src/HOL/Int.thy
 author blanchet Tue Nov 07 15:16:42 2017 +0100 (19 months ago) changeset 67022 49309fe530fd parent 66912 a99a7cbf0fb5 child 67116 7397a6df81d8 permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
```     1 (*  Title:      HOL/Int.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>
```
```     7
```
```     8 theory Int
```
```     9   imports Equiv_Relations Power Quotient Fun_Def
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Definition of integers as a quotient type\<close>
```
```    13
```
```    14 definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
```
```    15   where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
```
```    16
```
```    17 lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
```
```    18   by (simp add: intrel_def)
```
```    19
```
```    20 quotient_type int = "nat \<times> nat" / "intrel"
```
```    21   morphisms Rep_Integ Abs_Integ
```
```    22 proof (rule equivpI)
```
```    23   show "reflp intrel" by (auto simp: reflp_def)
```
```    24   show "symp intrel" by (auto simp: symp_def)
```
```    25   show "transp intrel" by (auto simp: transp_def)
```
```    26 qed
```
```    27
```
```    28 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    29   "(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
```
```    30   by (induct z) auto
```
```    31
```
```    32
```
```    33 subsection \<open>Integers form a commutative ring\<close>
```
```    34
```
```    35 instantiation int :: comm_ring_1
```
```    36 begin
```
```    37
```
```    38 lift_definition zero_int :: "int" is "(0, 0)" .
```
```    39
```
```    40 lift_definition one_int :: "int" is "(1, 0)" .
```
```    41
```
```    42 lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    43   is "\<lambda>(x, y) (u, v). (x + u, y + v)"
```
```    44   by clarsimp
```
```    45
```
```    46 lift_definition uminus_int :: "int \<Rightarrow> int"
```
```    47   is "\<lambda>(x, y). (y, x)"
```
```    48   by clarsimp
```
```    49
```
```    50 lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    51   is "\<lambda>(x, y) (u, v). (x + v, y + u)"
```
```    52   by clarsimp
```
```    53
```
```    54 lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    55   is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
```
```    56 proof (clarsimp)
```
```    57   fix s t u v w x y z :: nat
```
```    58   assume "s + v = u + t" and "w + z = y + x"
```
```    59   then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
```
```    60     (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
```
```    61     by simp
```
```    62   then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
```
```    63     by (simp add: algebra_simps)
```
```    64 qed
```
```    65
```
```    66 instance
```
```    67   by standard (transfer; clarsimp simp: algebra_simps)+
```
```    68
```
```    69 end
```
```    70
```
```    71 abbreviation int :: "nat \<Rightarrow> int"
```
```    72   where "int \<equiv> of_nat"
```
```    73
```
```    74 lemma int_def: "int n = Abs_Integ (n, 0)"
```
```    75   by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq)
```
```    76
```
```    77 lemma int_transfer [transfer_rule]: "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
```
```    78   by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)
```
```    79
```
```    80 lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
```
```    81   by transfer clarsimp
```
```    82
```
```    83
```
```    84 subsection \<open>Integers are totally ordered\<close>
```
```    85
```
```    86 instantiation int :: linorder
```
```    87 begin
```
```    88
```
```    89 lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    90   is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
```
```    91   by auto
```
```    92
```
```    93 lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    94   is "\<lambda>(x, y) (u, v). x + v < u + y"
```
```    95   by auto
```
```    96
```
```    97 instance
```
```    98   by standard (transfer, force)+
```
```    99
```
```   100 end
```
```   101
```
```   102 instantiation int :: distrib_lattice
```
```   103 begin
```
```   104
```
```   105 definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   106
```
```   107 definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   108
```
```   109 instance
```
```   110   by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)
```
```   111
```
```   112 end
```
```   113
```
```   114
```
```   115 subsection \<open>Ordering properties of arithmetic operations\<close>
```
```   116
```
```   117 instance int :: ordered_cancel_ab_semigroup_add
```
```   118 proof
```
```   119   fix i j k :: int
```
```   120   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   121     by transfer clarsimp
```
```   122 qed
```
```   123
```
```   124 text \<open>Strict Monotonicity of Multiplication.\<close>
```
```   125
```
```   126 text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close>
```
```   127 lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j"
```
```   128   for i j :: int
```
```   129 proof (induct k)
```
```   130   case 0
```
```   131   then show ?case by simp
```
```   132 next
```
```   133   case (Suc k)
```
```   134   then show ?case
```
```   135     by (cases "k = 0") (simp_all add: distrib_right add_strict_mono)
```
```   136 qed
```
```   137
```
```   138 lemma zero_le_imp_eq_int: "0 \<le> k \<Longrightarrow> \<exists>n. k = int n"
```
```   139   for k :: int
```
```   140   apply transfer
```
```   141   apply clarsimp
```
```   142   apply (rule_tac x="a - b" in exI)
```
```   143   apply simp
```
```   144   done
```
```   145
```
```   146 lemma zero_less_imp_eq_int: "0 < k \<Longrightarrow> \<exists>n>0. k = int n"
```
```   147   for k :: int
```
```   148   apply transfer
```
```   149   apply clarsimp
```
```   150   apply (rule_tac x="a - b" in exI)
```
```   151   apply simp
```
```   152   done
```
```   153
```
```   154 lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   155   for i j k :: int
```
```   156   by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)
```
```   157
```
```   158
```
```   159 text \<open>The integers form an ordered integral domain.\<close>
```
```   160
```
```   161 instantiation int :: linordered_idom
```
```   162 begin
```
```   163
```
```   164 definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
```
```   165
```
```   166 definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
```
```   167
```
```   168 instance
```
```   169 proof
```
```   170   fix i j k :: int
```
```   171   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   172     by (rule zmult_zless_mono2)
```
```   173   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
```
```   174     by (simp only: zabs_def)
```
```   175   show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   176     by (simp only: zsgn_def)
```
```   177 qed
```
```   178
```
```   179 end
```
```   180
```
```   181 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z"
```
```   182   for w z :: int
```
```   183   by transfer clarsimp
```
```   184
```
```   185 lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
```
```   186   for w z :: int
```
```   187   apply transfer
```
```   188   apply auto
```
```   189   apply (rename_tac a b c d)
```
```   190   apply (rule_tac x="c+b - Suc(a+d)" in exI)
```
```   191   apply arith
```
```   192   done
```
```   193
```
```   194 lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   195   for z :: int
```
```   196 proof
```
```   197   assume ?rhs
```
```   198   then show ?lhs by simp
```
```   199 next
```
```   200   assume ?lhs
```
```   201   with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp
```
```   202   then have "\<bar>z\<bar> \<le> 0" by simp
```
```   203   then show ?rhs by simp
```
```   204 qed
```
```   205
```
```   206 lemmas int_distrib =
```
```   207   distrib_right [of z1 z2 w]
```
```   208   distrib_left [of w z1 z2]
```
```   209   left_diff_distrib [of z1 z2 w]
```
```   210   right_diff_distrib [of w z1 z2]
```
```   211   for z1 z2 w :: int
```
```   212
```
```   213
```
```   214 subsection \<open>Embedding of the Integers into any \<open>ring_1\<close>: \<open>of_int\<close>\<close>
```
```   215
```
```   216 context ring_1
```
```   217 begin
```
```   218
```
```   219 lift_definition of_int :: "int \<Rightarrow> 'a"
```
```   220   is "\<lambda>(i, j). of_nat i - of_nat j"
```
```   221   by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
```
```   222       of_nat_add [symmetric] simp del: of_nat_add)
```
```   223
```
```   224 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   225   by transfer simp
```
```   226
```
```   227 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   228   by transfer simp
```
```   229
```
```   230 lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
```
```   231   by transfer (clarsimp simp add: algebra_simps)
```
```   232
```
```   233 lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
```
```   234   by (transfer fixing: uminus) clarsimp
```
```   235
```
```   236 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   237   using of_int_add [of w "- z"] by simp
```
```   238
```
```   239 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   240   by (transfer fixing: times) (clarsimp simp add: algebra_simps)
```
```   241
```
```   242 lemma mult_of_int_commute: "of_int x * y = y * of_int x"
```
```   243   by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)
```
```   244
```
```   245 text \<open>Collapse nested embeddings.\<close>
```
```   246 lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
```
```   247   by (induct n) auto
```
```   248
```
```   249 lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
```
```   250   by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
```
```   251
```
```   252 lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
```
```   253   by simp
```
```   254
```
```   255 lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
```
```   256   by (induct n) simp_all
```
```   257
```
```   258 lemma of_int_of_bool [simp]:
```
```   259   "of_int (of_bool P) = of_bool P"
```
```   260   by auto
```
```   261
```
```   262 end
```
```   263
```
```   264 context ring_char_0
```
```   265 begin
```
```   266
```
```   267 lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   268   by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
```
```   269
```
```   270 text \<open>Special cases where either operand is zero.\<close>
```
```   271 lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
```
```   272   using of_int_eq_iff [of z 0] by simp
```
```   273
```
```   274 lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
```
```   275   using of_int_eq_iff [of 0 z] by simp
```
```   276
```
```   277 lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
```
```   278   using of_int_eq_iff [of z 1] by simp
```
```   279
```
```   280 lemma numeral_power_eq_of_int_cancel_iff [simp]:
```
```   281   "numeral x ^ n = of_int y \<longleftrightarrow> numeral x ^ n = y"
```
```   282   using of_int_eq_iff[of "numeral x ^ n" y, unfolded of_int_numeral of_int_power] .
