src/HOL/NatBin.thy
 author urbanc Tue Jun 05 09:56:19 2007 +0200 (2007-06-05) changeset 23243 a37d3e6e8323 parent 23164 69e55066dbca child 23277 aa158e145ea3 permissions -rw-r--r--
included Class.thy in the compiling process for Nominal/Examples
```     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 inj_int [THEN injD], 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 zadd_int
```
```    69                  zmult_int)
```
```    70 done
```
```    71
```
```    72 (*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```    73 lemma nat_mod_distrib:
```
```    74      "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"
```
```    75 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
```
```    76 apply (auto elim!: nonneg_eq_int)
```
```    77 apply (rename_tac m m')
```
```    78 apply (subgoal_tac "0 <= int m mod int m'")
```
```    79  prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign)
```
```    80 apply (rule inj_int [THEN injD], simp)
```
```    81 apply (rule_tac q = "int (m div m') " in quorem_mod)
```
```    82  prefer 2 apply force
```
```    83 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int zmult_int)
```
```    84 done
```
```    85
```
```    86 text{*Suggested by Matthias Daum*}
```
```    87 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```    88 apply (subgoal_tac "nat x div nat k < nat x")
```
```    89  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
```
```    90 apply (rule Divides.div_less_dividend, simp_all)
```
```    91 done
```
```    92
```
```    93 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
```
```    94
```
```    95 (*"neg" is used in rewrite rules for binary comparisons*)
```
```    96 lemma int_nat_number_of [simp]:
```
```    97      "int (number_of v :: nat) =
```
```    98          (if neg (number_of v :: int) then 0
```
```    99           else (number_of v :: int))"
```
```   100 by (simp del: nat_number_of
```
```   101 	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)
```
```   102
```
```   103
```
```   104 subsubsection{*Successor *}
```
```   105
```
```   106 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   107 apply (rule sym)
```
```   108 apply (simp add: nat_eq_iff int_Suc)
```
```   109 done
```
```   110
```
```   111 lemma Suc_nat_number_of_add:
```
```   112      "Suc (number_of v + n) =
```
```   113         (if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)"
```
```   114 by (simp del: nat_number_of
```
```   115          add: nat_number_of_def neg_nat
```
```   116               Suc_nat_eq_nat_zadd1 number_of_succ)
```
```   117
```
```   118 lemma Suc_nat_number_of [simp]:
```
```   119      "Suc (number_of v) =
```
```   120         (if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))"
```
```   121 apply (cut_tac n = 0 in Suc_nat_number_of_add)
```
```   122 apply (simp cong del: if_weak_cong)
```
```   123 done
```
```   124
```
```   125
```
```   126 subsubsection{*Addition *}
```
```   127
```
```   128 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   129 lemma add_nat_number_of [simp]:
```
```   130      "(number_of v :: nat) + number_of v' =
```
```   131          (if neg (number_of v :: int) then number_of v'
```
```   132           else if neg (number_of v' :: int) then number_of v
```
```   133           else number_of (v + v'))"
```
```   134 by (force dest!: neg_nat
```
```   135           simp del: nat_number_of
```
```   136           simp add: nat_number_of_def nat_add_distrib [symmetric])
```
```   137
```
```   138
```
```   139 subsubsection{*Subtraction *}
```
```   140
```
```   141 lemma diff_nat_eq_if:
```
```   142      "nat z - nat z' =
```
```   143         (if neg z' then nat z
```
```   144          else let d = z-z' in
```
```   145               if neg d then 0 else nat d)"
```
```   146 apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
```
```   147 done
```
```   148
```
```   149 lemma diff_nat_number_of [simp]:
```
```   150      "(number_of v :: nat) - number_of v' =
```
```   151         (if neg (number_of v' :: int) then number_of v
```
```   152          else let d = number_of (v + uminus v') in
```
```   153               if neg d then 0 else nat d)"
```
```   154 by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def)
```
```   155
```
```   156
```
```   157
```
```   158 subsubsection{*Multiplication *}
```
```   159
```
```   160 lemma mult_nat_number_of [simp]:
```
```   161      "(number_of v :: nat) * number_of v' =
```
```   162        (if neg (number_of v :: int) then 0 else number_of (v * v'))"
```
```   163 by (force dest!: neg_nat
```
```   164           simp del: nat_number_of
```
```   165           simp add: nat_number_of_def nat_mult_distrib [symmetric])
```
```   166
```
```   167
```
```   168
```
```   169 subsubsection{*Quotient *}
```
```   170
```
