src/HOL/Int.thy
 author haftmann Mon Jun 05 15:59:41 2017 +0200 (2017-06-05) changeset 66010 2f7d39285a1a parent 64996 b316cd527a11 child 66035 de6cd60b1226 permissions -rw-r--r--
executable domain membership checks
```     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 end
```
```   259
```
```   260 context ring_char_0
```
```   261 begin
```
```   262
```
```   263 lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   264   by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
```
```   265
```
```   266 text \<open>Special cases where either operand is zero.\<close>
```
```   267 lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
```
```   268   using of_int_eq_iff [of z 0] by simp
```
```   269
```
```   270 lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
```
```   271   using of_int_eq_iff [of 0 z] by simp
```
```   272
```
```   273 lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
```
```   274   using of_int_eq_iff [of z 1] by simp
```
```   275
```
```   276 end
```
```   277
```
```   278 context linordered_idom
```
```   279 begin
```
```   280
```
```   281 text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
```
```   282 subclass ring_char_0 ..
```
```   283
```
```   284 lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   285   by (transfer fixing: less_eq)
```
```   286     (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
```
```   287
```
```   288 lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   289   by (simp add: less_le order_less_le)
```
```   290
```
```   291 lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
```
```   292   using of_int_le_iff [of 0 z] by simp
```
```   293
```
```   294 lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
```
```   295   using of_int_le_iff [of z 0] by simp
```
```   296
```
```   297 lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
```
```   298   using of_int_less_iff [of 0 z] by simp
```
```   299
```
```   300 lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
```
```   301   using of_int_less_iff [of z 0] by simp
```
```   302
```
```   303 lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
```
```   304   using of_int_le_iff [of 1 z] by simp
```
```   305
```
```   306 lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
```
```   307   using of_int_le_iff [of z 1] by simp
```
```   308
```
```   309 lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
```
```   310   using of_int_less_iff [of 1 z] by simp
```
```   311
```
```   312 lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
```
```   313   using of_int_less_iff [of z 1] by simp
```
```   314
```
```   315 lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
```
```   316   by simp
```
```   317
```
```   318 lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
```
```   319   by simp
```
```   320
```
```   321 lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
```
```   322   by (auto simp add: abs_if)
```
```   323
```
```   324 lemma of_int_lessD:
```
```   325   assumes "\<bar>of_int n\<bar> < x"
```
```   326   shows "n = 0 \<or> x > 1"
```
```   327 proof (cases "n = 0")
```
```   328   case True
```
```   329   then show ?thesis by simp
```
```   330 next
```
```   331   case False
```
```   332   then have "\<bar>n\<bar> \<noteq> 0" by simp
```
```   333   then have "\<bar>n\<bar> > 0" by simp
```
```   334   then have "\<bar>n\<bar> \<ge> 1"
```
```   335     using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
```
```   336   then have "\<bar>of_int n\<bar> \<ge> 1"
```
```   337     unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
```
```   338   then have "1 < x" using assms by (rule le_less_trans)
```
```   339   then show ?thesis ..
```
```   340 qed
```
```   341
```
```   342 lemma of_int_leD:
```
```   343   assumes "\<bar>of_int n\<bar> \<le> x"
```
```   344   shows "n = 0 \<or> 1 \<le> x"
```
```   345 proof (cases "n = 0")
```
```   346   case True
```
```   347   then show ?thesis by simp
```
```   348 next
```
```   349   case False
```
```   350   then have "\<bar>n\<bar> \<noteq> 0" by simp
```
```   351   then have "\<bar>n\<bar> > 0" by simp
```
```   352   then have "\<bar>n\<bar> \<ge> 1"
```
```   353     using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
```
```   354   then have "\<bar>of_int n\<bar> \<ge> 1"
```
```   355     unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
```
```   356   then have "1 \<le> x" using assms by (rule order_trans)
```
```   357   then show ?thesis ..
```
```   358 qed
```
```   359
```
```   360 end
```
```   361
```
```   362 text \<open>Comparisons involving @{term of_int}.\<close>
```
```   363
```
```   364 lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
```
```   365   using of_int_eq_iff by fastforce
```
```   366
```
```   367 lemma of_int_le_numeral_iff [simp]:
```
```   368   "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
```
```   369   using of_int_le_iff [of z "numeral n"] by simp
```
```   370
```
```   371 lemma of_int_numeral_le_iff [simp]:
```
```   372   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
```
```   373   using of_int_le_iff [of "numeral n"] by simp
```
```   374
```
```   375 lemma of_int_less_numeral_iff [simp]:
```
```   376   "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
```
```   377   using of_int_less_iff [of z "numeral n"] by simp
```
```   378
```
```   379 lemma of_int_numeral_less_iff [simp]:
```
```   380   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
```
```   381   using of_int_less_iff [of "numeral n" z] by simp
```
```   382
```
```   383 lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
```
```   384   by (metis of_int_of_nat_eq of_int_less_iff)
```
```   385
```
```   386 lemma of_int_eq_id [simp]: "of_int = id"
```
```   387 proof
```
```   388   show "of_int z = id z" for z
```
```   389     by (cases z rule: int_diff_cases) simp
```
```   390 qed
```
```   391
```
```   392 instance int :: no_top
```
```   393   apply standard
```
```   394   apply (rule_tac x="x + 1" in exI)
```
```   395   apply simp
```
```   396   done
```
```   397
```
```   398 instance int :: no_bot
```
```   399   apply standard
```
```   400   apply (rule_tac x="x - 1" in exI)
```
```   401   apply simp
```
```   402   done
```
```   403
```
```   404
```
```   405 subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
```
```   406
```
```   407 lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
```
```   408   by auto
```
```   409
```
```   410 lemma nat_int [simp]: "nat (int n) = n"
```
```   411   by transfer simp
```
```   412
```
```   413 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   414   by transfer clarsimp
```
```   415
```
```   416 lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
```
```   417   by simp
```
```   418
```
```   419 lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
```
```   420   by transfer clarsimp
```
```   421
```
```   422 lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
```
```   423   by transfer (clarsimp, arith)
```
```   424
```
```   425 text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
```
```   426 lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   427   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   428
```
```   429 lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   430   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   431
```
```   432 lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
```
```   433   by transfer (clarsimp, arith)
```
```   434
```
```   435 lemma nonneg_int_cases:
```
```   436   assumes "0 \<le> k"
```
```   437   obtains n where "k = int n"
```
```   438 proof -
```
```   439   from assms have "k = int (nat k)"
```
```   440     by simp
```
```   441   then show thesis
```
```   442     by (rule that)
```
```   443 qed
```
```   444
```
```   445 lemma pos_int_cases:
```
```   446   assumes "0 < k"
```
```   447   obtains n where "k = int n" and "n > 0"
```
```   448 proof -
```
```   449   from assms have "0 \<le> k"
```
```   450     by simp
```
```   451   then obtain n where "k = int n"
```
```   452     by (rule nonneg_int_cases)
```
```   453   moreover have "n > 0"
```
```   454     using \<open>k = int n\<close> assms by simp
```
```   455   ultimately show thesis
```
```   456     by (rule that)
```
```   457 qed
```
```   458
```
```   459 lemma nonpos_int_cases:
```
```   460   assumes "k \<le> 0"
```
```   461   obtains n where "k = - int n"
```
```   462 proof -
```
```   463   from assms have "- k \<ge> 0"
```
```   464     by simp
```
```   465   then obtain n where "- k = int n"
```
```   466     by (rule nonneg_int_cases)
```
```   467   then have "k = - int n"
```
```   468     by simp
```
```   469   then show thesis
```
```   470     by (rule that)
```
```   471 qed
```
```   472
```
```   473 lemma neg_int_cases:
```
```   474   assumes "k < 0"
```
```   475   obtains n where "k = - int n" and "n > 0"
```
```   476 proof -
```
```   477   from assms have "- k > 0"
```
```   478     by simp
```
```   479   then obtain n where "- k = int n" and "- k > 0"
```
```   480     by (blast elim: pos_int_cases)
```
```   481   then have "k = - int n" and "n > 0"
```