```
```   283
```
```   284 lemma of_int_eq_numeral_power_cancel_iff [simp]:
```
```   285   "of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
```
```   286   using numeral_power_eq_of_int_cancel_iff [of x n y] by (metis (mono_tags))
```
```   287
```
```   288 lemma neg_numeral_power_eq_of_int_cancel_iff [simp]:
```
```   289   "(- numeral x) ^ n = of_int y \<longleftrightarrow> (- numeral x) ^ n = y"
```
```   290   using of_int_eq_iff[of "(- numeral x) ^ n" y]
```
```   291   by simp
```
```   292
```
```   293 lemma of_int_eq_neg_numeral_power_cancel_iff [simp]:
```
```   294   "of_int y = (- numeral x) ^ n \<longleftrightarrow> y = (- numeral x) ^ n"
```
```   295   using neg_numeral_power_eq_of_int_cancel_iff[of x n y] by (metis (mono_tags))
```
```   296
```
```   297 lemma of_int_eq_of_int_power_cancel_iff[simp]: "(of_int b) ^ w = of_int x \<longleftrightarrow> b ^ w = x"
```
```   298   by (metis of_int_power of_int_eq_iff)
```
```   299
```
```   300 lemma of_int_power_eq_of_int_cancel_iff[simp]: "of_int x = (of_int b) ^ w \<longleftrightarrow> x = b ^ w"
```
```   301   by (metis of_int_eq_of_int_power_cancel_iff)
```
```   302
```
```   303 end
```
```   304
```
```   305 context linordered_idom
```
```   306 begin
```
```   307
```
```   308 text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
```
```   309 subclass ring_char_0 ..
```
```   310
```
```   311 lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   312   by (transfer fixing: less_eq)
```
```   313     (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
```
```   314
```
```   315 lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   316   by (simp add: less_le order_less_le)
```
```   317
```
```   318 lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
```
```   319   using of_int_le_iff [of 0 z] by simp
```
```   320
```
```   321 lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
```
```   322   using of_int_le_iff [of z 0] by simp
```
```   323
```
```   324 lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
```
```   325   using of_int_less_iff [of 0 z] by simp
```
```   326
```
```   327 lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
```
```   328   using of_int_less_iff [of z 0] by simp
```
```   329
```
```   330 lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
```
```   331   using of_int_le_iff [of 1 z] by simp
```
```   332
```
```   333 lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
```
```   334   using of_int_le_iff [of z 1] by simp
```
```   335
```
```   336 lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
```
```   337   using of_int_less_iff [of 1 z] by simp
```
```   338
```
```   339 lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
```
```   340   using of_int_less_iff [of z 1] by simp
```
```   341
```
```   342 lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
```
```   343   by simp
```
```   344
```
```   345 lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
```
```   346   by simp
```
```   347
```
```   348 lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
```
```   349   by (auto simp add: abs_if)
```
```   350
```
```   351 lemma of_int_lessD:
```
```   352   assumes "\<bar>of_int n\<bar> < x"
```
```   353   shows "n = 0 \<or> x > 1"
```
```   354 proof (cases "n = 0")
```
```   355   case True
```
```   356   then show ?thesis by simp
```
```   357 next
```
```   358   case False
```
```   359   then have "\<bar>n\<bar> \<noteq> 0" by simp
```
```   360   then have "\<bar>n\<bar> > 0" by simp
```
```   361   then have "\<bar>n\<bar> \<ge> 1"
```
```   362     using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
```
```   363   then have "\<bar>of_int n\<bar> \<ge> 1"
```
```   364     unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
```
```   365   then have "1 < x" using assms by (rule le_less_trans)
```
```   366   then show ?thesis ..
```
```   367 qed
```
```   368
```
```   369 lemma of_int_leD:
```
```   370   assumes "\<bar>of_int n\<bar> \<le> x"
```
```   371   shows "n = 0 \<or> 1 \<le> x"
```
```   372 proof (cases "n = 0")
```
```   373   case True
```
```   374   then show ?thesis by simp
```
```   375 next
```
```   376   case False
```
```   377   then have "\<bar>n\<bar> \<noteq> 0" by simp
```
```   378   then have "\<bar>n\<bar> > 0" by simp
```
```   379   then have "\<bar>n\<bar> \<ge> 1"
```
```   380     using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
```
```   381   then have "\<bar>of_int n\<bar> \<ge> 1"
```
```   382     unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
```
```   383   then have "1 \<le> x" using assms by (rule order_trans)
```
```   384   then show ?thesis ..
```
```   385 qed
```
```   386
```
```   387 lemma numeral_power_le_of_int_cancel_iff [simp]:
```
```   388   "numeral x ^ n \<le> of_int a \<longleftrightarrow> numeral x ^ n \<le> a"
```
```   389   by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_le_iff)
```
```   390
```
```   391 lemma of_int_le_numeral_power_cancel_iff [simp]:
```
```   392   "of_int a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
```
```   393   by (metis (mono_tags) local.numeral_power_eq_of_int_cancel_iff of_int_le_iff)
```
```   394
```
```   395 lemma numeral_power_less_of_int_cancel_iff [simp]:
```
```   396   "numeral x ^ n < of_int a \<longleftrightarrow> numeral x ^ n < a"
```
```   397   by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
```
```   398
```
```   399 lemma of_int_less_numeral_power_cancel_iff [simp]:
```
```   400   "of_int a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
```
```   401   by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
```
```   402
```
```   403 lemma neg_numeral_power_le_of_int_cancel_iff [simp]:
```
```   404   "(- numeral x) ^ n \<le> of_int a \<longleftrightarrow> (- numeral x) ^ n \<le> a"
```
```   405   by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
```
```   406
```
```   407 lemma of_int_le_neg_numeral_power_cancel_iff [simp]:
```
```   408   "of_int a \<le> (- numeral x) ^ n \<longleftrightarrow> a \<le> (- numeral x) ^ n"
```
```   409   by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
```
```   410
```
```   411 lemma neg_numeral_power_less_of_int_cancel_iff [simp]:
```
```   412   "(- numeral x) ^ n < of_int a \<longleftrightarrow> (- numeral x) ^ n < a"
```
```   413   using of_int_less_iff[of "(- numeral x) ^ n" a]
```
```   414   by simp
```
```   415
```
```   416 lemma of_int_less_neg_numeral_power_cancel_iff [simp]:
```
```   417   "of_int a < (- numeral x) ^ n \<longleftrightarrow> a < (- numeral x::int) ^ n"
```
```   418   using of_int_less_iff[of a "(- numeral x) ^ n"]
```
```   419   by simp
```
```   420
```
```   421 lemma of_int_le_of_int_power_cancel_iff[simp]: "(of_int b) ^ w \<le> of_int x \<longleftrightarrow> b ^ w \<le> x"
```
```   422   by (metis (mono_tags) of_int_le_iff of_int_power)
```
```   423
```
```   424 lemma of_int_power_le_of_int_cancel_iff[simp]: "of_int x \<le> (of_int b) ^ w\<longleftrightarrow> x \<le> b ^ w"
```
```   425   by (metis (mono_tags) of_int_le_iff of_int_power)
```
```   426
```
```   427 lemma of_int_less_of_int_power_cancel_iff[simp]: "(of_int b) ^ w < of_int x \<longleftrightarrow> b ^ w < x"
```
```   428   by (metis (mono_tags) of_int_less_iff of_int_power)
```
```   429
```
```   430 lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w\<longleftrightarrow> x < b ^ w"
```
```   431   by (metis (mono_tags) of_int_less_iff of_int_power)
```
```   432
```
```   433 end
```
```   434
```
```   435 text \<open>Comparisons involving @{term of_int}.\<close>
```
```   436
```
```   437 lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
```
```   438   using of_int_eq_iff by fastforce
```
```   439
```
```   440 lemma of_int_le_numeral_iff [simp]:
```
```   441   "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
```
```   442   using of_int_le_iff [of z "numeral n"] by simp
```
```   443
```
```   444 lemma of_int_numeral_le_iff [simp]:
```
```   445   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
```
```   446   using of_int_le_iff [of "numeral n"] by simp
```
```   447
```
```   448 lemma of_int_less_numeral_iff [simp]:
```
```   449   "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
```
```   450   using of_int_less_iff [of z "numeral n"] by simp
```
```   451
```
```   452 lemma of_int_numeral_less_iff [simp]:
```
```   453   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
```
```   454   using of_int_less_iff [of "numeral n" z] by simp
```
```   455
```
```   456 lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
```
```   457   by (metis of_int_of_nat_eq of_int_less_iff)
```
```   458
```
```   459 lemma of_int_eq_id [simp]: "of_int = id"
```
```   460 proof
```
```   461   show "of_int z = id z" for z
```
```   462     by (cases z rule: int_diff_cases) simp
```
```   463 qed
```
```   464
```
```   465 instance int :: no_top
```
```   466   apply standard
```
```   467   apply (rule_tac x="x + 1" in exI)
```
```   468   apply simp
```
```   469   done
```
```   470
```
```   471 instance int :: no_bot
```
```   472   apply standard
```
```   473   apply (rule_tac x="x - 1" in exI)
```
```   474   apply simp
```
```   475   done
```
```   476
```
```   477
```
```   478 subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
```
```   479
```
```   480 lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
```
```   481   by auto
```
```   482
```
```   483 lemma nat_int [simp]: "nat (int n) = n"
```
```   484   by transfer simp
```
```   485
```
```   486 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   487   by transfer clarsimp
```
```   488
```
```   489 lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
```
```   490   by simp
```
```   491
```
```   492 lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
