src/HOL/Int.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 64014 ca1239a3277b child 64272 f76b6dda2e56 permissions -rw-r--r--
setsum -> sum
```     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
```
```   361 end
```
```   362
```
```   363 text \<open>Comparisons involving @{term of_int}.\<close>
```
```   364
```
```   365 lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
```
```   366   using of_int_eq_iff by fastforce
```
```   367
```
```   368 lemma of_int_le_numeral_iff [simp]:
```
```   369   "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
```
```   370   using of_int_le_iff [of z "numeral n"] by simp
```
```   371
```
```   372 lemma of_int_numeral_le_iff [simp]:
```
```   373   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
```
```   374   using of_int_le_iff [of "numeral n"] by simp
```
```   375
```
```   376 lemma of_int_less_numeral_iff [simp]:
```
```   377   "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
```
```   378   using of_int_less_iff [of z "numeral n"] by simp
```
```   379
```
```   380 lemma of_int_numeral_less_iff [simp]:
```
```   381   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
```
```   382   using of_int_less_iff [of "numeral n" z] by simp
```
```   383
```
```   384 lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
```
```   385   by (metis of_int_of_nat_eq of_int_less_iff)
```
```   386
```
```   387 lemma of_int_eq_id [simp]: "of_int = id"
```
```   388 proof
```
```   389   show "of_int z = id z" for z
```
```   390     by (cases z rule: int_diff_cases) simp
```
```   391 qed
```
```   392
```
```   393 instance int :: no_top
```
```   394   apply standard
```
```   395   apply (rule_tac x="x + 1" in exI)
```
```   396   apply simp
```
```   397   done
```
```   398
```
```   399 instance int :: no_bot
```
```   400   apply standard
```
```   401   apply (rule_tac x="x - 1" in exI)
```
```   402   apply simp
```
```   403   done
```
```   404
```
```   405
```
```   406 subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
```
```   407
```
```   408 lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
```
```   409   by auto
```
```   410
```
```   411 lemma nat_int [simp]: "nat (int n) = n"
```
```   412   by transfer simp
```
```   413
```
```   414 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   415   by transfer clarsimp
```
```   416
```
```   417 lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
```
```   418   by simp
```
```   419
```
```   420 lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
```
```   421   by transfer clarsimp
```
```   422
```
```   423 lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
```
```   424   by transfer (clarsimp, arith)
```
```   425
```
```   426 text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
```
```   427 lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   428   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   429
```
```   430 lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
```
```   431   by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   432
```
```   433 lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
```
```   434   by transfer (clarsimp, arith)
```
```   435
```
```   436 lemma nonneg_eq_int:
```
```   437   fixes z :: int
```
```   438   assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
```
```   439   shows P
```
```   440   using assms by (blast dest: nat_0_le sym)
```
```   441
```
```   442 lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   443   by transfer (clarsimp simp add: le_imp_diff_is_add)
```
```   444
```
```   445 lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   446   using nat_eq_iff [of w m] by auto
```
```   447
```
```   448 lemma nat_0 [simp]: "nat 0 = 0"
```
```   449   by (simp add: nat_eq_iff)
```
```   450
```
```   451 lemma nat_1 [simp]: "nat 1 = Suc 0"
```
```   452   by (simp add: nat_eq_iff)
```
```   453
```
```   454 lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
```
```   455   by (simp add: nat_eq_iff)
```
```   456
```
```   457 lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
```
```   458   by simp
```
```   459
```
```   460 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   461   by simp
```
```   462
```
```   463 lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
```
```   464   by transfer (clarsimp, arith)
```
```   465
```
```   466 lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
```
```   467   by transfer (clarsimp simp add: le_diff_conv)
```
```   468
```
```   469 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
```
```   470   by transfer auto
```
```   471
```
```   472 lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
```
```   473   for i :: int
```
```   474   by transfer clarsimp