```   482     by simp_all
```
```   483   then show thesis
```
```   484     by (rule that)
```
```   485 qed
```
```   486
```
```   487 lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   488   by transfer (clarsimp simp add: le_imp_diff_is_add)
```
```   489
```
```   490 lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   491   using nat_eq_iff [of w m] by auto
```
```   492
```
```   493 lemma nat_0 [simp]: "nat 0 = 0"
```
```   494   by (simp add: nat_eq_iff)
```
```   495
```
```   496 lemma nat_1 [simp]: "nat 1 = Suc 0"
```
```   497   by (simp add: nat_eq_iff)
```
```   498
```
```   499 lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
```
```   500   by (simp add: nat_eq_iff)
```
```   501
```
```   502 lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
```
```   503   by simp
```
```   504
```
```   505 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   506   by simp
```
```   507
```
```   508 lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
```
```   509   by transfer (clarsimp, arith)
```
```   510
```
```   511 lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
```
```   512   by transfer (clarsimp simp add: le_diff_conv)
```
```   513
```
```   514 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
```
```   515   by transfer auto
```
```   516
```
```   517 lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
```
```   518   for i :: int
```
```   519   by transfer clarsimp
```
```   520
```
```   521 lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
```
```   522   by (auto simp add: nat_eq_iff2)
```
```   523
```
```   524 lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
```
```   525   using zless_nat_conj [of 0] by auto
```
```   526
```
```   527 lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
```
```   528   by transfer clarsimp
```
```   529
```
```   530 lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
```
```   531   by transfer clarsimp
```
```   532
```
```   533 lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
```
```   534   by (rule nat_diff_distrib') auto
```
```   535
```
```   536 lemma nat_zminus_int [simp]: "nat (- int n) = 0"
```
```   537   by transfer simp
```
```   538
```
```   539 lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
```
```   540   by transfer auto
```
```   541
```
```   542 lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
```
```   543   by transfer (clarsimp simp add: less_diff_conv)
```
```   544
```
```   545 lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   546   by transfer (clarsimp simp add: of_nat_diff)
```
```   547
```
```   548 lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
```
```   549   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
```
```   550
```
```   551
```
```   552 text \<open>For termination proofs:\<close>
```
```   553 lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
```
```   554
```
```   555
```
```   556 subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
```
```   557
```
```   558 lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
```
```   559   by (simp add: order_less_le del: of_nat_Suc)
```
```   560
```
```   561 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   562   by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   563
```
```   564 lemma negative_zle_0: "- int n \<le> 0"
```
```   565   by (simp add: minus_le_iff)
```
```   566
```
```   567 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   568   by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   569
```
```   570 lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
```
```   571   by (subst le_minus_iff) (simp del: of_nat_Suc)
```
```   572
```
```   573 lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   574   by transfer simp
```
```   575
```
```   576 lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
```
```   577   by (simp add: linorder_not_less)
```
```   578
```
```   579 lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   580   by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   581
```
```   582 lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
```
```   583   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   584 proof
```
```   585   assume ?rhs
```
```   586   then show ?lhs by auto
```
```   587 next
```
```   588   assume ?lhs
```
```   589   then have "0 \<le> z - w" by simp
```
```   590   then obtain n where "z - w = int n"
```
```   591     using zero_le_imp_eq_int [of "z - w"] by blast
```
```   592   then have "z = w + int n" by simp
```
```   593   then show ?rhs ..
```
```   594 qed
```
```   595
```
```   596 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
```
```   597   by simp
```
```   598
```
```   599 text \<open>
```
```   600   This version is proved for all ordered rings, not just integers!
```
```   601   It is proved here because attribute \<open>arith_split\<close> is not available
```
```   602   in theory \<open>Rings\<close>.
```
```   603   But is it really better than just rewriting with \<open>abs_if\<close>?
```
```   604 \<close>
```
```   605 lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
```
```   606   for a :: "'a::linordered_idom"
```
```   607   by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   608
```
```   609 lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
```
```   610   apply transfer
```
```   611   apply clarsimp
```
```   612   apply (rule_tac x="b - Suc a" in exI)
```
```   613   apply arith
```
```   614   done
```
```   615
```
```   616
```
```   617 subsection \<open>Cases and induction\<close>
```
```   618
```
```   619 text \<open>
```
```   620   Now we replace the case analysis rule by a more conventional one:
```
```   621   whether an integer is negative or not.
```
```   622 \<close>
```
```   623
```
```   624 text \<open>This version is symmetric in the two subgoals.\<close>
```
```   625 lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
```
```   626   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   627   by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
```
```   628
```
```   629 text \<open>This is the default, with a negative case.\<close>
```
```   630 lemma int_cases [case_names nonneg neg, cases type: int]:
```
```   631   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   632   apply (cases "z < 0")
```
```   633    apply (blast dest!: negD)
```
```   634   apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   635   apply auto
```
```   636   apply (blast dest: nat_0_le [THEN sym])
```
```   637   done
```
```   638
```
```   639 lemma int_cases3 [case_names zero pos neg]:
```
```   640   fixes k :: int
```
```   641   assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   642     and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   643   shows "P"
```
```   644 proof (cases k "0::int" rule: linorder_cases)
```
```   645   case equal
```
```   646   with assms(1) show P by simp
```
```   647 next
```
```   648   case greater
```
```   649   then have *: "nat k > 0" by simp
```
```   650   moreover from * have "k = int (nat k)" by auto
```
```   651   ultimately show P using assms(2) by blast
```
```   652 next
```
```   653   case less
```
```   654   then have *: "nat (- k) > 0" by simp
```
```   655   moreover from * have "k = - int (nat (- k))" by auto
```
```   656   ultimately show P using assms(3) by blast
```
```   657 qed
```
```   658
```
```   659 lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
```
```   660   "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
```
```   661   by (cases z) auto
```
```   662
```
```   663 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```   664   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   665   by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
```
```   666
```
```   667 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
```
```   668   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   669   by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
```
```   670
```
```   671 text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
```
```   672
```
```   673 lemmas max_number_of [simp] =
```
```   674   max_def [of "numeral u" "numeral v"]
```
```   675   max_def [of "numeral u" "- numeral v"]
```
```   676   max_def [of "- numeral u" "numeral v"]
```
```   677   max_def [of "- numeral u" "- numeral v"] for u v
```
```   678
```
```   679 lemmas min_number_of [simp] =
```
```   680   min_def [of "numeral u" "numeral v"]
```
```   681   min_def [of "numeral u" "- numeral v"]
```
```   682   min_def [of "- numeral u" "numeral v"]
```
```   683   min_def [of "- numeral u" "- numeral v"] for u v
```
```   684
```
```   685
```
```   686 subsubsection \<open>Binary comparisons\<close>
```
```   687
```
```   688 text \<open>Preliminaries\<close>
```
```   689
```
```   690 lemma le_imp_0_less:
```
```   691   fixes z :: int
```
```   692   assumes le: "0 \<le> z"
```
```   693   shows "0 < 1 + z"
```
```   694 proof -
```
```   695   have "0 \<le> z" by fact
```
```   696   also have "\<dots> < z + 1" by (rule less_add_one)
```
```   697   also have "\<dots> = 1 + z" by (simp add: ac_simps)
```
```   698   finally show "0 < 1 + z" .
