Suc n else Suc(max n (nat pv)))"
```
```   565 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   566             split add: split_if nat.split)
```
```   567 apply (rule_tac x = "number_of v" in spec)
```
```   568 apply auto
```
```   569 done
```
```   570
```
```   571 lemma max_Suc_number_of [simp]:
```
```   572      "max (number_of v) (Suc n) =
```
```   573         (let pv = number_of (Numeral.pred v) in
```
```   574          if neg pv then Suc n else Suc(max (nat pv) n))"
```
```   575 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   576             split add: split_if nat.split)
```
```   577 apply (rule_tac x = "number_of v" in spec)
```
```   578 apply auto
```
```   579 done
```
```   580
```
```   581 lemma min_number_of_Suc [simp]:
```
```   582      "min (Suc n) (number_of v) =
```
```   583         (let pv = number_of (Numeral.pred v) in
```
```   584          if neg pv then 0 else Suc(min n (nat pv)))"
```
```   585 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   586             split add: split_if nat.split)
```
```   587 apply (rule_tac x = "number_of v" in spec)
```
```   588 apply auto
```
```   589 done
```
```   590
```
```   591 lemma min_Suc_number_of [simp]:
```
```   592      "min (number_of v) (Suc n) =
```
```   593         (let pv = number_of (Numeral.pred v) in
```
```   594          if neg pv then 0 else Suc(min (nat pv) n))"
```
```   595 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
```
```   596             split add: split_if nat.split)
```
```   597 apply (rule_tac x = "number_of v" in spec)
```
```   598 apply auto
```
```   599 done
```
```   600
```
```   601 subsection{*Literal arithmetic involving powers*}
```
```   602
```
```   603 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
```
```   604 apply (induct "n")
```
```   605 apply (simp_all (no_asm_simp) add: nat_mult_distrib)
```
```   606 done
```
```   607
```
```   608 lemma power_nat_number_of:
```
```   609      "(number_of v :: nat) ^ n =
```
```   610        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
```
```   611 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
```
```   612          split add: split_if cong: imp_cong)
```
```   613
```
```   614
```
```   615 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
```
```   616 declare power_nat_number_of_number_of [simp]
```
```   617
```
```   618
```
```   619
```
```   620 text{*For arbitrary rings*}
```
```   621
```
```   622 lemma power_number_of_even:
```
```   623   fixes z :: "'a::{number_ring,recpower}"
```
```   624   shows "z ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"
```
```   625 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
```
```   626 apply (rule_tac x = "number_of w" in spec, clarify)
```
```   627 apply (case_tac " (0::int) <= x")
```
```   628 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
```
```   629 done
```
```   630
```
```   631 lemma power_number_of_odd:
```
```   632   fixes z :: "'a::{number_ring,recpower}"
```
```   633   shows "z ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w
```
```   634      then (let w = z ^ (number_of w) in z * w * w) else 1)"
```
```   635 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
```
```   636 apply (rule_tac x = "number_of w" in spec, auto)
```
```   637 apply (simp only: nat_add_distrib nat_mult_distrib)
```
```   638 apply simp
```
```   639 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
```
```   640 done
```
```   641
```
```   642 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
```
```   643 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
```
```   644
```
```   645 lemmas power_number_of_even_number_of [simp] =
```
```   646     power_number_of_even [of "number_of v", standard]
```
```   647
```
```   648 lemmas power_number_of_odd_number_of [simp] =
```
```   649     power_number_of_odd [of "number_of v", standard]
```
```   650
```
```   651
```
```   652
```
```   653 ML
```
```   654 {*
```
```   655 val numerals = thms"numerals";
```
```   656 val numeral_ss = simpset() addsimps numerals;
```
```   657
```
```   658 val nat_bin_arith_setup =
```
```   659  LinArith.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 [Suc_nat_number_of, int_nat_number_of,
```
```   665                                   not_neg_number_of_Pls,
```
```   666                                   neg_number_of_Min,neg_number_of_BIT]})
```
```   667 *}
```
```   668
```
```   669 declaration {* K nat_bin_arith_setup *}
```
```   670
```
```   671 (* Enable arith to deal with div/mod k where k is a numeral: *)
```
```   672 declare split_div[of _ _ "number_of k", standard, arith_split]
```
```   673 declare split_mod[of _ _ "number_of k", standard, arith_split]
```
```   674
```
```   675 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
```
```   676   by (simp add: number_of_Pls nat_number_of_def)
```
```   677
```
```   678 lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
```
```   679   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
```
```   680   done
```
```   681
```
```   682 lemma nat_number_of_BIT_1:
```
```   683   "number_of (w BIT bit.B1) =
```
```   684     (if neg (number_of w :: int) then 0
```
```   685      else let n = number_of w in Suc (n + n))"
```