```
```   475
```
```   476 lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
```
```   477   by (auto simp add: nat_eq_iff2)
```
```   478
```
```   479 lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
```
```   480   using zless_nat_conj [of 0] by auto
```
```   481
```
```   482 lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
```
```   483   by transfer clarsimp
```
```   484
```
```   485 lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
```
```   486   by transfer clarsimp
```
```   487
```
```   488 lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
```
```   489   by (rule nat_diff_distrib') auto
```
```   490
```
```   491 lemma nat_zminus_int [simp]: "nat (- int n) = 0"
```
```   492   by transfer simp
```
```   493
```
```   494 lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
```
```   495   by transfer auto
```
```   496
```
```   497 lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
```
```   498   by transfer (clarsimp simp add: less_diff_conv)
```
```   499
```
```   500 lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   501   by transfer (clarsimp simp add: of_nat_diff)
```
```   502
```
```   503 lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
```
```   504   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
```
```   505
```
```   506
```
```   507 text \<open>For termination proofs:\<close>
```
```   508 lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
```
```   509
```
```   510
```
```   511 subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
```
```   512
```
```   513 lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
```
```   514   by (simp add: order_less_le del: of_nat_Suc)
```
```   515
```
```   516 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   517   by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   518
```
```   519 lemma negative_zle_0: "- int n \<le> 0"
```
```   520   by (simp add: minus_le_iff)
```
```   521
```
```   522 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   523   by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   524
```
```   525 lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
```
```   526   by (subst le_minus_iff) (simp del: of_nat_Suc)
```
```   527
```
```   528 lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   529   by transfer simp
```
```   530
```
```   531 lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
```
```   532   by (simp add: linorder_not_less)
```
```   533
```
```   534 lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
```
```   535   by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   536
```
```   537 lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
```
```   538   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   539 proof
```
```   540   assume ?rhs
```
```   541   then show ?lhs by auto
```
```   542 next
```
```   543   assume ?lhs
```
```   544   then have "0 \<le> z - w" by simp
```
```   545   then obtain n where "z - w = int n"
```
```   546     using zero_le_imp_eq_int [of "z - w"] by blast
```
```   547   then have "z = w + int n" by simp
```
```   548   then show ?rhs ..
```
```   549 qed
```
```   550
```
```   551 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
```
```   552   by simp
```
```   553
```
```   554 text \<open>
```
```   555   This version is proved for all ordered rings, not just integers!
```
```   556   It is proved here because attribute \<open>arith_split\<close> is not available
```
```   557   in theory \<open>Rings\<close>.
```
```   558   But is it really better than just rewriting with \<open>abs_if\<close>?
```
```   559 \<close>
```
```   560 lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
```
```   561   for a :: "'a::linordered_idom"
```
```   562   by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   563
```
```   564 lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
```
```   565   apply transfer
```
```   566   apply clarsimp
```
```   567   apply (rule_tac x="b - Suc a" in exI)
```
```   568   apply arith
```
```   569   done
```
```   570
```
```   571
```
```   572 subsection \<open>Cases and induction\<close>
```
```   573
```
```   574 text \<open>
```
```   575   Now we replace the case analysis rule by a more conventional one:
```
```   576   whether an integer is negative or not.
```
```   577 \<close>
```
```   578
```
```   579 text \<open>This version is symmetric in the two subgoals.\<close>
```
```   580 lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
```
```   581   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   582   by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
```
```   583
```
```   584 text \<open>This is the default, with a negative case.\<close>
```