```
```   699 qed
```
```   700
```
```   701 lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
```
```   702   for z :: int
```
```   703 proof (cases z)
```
```   704   case (nonneg n)
```
```   705   then show ?thesis
```
```   706     by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
```
```   707 next
```
```   708   case (neg n)
```
```   709   then show ?thesis
```
```   710     by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   711         add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   712 qed
```
```   713
```
```   714
```
```   715 subsubsection \<open>Comparisons, for Ordered Rings\<close>
```
```   716
```
```   717 lemmas double_eq_0_iff = double_zero
```
```   718
```
```   719 lemma odd_nonzero: "1 + z + z \<noteq> 0"
```
```   720   for z :: int
```
```   721 proof (cases z)
```
```   722   case (nonneg n)
```
```   723   have le: "0 \<le> z + z"
```
```   724     by (simp add: nonneg add_increasing)
```
```   725   then show ?thesis
```
```   726     using  le_imp_0_less [OF le] by (auto simp: add.assoc)
```
```   727 next
```
```   728   case (neg n)
```
```   729   show ?thesis
```
```   730   proof
```
```   731     assume eq: "1 + z + z = 0"
```
```   732     have "0 < 1 + (int n + int n)"
```
```   733       by (simp add: le_imp_0_less add_increasing)
```
```   734     also have "\<dots> = - (1 + z + z)"
```
```   735       by (simp add: neg add.assoc [symmetric])
```
```   736     also have "\<dots> = 0" by (simp add: eq)
```
```   737     finally have "0<0" ..
```
```   738     then show False by blast
```
```   739   qed
```
```   740 qed
```
```   741
```
```   742
```
```   743 subsection \<open>The Set of Integers\<close>
```
```   744
```
```   745 context ring_1
```
```   746 begin
```
```   747
```
```   748 definition Ints :: "'a set"  ("\<int>")
```
```   749   where "\<int> = range of_int"
```
```   750
```
```   751 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```   752   by (simp add: Ints_def)
```
```   753
```
```   754 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```   755   using Ints_of_int [of "of_nat n"] by simp
```
```   756
```
```   757 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```   758   using Ints_of_int [of "0"] by simp
```
```   759
```
```   760 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```   761   using Ints_of_int [of "1"] by simp
```
```   762
```
```   763 lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
```
```   764   by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
```
```   765
```
```   766 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```   767   apply (auto simp add: Ints_def)
```
```   768   apply (rule range_eqI)
```
```   769   apply (rule of_int_add [symmetric])
```
```   770   done
```
```   771
```
```   772 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```   773   apply (auto simp add: Ints_def)
```
```   774   apply (rule range_eqI)
```
```   775   apply (rule of_int_minus [symmetric])
```
```   776   done
```
```   777
```
```   778 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```   779   apply (auto simp add: Ints_def)
```
```   780   apply (rule range_eqI)
```
```   781   apply (rule of_int_diff [symmetric])
```
```   782   done
```
```   783
```
```   784 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```   785   apply (auto simp add: Ints_def)
```
```   786   apply (rule range_eqI)
```
```   787   apply (rule of_int_mult [symmetric])
```
```   788   done
```
```   789
```
```   790 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```   791   by (induct n) simp_all
```
```   792
```
```   793 lemma Ints_cases [cases set: Ints]:
```
```   794   assumes "q \<in> \<int>"
```
```   795   obtains (of_int) z where "q = of_int z"
```
```   796   unfolding Ints_def
```
```   797 proof -
```
```   798   from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
```
```   799   then obtain z where "q = of_int z" ..
```
```   800   then show thesis ..
```
```   801 qed
```
```   802
```
```   803 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   804   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```   805   by (rule Ints_cases) auto
```
```   806
```
```   807 lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
```
```   808   unfolding Nats_def Ints_def
```
```   809   by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
```
```   810
```
```   811 lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
```
```   812 proof (intro subsetI equalityI)
```
```   813   fix x :: 'a
```
```   814   assume "x \<in> {of_int n |n. n \<ge> 0}"
```
```   815   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   816     by (auto elim!: Ints_cases)
```
```   817   then have "x = of_nat (nat n)"
```
```   818     by (subst of_nat_nat) simp_all
```
```   819   then show "x \<in> \<nat>"
```
```   820     by simp
```
```   821 next
```
```   822   fix x :: 'a
```
```   823   assume "x \<in> \<nat>"
```
```   824   then obtain n where "x = of_nat n"
```
```   825     by (auto elim!: Nats_cases)
```
```   826   then have "x = of_int (int n)" by simp
```
```   827   also have "int n \<ge> 0" by simp
```
```   828   then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
```
```   829   finally show "x \<in> {of_int n |n. n \<ge> 0}" .
```
```   830 qed
```
```   831
```
```   832 end
```
```   833
```
```   834 lemma (in linordered_idom) Ints_abs [simp]:
```
```   835   shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
```
```   836   by (auto simp: abs_if)
```
```   837
```
```   838 lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
```
```   839 proof (intro subsetI equalityI)
```
```   840   fix x :: 'a
```
```   841   assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
```
```   842   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   843     by (auto elim!: Ints_cases)
```
```   844   then have "x = of_nat (nat n)"
```
```   845     by (subst of_nat_nat) simp_all
```
```   846   then show "x \<in> \<nat>"
```
```   847     by simp
```
```   848 qed (auto elim!: Nats_cases)
```
```   849
```
```   850 lemma (in idom_divide) of_int_divide_in_Ints:
```
```   851   "of_int a div of_int b \<in> \<int>" if "b dvd a"
```
```   852 proof -
```
```   853   from that obtain c where "a = b * c" ..
```
```   854   then show ?thesis
```
```   855     by (cases "of_int b = 0") simp_all
```
```   856 qed
```
```   857
```
```   858 text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
```
```   859
```
```   860 lemma Ints_double_eq_0_iff:
```
```   861   fixes a :: "'a::ring_char_0"
```
```   862   assumes in_Ints: "a \<in> \<int>"
```
```   863   shows "a + a = 0 \<longleftrightarrow> a = 0"
```
```   864     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   865 proof -
```
```   866   from in_Ints have "a \<in> range of_int"
```
```   867     unfolding Ints_def [symmetric] .
```
```   868   then obtain z where a: "a = of_int z" ..
```
```   869   show ?thesis
```
```   870   proof
```
```   871     assume ?rhs
```
```   872     then show ?lhs by simp
```
```   873   next
```
```   874     assume ?lhs
```
```   875     with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
```
```   876     then have "z + z = 0" by (simp only: of_int_eq_iff)
```
```   877     then have "z = 0" by (simp only: double_eq_0_iff)
```
```   878     with a show ?rhs by simp
```
```   879   qed
```
```   880 qed
```
```   881
```
```   882 lemma Ints_odd_nonzero:
```
```   883   fixes a :: "'a::ring_char_0"
```
```   884   assumes in_Ints: "a \<in> \<int>"
```
```   885   shows "1 + a + a \<noteq> 0"
```
```   886 proof -
```
```   887   from in_Ints have "a \<in> range of_int"
```
```   888     unfolding Ints_def [symmetric] .