```   171 lemma div_nat_number_of [simp]:
```
```   172      "(number_of v :: nat)  div  number_of v' =
```
```   173           (if neg (number_of v :: int) then 0
```
```   174            else nat (number_of v div number_of v'))"
```
```   175 by (force dest!: neg_nat
```
```   176           simp del: nat_number_of
```
```   177           simp add: nat_number_of_def nat_div_distrib [symmetric])
```
```   178
```
```   179 lemma one_div_nat_number_of [simp]:
```
```   180      "(Suc 0)  div  number_of v' = (nat (1 div number_of v'))"
```
```   181 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   182
```
```   183
```
```   184 subsubsection{*Remainder *}
```
```   185
```
```   186 lemma mod_nat_number_of [simp]:
```
```   187      "(number_of v :: nat)  mod  number_of v' =
```
```   188         (if neg (number_of v :: int) then 0
```
```   189          else if neg (number_of v' :: int) then number_of v
```
```   190          else nat (number_of v mod number_of v'))"
```
```   191 by (force dest!: neg_nat
```
```   192           simp del: nat_number_of
```
```   193           simp add: nat_number_of_def nat_mod_distrib [symmetric])
```
```   194
```
```   195 lemma one_mod_nat_number_of [simp]:
```
```   196      "(Suc 0)  mod  number_of v' =
```
```   197         (if neg (number_of v' :: int) then Suc 0
```
```   198          else nat (1 mod number_of v'))"
```
```   199 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
```
```   200
```
```   201
```
```   202 subsubsection{* Divisibility *}
```
```   203
```
```   204 lemmas dvd_eq_mod_eq_0_number_of =
```
```   205   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
```
```   206
```
```   207 declare dvd_eq_mod_eq_0_number_of [simp]
```
```   208
```
```   209 ML
```
```   210 {*
```
```   211 val nat_number_of_def = thm"nat_number_of_def";
```
```   212
```
```   213 val nat_number_of = thm"nat_number_of";
```
```   214 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
```
```   215 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
```
```   216 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
```
```   217 val numeral_2_eq_2 = thm"numeral_2_eq_2";
```
```   218 val nat_div_distrib = thm"nat_div_distrib";
```
```   219 val nat_mod_distrib = thm"nat_mod_distrib";
```
```   220 val int_nat_number_of = thm"int_nat_number_of";
```
```   221 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
```
```   222 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
```
```   223 val Suc_nat_number_of = thm"Suc_nat_number_of";
```
```   224 val add_nat_number_of = thm"add_nat_number_of";
```
```   225 val diff_nat_eq_if = thm"diff_nat_eq_if";
```
```   226 val diff_nat_number_of = thm"diff_nat_number_of";
```
```   227 val mult_nat_number_of = thm"mult_nat_number_of";
```
```   228 val div_nat_number_of = thm"div_nat_number_of";
```
```   229 val mod_nat_number_of = thm"mod_nat_number_of";
```
```   230 *}
```
```   231
```
```   232
```
```   233 subsection{*Comparisons*}
```
```   234
```
```   235 subsubsection{*Equals (=) *}
```
```   236
```
```   237 lemma eq_nat_nat_iff:
```
```   238      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
```
```   239 by (auto elim!: nonneg_eq_int)
```
```   240
```
```   241 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   242 lemma eq_nat_number_of [simp]:
```
```   243      "((number_of v :: nat) = number_of v') =
```
```   244       (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))
```
```   245        else if neg (number_of v' :: int) then iszero (number_of v :: int)
```
```   246        else iszero (number_of (v + uminus v') :: int))"
```
```   247 apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
```
```   248                   eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
```
```   249             split add: split_if cong add: imp_cong)
```
```   250 apply (simp only: nat_eq_iff nat_eq_iff2)
```
```   251 apply (simp add: not_neg_eq_ge_0 [symmetric])
```
```   252 done
```
```   253
```
```   254
```
```   255 subsubsection{*Less-than (<) *}
```
```   256
```
```   257 (*"neg" is used in rewrite rules for binary comparisons*)
```
```   258 lemma less_nat_number_of [simp]:
```
```   259      "((number_of v :: nat) < number_of v') =
```
```   260          (if neg (number_of v :: int) then neg (number_of (uminus v') :: int)
```
```   261           else neg (number_of (v + uminus v') :: int))"
```
```   262 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
```
```   263                 nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
```
```   264          cong add: imp_cong, simp add: Pls_def)
```
```   265
```
```   266
```
```   267 (*Maps #n to n for n = 0, 1, 2*)
```
```   268 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
```
```   269
```
```   270
```
```   271 subsection{*Powers with Numeric Exponents*}