```   585 lemma int_cases [case_names nonneg neg, cases type: int]:
```
```   586   "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
```
```   587   apply (cases "z < 0")
```
```   588    apply (blast dest!: negD)
```
```   589   apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   590   apply auto
```
```   591   apply (blast dest: nat_0_le [THEN sym])
```
```   592   done
```
```   593
```
```   594 lemma int_cases3 [case_names zero pos neg]:
```
```   595   fixes k :: int
```
```   596   assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   597     and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   598   shows "P"
```
```   599 proof (cases k "0::int" rule: linorder_cases)
```
```   600   case equal
```
```   601   with assms(1) show P by simp
```
```   602 next
```
```   603   case greater
```
```   604   then have *: "nat k > 0" by simp
```
```   605   moreover from * have "k = int (nat k)" by auto
```
```   606   ultimately show P using assms(2) by blast
```
```   607 next
```
```   608   case less
```
```   609   then have *: "nat (- k) > 0" by simp
```
```   610   moreover from * have "k = - int (nat (- k))" by auto
```
```   611   ultimately show P using assms(3) by blast
```
```   612 qed
```
```   613
```
```   614 lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
```
```   615   "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
```
```   616   by (cases z) auto
```
```   617
```
```   618 lemma nonneg_int_cases:
```
```   619   assumes "0 \<le> k"
```
```   620   obtains n where "k = int n"
```
```   621   using assms by (rule nonneg_eq_int)
```
```   622
```
```   623 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```   624   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   625   by (fact Let_numeral) \<comment> \<open>FIXME drop\<close>
```
```   626
```
```   627 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
```
```   628   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```   629   by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close>
```
```   630
```
```   631 text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
```
```   632
```
```   633 lemmas max_number_of [simp] =
```
```   634   max_def [of "numeral u" "numeral v"]
```
```   635   max_def [of "numeral u" "- numeral v"]
```
```   636   max_def [of "- numeral u" "numeral v"]
```
```   637   max_def [of "- numeral u" "- numeral v"] for u v
```
```   638
```
```   639 lemmas min_number_of [simp] =
```
```   640   min_def [of "numeral u" "numeral v"]
```
```   641   min_def [of "numeral u" "- numeral v"]
```
```   642   min_def [of "- numeral u" "numeral v"]
```
```   643   min_def [of "- numeral u" "- numeral v"] for u v
```
```   644
```
```   645
```
```   646 subsubsection \<open>Binary comparisons\<close>
```
```   647
```
```   648 text \<open>Preliminaries\<close>
```
```   649
```
```   650 lemma le_imp_0_less:
```
```   651   fixes z :: int
```
```   652   assumes le: "0 \<le> z"
```
```   653   shows "0 < 1 + z"
```
```   654 proof -
```
```   655   have "0 \<le> z" by fact
```
```   656   also have "\<dots> < z + 1" by (rule less_add_one)
```
```   657   also have "\<dots> = 1 + z" by (simp add: ac_simps)
```
```   658   finally show "0 < 1 + z" .
```
```   659 qed
```
```   660
```
```   661 lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
```
```   662   for z :: int
```
```   663 proof (cases z)
```
```   664   case (nonneg n)
```
```   665   then show ?thesis
```
```   666     by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
```
```   667 next
```
```   668   case (neg n)
```
```   669   then show ?thesis
```
```   670     by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   671         add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   672 qed
```
```   673
```
```   674
```
```   675 subsubsection \<open>Comparisons, for Ordered Rings\<close>
```
```   676
```
```   677 lemmas double_eq_0_iff = double_zero
```
```   678
```
```   679 lemma odd_nonzero: "1 + z + z \<noteq> 0"
```
```   680   for z :: int
```
```   681 proof (cases z)
```
```   682   case (nonneg n)
```
```   683   have le: "0 \<le> z + z"
```
```   684     by (simp add: nonneg add_increasing)
```
```   685   then show ?thesis
```
```   686     using  le_imp_0_less [OF le] by (auto simp: add.assoc)
```
```   687 next
```
```   688   case (neg n)
```
```   689   show ?thesis
```
```   690   proof
```
```   691     assume eq: "1 + z + z = 0"
```
```   692     have "0 < 1 + (int n + int n)"
```
```   693       by (simp add: le_imp_0_less add_increasing)
```
```   694     also have "\<dots> = - (1 + z + z)"
```
```   695       by (simp add: neg add.assoc [symmetric])
```
```   696     also have "\<dots> = 0" by (simp add: eq)
```
```   697     finally have "0<0" ..