```   686   apply (simp only: nat_number_of_def Let_def split: split_if)
```
```   687   apply (intro conjI impI)
```
```   688    apply (simp add: neg_nat neg_number_of_BIT)
```
```   689   apply (rule int_int_eq [THEN iffD1])
```
```   690   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
```
```   691   apply (simp only: number_of_BIT zadd_assoc split: bit.split)
```
```   692   apply simp
```
```   693   done
```
```   694
```
```   695 lemma nat_number_of_BIT_0:
```
```   696     "number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
```
```   697   apply (simp only: nat_number_of_def Let_def)
```
```   698   apply (cases "neg (number_of w :: int)")
```
```   699    apply (simp add: neg_nat neg_number_of_BIT)
```
```   700   apply (rule int_int_eq [THEN iffD1])
```
```   701   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
```
```   702   apply (simp only: number_of_BIT zadd_assoc)
```
```   703   apply simp
```
```   704   done
```
```   705
```
```   706 lemmas nat_number =
```
```   707   nat_number_of_Pls nat_number_of_Min
```
```   708   nat_number_of_BIT_1 nat_number_of_BIT_0
```
```   709
```
```   710 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
```
```   711   by (simp add: Let_def)
```
```   712
```
```   713 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
```
```   714 by (simp add: power_mult power_Suc);
```
```   715
```
```   716 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
```
```   717 by (simp add: power_mult power_Suc);
```
```   718
```
```   719
```
```   720 subsection{*Literal arithmetic and @{term of_nat}*}
```
```   721
```
```   722 lemma of_nat_double:
```
```   723      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
```
```   724 by (simp only: mult_2 nat_add_distrib of_nat_add)
```
```   725
```
```   726 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
```
```   727 by (simp only: nat_number_of_def)
```
```   728
```
```   729 lemma of_nat_number_of_lemma:
```
```   730      "of_nat (number_of v :: nat) =
```
```   731          (if 0 \<le> (number_of v :: int)
```
```   732           then (number_of v :: 'a :: number_ring)
```
```   733           else 0)"
```
```   734 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
```
```   735
```
```   736 lemma of_nat_number_of_eq [simp]:
```
```   737      "of_nat (number_of v :: nat) =
```
```   738          (if neg (number_of v :: int) then 0
```
```   739           else (number_of v :: 'a :: number_ring))"
```
```   740 by (simp only: of_nat_number_of_lemma neg_def, simp)
```
```   741
```
```   742
```
```   743 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
```
```   744
```
```   745 lemma nat_number_of_add_left:
```
```   746      "number_of v + (number_of v' + (k::nat)) =
```
```   747          (if neg (number_of v :: int) then number_of v' + k
```
```   748           else if neg (number_of v' :: int) then number_of v + k
```
```   749           else number_of (v + v') + k)"
```
```   750 by simp
```
```   751
```
```   752 lemma nat_number_of_mult_left:
```
```   753      "number_of v * (number_of v' * (k::nat)) =
```
```   754          (if neg (number_of v :: int) then 0
```
```   755           else number_of (v * v') * k)"
```
```   756 by simp
```
```   757
```
```   758
```
```   759 subsubsection{*For @{text combine_numerals}*}
```
```   760
```
```   761 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
```
```   762 by (simp add: add_mult_distrib)
```
```   763
```
```   764
```
```   765 subsubsection{*For @{text cancel_numerals}*}
```
```   766
```
```   767 lemma nat_diff_add_eq1:
```
```   768      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
```
```   769 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   770
```
```   771 lemma nat_diff_add_eq2:
```
```   772      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
```
```   773 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   774
```
```   775 lemma nat_eq_add_iff1:
```
```   776      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
```
```   777 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   778
```
```   779 lemma nat_eq_add_iff2:
```
```   780      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
```
```   781 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   782
```
```   783 lemma nat_less_add_iff1:
```
```   784      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
```
```   785 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   786
```
```   787 lemma nat_less_add_iff2:
```
```   788      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
```
```   789 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   790
```
```   791 lemma nat_le_add_iff1:
```
```   792      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
```
```   793 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   794
```
```   795 lemma nat_le_add_iff2:
```
```   796      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
```
```   797 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   798
```
```   799
```
```   800 subsubsection{*For @{text cancel_numeral_factors} *}
```
```   801
```
```   802 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
```
```   803 by auto
```
```   804
```
```   805 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
```
```   806 by auto
```
```   807
```