```
```   889   then obtain z where a: "a = of_int z" ..
```
```   890   show ?thesis
```
```   891   proof
```
```   892     assume "1 + a + a = 0"
```
```   893     with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
```
```   894     then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   895     with odd_nonzero show False by blast
```
```   896   qed
```
```   897 qed
```
```   898
```
```   899 lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
```
```   900   using of_nat_in_Nats [of "numeral w"] by simp
```
```   901
```
```   902 lemma Ints_odd_less_0:
```
```   903   fixes a :: "'a::linordered_idom"
```
```   904   assumes in_Ints: "a \<in> \<int>"
```
```   905   shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
```
```   906 proof -
```
```   907   from in_Ints have "a \<in> range of_int"
```
```   908     unfolding Ints_def [symmetric] .
```
```   909   then obtain z where a: "a = of_int z" ..
```
```   910   with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
```
```   911     by simp
```
```   912   also have "\<dots> \<longleftrightarrow> z < 0"
```
```   913     by (simp only: of_int_less_iff odd_less_0_iff)
```
```   914   also have "\<dots> \<longleftrightarrow> a < 0"
```
```   915     by (simp add: a)
```
```   916   finally show ?thesis .
```
```   917 qed
```
```   918
```
```   919
```
```   920 subsection \<open>@{term sum} and @{term prod}\<close>
```
```   921
```
```   922 lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   923   by (induct A rule: infinite_finite_induct) auto
```
```   924
```
```   925 lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   926   by (induct A rule: infinite_finite_induct) auto
```
```   927
```
```   928 lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   929   by (induct A rule: infinite_finite_induct) auto
```
```   930
```
```   931 lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   932   by (induct A rule: infinite_finite_induct) auto
```
```   933
```
```   934
```
```   935 text \<open>Legacy theorems\<close>
```
```   936
```
```   937 lemmas int_sum = of_nat_sum [where 'a=int]
```
```   938 lemmas int_prod = of_nat_prod [where 'a=int]
```
```   939 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```   940 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```   941 lemmas nonneg_eq_int = nonneg_int_cases
```
```   942
```
```   943
```
```   944 subsection \<open>Setting up simplification procedures\<close>
```
```   945
```
```   946 lemmas of_int_simps =
```
```   947   of_int_0 of_int_1 of_int_add of_int_mult
```
```   948
```
```   949 ML_file "Tools/int_arith.ML"
```
```   950 declaration \<open>K Int_Arith.setup\<close>
```
```   951
```
```   952 simproc_setup fast_arith
```
```   953   ("(m::'a::linordered_idom) < n" |
```
```   954     "(m::'a::linordered_idom) \<le> n" |
```
```   955     "(m::'a::linordered_idom) = n") =
```
```   956   \<open>K Lin_Arith.simproc\<close>
```
```   957
```
```   958
```
```   959 subsection\<open>More Inequality Reasoning\<close>
```
```   960
```
```   961 lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
```
```   962   for w z :: int
```
```   963   by arith
```
```   964
```
```   965 lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
```
```   966   for w z :: int
```
```   967   by arith
```
```   968
```
```   969 lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
```
```   970   for w z :: int
```
```   971   by arith
```
```   972
```
```   973 lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
```
```   974   for w z :: int
```
```   975   by arith
```
```   976
```
```   977 lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
```
```   978   for z :: int
```
```   979   by arith
```
```   980
```
```   981 lemma Ints_nonzero_abs_ge1:
```
```   982   fixes x:: "'a :: linordered_idom"
```
```   983     assumes "x \<in> Ints" "x \<noteq> 0"
```
```   984     shows "1 \<le> abs x"
```
```   985 proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
```
```   986   fix z::int
```
```   987   assume "x = of_int z"
```
```   988     with \<open>x \<noteq> 0\<close>
```
```   989   show "1 \<le> \<bar>x\<bar>"
```
```   990     apply (auto simp add: abs_if)
```
```   991     by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
```
```   992 qed
```
```   993
```
```   994 lemma Ints_nonzero_abs_less1:
```
```   995   fixes x:: "'a :: linordered_idom"
```
```   996   shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
```
```   997     using Ints_nonzero_abs_ge1 [of x] by auto
```
```   998
```
```   999
```
```  1000 subsection \<open>The functions @{term nat} and @{term int}\<close>
```
```  1001
```
```  1002 text \<open>Simplify the term @{term "w + - z"}.\<close>
```
```  1003
```
```  1004 lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
```
```  1005   using zless_nat_conj [of 1 z] by auto
```
```  1006
```
```  1007 text \<open>
```
```  1008   This simplifies expressions of the form @{term "int n = z"} where
```
```  1009   \<open>z\<close> is an integer literal.
```
```  1010 \<close>
```
```  1011 lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
```
```  1012
```
```  1013 lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
```
```  1014   (is "?P = (?L \<and> ?R)")
```
```  1015   for i :: int
```
```  1016 proof (cases "i < 0")
```
```  1017   case True
```
```  1018   then show ?thesis by auto
```
```  1019 next
```
```  1020   case False
```
```  1021   have "?P = ?L"
```
```  1022   proof
```
```  1023     assume ?P
```
```  1024     then show ?L using False by auto
```
```  1025   next
```
```  1026     assume ?L
```
```  1027     then show ?P using False by simp
```
```  1028   qed
```
```  1029   with False show ?thesis by simp
```
```  1030 qed
```
```  1031
```
```  1032 lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
```
```  1033   by auto
```
```  1034
```
```  1035 lemma nat_int_add: "nat (int a + int b) = a + b"
```
```  1036   by auto
```
```  1037
```
```  1038 context ring_1
```
```  1039 begin
```
```  1040
```
```  1041 lemma of_int_of_nat [nitpick_simp]:
```
```  1042   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```  1043 proof (cases "k < 0")
```
```  1044   case True
```
```  1045   then have "0 \<le> - k" by simp
```
```  1046   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```  1047   with True show ?thesis by simp
```
```  1048 next
```
```  1049   case False
```
```  1050   then show ?thesis by (simp add: not_less)
```
```  1051 qed
```
```  1052
```
```  1053 end
```
```  1054
```
```  1055 lemma transfer_rule_of_int:
```
```  1056   fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
```
```  1057   assumes [transfer_rule]: "R 0 0" "R 1 1"
```
```  1058     "rel_fun R (rel_fun R R) plus plus"
```
```  1059     "rel_fun R R uminus uminus"
```
```  1060   shows "rel_fun HOL.eq R of_int of_int"
```
```  1061 proof -
```
```  1062   note transfer_rule_of_nat [transfer_rule]
```
```  1063   have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
```
```  1064     by transfer_prover
```
```  1065   show ?thesis
```
```  1066     by (unfold of_int_of_nat [abs_def]) transfer_prover
```
```  1067 qed
```
```  1068
```
```  1069 lemma nat_mult_distrib:
```
```  1070   fixes z z' :: int
```
```  1071   assumes "0 \<le> z"
```
```  1072   shows "nat (z * z') = nat z * nat z'"
```
```  1073 proof (cases "0 \<le> z'")
```
```  1074   case False
```
```  1075   with assms have "z * z' \<le> 0"
```
```  1076     by (simp add: not_le mult_le_0_iff)
```
```  1077   then have "nat (z * z') = 0" by simp
```
```  1078   moreover from False have "nat z' = 0" by simp
```
```  1079   ultimately show ?thesis by simp
```
```  1080 next
```
```  1081   case True
```
```  1082   with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1083   show ?thesis
```
```  1084     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1085       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1086          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1087 qed
```
```  1088
```
```  1089 lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
```
```  1090   for z z' :: int
```
```  1091   apply (rule trans)
```
```  1092    apply (rule_tac  nat_mult_distrib)
```
```  1093    apply auto
```
```  1094   done
```
```  1095
```
```  1096 lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
```
```  1097   by (cases "z = 0 \<or> w = 0")
```