```
```   698     then show False by blast
```
```   699   qed
```
```   700 qed
```
```   701
```
```   702
```
```   703 subsection \<open>The Set of Integers\<close>
```
```   704
```
```   705 context ring_1
```
```   706 begin
```
```   707
```
```   708 definition Ints :: "'a set"  ("\<int>")
```
```   709   where "\<int> = range of_int"
```
```   710
```
```   711 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```   712   by (simp add: Ints_def)
```
```   713
```
```   714 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```   715   using Ints_of_int [of "of_nat n"] by simp
```
```   716
```
```   717 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```   718   using Ints_of_int [of "0"] by simp
```
```   719
```
```   720 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```   721   using Ints_of_int [of "1"] by simp
```
```   722
```
```   723 lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
```
```   724   by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
```
```   725
```
```   726 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```   727   apply (auto simp add: Ints_def)
```
```   728   apply (rule range_eqI)
```
```   729   apply (rule of_int_add [symmetric])
```
```   730   done
```
```   731
```
```   732 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```   733   apply (auto simp add: Ints_def)
```
```   734   apply (rule range_eqI)
```
```   735   apply (rule of_int_minus [symmetric])
```
```   736   done
```
```   737
```
```   738 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```   739   apply (auto simp add: Ints_def)
```
```   740   apply (rule range_eqI)
```
```   741   apply (rule of_int_diff [symmetric])
```
```   742   done
```
```   743
```
```   744 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```   745   apply (auto simp add: Ints_def)
```
```   746   apply (rule range_eqI)
```
```   747   apply (rule of_int_mult [symmetric])
```
```   748   done
```
```   749
```
```   750 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```   751   by (induct n) simp_all
```
```   752
```
```   753 lemma Ints_cases [cases set: Ints]:
```
```   754   assumes "q \<in> \<int>"
```
```   755   obtains (of_int) z where "q = of_int z"
```
```   756   unfolding Ints_def
```
```   757 proof -
```
```   758   from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
```
```   759   then obtain z where "q = of_int z" ..
```
```   760   then show thesis ..
```
```   761 qed
```
```   762
```
```   763 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   764   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```   765   by (rule Ints_cases) auto
```
```   766
```
```   767 lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
```
```   768   unfolding Nats_def Ints_def
```
```   769   by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
```
```   770
```
```   771 lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
```
```   772 proof (intro subsetI equalityI)
```
```   773   fix x :: 'a
```
```   774   assume "x \<in> {of_int n |n. n \<ge> 0}"
```
```   775   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   776     by (auto elim!: Ints_cases)
```
```   777   then have "x = of_nat (nat n)"
```
```   778     by (subst of_nat_nat) simp_all
```
```   779   then show "x \<in> \<nat>"
```
```   780     by simp
```
```   781 next
```
```   782   fix x :: 'a
```
```   783   assume "x \<in> \<nat>"
```
```   784   then obtain n where "x = of_nat n"
```
```   785     by (auto elim!: Nats_cases)
```
```   786   then have "x = of_int (int n)" by simp
```
```   787   also have "int n \<ge> 0" by simp
```
```   788   then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
```
```   789   finally show "x \<in> {of_int n |n. n \<ge> 0}" .
```
```   790 qed
```
```   791
```
```   792 end
```
```   793
```
```   794 lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
```
```   795 proof (intro subsetI equalityI)
```
```   796   fix x :: 'a
```
```   797   assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
```
```   798   then obtain n where "x = of_int n" "n \<ge> 0"
```
```   799     by (auto elim!: Ints_cases)
```
```   800   then have "x = of_nat (nat n)"
```
```   801     by (subst of_nat_nat) simp_all
```
```   802   then show "x \<in> \<nat>"
```
```   803     by simp
```
```   804 qed (auto elim!: Nats_cases)
```
```   805
```
```   806
```
```   807 text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
```
```   808
```
```   809 lemma Ints_double_eq_0_iff:
```
```   810   fixes a :: "'a::ring_char_0"
```
```   811   assumes in_Ints: "a \<in> \<int>"
```
```   812   shows "a + a = 0 \<longleftrightarrow> a = 0"
```
```   813     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   814 proof -
```
```   815   from in_Ints have "a \<in> range of_int"
```
```   816     unfolding Ints_def [symmetric] .
```
```   817   then obtain z where a: "a = of_int z" ..
```
```   818   show ?thesis
```
```   819   proof
```
```   820     assume ?rhs
```
```   821     then show ?lhs by simp
```
```   822   next
```
```   823     assume ?lhs
```
```   824     with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
```
```   825     then have "z + z = 0" by (simp only: of_int_eq_iff)
```
```   826     then have "z = 0" by (simp only: double_eq_0_iff)
```
```   827     with a show ?rhs by simp
```
```   828   qed
```
```   829 qed
```
```   830
```
```   831 lemma Ints_odd_nonzero:
```
```   832   fixes a :: "'a::ring_char_0"
```
```   833   assumes in_Ints: "a \<in> \<int>"
```
```   834   shows "1 + a + a \<noteq> 0"
```
```   835 proof -
```
```   836   from in_Ints have "a \<in> range of_int"
```
```   837     unfolding Ints_def [symmetric] .