```
```   493   by transfer clarsimp
```
```   494
```
```   495 lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
```
```   496   by transfer (clarsimp, arith)
```
```   497
```
```   498 text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
```
```   499 lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   500   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   501
```
```   502 lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   503   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   504
```
```   505 lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
```
```   506   by transfer (clarsimp, arith)
```
```   507
```
```   508 lemma nonneg_int_cases:
```
```   509   assumes "0 \<le> k"
```
```   510   obtains n where "k = int n"
```
```   511 proof -
```
```   512   from assms have "k = int (nat k)"
```
```   513     by simp
```
```   514   then show thesis
```
```   515     by (rule that)
```
```   516 qed
```
```   517
```
```   518 lemma pos_int_cases:
```
```   519   assumes "0 < k"
```
```   520   obtains n where "k = int n" and "n > 0"
```
```   521 proof -
```
```   522   from assms have "0 \<le> k"
```
```   523     by simp
```
```   524   then obtain n where "k = int n"
```
```   525     by (rule nonneg_int_cases)
```
```   526   moreover have "n > 0"
```
```   527     using \<open>k = int n\<close> assms by simp
```
```   528   ultimately show thesis
```
```   529     by (rule that)
```
```   530 qed
```
```   531
```
```   532 lemma nonpos_int_cases:
```
```   533   assumes "k \<le> 0"
```
```   534   obtains n where "k = - int n"
```
```   535 proof -
```
```   536   from assms have "- k \<ge> 0"
```
```   537     by simp
```
```   538   then obtain n where "- k = int n"
```
```   539     by (rule nonneg_int_cases)
```
```   540   then have "k = - int n"
```
```   541     by simp
```
```   542   then show thesis
```
```   543     by (rule that)
```
```   544 qed
```
```   545
```
```   546 lemma neg_int_cases:
```
```   547   assumes "k < 0"
```
```   548   obtains n where "k = - int n" and "n > 0"
```
```   549 proof -
```
```   550   from assms have "- k > 0"
```
```   551     by simp
```
```   552   then obtain n where "- k = int n" and "- k > 0"
```
```   553     by (blast elim: pos_int_cases)
```
```   554   then have "k = - int n" and "n > 0"
```
```   555     by simp_all
```
```   556   then show thesis
```
```   557     by (rule that)
```
```   558 qed
```
```   559
```
```   560 lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   561   by transfer (clarsimp simp add: le_imp_diff_is_add)
```
```   562
```
```   563 lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   564   using nat_eq_iff [of w m] by auto
```
```   565
```
```   566 lemma nat_0 [simp]: "nat 0 = 0"
```
```   567   by (simp add: nat_eq_iff)
```
```   568
```
```   569 lemma nat_1 [simp]: "nat 1 = Suc 0"
```
```   570   by (simp add: nat_eq_iff)
```
```   571
```
```   572 lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
```
```   573   by (simp add: nat_eq_iff)
```
```   574
```
```   575 lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
```
```   576   by simp
```
```   577
```
```   578 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   579   by simp
```
```   580
```
```   581 lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
```
```   582   by transfer (clarsimp, arith)
```
```   583
```
```   584 lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
```
```   585   by transfer (clarsimp simp add: le_diff_conv)
```
```   586
```
```   587 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
```
```   588   by transfer auto
```
```   589
```
```   590 lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
```
```   591   for i :: int
```
```   592   by transfer clarsimp
```
```   593
```
```   594 lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
```
```   595   by (auto simp add: nat_eq_iff2)
```
```   596
```
```   597 lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
```
```   598   using zless_nat_conj [of 0] by auto
```
```   599
```
```   600 lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
```
```   601   by transfer clarsimp
```
```   602
```
```   603 lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
```
```   604   by transfer clarsimp
```
```   605
```
```   606 lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
```
```   607   by (rule nat_diff_distrib') auto
```
```   608
```
```   609 lemma nat_zminus_int [simp]: "nat (- int n) = 0"
```
```   610   by transfer simp
```
```   611
```
```   612 lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
```
```   613   by transfer auto
```
```   614
```
```   615 lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
```
```   616   by transfer (clarsimp simp add: less_diff_conv)
```
```   617
```
```   618 lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   619   by transfer (clarsimp simp add: of_nat_diff)
```
```   620
```
```   621 lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
```
```   622   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
```
```   623
```
```   624 lemma nat_abs_triangle_ineq:
```
```   625   "nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>"
```
```   626   by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq)
```
```   627
```
```   628 lemma nat_of_bool [simp]:
```
```   629   "nat (of_bool P) = of_bool P"
```
```   630   by auto
```
```   631
```
```   632 lemma split_nat [arith_split]: "P (nat i) \<longleftrightarrow> ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
```
```   633   (is "?P = (?L \<and> ?R)")
```
```   634   for i :: int
```
```   635 proof (cases "i < 0")
```
```   636   case True
```
```   637   then show ?thesis
```
```   638     by auto
```
```   639 next
```
```   640   case False
```
```   641   have "?P = ?L"
```
```   642   proof
```
```   643     assume ?P
```
```   644     then show ?L using False by auto
```
```   645   next
```
```   646     assume ?L
```
```   647     moreover from False have "int (nat i) = i"
```
```   648       by (simp add: not_less)
```
```   649     ultimately show ?P
```
```   650       by simp
```
```   651   qed
```
```   652   with False show ?thesis by simp
```
```   653 qed
```
```   654
```
```   655 lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
```
```   656   by (auto split: split_nat)
```
```   657
```
```   658 lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
```
```   659 proof
```
```   660   assume "\<exists>x. P x"
```
```   661   then obtain x where "P x" ..
```
```   662   then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
```
```   663   then show "\<exists>x\<ge>0. P (nat x)" ..
```
```   664 next
```
```   665   assume "\<exists>x\<ge>0. P (nat x)"
```
```   666   then show "\<exists>x. P x" by auto
```
```   667 qed
```
```   668
```
```   669
```
```   670 text \<open>For termination proofs:\<close>
```
```   671 lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
```
```   672
```
```   673
```
```   674 subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
```
```   675
```
```   676 lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
```
```   677   by (simp add: order_less_le del: of_nat_Suc)
```
```   678
```
```   679 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   680   by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   681
```
```   682 lemma negative_zle_0: "- int n \<le> 0"
```
```   683   by (simp add: minus_le_iff)
```
```   684
```
```   685 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   686   by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   687
```
```   688 lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
```
```   689   by (subst le_minus_iff) (simp del: of_nat_Suc)
```
```   690
```
```   691 lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   692   by transfer simp
```
```   693
```
```   694 lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
```
```   695   by (simp add: linorder_not_less)
```
```   696
```
```   697 lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   698   by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   699
```
```   700 lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
```
```   701   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   702 proof
```
```   703   assume ?rhs
```
```   704   then show ?lhs by auto
```
```   705 next
```
```   706   assume ?lhs
```
```   707   then have "0 \<le> z - w" by simp
```
```   708   then obtain n where "z - w = int n"
```
```   709     using zero_le_imp_eq_int [of "z - w"] by blast
```
```   710   then have "z = w + int n" by simp
```
```   711   then show ?rhs ..
```
```   712 qed
```
```   713
```
```   714 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
```
```   715   by simp
```
```   716
```
```   717 text \<open>
```
```   718   This version is proved for all ordered rings, not just integers!
```
```   719   It is proved here because attribute \<open>arith_split\<close> is not available
```
```   720   in theory \<open>Rings\<close>.
```
```   721   But is it really better than just rewriting with \<open>abs_if\<close>?
```
```   722 \<close>
```
```   723 lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
```
```   724   for a :: "'a::linordered_idom"
```
```   725   by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   726
```
```   727 lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
```
```   728   apply transfer
```
```   729   apply clarsimp
```
```   730   apply (rule_tac x="b - Suc a" in exI)
```
```   731   apply arith
```
```   732   done
```
```   733
```
```   734
```
```   735 subsection \<open>Cases and induction\<close>
```
```   736
```
```   737 text \<open>
```
```   738   Now we replace the case analysis rule by a more conventional one:
```
```   739   whether an integer is negative or not.