```  1098     (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1099       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1100
```
```  1101 lemma int_in_range_abs [simp]: "int n \<in> range abs"
```
```  1102 proof (rule range_eqI)
```
```  1103   show "int n = \<bar>int n\<bar>" by simp
```
```  1104 qed
```
```  1105
```
```  1106 lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
```
```  1107 proof -
```
```  1108   have "\<bar>k\<bar> \<in> \<nat>" for k :: int
```
```  1109     by (cases k) simp_all
```
```  1110   moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
```
```  1111     using that by induct simp
```
```  1112   ultimately show ?thesis by blast
```
```  1113 qed
```
```  1114
```
```  1115 lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
```
```  1116   for z :: int
```
```  1117   by (rule sym) (simp add: nat_eq_iff)
```
```  1118
```
```  1119 lemma diff_nat_eq_if:
```
```  1120   "nat z - nat z' =
```
```  1121     (if z' < 0 then nat z
```
```  1122      else
```
```  1123       let d = z - z'
```
```  1124       in if d < 0 then 0 else nat d)"
```
```  1125   by (simp add: Let_def nat_diff_distrib [symmetric])
```
```  1126
```
```  1127 lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
```
```  1128   using diff_nat_numeral [of v Num.One] by simp
```
```  1129
```
```  1130
```
```  1131 subsection \<open>Induction principles for int\<close>
```
```  1132
```
```  1133 text \<open>Well-founded segments of the integers.\<close>
```
```  1134
```
```  1135 definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
```
```  1136   where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
```
```  1137
```
```  1138 lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1139 proof -
```
```  1140   have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
```
```  1141     by (auto simp add: int_ge_less_than_def)
```
```  1142   then show ?thesis
```
```  1143     by (rule wf_subset [OF wf_measure])
```
```  1144 qed
```
```  1145
```
```  1146 text \<open>
```
```  1147   This variant looks odd, but is typical of the relations suggested
```
```  1148   by RankFinder.\<close>
```
```  1149
```
```  1150 definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
```
```  1151   where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
```
```  1152
```
```  1153 lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1154 proof -
```
```  1155   have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
```
```  1156     by (auto simp add: int_ge_less_than2_def)
```
```  1157   then show ?thesis
```
```  1158     by (rule wf_subset [OF wf_measure])
```
```  1159 qed
```
```  1160
```
```  1161 (* `set:int': dummy construction *)
```
```  1162 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1163   fixes i :: int
```
```  1164   assumes ge: "k \<le> i"
```
```  1165     and base: "P k"
```
```  1166     and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1167   shows "P i"
```
```  1168 proof -
```
```  1169   have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
```
```  1170   proof (induct n)
```
```  1171     case 0
```
```  1172     then have "i = k" by arith
```
```  1173     with base show "P i" by simp
```
```  1174   next
```
```  1175     case (Suc n)
```
```  1176     then have "n = nat ((i - 1) - k)" by arith
```
```  1177     moreover have k: "k \<le> i - 1" using Suc.prems by arith
```
```  1178     ultimately have "P (i - 1)" by (rule Suc.hyps)
```
```  1179     from step [OF k this] show ?case by simp
```
```  1180   qed
```
```  1181   with ge show ?thesis by fast
```
```  1182 qed
```
```  1183
```
```  1184 (* `set:int': dummy construction *)
```
```  1185 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1186   fixes i k :: int
```
```  1187   assumes gr: "k < i"
```
```  1188     and base: "P (k + 1)"
```
```  1189     and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1190   shows "P i"
```
```  1191   apply (rule int_ge_induct[of "k + 1"])
```
```  1192   using gr apply arith
```
```  1193    apply (rule base)
```
```  1194   apply (rule step)
```
```  1195    apply simp_all
```
```  1196   done
```
```  1197
```
```  1198 theorem int_le_induct [consumes 1, case_names base step]:
```
```  1199   fixes i k :: int
```
```  1200   assumes le: "i \<le> k"
```
```  1201     and base: "P k"
```
```  1202     and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1203   shows "P i"
```
```  1204 proof -
```
```  1205   have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
```
```  1206   proof (induct n)
```
```  1207     case 0
```
```  1208     then have "i = k" by arith
```
```  1209     with base show "P i" by simp
```
```  1210   next
```
```  1211     case (Suc n)
```
```  1212     then have "n = nat (k - (i + 1))" by arith
```
```  1213     moreover have k: "i + 1 \<le> k" using Suc.prems by arith
```
```  1214     ultimately have "P (i + 1)" by (rule Suc.hyps)
```
```  1215     from step[OF k this] show ?case by simp
```
```  1216   qed
```
```  1217   with le show ?thesis by fast
```
```  1218 qed
```
```  1219
```
```  1220 theorem int_less_induct [consumes 1, case_names base step]:
```
```  1221   fixes i k :: int
```
```  1222   assumes less: "i < k"
```
```  1223     and base: "P (k - 1)"
```
```  1224     and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1225   shows "P i"
```
```  1226   apply (rule int_le_induct[of _ "k - 1"])
```
```  1227   using less apply arith
```
```  1228    apply (rule base)
```
```  1229   apply (rule step)
```
```  1230    apply simp_all
```
```  1231   done
```
```  1232
```
```  1233 theorem int_induct [case_names base step1 step2]:
```
```  1234   fixes k :: int
```
```  1235   assumes base: "P k"
```
```  1236     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1237     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1238   shows "P i"
```
```  1239 proof -
```
```  1240   have "i \<le> k \<or> i \<ge> k" by arith
```
```  1241   then show ?thesis
```
```  1242   proof
```
```  1243     assume "i \<ge> k"
```
```  1244     then show ?thesis
```
```  1245       using base by (rule int_ge_induct) (fact step1)
```
```  1246   next
```
```  1247     assume "i \<le> k"
```
```  1248     then show ?thesis
```
```  1249       using base by (rule int_le_induct) (fact step2)
```
```  1250   qed
```
```  1251 qed
```
```  1252
```
```  1253
```
```  1254 subsection \<open>Intermediate value theorems\<close>
```
```  1255
```
```  1256 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)"
```
```  1257   for n :: nat and k :: int
```
```  1258   unfolding One_nat_def
```
```  1259   apply (induct n)
```
```  1260    apply simp
```
```  1261   apply (intro strip)
```
```  1262   apply (erule impE)
```
```  1263    apply simp
```
```  1264   apply (erule_tac x = n in allE)
```
```  1265   apply simp
```
```  1266   apply (case_tac "k = f (Suc n)")
```
```  1267    apply force
```
```  1268   apply (erule impE)
```
```  1269    apply (simp add: abs_if split: if_split_asm)
```
```  1270   apply (blast intro: le_SucI)
```
```  1271   done
```
```  1272
```
```  1273 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1274
```
```  1275 lemma nat_intermed_int_val:
```
```  1276   "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
```
```  1277     f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
```
```  1278     for f :: "nat \<Rightarrow> int" and k :: int
```
```  1279   apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
```
```  1280   unfolding One_nat_def
```
```  1281   apply simp
```
```  1282   apply (erule exE)
```
```  1283   apply (rule_tac x = "i+m" in exI)
```
```  1284   apply arith
```
```  1285   done
```
```  1286
```
```  1287
```
```  1288 subsection \<open>Products and 1, by T. M. Rasmussen\<close>
```
```  1289
```
```  1290 lemma abs_zmult_eq_1:
```
```  1291   fixes m n :: int
```
```  1292   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1293   shows "\<bar>m\<bar> = 1"
```
```  1294 proof -
```
```  1295   from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
```
```  1296   have "\<not> 2 \<le> \<bar>m\<bar>"
```
```  1297   proof
```
```  1298     assume "2 \<le> \<bar>m\<bar>"
```
```  1299     then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
```
```  1300     also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
```
```  1301     also from mn have "\<dots> = 1" by simp
```
```  1302     finally have "2 * \<bar>n\<bar> \<le> 1" .