```
```   838   then obtain z where a: "a = of_int z" ..
```
```   839   show ?thesis
```
```   840   proof
```
```   841     assume "1 + a + a = 0"
```
```   842     with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
```
```   843     then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   844     with odd_nonzero show False by blast
```
```   845   qed
```
```   846 qed
```
```   847
```
```   848 lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
```
```   849   using of_nat_in_Nats [of "numeral w"] by simp
```
```   850
```
```   851 lemma Ints_odd_less_0:
```
```   852   fixes a :: "'a::linordered_idom"
```
```   853   assumes in_Ints: "a \<in> \<int>"
```
```   854   shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
```
```   855 proof -
```
```   856   from in_Ints have "a \<in> range of_int"
```
```   857     unfolding Ints_def [symmetric] .
```
```   858   then obtain z where a: "a = of_int z" ..
```
```   859   with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
```
```   860     by simp
```
```   861   also have "\<dots> \<longleftrightarrow> z < 0"
```
```   862     by (simp only: of_int_less_iff odd_less_0_iff)
```
```   863   also have "\<dots> \<longleftrightarrow> a < 0"
```
```   864     by (simp add: a)
```
```   865   finally show ?thesis .
```
```   866 qed
```
```   867
```
```   868
```
```   869 subsection \<open>@{term sum} and @{term setprod}\<close>
```
```   870
```
```   871 lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   872   by (induct A rule: infinite_finite_induct) auto
```
```   873
```
```   874 lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   875   by (induct A rule: infinite_finite_induct) auto
```
```   876
```
```   877 lemma of_nat_setprod [simp]: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   878   by (induct A rule: infinite_finite_induct) auto
```
```   879
```
```   880 lemma of_int_setprod [simp]: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   881   by (induct A rule: infinite_finite_induct) auto
```
```   882
```
```   883 lemmas int_sum = of_nat_sum [where 'a=int]
```
```   884 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```   885
```
```   886
```
```   887 text \<open>Legacy theorems\<close>
```
```   888
```
```   889 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```   890 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```   891
```
```   892
```
```   893 subsection \<open>Setting up simplification procedures\<close>
```
```   894
```
```   895 lemmas of_int_simps =
```
```   896   of_int_0 of_int_1 of_int_add of_int_mult
```
```   897
```
```   898 ML_file "Tools/int_arith.ML"
```
```   899 declaration \<open>K Int_Arith.setup\<close>
```
```   900
```
```   901 simproc_setup fast_arith
```
```   902   ("(m::'a::linordered_idom) < n" |
```
```   903     "(m::'a::linordered_idom) \<le> n" |
```
```   904     "(m::'a::linordered_idom) = n") =
```
```   905   \<open>K Lin_Arith.simproc\<close>
```
```   906
```
```   907
```
```   908 subsection\<open>More Inequality Reasoning\<close>
```
```   909
```
```   910 lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
```
```   911   for w z :: int
```
```   912   by arith
```
```   913
```
```   914 lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
```
```   915   for w z :: int
```
```   916   by arith
```
```   917
```
```   918 lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
```
```   919   for w z :: int
```
```   920   by arith
```
```   921
```
```   922 lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
```
```   923   for w z :: int
```
```   924   by arith
```
```   925
```
```   926 lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
```
```   927   for z :: int
```
```   928   by arith
```
```   929
```
```   930
```
```   931 subsection \<open>The functions @{term nat} and @{term int}\<close>
```
```   932
```
```   933 text \<open>Simplify the term @{term "w + - z"}.\<close>
```
```   934
```
```   935 lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
```
```   936   using zless_nat_conj [of 1 z] by auto
```
```   937
```
```   938 text \<open>
```
```   939   This simplifies expressions of the form @{term "int n = z"} where
```
```   940   \<open>z\<close> is an integer literal.