```
```   740 \<close>
```
```   741
```
```   742 text \<open>This version is symmetric in the two subgoals.\<close>
```
```   743 lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
```
```   744   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   745   by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
```
```   746
```
```   747 text \<open>This is the default, with a negative case.\<close>
```
```   748 lemma int_cases [case_names nonneg neg, cases type: int]:
```
```   749   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   750   apply (cases "z < 0")
```
```   751    apply (blast dest!: negD)
```
```   752   apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   753   apply auto
```
```   754   apply (blast dest: nat_0_le [THEN sym])
```
```   755   done
```
```   756
```
```   757 lemma int_cases3 [case_names zero pos neg]:
```
```   758   fixes k :: int
```
```   759   assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   760     and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   761   shows "P"
```
```   762 proof (cases k "0::int" rule: linorder_cases)
```
```   763   case equal
```
```   764   with assms(1) show P by simp
```
```   765 next
```
```   766   case greater
```
```   767   then have *: "nat k > 0" by simp
```
```   768   moreover from * have "k = int (nat k)" by auto
```
```   769   ultimately show P using assms(2) by blast
```
```   770 next
```
```   771   case less
```
```   772   then have *: "nat (- k) > 0" by simp
```
```   773   moreover from * have "k = - int (nat (- k))" by auto
```
```   774   ultimately show P using assms(3) by blast
```
```   775 qed
```
```   776
```
```   777 lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
```
```   778   "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
```
```   779   by (cases z) auto
```
```   780
```
```   781 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```   782   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   783   by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
```
```   784
```
```   785 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
```
```   786   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   787   by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
```
```   788
```
```   789 lemma sgn_mult_dvd_iff [simp]:
```
```   790   "sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
```
```   791   by (cases r rule: int_cases3) auto
```
```   792
```
```   793 lemma mult_sgn_dvd_iff [simp]:
```
```   794   "l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
```
```   795   using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)
```
```   796
```
```   797 lemma dvd_sgn_mult_iff [simp]:
```
```   798   "l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
```
```   799   by (cases r rule: int_cases3) simp_all
```
```   800
```
```   801 lemma dvd_mult_sgn_iff [simp]:
```
```   802   "l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
```
```   803   using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)
```
```   804
```
```   805 lemma int_sgnE:
```
```   806   fixes k :: int
```
```   807   obtains n and l where "k = sgn l * int n"
```
```   808 proof -
```
```   809   have "k = sgn k * int (nat \<bar>k\<bar>)"
```
```   810     by (simp add: sgn_mult_abs)
```
```   811   then show ?thesis ..
```
```   812 qed
```
```   813
```
```   814 text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
```
```   815
```
```   816 lemmas max_number_of [simp] =
```
```   817   max_def [of "numeral u" "numeral v"]
```
```   818   max_def [of "numeral u" "- numeral v"]
```
```   819   max_def [of "- numeral u" "numeral v"]
```
```   820   max_def [of "- numeral u" "- numeral v"] for u v
```
```   821
```
```   822 lemmas min_number_of [simp] =
```
```   823   min_def [of "numeral u" "numeral v"]
```
```   824   min_def [of "numeral u" "- numeral v"]
```
```   825   min_def [of "- numeral u" "numeral v"]
```
```   826   min_def [of "- numeral u" "- numeral v"] for u v
```
```   827
```
```   828
```
```   829 subsubsection \<open>Binary comparisons\<close>
```
```   830
```
```   831 text \<open>Preliminaries\<close>
```
```   832
```
```   833 lemma le_imp_0_less:
```
```   834   fixes z :: int
```
```   835   assumes le: "0 \<le> z"
```
```   836   shows "0 < 1 + z"
```
```   837 proof -
```
```   838   have "0 \<le> z" by fact
```
```   839   also have "\<dots> < z + 1" by (rule less_add_one)
```
```   840   also have "\<dots> = 1 + z" by (simp add: ac_simps)
```
```   841   finally show "0 < 1 + z" .
```
```   842 qed
```
```   843
```
```   844 lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
```
```   845   for z :: int
```
```   846 proof (cases z)
```
```   847   case (nonneg n)
```
```   848   then show ?thesis
```
```   849     by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
```
```   850 next
```
```   851   case (neg n)
```
```   852   then show ?thesis
```
```   853     by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   854         add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   855 qed
```
```   856
```
```   857
```
```   858 subsubsection \<open>Comparisons, for Ordered Rings\<close>
```
```   859
```
```   860 lemmas double_eq_0_iff = double_zero
```
```   861
```
```   862 lemma odd_nonzero: "1 + z + z \<noteq> 0"
```
```   863   for z :: int
```
```   864 proof (cases z)
```
```   865   case (nonneg n)
```
```   866   have le: "0 \<le> z + z"
```
```   867     by (simp add: nonneg add_increasing)
```
```   868   then show ?thesis
```
```   869     using  le_imp_0_less [OF le] by (auto simp: add.assoc)
```
```   870 next
```
```   871   case (neg n)
```
```   872   show ?thesis
```
```   873   proof
```
```   874     assume eq: "1 + z + z = 0"
```
```   875     have "0 < 1 + (int n + int n)"
```
```   876       by (simp add: le_imp_0_less add_increasing)
```
```   877     also have "\<dots> = - (1 + z + z)"
```
```   878       by (simp add: neg add.assoc [symmetric])
```
```   879     also have "\<dots> = 0" by (simp add: eq)
```
```   880     finally have "0<0" ..
```
```   881     then show False by blast
```
```   882   qed
```
```   883 qed
```
```   884
```
```   885
```
```   886 subsection \<open>The Set of Integers\<close>
```
```   887
```
```   888 context ring_1
```
```   889 begin
```
```   890
```
```   891 definition Ints :: "'a set"  ("\<int>")
```
```   892   where "\<int> = range of_int"
```
```   893
```
```   894 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```   895   by (simp add: Ints_def)
```
```   896
```
```   897 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```   898   using Ints_of_int [of "of_nat n"] by simp
```
```   899
```
```   900 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```   901   using Ints_of_int [of "0"] by simp
```
```   902
```
```   903 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```   904   using Ints_of_int [of "1"] by simp
```
```   905
```
```   906 lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
```
```   907   by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
```
```   908
```
```   909 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```   910   apply (auto simp add: Ints_def)
```
```   911   apply (rule range_eqI)
```
```   912   apply (rule of_int_add [symmetric])
```
```   913   done
```
```   914
```
```   915 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```   916   apply (auto simp add: Ints_def)
```
```   917   apply (rule range_eqI)
```
```   918   apply (rule of_int_minus [symmetric])
```
```   919   done
```
```   920
```
```   921 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```   922   apply (auto simp add: Ints_def)
```
```   923   apply (rule range_eqI)
```
```   924   apply (rule of_int_diff [symmetric])
```
```   925   done
```
```   926
```
```   927 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```   928   apply (auto simp add: Ints_def)
```
```   929   apply (rule range_eqI)
```
```   930   apply (rule of_int_mult [symmetric])
```
```   931   done
```
```   932
```
```   933 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```   934   by (induct n) simp_all
```
```   935
```
```   936 lemma Ints_cases [cases set: Ints]:
```
```   937   assumes "q \<in> \<int>"
```
```   938   obtains (of_int) z where "q = of_int z"
```
```   939   unfolding Ints_def
```
```   940 proof -
```
```   941   from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
```
```   942   then obtain z where "q = of_int z" ..
```
```   943   then show thesis ..
```
```   944 qed
```
```   945
```
```   946 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   947   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```   948   by (rule Ints_cases) auto
```
```   949
```
```   950 lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
```
```   951   unfolding Nats_def Ints_def
```
```   952   by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
```
```   953
```
```   954 lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
```
```   955 proof (intro subsetI equalityI)
```
```   956   fix x :: 'a
```
```   957   assume "x \<in> {of_int n |n. n \<ge> 0}"
```
```   958   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   959     by (auto elim!: Ints_cases)
```
```   960   then have "x = of_nat (nat n)"
```
```   961     by (subst of_nat_nat) simp_all
```
```   962   then show "x \<in> \<nat>"
```
```   963     by simp
```
```   964 next
```
```   965   fix x :: 'a
```
```   966   assume "x \<in> \<nat>"
```
```   967   then obtain n where "x = of_nat n"
```
```   968     by (auto elim!: Nats_cases)
```
```   969   then have "x = of_int (int n)" by simp
```
```   970   also have "int n \<ge> 0" by simp
```
```   971   then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
```
```   972   finally show "x \<in> {of_int n |n. n \<ge> 0}" .
```
```   973 qed
```
```   974
```
```   975 end
```
```   976
```
```   977 lemma (in linordered_idom) Ints_abs [simp]:
```
```   978   shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
```
```   979   by (auto simp: abs_if)
```
```   980
```
```   981 lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
```
```   982 proof (intro subsetI equalityI)
```
```   983   fix x :: 'a
```
```   984   assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
```
```   985   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   986     by (auto elim!: Ints_cases)
```
```   987   then have "x = of_nat (nat n)"
```
```   988     by (subst of_nat_nat) simp_all
```
```   989   then show "x \<in> \<nat>"
```
```   990     by simp
```
```   991 qed (auto elim!: Nats_cases)
```
```   992
```
```   993 lemma (in idom_divide) of_int_divide_in_Ints:
```
```   994   "of_int a div of_int b \<in> \<int>" if "b dvd a"
```
```   995 proof -
```
```   996   from that obtain c where "a = b * c" ..
```
```   997   then show ?thesis
```
```   998     by (cases "of_int b = 0") simp_all
```
```   999 qed
```
```  1000
```
```  1001 text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
```
```  1002
```
```  1003 lemma Ints_double_eq_0_iff:
```
```  1004   fixes a :: "'a::ring_char_0"
```
```  1005   assumes in_Ints: "a \<in> \<int>"
```
```  1006   shows "a + a = 0 \<longleftrightarrow> a = 0"
```
```  1007     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1008 proof -
```
```  1009   from in_Ints have "a \<in> range of_int"
```
```  1010     unfolding Ints_def [symmetric] .
```
```  1011   then obtain z where a: "a = of_int z" ..