```
```   272
```
```   273 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
```
```   274 We cannot prove general results about the numeral @{term "-1"}, so we have to
```
```   275 use @{term "- 1"} instead.*}
```
```   276
```
```   277 lemma power2_eq_square: "(a::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = a * a"
```
```   278   by (simp add: numeral_2_eq_2 Power.power_Suc)
```
```   279
```
```   280 lemma zero_power2 [simp]: "(0::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 0"
```
```   281   by (simp add: power2_eq_square)
```
```   282
```
```   283 lemma one_power2 [simp]: "(1::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 1"
```
```   284   by (simp add: power2_eq_square)
```
```   285
```
```   286 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
```
```   287   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
```
```   288   apply (erule ssubst)
```
```   289   apply (simp add: power_Suc mult_ac)
```
```   290   apply (unfold nat_number_of_def)
```
```   291   apply (subst nat_eq_iff)
```
```   292   apply simp
```
```   293 done
```
```   294
```
```   295 text{*Squares of literal numerals will be evaluated.*}
```
```   296 lemmas power2_eq_square_number_of =
```
```   297     power2_eq_square [of "number_of w", standard]
```
```   298 declare power2_eq_square_number_of [simp]
```
```   299
```
```   300
```
```   301 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   302   by (simp add: power2_eq_square)
```
```   303
```
```   304 lemma zero_less_power2[simp]:
```
```   305      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
```
```   306   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   307
```
```   308 lemma power2_less_0[simp]:
```
```   309   fixes a :: "'a::{ordered_idom,recpower}"
```
```   310   shows "~ (a\<twosuperior> < 0)"
```
```   311 by (force simp add: power2_eq_square mult_less_0_iff)
```
```   312
```
```   313 lemma zero_eq_power2[simp]:
```
```   314      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
```
```   315   by (force simp add: power2_eq_square mult_eq_0_iff)
```
```   316
```
```   317 lemma abs_power2[simp]:
```
```   318      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   319   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   320
```
```   321 lemma power2_abs[simp]:
```
```   322      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
```
```   323   by (simp add: power2_eq_square abs_mult_self)
```
```   324
```
```   325 lemma power2_minus[simp]:
```
```   326      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
```
```   327   by (simp add: power2_eq_square)
```
```   328
```
```   329 lemma power2_le_imp_le:
```
```   330   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   331   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
```
```   332 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   333
```
```   334 lemma power2_less_imp_less:
```
```   335   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   336   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
```
```   337 by (rule power_less_imp_less_base)
```
```   338
```
```   339 lemma power2_eq_imp_eq:
```
```   340   fixes x y :: "'a::{ordered_semidom,recpower}"
```
```   341   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
```
```   342 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
```
```   343
```
```   344 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
```
```   345 apply (induct "n")
```
```   346 apply (auto simp add: power_Suc power_add)
```
```   347 done
```
```   348
```
```   349 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
```
```   350 by (subst mult_commute) (simp add: power_mult)
```
```   351
```
```   352 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
```
```   353 by (simp add: power_even_eq)
```
```   354
```
```   355 lemma power_minus_even [simp]:
```
```   356      "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
```
```   357 by (simp add: power_minus1_even power_minus [of a])
```
```   358
```
```   359 lemma zero_le_even_power'[simp]:
```
```   360      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
```
```   361 proof (induct "n")
```
```   362   case 0
```
```   363     show ?case by (simp add: zero_le_one)
```
```   364 next
```
```   365   case (Suc n)
```
```   366     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   367       by (simp add: mult_ac power_add power2_eq_square)
```
```   368     thus ?case
```
```   369       by (simp add: prems zero_le_mult_iff)
```
```   370 qed
```
```   371
```
```   372 lemma odd_power_less_zero:
```
```   373      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
```
```   374 proof (induct "n")
```
```   375   case 0
```
```   376     show ?case by (simp add: Power.power_Suc)
```
```   377 next
```
```   378   case (Suc n)