```
```  1303     with 0 show "False" by arith
```
```  1304   qed
```
```  1305   with 0 show ?thesis by auto
```
```  1306 qed
```
```  1307
```
```  1308 lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
```
```  1309   for m n :: int
```
```  1310   using abs_zmult_eq_1 [of m n] by arith
```
```  1311
```
```  1312 lemma pos_zmult_eq_1_iff:
```
```  1313   fixes m n :: int
```
```  1314   assumes "0 < m"
```
```  1315   shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
```
```  1316 proof -
```
```  1317   from assms have "m * n = 1 \<Longrightarrow> m = 1"
```
```  1318     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1319   then show ?thesis
```
```  1320     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1321 qed
```
```  1322
```
```  1323 lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
```
```  1324   for m n :: int
```
```  1325   apply (rule iffI)
```
```  1326    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1327    apply (simp add: mult.commute [of m])
```
```  1328    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1329    apply auto
```
```  1330   done
```
```  1331
```
```  1332 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1333 proof
```
```  1334   assume "finite (UNIV::int set)"
```
```  1335   moreover have "inj (\<lambda>i::int. 2 * i)"
```
```  1336     by (rule injI) simp
```
```  1337   ultimately have "surj (\<lambda>i::int. 2 * i)"
```
```  1338     by (rule finite_UNIV_inj_surj)
```
```  1339   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1340   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1341 qed
```
```  1342
```
```  1343
```
```  1344 subsection \<open>Further theorems on numerals\<close>
```
```  1345
```
```  1346 subsubsection \<open>Special Simplification for Constants\<close>
```
```  1347
```
```  1348 text \<open>These distributive laws move literals inside sums and differences.\<close>
```
```  1349
```
```  1350 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
```
```  1351 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
```
```  1352 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
```
```  1353 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
```
```  1354
```
```  1355 text \<open>These are actually for fields, like real: but where else to put them?\<close>
```
```  1356
```
```  1357 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
```
```  1358 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
```
```  1359 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
```
```  1360 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
```
```  1361
```
```  1362
```
```  1363 text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
```
```  1364   strange, but then other simprocs simplify the quotient.\<close>
```
```  1365
```
```  1366 lemmas inverse_eq_divide_numeral [simp] =
```
```  1367   inverse_eq_divide [of "numeral w"] for w
```
```  1368
```
```  1369 lemmas inverse_eq_divide_neg_numeral [simp] =
```
```  1370   inverse_eq_divide [of "- numeral w"] for w
```
```  1371
```
```  1372 text \<open>These laws simplify inequalities, moving unary minus from a term
```
```  1373   into the literal.\<close>
```
```  1374
```
```  1375 lemmas equation_minus_iff_numeral [no_atp] =
```
```  1376   equation_minus_iff [of "numeral v"] for v
```
```  1377
```
```  1378 lemmas minus_equation_iff_numeral [no_atp] =
```
```  1379   minus_equation_iff [of _ "numeral v"] for v
```
```  1380
```
```  1381 lemmas le_minus_iff_numeral [no_atp] =
```
```  1382   le_minus_iff [of "numeral v"] for v
```
```  1383
```
```  1384 lemmas minus_le_iff_numeral [no_atp] =
```
```  1385   minus_le_iff [of _ "numeral v"] for v
```
```  1386
```
```  1387 lemmas less_minus_iff_numeral [no_atp] =
```
```  1388   less_minus_iff [of "numeral v"] for v
```
```  1389
```
```  1390 lemmas minus_less_iff_numeral [no_atp] =
```
```  1391   minus_less_iff [of _ "numeral v"] for v
```
```  1392
```
```  1393 (* FIXME maybe simproc *)
```
```  1394
```
```  1395
```
```  1396 text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
```
```  1397
```
```  1398 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
```
```  1399 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
```
```  1400 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
```
```  1401 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
```
```  1402
```
```  1403
```
```  1404 text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
```
```  1405
```
```  1406 named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
```
```  1407
```
```  1408 lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1409   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1410   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1411
```
```  1412 lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
```
```  1413   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
```
```  1414   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1415
```
```  1416 lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1417   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1418   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1419
```
```  1420 lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
```
```  1421   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
```
```  1422   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1423
```
```  1424 lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1425   eq_divide_eq [of _ _ "numeral w"]
```
```  1426   eq_divide_eq [of _ _ "- numeral w"] for w
```
```  1427
```
```  1428 lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
```
```  1429   divide_eq_eq [of _ "numeral w"]
```
```  1430   divide_eq_eq [of _ "- numeral w"] for w
```
```  1431
```
```  1432
```
```  1433 subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
```
```  1434
```
```  1435 text \<open>Simplify quotients that are compared with a literal constant.\<close>
```
```  1436
```
```  1437 lemmas le_divide_eq_numeral [divide_const_simps] =
```
```  1438   le_divide_eq [of "numeral w"]
```
```  1439   le_divide_eq [of "- numeral w"] for w
```
```  1440
```
```  1441 lemmas divide_le_eq_numeral [divide_const_simps] =
```
```  1442   divide_le_eq [of _ _ "numeral w"]
```
```  1443   divide_le_eq [of _ _ "- numeral w"] for w
```
```  1444
```
```  1445 lemmas less_divide_eq_numeral [divide_const_simps] =
```
```  1446   less_divide_eq [of "numeral w"]
```
```  1447   less_divide_eq [of "- numeral w"] for w
```
```  1448
```
```  1449 lemmas divide_less_eq_numeral [divide_const_simps] =
```
```  1450   divide_less_eq [of _ _ "numeral w"]
```
```  1451   divide_less_eq [of _ _ "- numeral w"] for w
```
```  1452
```
```  1453 lemmas eq_divide_eq_numeral [divide_const_simps] =
```
```  1454   eq_divide_eq [of "numeral w"]
```
```  1455   eq_divide_eq [of "- numeral w"] for w
```
```  1456
```
```  1457 lemmas divide_eq_eq_numeral [divide_const_simps] =
```
```  1458   divide_eq_eq [of _ _ "numeral w"]
```
```  1459   divide_eq_eq [of _ _ "- numeral w"] for w
```
```  1460
```
```  1461
```
```  1462 text \<open>Not good as automatic simprules because they cause case splits.\<close>
```
```  1463 lemmas [divide_const_simps] =
```
```  1464   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  1465
```
```  1466
```
```  1467 subsection \<open>The divides relation\<close>
```
```  1468
```
```  1469 lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
```
```  1470   for m n :: int
```
```  1471   by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  1472
```
```  1473 lemma zdvd_antisym_abs:
```
```  1474   fixes a b :: int
```
```  1475   assumes "a dvd b" and "b dvd a"
```
```  1476   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  1477 proof (cases "a = 0")
```
```  1478   case True
```
```  1479   with assms show ?thesis by simp
```
```  1480 next
```
```  1481   case False
```
```  1482   from \<open>a dvd b\<close> obtain k where k: "b = a * k"
```
```  1483     unfolding dvd_def by blast
```
```  1484   from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
```
```  1485     unfolding dvd_def by blast
```
```  1486   from k k' have "a = a * k * k'" by simp
```
```  1487   with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
```
```  1488     using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
```
```  1489   then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
```
```  1490     by (simp add: zmult_eq_1_iff)
```
```  1491   with k k' show ?thesis by auto
```
```  1492 qed
```
```  1493
```
```  1494 lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
```
```  1495   for k m n :: int
```
```  1496   using dvd_add_right_iff [of k "- n" m] by simp
```
```  1497
```
```  1498 lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
```
```  1499   for k m n :: int
```
```  1500   using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
```
```  1501
```
```  1502 lemma dvd_imp_le_int:
```
```  1503   fixes d i :: int
```
```  1504   assumes "i \<noteq> 0" and "d dvd i"
```
```  1505   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  1506 proof -
```
```  1507   from \<open>d dvd i\<close> obtain k where "i = d * k" ..