```   808 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
```
```   809 by auto
```
```   810
```
```   811 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
```
```   812 by auto
```
```   813
```
```   814 lemma nat_mult_dvd_cancel_disj[simp]:
```
```   815   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
```
```   816 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
```
```   817
```
```   818 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
```
```   819 by(auto)
```
```   820
```
```   821
```
```   822 subsubsection{*For @{text cancel_factor} *}
```
```   823
```
```   824 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
```
```   825 by auto
```
```   826
```
```   827 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
```
```   828 by auto
```
```   829
```
```   830 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
```
```   831 by auto
```
```   832
```
```   833 lemma nat_mult_div_cancel_disj[simp]:
```
```   834      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
```
```   835 by (simp add: nat_mult_div_cancel1)
```
```   836
```
```   837
```
```   838 subsection {* legacy ML bindings *}
```
```   839
```
```   840 ML
```
```   841 {*
```
```   842 val eq_nat_nat_iff = thm"eq_nat_nat_iff";
```
```   843 val eq_nat_number_of = thm"eq_nat_number_of";
```
```   844 val less_nat_number_of = thm"less_nat_number_of";
```
```   845 val power2_eq_square = thm "power2_eq_square";
```
```   846 val zero_le_power2 = thm "zero_le_power2";
```
```   847 val zero_less_power2 = thm "zero_less_power2";
```
```   848 val zero_eq_power2 = thm "zero_eq_power2";
```
```   849 val abs_power2 = thm "abs_power2";
```
```   850 val power2_abs = thm "power2_abs";
```
```   851 val power2_minus = thm "power2_minus";
```
```   852 val power_minus1_even = thm "power_minus1_even";
```
```   853 val power_minus_even = thm "power_minus_even";
```
```   854 val odd_power_less_zero = thm "odd_power_less_zero";
```
```   855 val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
```
```   856
```
```   857 val Suc_pred' = thm"Suc_pred'";
```
```   858 val expand_Suc = thm"expand_Suc";
```
```   859 val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
```
```   860 val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
```
```   861 val add_eq_if = thm"add_eq_if";
```
```   862 val mult_eq_if = thm"mult_eq_if";
```
```   863 val power_eq_if = thm"power_eq_if";
```
```   864 val eq_number_of_0 = thm"eq_number_of_0";
```
```   865 val eq_0_number_of = thm"eq_0_number_of";
```
```   866 val less_0_number_of = thm"less_0_number_of";
```
```   867 val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
```
```   868 val eq_number_of_Suc = thm"eq_number_of_Suc";
```
```   869 val Suc_eq_number_of = thm"Suc_eq_number_of";
```
```   870 val less_number_of_Suc = thm"less_number_of_Suc";
```
```   871 val less_Suc_number_of = thm"less_Suc_number_of";
```
```   872 val le_number_of_Suc = thm"le_number_of_Suc";
```
```   873 val le_Suc_number_of = thm"le_Suc_number_of";
```
```   874 val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
```
```   875 val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
```
```   876 val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
```
```   877 val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
```
```   878 val of_nat_number_of_eq = thm"of_nat_number_of_eq";
```
```   879 val nat_power_eq = thm"nat_power_eq";
```
```   880 val power_nat_number_of = thm"power_nat_number_of";
```
```   881 val zpower_number_of_even = thm"zpower_number_of_even";
```
```   882 val zpower_number_of_odd = thm"zpower_number_of_odd";
```
```   883 val nat_number_of_Pls = thm"nat_number_of_Pls";
```
```   884 val nat_number_of_Min = thm"nat_number_of_Min";
```
```   885 val Let_Suc = thm"Let_Suc";
```
```   886
```
```   887 val nat_number = thms"nat_number";
```
```   888
```
```   889 val nat_number_of_add_left = thm"nat_number_of_add_left";
```
```   890 val nat_number_of_mult_left = thm"nat_number_of_mult_left";
```
```   891 val left_add_mult_distrib = thm"left_add_mult_distrib";
```
```   892 val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
```
```   893 val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
```
```   894 val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
```
```   895 val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
```
```   896 val nat_less_add_iff1 = thm"nat_less_add_iff1";
```
```   897 val nat_less_add_iff2 = thm"nat_less_add_iff2";
```
```   898 val nat_le_add_iff1 = thm"nat_le_add_iff1";
```
```   899 val nat_le_add_iff2 = thm"nat_le_add_iff2";
```
```   900 val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
```
```   901 val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
```
```   902 val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
```
```   903 val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
```
```   904 val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
```
```   905 val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
```
```   906 val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
```
```   907 val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
```
```   908
```
```   909 val power_minus_even = thm"power_minus_even";
```
```   910 *}
```
```   911
```
```   912 end
```