```
```  1012   show ?thesis
```
```  1013   proof
```
```  1014     assume ?rhs
```
```  1015     then show ?lhs by simp
```
```  1016   next
```
```  1017     assume ?lhs
```
```  1018     with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
```
```  1019     then have "z + z = 0" by (simp only: of_int_eq_iff)
```
```  1020     then have "z = 0" by (simp only: double_eq_0_iff)
```
```  1021     with a show ?rhs by simp
```
```  1022   qed
```
```  1023 qed
```
```  1024
```
```  1025 lemma Ints_odd_nonzero:
```
```  1026   fixes a :: "'a::ring_char_0"
```
```  1027   assumes in_Ints: "a \<in> \<int>"
```
```  1028   shows "1 + a + a \<noteq> 0"
```
```  1029 proof -
```
```  1030   from in_Ints have "a \<in> range of_int"
```
```  1031     unfolding Ints_def [symmetric] .
```
```  1032   then obtain z where a: "a = of_int z" ..
```
```  1033   show ?thesis
```
```  1034   proof
```
```  1035     assume "1 + a + a = 0"
```
```  1036     with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
```
```  1037     then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```  1038     with odd_nonzero show False by blast
```
```  1039   qed
```
```  1040 qed
```
```  1041
```
```  1042 lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
```
```  1043   using of_nat_in_Nats [of "numeral w"] by simp
```
```  1044
```
```  1045 lemma Ints_odd_less_0:
```
```  1046   fixes a :: "'a::linordered_idom"
```
```  1047   assumes in_Ints: "a \<in> \<int>"
```
```  1048   shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
```
```  1049 proof -
```
```  1050   from in_Ints have "a \<in> range of_int"
```
```  1051     unfolding Ints_def [symmetric] .
```
```  1052   then obtain z where a: "a = of_int z" ..
```
```  1053   with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
```
```  1054     by simp
```
```  1055   also have "\<dots> \<longleftrightarrow> z < 0"
```
```  1056     by (simp only: of_int_less_iff odd_less_0_iff)
```
```  1057   also have "\<dots> \<longleftrightarrow> a < 0"
```
```  1058     by (simp add: a)
```
```  1059   finally show ?thesis .
```
```  1060 qed
```
```  1061
```
```  1062
```
```  1063 subsection \<open>@{term sum} and @{term prod}\<close>
```
```  1064
```
```  1065 lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```  1066   by (induct A rule: infinite_finite_induct) auto
```
```  1067
```
```  1068 lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```  1069   by (induct A rule: infinite_finite_induct) auto
```
```  1070
```
```  1071 lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```  1072   by (induct A rule: infinite_finite_induct) auto
```
```  1073
```
```  1074 lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```  1075   by (induct A rule: infinite_finite_induct) auto
```
```  1076
```
```  1077
```
```  1078 text \<open>Legacy theorems\<close>
```
```  1079
```
```  1080 lemmas int_sum = of_nat_sum [where 'a=int]
```
```  1081 lemmas int_prod = of_nat_prod [where 'a=int]
```
```  1082 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```  1083 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```  1084 lemmas nonneg_eq_int = nonneg_int_cases
```
```  1085
```
```  1086
```
```  1087 subsection \<open>Setting up simplification procedures\<close>
```
```  1088
```
```  1089 lemmas of_int_simps =
```
```  1090   of_int_0 of_int_1 of_int_add of_int_mult
```
```  1091
```
```  1092 ML_file "Tools/int_arith.ML"
```
```  1093 declaration \<open>K Int_Arith.setup\<close>
```
```  1094
```
```  1095 simproc_setup fast_arith
```
```  1096   ("(m::'a::linordered_idom) < n" |
```
```  1097     "(m::'a::linordered_idom) \<le> n" |
```
```  1098     "(m::'a::linordered_idom) = n") =
```
```  1099   \<open>K Lin_Arith.simproc\<close>
```
```  1100
```
```  1101
```
```  1102 subsection\<open>More Inequality Reasoning\<close>
```
```  1103
```
```  1104 lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
```
```  1105   for w z :: int
```
```  1106   by arith
```
```  1107
```
```  1108 lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
```
```  1109   for w z :: int
```
```  1110   by arith
```
```  1111
```
```  1112 lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
```
```  1113   for w z :: int
```
```  1114   by arith
```
```  1115
```
```  1116 lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
```
```  1117   for w z :: int
```
```  1118   by arith
```
```  1119
```
```  1120 lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
```
```  1121   for z :: int
```
```  1122   by arith
```
```  1123
```
```  1124 lemma Ints_nonzero_abs_ge1:
```
```  1125   fixes x:: "'a :: linordered_idom"
```
```  1126     assumes "x \<in> Ints" "x \<noteq> 0"
```
```  1127     shows "1 \<le> abs x"
```
```  1128 proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
```
```  1129   fix z::int
```
```  1130   assume "x = of_int z"
```
```  1131     with \<open>x \<noteq> 0\<close>
```
```  1132   show "1 \<le> \<bar>x\<bar>"
```
```  1133     apply (auto simp add: abs_if)
```
```  1134     by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
```
```  1135 qed
```
```  1136
```
```  1137 lemma Ints_nonzero_abs_less1:
```
```  1138   fixes x:: "'a :: linordered_idom"
```
```  1139   shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
```
```  1140     using Ints_nonzero_abs_ge1 [of x] by auto
```
```  1141
```
```  1142
```
```  1143 subsection \<open>The functions @{term nat} and @{term int}\<close>
```
```  1144
```
```  1145 text \<open>Simplify the term @{term "w + - z"}.\<close>
```
```  1146
```
```  1147 lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
```
```  1148   using zless_nat_conj [of 1 z] by auto
```
```  1149
```
```  1150 text \<open>
```
```  1151   This simplifies expressions of the form @{term "int n = z"} where
```
```  1152   \<open>z\<close> is an integer literal.
```
```  1153 \<close>
```
```  1154 lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
```
```  1155
```
```  1156 lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
```
```  1157   by auto
```
```  1158
```
```  1159 lemma nat_int_add: "nat (int a + int b) = a + b"
```
```  1160   by auto
```
```  1161
```
```  1162 context ring_1
```
```  1163 begin
```
```  1164
```
```  1165 lemma of_int_of_nat [nitpick_simp]:
```
```  1166   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```  1167 proof (cases "k < 0")
```
```  1168   case True
```
```  1169   then have "0 \<le> - k" by simp
```
```  1170   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```  1171   with True show ?thesis by simp
```
```  1172 next
```
```  1173   case False
```
```  1174   then show ?thesis by (simp add: not_less)
```
```  1175 qed
```
```  1176
```
```  1177 end
```
```  1178
```
```  1179 lemma transfer_rule_of_int:
```
```  1180   fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
```
```  1181   assumes [transfer_rule]: "R 0 0" "R 1 1"
```
```  1182     "rel_fun R (rel_fun R R) plus plus"
```
```  1183     "rel_fun R R uminus uminus"
```
```  1184   shows "rel_fun HOL.eq R of_int of_int"
```
```  1185 proof -
```
```  1186   note transfer_rule_of_nat [transfer_rule]
```
```  1187   have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
```
```  1188     by transfer_prover
```
```  1189   show ?thesis
```
```  1190     by (unfold of_int_of_nat [abs_def]) transfer_prover
```
```  1191 qed
```
```  1192
```
```  1193 lemma nat_mult_distrib:
```
```  1194   fixes z z' :: int
```
```  1195   assumes "0 \<le> z"
```
```  1196   shows "nat (z * z') = nat z * nat z'"
```
```  1197 proof (cases "0 \<le> z'")
```
```  1198   case False
```
```  1199   with assms have "z * z' \<le> 0"
```
```  1200     by (simp add: not_le mult_le_0_iff)
```
```  1201   then have "nat (z * z') = 0" by simp
```
```  1202   moreover from False have "nat z' = 0" by simp
```
```  1203   ultimately show ?thesis by simp
```
```  1204 next
```
```  1205   case True
```
```  1206   with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1207   show ?thesis
```
```  1208     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1209       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1210          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1211 qed
```
```  1212
```
```  1213 lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
```
```  1214   for z z' :: int
```
```  1215   apply (rule trans)
```
```  1216    apply (rule_tac  nat_mult_distrib)
```
```  1217    apply auto
```
```  1218   done
```
```  1219
```
```  1220 lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
```
```  1221   by (cases "z = 0 \<or> w = 0")