```
```   379     have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   380       by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
```
```   381     thus ?case
```
```   382       by (simp add: prems mult_less_0_iff mult_neg_neg)
```
```   383 qed
```
```   384
```
```   385 lemma odd_0_le_power_imp_0_le:
```
```   386      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
```
```   387 apply (insert odd_power_less_zero [of a n])
```
```   388 apply (force simp add: linorder_not_less [symmetric])
```
```   389 done
```
```   390
```
```   391 text{*Simprules for comparisons where common factors can be cancelled.*}
```
```   392 lemmas zero_compare_simps =
```
```   393     add_strict_increasing add_strict_increasing2 add_increasing
```
```   394     zero_le_mult_iff zero_le_divide_iff
```
```   395     zero_less_mult_iff zero_less_divide_iff
```
```   396     mult_le_0_iff divide_le_0_iff
```
```   397     mult_less_0_iff divide_less_0_iff
```
```   398     zero_le_power2 power2_less_0
```
```   399
```
```   400 subsubsection{*Nat *}
```
```   401
```
```   402 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
```
```   403 by (simp add: numerals)
```
```   404
```
```   405 (*Expresses a natural number constant as the Suc of another one.
```
```   406   NOT suitable for rewriting because n recurs in the condition.*)
```
```   407 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
```
```   408
```
```   409 subsubsection{*Arith *}
```
```   410
```
```   411 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
```
```   412 by (simp add: numerals)
```
```   413
```
```   414 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
```
```   415 by (simp add: numerals)
```
```   416
```
```   417 (* These two can be useful when m = number_of... *)
```
```   418
```
```   419 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
```
```   420 apply (case_tac "m")
```
```   421 apply (simp_all add: numerals)
```
```   422 done
```
```   423
```
```   424 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
```
```   425 apply (case_tac "m")
```
```   426 apply (simp_all add: numerals)
```
```   427 done
```
```   428
```
```   429 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   430 apply (case_tac "m")
```
```   431 apply (simp_all add: numerals)
```
```   432 done
```
```   433
```
```   434
```
```   435 subsection{*Comparisons involving (0::nat) *}
```
```   436
```
```   437 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
```
```   438
```
```   439 lemma eq_number_of_0 [simp]:
```
```   440      "(number_of v = (0::nat)) =
```
```   441       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
```
```   442 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
```
```   443
```
```   444 lemma eq_0_number_of [simp]:
```
```   445      "((0::nat) = number_of v) =
```
```   446       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
```
```   447 by (rule trans [OF eq_sym_conv eq_number_of_0])
```
```   448
```
```   449 lemma less_0_number_of [simp]:
```
```   450      "((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
```
```   451 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
```
```   452
```
```   453
```
```   454 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
```
```   455 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
```
```   456
```
```   457
```
```   458
```
```   459 subsection{*Comparisons involving  @{term Suc} *}
```
```   460
```
```   461 lemma eq_number_of_Suc [simp]:
```
```   462      "(number_of v = Suc n) =
```
```   463         (let pv = number_of (Numeral.pred v) in
```
```   464          if neg pv then False else nat pv = n)"
```
```   465 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   466                   number_of_pred nat_number_of_def
```
```   467             split add: split_if)
```
```   468 apply (rule_tac x = "number_of v" in spec)
```
```   469 apply (auto simp add: nat_eq_iff)
```
```   470 done
```
```   471
```
```   472 lemma Suc_eq_number_of [simp]:
```
```   473      "(Suc n = number_of v) =
```
```   474         (let pv = number_of (Numeral.pred v) in
```
```   475          if neg pv then False else nat pv = n)"
```
```   476 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
```
```   477
```
```   478 lemma less_number_of_Suc [simp]:
```
```   479      "(number_of v < Suc n) =
```
```   480         (let pv = number_of (Numeral.pred v) in
```
```   481          if neg pv then True else nat pv < n)"
```
```   482 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   483                   number_of_pred nat_number_of_def
```
```   484             split add: split_if)
```
```   485 apply (rule_tac x = "number_of v" in spec)
```
```   486 apply (auto simp add: nat_less_iff)
```
```   487 done
```
```   488
```
```   489 lemma less_Suc_number_of [simp]:
```
```   490      "(Suc n < number_of v) =
```
```   491         (let pv = number_of (Numeral.pred v) in
```
```   492          if neg pv then False else n < nat pv)"
```
```   493 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
```
```   494                   number_of_pred nat_number_of_def
```
```   495             split add: split_if)
```
```   496 apply (rule_tac x = "number_of v" in spec)
```
```   497 apply (auto simp add: zless_nat_eq_int_zless)
```
```   498 done
```
```   499
```
```   500 lemma le_number_of_Suc [simp]:
```
```   501      "(number_of v <= Suc n) =
```
```   502         (let pv = number_of (Numeral.pred v) in
```
```   503          if neg pv then True else nat pv <= n)"
```
```   504 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
```
```   505
```
```   506 lemma le_Suc_number_of [simp]:
```
```   507      "(Suc n <= number_of v) =
```
```   508         (let pv = number_of (Numeral.pred v) in
```
```   509          if neg pv then False else n <= nat pv)"
```
```   510 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
```
```   511
```
```   512
```
```   513 (* Push int(.) inwards: *)
```
```   514 declare zadd_int [symmetric, simp]
```
```   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 the integers*}
```
```   621
```
```   622 lemma zpower_number_of_even:
```
```   623   "(z::int) ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"
```
```   624 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
```
```   625 apply (rule_tac x = "number_of w" in spec, clarify)
```
```   626 apply (case_tac " (0::int) <= x")
```
```   627 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
```
```   628 done
```
```   629
```
```   630 lemma zpower_number_of_odd:
```
```   631   "(z::int) ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w
```
```   632      then (let w = z ^ (number_of w) in z * w * w) else 1)"
```
```   633 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
```
```   634 apply (rule_tac x = "number_of w" in spec, auto)
```
```   635 apply (simp only: nat_add_distrib nat_mult_distrib)
```
```   636 apply simp
```
```   637 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat)
```
```   638 done
```
```   639
```
```   640 lemmas zpower_number_of_even_number_of =
```
```   641     zpower_number_of_even [of "number_of v", standard]
```
```   642 declare zpower_number_of_even_number_of [simp]
```
```   643
```
```   644 lemmas zpower_number_of_odd_number_of =
```
```   645     zpower_number_of_odd [of "number_of v", standard]
```
```   646 declare zpower_number_of_odd_number_of [simp]
```
```   647
```
```   648
```
```   649
```
```   650
```
```   651 ML
```
```   652 {*
```
```   653 val numerals = thms"numerals";
```
```   654 val numeral_ss = simpset() addsimps numerals;
```
```   655
```
```   656 val nat_bin_arith_setup =
```
```   657  Fast_Arith.map_data
```
```   658    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
```
```   659      {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
```
```   660       inj_thms = inj_thms,
```
```   661       lessD = lessD, neqE = neqE,
```
```   662       simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
```
```   663                                   not_neg_number_of_Pls,
```
```   664                                   neg_number_of_Min,neg_number_of_BIT]})
```
```   665 *}
```
```   666
```
```   667 setup nat_bin_arith_setup
```
```   668
```
```   669 (* Enable arith to deal with div/mod k where k is a numeral: *)
```
```   670 declare split_div[of _ _ "number_of k", standard, arith_split]
```
```   671 declare split_mod[of _ _ "number_of k", standard, arith_split]
```
```   672
```
```   673 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
```
```   674   by (simp add: number_of_Pls nat_number_of_def)
```
```   675
```
```   676 lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
```
```   677   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
```
```   678   done
```
```   679
```
```   680 lemma nat_number_of_BIT_1:
```
```   681   "number_of (w BIT bit.B1) =
```
```   682     (if neg (number_of w :: int) then 0
```
```   683      else let n = number_of w in Suc (n + n))"
```
```   684   apply (simp only: nat_number_of_def Let_def split: split_if)
```
```   685   apply (intro conjI impI)
```
```   686    apply (simp add: neg_nat neg_number_of_BIT)
```
```   687   apply (rule int_int_eq [THEN iffD1])
```
```   688   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
```
```   689   apply (simp only: number_of_BIT zadd_assoc split: bit.split)
```
```   690   apply simp
```
```   691   done
```
```   692
```
```   693 lemma nat_number_of_BIT_0:
```
```   694     "number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
```
```   695   apply (simp only: nat_number_of_def Let_def)
```