```
```  1508   with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
```
```  1509   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  1510   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  1511   with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
```
```  1512 qed
```
```  1513
```
```  1514 lemma zdvd_not_zless:
```
```  1515   fixes m n :: int
```
```  1516   assumes "0 < m" and "m < n"
```
```  1517   shows "\<not> n dvd m"
```
```  1518 proof
```
```  1519   from assms have "0 < n" by auto
```
```  1520   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  1521   with \<open>0 < m\<close> have "0 < n * k" by auto
```
```  1522   with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
```
```  1523   with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
```
```  1524   with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
```
```  1525 qed
```
```  1526
```
```  1527 lemma zdvd_mult_cancel:
```
```  1528   fixes k m n :: int
```
```  1529   assumes d: "k * m dvd k * n"
```
```  1530     and "k \<noteq> 0"
```
```  1531   shows "m dvd n"
```
```  1532 proof -
```
```  1533   from d obtain h where h: "k * n = k * m * h"
```
```  1534     unfolding dvd_def by blast
```
```  1535   have "n = m * h"
```
```  1536   proof (rule ccontr)
```
```  1537     assume "\<not> ?thesis"
```
```  1538     with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
```
```  1539     with h show False
```
```  1540       by (simp add: mult.assoc)
```
```  1541   qed
```
```  1542   then show ?thesis by simp
```
```  1543 qed
```
```  1544
```
```  1545 theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
```
```  1546 proof -
```
```  1547   have "x dvd y" if "int y = int x * k" for k
```
```  1548   proof (cases k)
```
```  1549     case (nonneg n)
```
```  1550     with that have "y = x * n"
```
```  1551       by (simp del: of_nat_mult add: of_nat_mult [symmetric])
```
```  1552     then show ?thesis ..
```
```  1553   next
```
```  1554     case (neg n)
```
```  1555     with that have "int y = int x * (- int (Suc n))"
```
```  1556       by simp
```
```  1557     also have "\<dots> = - (int x * int (Suc n))"
```
```  1558       by (simp only: mult_minus_right)
```
```  1559     also have "\<dots> = - int (x * Suc n)"
```
```  1560       by (simp only: of_nat_mult [symmetric])
```
```  1561     finally have "- int (x * Suc n) = int y" ..
```
```  1562     then show ?thesis
```
```  1563       by (simp only: negative_eq_positive) auto
```
```  1564   qed
```
```  1565   then show ?thesis
```
```  1566     by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  1567 qed
```
```  1568
```
```  1569 lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
```
```  1570   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1571   for x :: int
```
```  1572 proof
```
```  1573   assume ?lhs
```
```  1574   then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  1575   then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  1576   then have "nat \<bar>x\<bar> = 1" by simp
```
```  1577   then show ?rhs by (cases "x < 0") auto
```
```  1578 next
```
```  1579   assume ?rhs
```
```  1580   then have "x = 1 \<or> x = - 1" by auto
```
```  1581   then show ?lhs by (auto intro: dvdI)
```
```  1582 qed
```
```  1583
```
```  1584 lemma zdvd_mult_cancel1:
```
```  1585   fixes m :: int
```
```  1586   assumes mp: "m \<noteq> 0"
```
```  1587   shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
```
```  1588     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1589 proof
```
```  1590   assume ?rhs
```
```  1591   then show ?lhs
```
```  1592     by (cases "n > 0") (auto simp add: minus_equation_iff)
```
```  1593 next
```
```  1594   assume ?lhs
```
```  1595   then have "m * n dvd m * 1" by simp
```
```  1596   from zdvd_mult_cancel[OF this mp] show ?rhs
```
```  1597     by (simp only: zdvd1_eq)
```
```  1598 qed
```
```  1599
```
```  1600 lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
```
```  1601   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1602
```
```  1603 lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
```
```  1604   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1605
```
```  1606 lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
```
```  1607   by (simp add: dvd_int_iff [symmetric])
```
```  1608
```
```  1609 lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
```
```  1610   by (auto simp add: dvd_int_iff)
```
```  1611
```
```  1612 lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  1613   by (auto elim!: nonneg_eq_int)
```
```  1614
```
```  1615 lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  1616   by (induct n) (simp_all add: nat_mult_distrib)
```
```  1617
```
```  1618 lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
```
```  1619   for n z :: int
```
```  1620   apply (cases n)
```
```  1621    apply (auto simp add: dvd_int_iff)
```
```  1622   apply (cases z)
```
```  1623    apply (auto simp add: dvd_imp_le)
```
```  1624   done
```
```  1625
```
```  1626 lemma zdvd_period:
```
```  1627   fixes a d :: int
```
```  1628   assumes "a dvd d"
```
```  1629   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  1630     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1631 proof -
```
```  1632   from assms obtain k where "d = a * k" by (rule dvdE)
```
```  1633   show ?thesis
```
```  1634   proof
```
```  1635     assume ?lhs
```
```  1636     then obtain l where "x + t = a * l" by (rule dvdE)
```
```  1637     then have "x = a * l - t" by simp
```
```  1638     with \<open>d = a * k\<close> show ?rhs by simp
```
```  1639   next
```
```  1640     assume ?rhs
```
```  1641     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
```
```  1642     then have "x = a * l - c * d - t" by simp
```
```  1643     with \<open>d = a * k\<close> show ?lhs by simp
```
```  1644   qed
```
```  1645 qed
```
```  1646
```
```  1647
```
```  1648 subsection \<open>Finiteness of intervals\<close>
```
```  1649
```
```  1650 lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
```
```  1651 proof (cases "a \<le> b")
```
```  1652   case True
```
```  1653   then show ?thesis
```
```  1654   proof (induct b rule: int_ge_induct)
```
```  1655     case base
```
```  1656     have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
```
```  1657     then show ?case by simp
```
```  1658   next
```
```  1659     case (step b)
```
```  1660     then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
```
```  1661     with step show ?case by simp
```
```  1662   qed
```
```  1663 next
```
```  1664   case False
```
```  1665   then show ?thesis
```
```  1666     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
```
```  1667 qed
```
```  1668
```
```  1669 lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
```
```  1670   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1671
```
```  1672 lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
```