```
```  1222     (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1223       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1224
```
```  1225 lemma int_in_range_abs [simp]: "int n \<in> range abs"
```
```  1226 proof (rule range_eqI)
```
```  1227   show "int n = \<bar>int n\<bar>" by simp
```
```  1228 qed
```
```  1229
```
```  1230 lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
```
```  1231 proof -
```
```  1232   have "\<bar>k\<bar> \<in> \<nat>" for k :: int
```
```  1233     by (cases k) simp_all
```
```  1234   moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
```
```  1235     using that by induct simp
```
```  1236   ultimately show ?thesis by blast
```
```  1237 qed
```
```  1238
```
```  1239 lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
```
```  1240   for z :: int
```
```  1241   by (rule sym) (simp add: nat_eq_iff)
```
```  1242
```
```  1243 lemma diff_nat_eq_if:
```
```  1244   "nat z - nat z' =
```
```  1245     (if z' < 0 then nat z
```
```  1246      else
```
```  1247       let d = z - z'
```
```  1248       in if d < 0 then 0 else nat d)"
```
```  1249   by (simp add: Let_def nat_diff_distrib [symmetric])
```
```  1250
```
```  1251 lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
```
```  1252   using diff_nat_numeral [of v Num.One] by simp
```
```  1253
```
```  1254
```
```  1255 subsection \<open>Induction principles for int\<close>
```
```  1256
```
```  1257 text \<open>Well-founded segments of the integers.\<close>
```
```  1258
```
```  1259 definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
```
```  1260   where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
```
```  1261
```
```  1262 lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1263 proof -
```
```  1264   have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
```
```  1265     by (auto simp add: int_ge_less_than_def)
```
```  1266   then show ?thesis
```
```  1267     by (rule wf_subset [OF wf_measure])
```
```  1268 qed
```
```  1269
```
```  1270 text \<open>
```
```  1271   This variant looks odd, but is typical of the relations suggested
```
```  1272   by RankFinder.\<close>
```
```  1273
```
```  1274 definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
```
```  1275   where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
```
```  1276
```
```  1277 lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1278 proof -
```
```  1279   have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
```
```  1280     by (auto simp add: int_ge_less_than2_def)
```
```  1281   then show ?thesis
```
```  1282     by (rule wf_subset [OF wf_measure])
```
```  1283 qed
```
```  1284
```
```  1285 (* `set:int': dummy construction *)
```
```  1286 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1287   fixes i :: int
```
```  1288   assumes ge: "k \<le> i"
```
```  1289     and base: "P k"
```
```  1290     and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1291   shows "P i"
```
```  1292 proof -
```
```  1293   have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
```
```  1294   proof (induct n)
```
```  1295     case 0
```
```  1296     then have "i = k" by arith
```
```  1297     with base show "P i" by simp
```
```  1298   next
```
```  1299     case (Suc n)
```
```  1300     then have "n = nat ((i - 1) - k)" by arith
```
```  1301     moreover have k: "k \<le> i - 1" using Suc.prems by arith
```
```  1302     ultimately have "P (i - 1)" by (rule Suc.hyps)
```
```  1303     from step [OF k this] show ?case by simp
```
```  1304   qed
```
```  1305   with ge show ?thesis by fast
```
```  1306 qed
```
```  1307
```
```  1308 (* `set:int': dummy construction *)
```
```  1309 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1310   fixes i k :: int
```
```  1311   assumes gr: "k < i"
```
```  1312     and base: "P (k + 1)"
```
```  1313     and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1314   shows "P i"
```
```  1315   apply (rule int_ge_induct[of "k + 1"])
```
```  1316   using gr apply arith
```
```  1317    apply (rule base)
```
```  1318   apply (rule step)
```
```  1319    apply simp_all
```
```  1320   done
```
```  1321
```
```  1322 theorem int_le_induct [consumes 1, case_names base step]:
```
```  1323   fixes i k :: int
```
```  1324   assumes le: "i \<le> k"
```
```  1325     and base: "P k"
```
```  1326     and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1327   shows "P i"
```
```  1328 proof -
```
```  1329   have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
```
```  1330   proof (induct n)
```
```  1331     case 0
```
```  1332     then have "i = k" by arith
```
```  1333     with base show "P i" by simp
```
```  1334   next
```
```  1335     case (Suc n)
```
```  1336     then have "n = nat (k - (i + 1))" by arith
```
```  1337     moreover have k: "i + 1 \<le> k" using Suc.prems by arith
```
```  1338     ultimately have "P (i + 1)" by (rule Suc.hyps)
```
```  1339     from step[OF k this] show ?case by simp
```
```  1340   qed
```
```  1341   with le show ?thesis by fast
```
```  1342 qed
```
```  1343
```
```  1344 theorem int_less_induct [consumes 1, case_names base step]:
```
```  1345   fixes i k :: int
```
```  1346   assumes less: "i < k"
```
```  1347     and base: "P (k - 1)"
```
```  1348     and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1349   shows "P i"
```
```  1350   apply (rule int_le_induct[of _ "k - 1"])
```
```  1351   using less apply arith
```
```  1352    apply (rule base)
```
```  1353   apply (rule step)
```
```  1354    apply simp_all
```
```  1355   done
```
```  1356
```
```  1357 theorem int_induct [case_names base step1 step2]:
```
```  1358   fixes k :: int
```
```  1359   assumes base: "P k"
```
```  1360     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1361     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1362   shows "P i"
```
```  1363 proof -
```
```  1364   have "i \<le> k \<or> i \<ge> k" by arith
```
```  1365   then show ?thesis
```
```  1366   proof
```
```  1367     assume "i \<ge> k"
```
```  1368     then show ?thesis
```
```  1369       using base by (rule int_ge_induct) (fact step1)
```
```  1370   next
```
```  1371     assume "i \<le> k"
```
```  1372     then show ?thesis
```
```  1373       using base by (rule int_le_induct) (fact step2)
```
```  1374   qed
```
```  1375 qed
```
```  1376
```
```  1377
```
```  1378 subsection \<open>Intermediate value theorems\<close>
```
```  1379
```
```  1380 lemma int_val_lemma: "(\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1) \<longrightarrow> f 0 \<le> k \<longrightarrow> k \<le> f n \<longrightarrow> (\<exists>i \<le> n. f i = k)"
```
```  1381   for n :: nat and k :: int
```
```  1382   unfolding One_nat_def
```
```  1383   apply (induct n)
```
```  1384    apply simp
```
```  1385   apply (intro strip)
```
```  1386   apply (erule impE)
```
```  1387    apply simp
```
```  1388   apply (erule_tac x = n in allE)
```
```  1389   apply simp
```
```  1390   apply (case_tac "k = f (Suc n)")
```
```  1391    apply force
```
```  1392   apply (erule impE)
```
```  1393    apply (simp add: abs_if split: if_split_asm)
```
```  1394   apply (blast intro: le_SucI)
```
```  1395   done
```
```  1396
```
```  1397 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1398
```
```  1399 lemma nat_intermed_int_val:
```
```  1400   "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
```
```  1401     f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
```
```  1402     for f :: "nat \<Rightarrow> int" and k :: int
```
```  1403   apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
```
```  1404   unfolding One_nat_def
```
```  1405   apply simp
```
```  1406   apply (erule exE)
```
```  1407   apply (rule_tac x = "i+m" in exI)
```
```  1408   apply arith
```
```  1409   done
```
```  1410
```
```  1411
```
```  1412 subsection \<open>Products and 1, by T. M. Rasmussen\<close>
```
```  1413
```
```  1414 lemma abs_zmult_eq_1:
```
```  1415   fixes m n :: int
```
```  1416   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1417   shows "\<bar>m\<bar> = 1"
```
```  1418 proof -
```
```  1419   from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
```
```  1420   have "\<not> 2 \<le> \<bar>m\<bar>"
```
```  1421   proof
```
```  1422     assume "2 \<le> \<bar>m\<bar>"
```
```  1423     then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
```
```  1424     also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
```
```  1425     also from mn have "\<dots> = 1" by simp
```
```  1426     finally have "2 * \<bar>n\<bar> \<le> 1" .