```   696   apply (cases "neg (number_of w :: int)")
```
```   697    apply (simp add: neg_nat neg_number_of_BIT)
```
```   698   apply (rule int_int_eq [THEN iffD1])
```
```   699   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
```
```   700   apply (simp only: number_of_BIT zadd_assoc)
```
```   701   apply simp
```
```   702   done
```
```   703
```
```   704 lemmas nat_number =
```
```   705   nat_number_of_Pls nat_number_of_Min
```
```   706   nat_number_of_BIT_1 nat_number_of_BIT_0
```
```   707
```
```   708 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
```
```   709   by (simp add: Let_def)
```
```   710
```
```   711 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
```
```   712 by (simp add: power_mult);
```
```   713
```
```   714 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
```
```   715 by (simp add: power_mult power_Suc);
```
```   716
```
```   717
```
```   718 subsection{*Literal arithmetic and @{term of_nat}*}
```
```   719
```
```   720 lemma of_nat_double:
```
```   721      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
```
```   722 by (simp only: mult_2 nat_add_distrib of_nat_add)
```
```   723
```
```   724 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
```
```   725 by (simp only: nat_number_of_def)
```
```   726
```
```   727 lemma of_nat_number_of_lemma:
```
```   728      "of_nat (number_of v :: nat) =
```
```   729          (if 0 \<le> (number_of v :: int)
```
```   730           then (number_of v :: 'a :: number_ring)
```
```   731           else 0)"
```
```   732 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
```
```   733
```
```   734 lemma of_nat_number_of_eq [simp]:
```
```   735      "of_nat (number_of v :: nat) =
```
```   736          (if neg (number_of v :: int) then 0
```
```   737           else (number_of v :: 'a :: number_ring))"
```
```   738 by (simp only: of_nat_number_of_lemma neg_def, simp)
```
```   739
```
```   740
```
```   741 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
```
```   742
```
```   743 lemma nat_number_of_add_left:
```
```   744      "number_of v + (number_of v' + (k::nat)) =
```
```   745          (if neg (number_of v :: int) then number_of v' + k
```
```   746           else if neg (number_of v' :: int) then number_of v + k
```
```   747           else number_of (v + v') + k)"
```
```   748 by simp
```
```   749
```
```   750 lemma nat_number_of_mult_left:
```
```   751      "number_of v * (number_of v' * (k::nat)) =
```
```   752          (if neg (number_of v :: int) then 0
```
```   753           else number_of (v * v') * k)"
```
```   754 by simp
```
```   755
```
```   756
```
```   757 subsubsection{*For @{text combine_numerals}*}
```
```   758
```
```   759 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
```
```   760 by (simp add: add_mult_distrib)
```
```   761
```
```   762
```
```   763 subsubsection{*For @{text cancel_numerals}*}
```
```   764
```
```   765 lemma nat_diff_add_eq1:
```
```   766      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
```
```   767 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   768
```
```   769 lemma nat_diff_add_eq2:
```
```   770      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
```
```   771 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```   772
```
```   773 lemma nat_eq_add_iff1:
```
```   774      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
```
```   775 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   776
```
```   777 lemma nat_eq_add_iff2:
```
```   778      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
```
```   779 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   780
```
```   781 lemma nat_less_add_iff1:
```
```   782      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
```
```   783 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   784
```
```   785 lemma nat_less_add_iff2:
```
```   786      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
```
```   787 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   788
```
```   789 lemma nat_le_add_iff1:
```
```   790      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
```
```   791 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   792
```
```   793 lemma nat_le_add_iff2:
```
```   794      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
```
```   795 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```   796
```
```   797
```
```   798 subsubsection{*For @{text cancel_numeral_factors} *}
```
```   799
```
```   800 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
```
```   801 by auto
```
```   802
```
```   803 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
```
```   804 by auto
```
```   805
```
```   806 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
```
```   807 by auto
```
```   808
```
```   809 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