```  1673   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1674
```
```  1675 lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
```
```  1676   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1677
```
```  1678
```
```  1679 subsection \<open>Configuration of the code generator\<close>
```
```  1680
```
```  1681 text \<open>Constructors\<close>
```
```  1682
```
```  1683 definition Pos :: "num \<Rightarrow> int"
```
```  1684   where [simp, code_abbrev]: "Pos = numeral"
```
```  1685
```
```  1686 definition Neg :: "num \<Rightarrow> int"
```
```  1687   where [simp, code_abbrev]: "Neg n = - (Pos n)"
```
```  1688
```
```  1689 code_datatype "0::int" Pos Neg
```
```  1690
```
```  1691
```
```  1692 text \<open>Auxiliary operations.\<close>
```
```  1693
```
```  1694 definition dup :: "int \<Rightarrow> int"
```
```  1695   where [simp]: "dup k = k + k"
```
```  1696
```
```  1697 lemma dup_code [code]:
```
```  1698   "dup 0 = 0"
```
```  1699   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```  1700   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```  1701   by (simp_all add: numeral_Bit0)
```
```  1702
```
```  1703 definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
```
```  1704   where [simp]: "sub m n = numeral m - numeral n"
```
```  1705
```
```  1706 lemma sub_code [code]:
```
```  1707   "sub Num.One Num.One = 0"
```
```  1708   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```  1709   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```  1710   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```  1711   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```  1712   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```  1713   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```  1714   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```  1715   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```  1716           apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
```
```  1717         apply (simp_all only: algebra_simps minus_diff_eq)
```
```  1718   apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
```
```  1719   apply (simp_all only: minus_add add.assoc left_minus)
```
```  1720   done
```
```  1721
```
```  1722 text \<open>Implementations.\<close>
```
```  1723
```
```  1724 lemma one_int_code [code]: "1 = Pos Num.One"
```
```  1725   by simp
```
```  1726
```
```  1727 lemma plus_int_code [code]:
```
```  1728   "k + 0 = k"
```
```  1729   "0 + l = l"
```
```  1730   "Pos m + Pos n = Pos (m + n)"
```
```  1731   "Pos m + Neg n = sub m n"
```
```  1732   "Neg m + Pos n = sub n m"
```
```  1733   "Neg m + Neg n = Neg (m + n)"
```
```  1734   for k l :: int
```
```  1735   by simp_all
```
```  1736
```
```  1737 lemma uminus_int_code [code]:
```
```  1738   "uminus 0 = (0::int)"
```
```  1739   "uminus (Pos m) = Neg m"
```
```  1740   "uminus (Neg m) = Pos m"
```
```  1741   by simp_all
```
```  1742
```
```  1743 lemma minus_int_code [code]:
```
```  1744   "k - 0 = k"
```
```  1745   "0 - l = uminus l"
```
```  1746   "Pos m - Pos n = sub m n"
```
```  1747   "Pos m - Neg n = Pos (m + n)"
```
```  1748   "Neg m - Pos n = Neg (m + n)"
```
```  1749   "Neg m - Neg n = sub n m"
```
```  1750   for k l :: int
```
```  1751   by simp_all
```
```  1752
```
```  1753 lemma times_int_code [code]:
```
```  1754   "k * 0 = 0"
```
```  1755   "0 * l = 0"
```
```  1756   "Pos m * Pos n = Pos (m * n)"
```
```  1757   "Pos m * Neg n = Neg (m * n)"
```
```  1758   "Neg m * Pos n = Neg (m * n)"
```
```  1759   "Neg m * Neg n = Pos (m * n)"
```
```  1760   for k l :: int
```
```  1761   by simp_all
```
```  1762
```
```  1763 instantiation int :: equal
```
```  1764 begin
```
```  1765
```
```  1766 definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
```
```  1767
```
```  1768 instance
```
```  1769   by standard (rule equal_int_def)
```
```  1770
```
```  1771 end
```
```  1772
```
```  1773 lemma equal_int_code [code]:
```
```  1774   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
```
```  1775   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```  1776   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```  1777   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```  1778   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```  1779   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```  1780   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```  1781   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```  1782   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```  1783   by (auto simp add: equal)
```
```  1784
```
```  1785 lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
```
```  1786   for k :: int
```
```  1787   by (fact equal_refl)
```
```  1788
```
```  1789 lemma less_eq_int_code [code]:
```
```  1790   "0 \<le> (0::int) \<longleftrightarrow> True"
```
```  1791   "0 \<le> Pos l \<longleftrightarrow> True"
```
```  1792   "0 \<le> Neg l \<longleftrightarrow> False"
```
```  1793   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```  1794   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```  1795   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```  1796   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```  1797   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```  1798   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```  1799   by simp_all
```
```  1800
```
```  1801 lemma less_int_code [code]:
```
```  1802   "0 < (0::int) \<longleftrightarrow> False"
```
```  1803   "0 < Pos l \<longleftrightarrow> True"
```
```  1804   "0 < Neg l \<longleftrightarrow> False"
```
```  1805   "Pos k < 0 \<longleftrightarrow> False"
```
```  1806   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```  1807   "Pos k < Neg l \<longleftrightarrow> False"
```
```  1808   "Neg k < 0 \<longleftrightarrow> True"
```
```  1809   "Neg k < Pos l \<longleftrightarrow> True"
```
```  1810   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```  1811   by simp_all
```
```  1812
```
```  1813 lemma nat_code [code]:
```
```  1814   "nat (Int.Neg k) = 0"
```
```  1815   "nat 0 = 0"
```
```  1816   "nat (Int.Pos k) = nat_of_num k"
```
```  1817   by (simp_all add: nat_of_num_numeral)
```
```  1818
```
```  1819 lemma (in ring_1) of_int_code [code]:
```
```  1820   "of_int (Int.Neg k) = - numeral k"
```
```  1821   "of_int 0 = 0"
```
```  1822   "of_int (Int.Pos k) = numeral k"
```
```  1823   by simp_all
```
```  1824
```
```  1825
```
```  1826 text \<open>Serializer setup.\<close>
```
```  1827
```
```  1828 code_identifier
```
```  1829   code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1830
```
```  1831 quickcheck_params [default_type = int]
```
```  1832
```
```  1833 hide_const (open) Pos Neg sub dup
```
```  1834
```
```  1835
```
```  1836 text \<open>De-register \<open>int\<close> as a quotient type:\<close>
```
```  1837
```
```  1838 lifting_update int.lifting
```
```  1839 lifting_forget int.lifting
```
```  1840
```
```  1841 end
```