```
```  1427     with 0 show "False" by arith
```
```  1428   qed
```
```  1429   with 0 show ?thesis by auto
```
```  1430 qed
```
```  1431
```
```  1432 lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
```
```  1433   for m n :: int
```
```  1434   using abs_zmult_eq_1 [of m n] by arith
```
```  1435
```
```  1436 lemma pos_zmult_eq_1_iff:
```
```  1437   fixes m n :: int
```
```  1438   assumes "0 < m"
```
```  1439   shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
```
```  1440 proof -
```
```  1441   from assms have "m * n = 1 \<Longrightarrow> m = 1"
```
```  1442     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1443   then show ?thesis
```
```  1444     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1445 qed
```
```  1446
```
```  1447 lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
```
```  1448   for m n :: int
```
```  1449   apply (rule iffI)
```
```  1450    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1451    apply (simp add: mult.commute [of m])
```
```  1452    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1453    apply auto
```
```  1454   done
```
```  1455
```
```  1456 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1457 proof
```
```  1458   assume "finite (UNIV::int set)"
```
```  1459   moreover have "inj (\<lambda>i::int. 2 * i)"
```
```  1460     by (rule injI) simp
```
```  1461   ultimately have "surj (\<lambda>i::int. 2 * i)"
```
```  1462     by (rule finite_UNIV_inj_surj)
```
```  1463   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1464   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1465 qed
```
```  1466
```
```  1467
```
```  1468 subsection \<open>Further theorems on numerals\<close>
```
```  1469
```
```  1470 subsubsection \<open>Special Simplification for Constants\<close>
```
```  1471
```
```  1472 text \<open>These distributive laws move literals inside sums and differences.\<close>
```
```  1473
```
```  1474 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
```
```  1475 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
```
```  1476 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
```
```  1477 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
```
```  1478
```
```  1479 text \<open>These are actually for fields, like real: but where else to put them?\<close>
```
```  1480
```
```  1481 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
```
```  1482 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
```
```  1483 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
```
```  1484 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
```
```  1485
```
```  1486
```
```  1487 text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
```
```  1488   strange, but then other simprocs simplify the quotient.\<close>
```
```  1489
```
```  1490 lemmas inverse_eq_divide_numeral [simp] =
```
```  1491   inverse_eq_divide [of "numeral w"] for w
```
```  1492
```
```  1493 lemmas inverse_eq_divide_neg_numeral [simp] =
```
```  1494   inverse_eq_divide [of "- numeral w"] for w
```
```  1495
```
```  1496 text \<open>These laws simplify inequalities, moving unary minus from a term
```
```  1497   into the literal.\<close>
```
```  1498
```
```  1499 lemmas equation_minus_iff_numeral [no_atp] =
```
```  1500   equation_minus_iff [of "numeral v"] for v
```
```  1501
```
```  1502 lemmas minus_equation_iff_numeral [no_atp] =
```
```  1503   minus_equation_iff [of _ "numeral v"] for v
```
```  1504
```
```  1505 lemmas le_minus_iff_numeral [no_atp] =
```
```  1506   le_minus_iff [of "numeral v"] for v
```
```  1507
```
```  1508 lemmas minus_le_iff_numeral [no_atp] =
```
```  1509   minus_le_iff [of _ "numeral v"] for v
```
```  1510
```
```  1511 lemmas less_minus_iff_numeral [no_atp] =
```
```  1512   less_minus_iff [of "numeral v"] for v
```
```  1513
```
```  1514 lemmas minus_less_iff_numeral [no_atp] =
```
```  1515   minus_less_iff [of _ "numeral v"] for v
```
```  1516
```
```  1517 (* FIXME maybe simproc *)
```
```  1518
```
```  1519
```
```  1520 text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
```
```  1521
```
```  1522 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
```
```  1523 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
```
```  1524 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
```
```  1525 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
```
```  1526
```
```  1527
```
```  1528 text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
```
```  1529
```
```  1530 named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
```
```  1531
```
```  1532 lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1533   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1534   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1535
```
```  1536 lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
```
```  1537   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
```
```  1538   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1539
```
```  1540 lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1541   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1542   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1543
```
```  1544 lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
```
```  1545   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
```
```  1546   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1547
```
```  1548 lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1549   eq_divide_eq [of _ _ "numeral w"]
```
```  1550   eq_divide_eq [of _ _ "- numeral w"] for w
```
```  1551
```
```  1552 lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
```
```  1553   divide_eq_eq [of _ "numeral w"]
```
```  1554   divide_eq_eq [of _ "- numeral w"] for w
```
```  1555
```
```  1556
```
```  1557 subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
```
```  1558
```
```  1559 text \<open>Simplify quotients that are compared with a literal constant.\<close>
```
```  1560
```
```  1561 lemmas le_divide_eq_numeral [divide_const_simps] =
```
```  1562   le_divide_eq [of "numeral w"]
```
```  1563   le_divide_eq [of "- numeral w"] for w
```
```  1564
```
```  1565 lemmas divide_le_eq_numeral [divide_const_simps] =
```
```  1566   divide_le_eq [of _ _ "numeral w"]
```
```  1567   divide_le_eq [of _ _ "- numeral w"] for w
```
```  1568
```
```  1569 lemmas less_divide_eq_numeral [divide_const_simps] =
```
```  1570   less_divide_eq [of "numeral w"]
```
```  1571   less_divide_eq [of "- numeral w"] for w
```
```  1572
```
```  1573 lemmas divide_less_eq_numeral [divide_const_simps] =
```
```  1574   divide_less_eq [of _ _ "numeral w"]
```
```  1575   divide_less_eq [of _ _ "- numeral w"] for w
```
```  1576
```
```  1577 lemmas eq_divide_eq_numeral [divide_const_simps] =
```
```  1578   eq_divide_eq [of "numeral w"]
```
```  1579   eq_divide_eq [of "- numeral w"] for w
```
```  1580
```
```  1581 lemmas divide_eq_eq_numeral [divide_const_simps] =
```
```  1582   divide_eq_eq [of _ _ "numeral w"]
```
```  1583   divide_eq_eq [of _ _ "- numeral w"] for w
```
```  1584
```
```  1585
```
```  1586 text \<open>Not good as automatic simprules because they cause case splits.\<close>
```
```  1587 lemmas [divide_const_simps] =
```
```  1588   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  1589
```
```  1590
```
```  1591 subsection \<open>The divides relation\<close>
```
```  1592
```
```  1593 lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
```
```  1594   for m n :: int
```
```  1595   by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  1596
```
```  1597 lemma zdvd_antisym_abs:
```
```  1598   fixes a b :: int
```
```  1599   assumes "a dvd b" and "b dvd a"
```
```  1600   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  1601 proof (cases "a = 0")
```
```  1602   case True
```
```  1603   with assms show ?thesis by simp
```
```  1604 next
```
```  1605   case False
```
```  1606   from \<open>a dvd b\<close> obtain k where k: "b = a * k"
```
```  1607     unfolding dvd_def by blast
```
```  1608   from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
```
```  1609     unfolding dvd_def by blast
```
```  1610   from k k' have "a = a * k * k'" by simp
```
```  1611   with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
```
```  1612     using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
```
```  1613   then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
```
```  1614     by (simp add: zmult_eq_1_iff)
```
```  1615   with k k' show ?thesis by auto
```
```  1616 qed
```
```  1617
```
```  1618 lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
```
```  1619   for k m n :: int
```
```  1620   using dvd_add_right_iff [of k "- n" m] by simp
```
```  1621
```
```  1622 lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
```
```  1623   for k m n :: int
```
```  1624   using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
```
```  1625
```
```  1626 lemma dvd_imp_le_int:
```
```  1627   fixes d i :: int
```
```  1628   assumes "i \<noteq> 0" and "d dvd i"
```
```  1629   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  1630 proof -
```
```  1631   from \<open>d dvd i\<close> obtain k where "i = d * k" ..
```
```  1632   with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
```
```  1633   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  1634   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  1635   with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
```
```  1636 qed
```
```  1637
```
```  1638 lemma zdvd_not_zless:
```
```  1639   fixes m n :: int
```
```  1640   assumes "0 < m" and "m < n"
```
```  1641   shows "\<not> n dvd m"
```
```  1642 proof
```
```  1643   from assms have "0 < n" by auto
```
```  1644   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  1645   with \<open>0 < m\<close> have "0 < n * k" by auto
```
```  1646   with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
```
```  1647   with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
```
```  1648   with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
```
```  1649 qed
```
```  1650
```
```  1651 lemma zdvd_mult_cancel:
```
```  1652   fixes k m n :: int
```
```  1653   assumes d: "k * m dvd k * n"
```
```  1654     and "k \<noteq> 0"
```
```  1655   shows "m dvd n"
```
```  1656 proof -
```
```  1657   from d obtain h where h: "k * n = k * m * h"
```
```  1658     unfolding dvd_def by blast
```
```  1659   have "n = m * h"
```
```  1660   proof (rule ccontr)
```
```  1661     assume "\<not> ?thesis"
```
```  1662     with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
```
```  1663     with h show False
```
```  1664       by (simp add: mult.assoc)
```
```  1665   qed
```
```  1666   then show ?thesis by simp
```
```  1667 qed
```
```  1668
```
```  1669 theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
```
```  1670 proof -
```
```  1671   have "x dvd y" if "int y = int x * k" for k
```
```  1672   proof (cases k)
```
```  1673     case (nonneg n)
```
```  1674     with that have "y = x * n"
```
```  1675       by (simp del: of_nat_mult add: of_nat_mult [symmetric])
```
```  1676     then show ?thesis ..
```
```  1677   next
```
```  1678     case (neg n)
```
```  1679     with that have "int y = int x * (- int (Suc n))"
```
```  1680       by simp
```
```  1681     also have "\<dots> = - (int x * int (Suc n))"
```
```  1682       by (simp only: mult_minus_right)
```
```  1683     also have "\<dots> = - int (x * Suc n)"
```
```  1684       by (simp only: of_nat_mult [symmetric])
```
```  1685     finally have "- int (x * Suc n) = int y" ..