```
```   810 by auto
```
```   811
```
```   812
```
```   813 subsubsection{*For @{text cancel_factor} *}
```
```   814
```
```   815 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
```
```   816 by auto
```
```   817
```
```   818 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
```
```   819 by auto
```
```   820
```
```   821 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
```
```   822 by auto
```
```   823
```
```   824 lemma nat_mult_div_cancel_disj:
```
```   825      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
```
```   826 by (simp add: nat_mult_div_cancel1)
```
```   827
```
```   828
```
```   829 subsection {* legacy ML bindings *}
```
```   830
```
```   831 ML
```
```   832 {*
```
```   833 val eq_nat_nat_iff = thm"eq_nat_nat_iff";
```
```   834 val eq_nat_number_of = thm"eq_nat_number_of";
```
```   835 val less_nat_number_of = thm"less_nat_number_of";
```
```   836 val power2_eq_square = thm "power2_eq_square";
```
```   837 val zero_le_power2 = thm "zero_le_power2";
```
```   838 val zero_less_power2 = thm "zero_less_power2";
```
```   839 val zero_eq_power2 = thm "zero_eq_power2";
```
```   840 val abs_power2 = thm "abs_power2";
```
```   841 val power2_abs = thm "power2_abs";
```
```   842 val power2_minus = thm "power2_minus";
```
```   843 val power_minus1_even = thm "power_minus1_even";
```
```   844 val power_minus_even = thm "power_minus_even";
```
```   845 val odd_power_less_zero = thm "odd_power_less_zero";
```
```   846 val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
```
```   847
```
```   848 val Suc_pred' = thm"Suc_pred'";
```
```   849 val expand_Suc = thm"expand_Suc";
```
```   850 val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
```
```   851 val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
```
```   852 val add_eq_if = thm"add_eq_if";
```
```   853 val mult_eq_if = thm"mult_eq_if";
```
```   854 val power_eq_if = thm"power_eq_if";
```
```   855 val eq_number_of_0 = thm"eq_number_of_0";
```
```   856 val eq_0_number_of = thm"eq_0_number_of";
```
```   857 val less_0_number_of = thm"less_0_number_of";
```
```   858 val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
```
```   859 val eq_number_of_Suc = thm"eq_number_of_Suc";
```
```   860 val Suc_eq_number_of = thm"Suc_eq_number_of";
```
```   861 val less_number_of_Suc = thm"less_number_of_Suc";
```
```   862 val less_Suc_number_of = thm"less_Suc_number_of";
```
```   863 val le_number_of_Suc = thm"le_number_of_Suc";
```
```   864 val le_Suc_number_of = thm"le_Suc_number_of";
```
```   865 val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
```
```   866 val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
```
```   867 val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
```
```   868 val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
```
```   869 val of_nat_number_of_eq = thm"of_nat_number_of_eq";
```
```   870 val nat_power_eq = thm"nat_power_eq";
```
```   871 val power_nat_number_of = thm"power_nat_number_of";
```
```   872 val zpower_number_of_even = thm"zpower_number_of_even";
```
```   873 val zpower_number_of_odd = thm"zpower_number_of_odd";
```
```   874 val nat_number_of_Pls = thm"nat_number_of_Pls";
```
```   875 val nat_number_of_Min = thm"nat_number_of_Min";
```
```   876 val Let_Suc = thm"Let_Suc";
```
```   877
```
```   878 val nat_number = thms"nat_number";
```
```   879
```
```   880 val nat_number_of_add_left = thm"nat_number_of_add_left";
```
```   881 val nat_number_of_mult_left = thm"nat_number_of_mult_left";
```
```   882 val left_add_mult_distrib = thm"left_add_mult_distrib";
```
```   883 val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
```
```   884 val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
```
```   885 val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
```
```   886 val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
```
```   887 val nat_less_add_iff1 = thm"nat_less_add_iff1";
```
```   888 val nat_less_add_iff2 = thm"nat_less_add_iff2";
```
```   889 val nat_le_add_iff1 = thm"nat_le_add_iff1";
```
```   890 val nat_le_add_iff2 = thm"nat_le_add_iff2";
```
```   891 val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
```
```   892 val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
```
```   893 val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
```
```   894 val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
```
```   895 val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
```
```   896 val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
```
```   897 val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
```
```   898 val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
```
```   899
```
```   900 val power_minus_even = thm"power_minus_even";
```
```   901 *}
```
```   902
```
```   903 end
```