```
```  1686     then show ?thesis
```
```  1687       by (simp only: negative_eq_positive) auto
```
```  1688   qed
```
```  1689   then show ?thesis
```
```  1690     by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  1691 qed
```
```  1692
```
```  1693 lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
```
```  1694   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1695   for x :: int
```
```  1696 proof
```
```  1697   assume ?lhs
```
```  1698   then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  1699   then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  1700   then have "nat \<bar>x\<bar> = 1" by simp
```
```  1701   then show ?rhs by (cases "x < 0") auto
```
```  1702 next
```
```  1703   assume ?rhs
```
```  1704   then have "x = 1 \<or> x = - 1" by auto
```
```  1705   then show ?lhs by (auto intro: dvdI)
```
```  1706 qed
```
```  1707
```
```  1708 lemma zdvd_mult_cancel1:
```
```  1709   fixes m :: int
```
```  1710   assumes mp: "m \<noteq> 0"
```
```  1711   shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
```
```  1712     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1713 proof
```
```  1714   assume ?rhs
```
```  1715   then show ?lhs
```
```  1716     by (cases "n > 0") (auto simp add: minus_equation_iff)
```
```  1717 next
```
```  1718   assume ?lhs
```
```  1719   then have "m * n dvd m * 1" by simp
```
```  1720   from zdvd_mult_cancel[OF this mp] show ?rhs
```
```  1721     by (simp only: zdvd1_eq)
```
```  1722 qed
```
```  1723
```
```  1724 lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
```
```  1725   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1726
```
```  1727 lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
```
```  1728   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1729
```
```  1730 lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
```
```  1731   by (simp add: dvd_int_iff [symmetric])
```
```  1732
```
```  1733 lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
```
```  1734   by (auto simp add: dvd_int_iff)
```
```  1735
```
```  1736 lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  1737   by (auto elim!: nonneg_eq_int)
```
```  1738
```
```  1739 lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  1740   by (induct n) (simp_all add: nat_mult_distrib)
```
```  1741
```
```  1742 lemma numeral_power_eq_nat_cancel_iff [simp]:
```
```  1743   "numeral x ^ n = nat y \<longleftrightarrow> numeral x ^ n = y"
```
```  1744   using nat_eq_iff2 by auto
```
```  1745
```
```  1746 lemma nat_eq_numeral_power_cancel_iff [simp]:
```
```  1747   "nat y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
```
```  1748   using numeral_power_eq_nat_cancel_iff[of x n y]
```
```  1749   by (metis (mono_tags))
```
```  1750
```
```  1751 lemma numeral_power_le_nat_cancel_iff [simp]:
```
```  1752   "numeral x ^ n \<le> nat a \<longleftrightarrow> numeral x ^ n \<le> a"
```
```  1753   using nat_le_eq_zle[of "numeral x ^ n" a]
```
```  1754   by (auto simp: nat_power_eq)
```
```  1755
```
```  1756 lemma nat_le_numeral_power_cancel_iff [simp]:
```
```  1757   "nat a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
```
```  1758   by (simp add: nat_le_iff)
```
```  1759
```
```  1760 lemma numeral_power_less_nat_cancel_iff [simp]:
```
```  1761   "numeral x ^ n < nat a \<longleftrightarrow> numeral x ^ n < a"
```
```  1762   using nat_less_eq_zless[of "numeral x ^ n" a]
```
```  1763   by (auto simp: nat_power_eq)
```
```  1764
```
```  1765 lemma nat_less_numeral_power_cancel_iff [simp]:
```
```  1766   "nat a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
```
```  1767   using nat_less_eq_zless[of a "numeral x ^ n"]
```
```  1768   by (cases "a < 0") (auto simp: nat_power_eq less_le_trans[where y=0])
```
```  1769
```
```  1770 lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
```
```  1771   for n z :: int
```
```  1772   apply (cases n)
```
```  1773    apply (auto simp add: dvd_int_iff)
```
```  1774   apply (cases z)
```
```  1775    apply (auto simp add: dvd_imp_le)
```
```  1776   done
```
```  1777
```
```  1778 lemma zdvd_period:
```
```  1779   fixes a d :: int
```
```  1780   assumes "a dvd d"
```
```  1781   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  1782     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1783 proof -
```
```  1784   from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)"
```
```  1785     by (simp add: dvd_add_left_iff)
```
```  1786   then show ?thesis
```
```  1787     by (simp add: ac_simps)
```
```  1788 qed
```
```  1789
```
```  1790
```
```  1791 subsection \<open>Finiteness of intervals\<close>
```
```  1792
```
```  1793 lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
```
```  1794 proof (cases "a \<le> b")
```
```  1795   case True
```
```  1796   then show ?thesis
```
```  1797   proof (induct b rule: int_ge_induct)
```
```  1798     case base
```
```  1799     have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
```
```  1800     then show ?case by simp
```
```  1801   next
```
```  1802     case (step b)
```
```  1803     then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
```
```  1804     with step show ?case by simp
```
```  1805   qed
```
```  1806 next
```
```  1807   case False
```
```  1808   then show ?thesis
```
```  1809     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
```
```  1810 qed
```
```  1811
```
```  1812 lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
```
```  1813   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1814
```
```  1815 lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
```
```  1816   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1817
```
```  1818 lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
```
```  1819   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1820
```
```  1821
```
```  1822 subsection \<open>Configuration of the code generator\<close>
```
```  1823
```
```  1824 text \<open>Constructors\<close>
```
```  1825
```
```  1826 definition Pos :: "num \<Rightarrow> int"
```
```  1827   where [simp, code_abbrev]: "Pos = numeral"
```
```  1828
```
```  1829 definition Neg :: "num \<Rightarrow> int"
```
```  1830   where [simp, code_abbrev]: "Neg n = - (Pos n)"
```
```  1831
```
```  1832 code_datatype "0::int" Pos Neg
```
```  1833
```
```  1834
```
```  1835 text \<open>Auxiliary operations.\<close>
```
```  1836
```
```  1837 definition dup :: "int \<Rightarrow> int"
```
```  1838   where [simp]: "dup k = k + k"
```
```  1839
```
```  1840 lemma dup_code [code]:
```
```  1841   "dup 0 = 0"
```
```  1842   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```  1843   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```  1844   by (simp_all add: numeral_Bit0)
```
```  1845
```
```  1846 definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
```
```  1847   where [simp]: "sub m n = numeral m - numeral n"
```
```  1848
```
```  1849 lemma sub_code [code]:
```
```  1850   "sub Num.One Num.One = 0"
```
```  1851   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```  1852   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```  1853   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```  1854   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```  1855   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```  1856   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```  1857   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```  1858   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```  1859   by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
```
```  1860
```
```  1861 text \<open>Implementations.\<close>
```
```  1862
```
```  1863 lemma one_int_code [code]: "1 = Pos Num.One"
```
```  1864   by simp
```
```  1865
```
```  1866 lemma plus_int_code [code]:
```
```  1867   "k + 0 = k"
```
```  1868   "0 + l = l"
```
```  1869   "Pos m + Pos n = Pos (m + n)"
```
```  1870   "Pos m + Neg n = sub m n"
```
```  1871   "Neg m + Pos n = sub n m"
```
```  1872   "Neg m + Neg n = Neg (m + n)"
```
```  1873   for k l :: int
```
```  1874   by simp_all
```
```  1875
```
```  1876 lemma uminus_int_code [code]:
```
```  1877   "uminus 0 = (0::int)"
```
```  1878   "uminus (Pos m) = Neg m"
```
```  1879   "uminus (Neg m) = Pos m"
```
```  1880   by simp_all
```
```  1881
```
```  1882 lemma minus_int_code [code]:
```
```  1883   "k - 0 = k"
```
```  1884   "0 - l = uminus l"
```
```  1885   "Pos m - Pos n = sub m n"
```
```  1886   "Pos m - Neg n = Pos (m + n)"
```
```  1887   "Neg m - Pos n = Neg (m + n)"
```
```  1888   "Neg m - Neg n = sub n m"
```
```  1889   for k l :: int
```
```  1890   by simp_all
```
```  1891
```
```  1892 lemma times_int_code [code]:
```
```  1893   "k * 0 = 0"
```
```  1894   "0 * l = 0"
```
```  1895   "Pos m * Pos n = Pos (m * n)"
```
```  1896   "Pos m * Neg n = Neg (m * n)"
```
```  1897   "Neg m * Pos n = Neg (m * n)"
```
```  1898   "Neg m * Neg n = Pos (m * n)"
```
```  1899   for k l :: int
```
```  1900   by simp_all
```
```  1901
```
```  1902 instantiation int :: equal
```
```  1903 begin
```
```  1904
```
```  1905 definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
```
```  1906
```
```  1907 instance
```
```  1908   by standard (rule equal_int_def)
```
```  1909
```
```  1910 end
```
```  1911
```
```  1912 lemma equal_int_code [code]:
```
```  1913   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
```
```  1914   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```  1915   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```  1916   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```  1917   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```  1918   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```  1919   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```  1920   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```  1921   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```  1922   by (auto simp add: equal)
```
```  1923
```
```  1924 lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
```
```  1925   for k :: int
```
```  1926   by (fact equal_refl)
```
```  1927
```
```  1928 lemma less_eq_int_code [code]:
```
```  1929   "0 \<le> (0::int) \<longleftrightarrow> True"
```
```  1930   "0 \<le> Pos l \<longleftrightarrow> True"
```
```  1931   "0 \<le> Neg l \<longleftrightarrow> False"
```
```  1932   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```  1933   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```  1934   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```  1935   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```  1936   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```  1937   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```  1938   by simp_all
```
```  1939
```
```  1940 lemma less_int_code [code]:
```
```  1941   "0 < (0::int) \<longleftrightarrow> False"
```
```  1942   "0 < Pos l \<longleftrightarrow> True"
```
```  1943   "0 < Neg l \<longleftrightarrow> False"
```
```  1944   "Pos k < 0 \<longleftrightarrow> False"
```
```  1945   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```  1946   "Pos k < Neg l \<longleftrightarrow> False"
```
```  1947   "Neg k < 0 \<longleftrightarrow> True"
```
```  1948   "Neg k < Pos l \<longleftrightarrow> True"
```
```  1949   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```  1950   by simp_all
```
```  1951
```
```  1952 lemma nat_code [code]:
```
```  1953   "nat (Int.Neg k) = 0"
```
```  1954   "nat 0 = 0"
```
```  1955   "nat (Int.Pos k) = nat_of_num k"
```
```  1956   by (simp_all add: nat_of_num_numeral)
```
```  1957
```
```  1958 lemma (in ring_1) of_int_code [code]:
```
```  1959   "of_int (Int.Neg k) = - numeral k"
```
```  1960   "of_int 0 = 0"
```
```  1961   "of_int (Int.Pos k) = numeral k"
```
```  1962   by simp_all
```
```  1963
```
```  1964
```
```  1965 text \<open>Serializer setup.\<close>
```
```  1966
```
```  1967 code_identifier
```
```  1968   code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1969
```
```  1970 quickcheck_params [default_type = int]
```
```  1971
```
```  1972 hide_const (open) Pos Neg sub dup
```
```  1973
```
```  1974
```
```  1975 text \<open>De-register \<open>int\<close> as a quotient type:\<close>
```
```  1976
```
```  1977 lifting_update int.lifting
```
```  1978 lifting_forget int.lifting
```
```  1979
```
```  1980 end
```