```
```   941 \<close>
```
```   942 lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
```
```   943
```
```   944 lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
```
```   945   (is "?P = (?L \<and> ?R)")
```
```   946   for i :: int
```
```   947 proof (cases "i < 0")
```
```   948   case True
```
```   949   then show ?thesis by auto
```
```   950 next
```
```   951   case False
```
```   952   have "?P = ?L"
```
```   953   proof
```
```   954     assume ?P
```
```   955     then show ?L using False by auto
```
```   956   next
```
```   957     assume ?L
```
```   958     then show ?P using False by simp
```
```   959   qed
```
```   960   with False show ?thesis by simp
```
```   961 qed
```
```   962
```
```   963 lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
```
```   964   by auto
```
```   965
```
```   966 lemma nat_int_add: "nat (int a + int b) = a + b"
```
```   967   by auto
```
```   968
```
```   969 context ring_1
```
```   970 begin
```
```   971
```
```   972 lemma of_int_of_nat [nitpick_simp]:
```
```   973   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```   974 proof (cases "k < 0")
```
```   975   case True
```
```   976   then have "0 \<le> - k" by simp
```
```   977   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```   978   with True show ?thesis by simp
```
```   979 next
```
```   980   case False
```
```   981   then show ?thesis by (simp add: not_less)
```
```   982 qed
```
```   983
```
```   984 end
```
```   985
```
```   986 lemma transfer_rule_of_int:
```
```   987   fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
```
```   988   assumes [transfer_rule]: "R 0 0" "R 1 1"
```
```   989     "rel_fun R (rel_fun R R) plus plus"
```
```   990     "rel_fun R R uminus uminus"
```
```   991   shows "rel_fun HOL.eq R of_int of_int"
```
```   992 proof -
```
```   993   note transfer_rule_of_nat [transfer_rule]
```
```   994   have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
```
```   995     by transfer_prover
```
```   996   show ?thesis
```
```   997     by (unfold of_int_of_nat [abs_def]) transfer_prover
```
```   998 qed
```
```   999
```
```  1000 lemma nat_mult_distrib:
```
```  1001   fixes z z' :: int
```
```  1002   assumes "0 \<le> z"
```
```  1003   shows "nat (z * z') = nat z * nat z'"
```
```  1004 proof (cases "0 \<le> z'")
```
```  1005   case False
```
```  1006   with assms have "z * z' \<le> 0"
```
```  1007     by (simp add: not_le mult_le_0_iff)
```
```  1008   then have "nat (z * z') = 0" by simp
```
```  1009   moreover from False have "nat z' = 0" by simp
```
```  1010   ultimately show ?thesis by simp
```
```  1011 next
```
```  1012   case True
```
```  1013   with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1014   show ?thesis
```
```  1015     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1016       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1017          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1018 qed
```
```  1019
```
```  1020 lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
```
```  1021   for z z' :: int
```
```  1022   apply (rule trans)
```
```  1023    apply (rule_tac  nat_mult_distrib)
```
```  1024    apply auto
```
```  1025   done
```
```  1026
```
```  1027 lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
```
```  1028   by (cases "z = 0 \<or> w = 0")
```
```  1029     (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1030       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1031
```
```  1032 lemma int_in_range_abs [simp]: "int n \<in> range abs"
```
```  1033 proof (rule range_eqI)
```
```  1034   show "int n = \<bar>int n\<bar>" by simp
```
```  1035 qed
```
```  1036
```
```  1037 lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
```
```  1038 proof -
```
```  1039   have "\<bar>k\<bar> \<in> \<nat>" for k :: int
```
```  1040     by (cases k) simp_all
```
```  1041   moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
```
```  1042     using that by induct simp
```
```  1043   ultimately show ?thesis by blast
```
```  1044 qed
```
```  1045
```
```  1046 lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
```
```  1047   for z :: int
```
```  1048   by (rule sym) (simp add: nat_eq_iff)
```
```  1049
```
```  1050 lemma diff_nat_eq_if:
```
```  1051   "nat z - nat z' =
```
```  1052     (if z' < 0 then nat z
```
```  1053      else
```
```  1054       let d = z - z'
```
```  1055       in if d < 0 then 0 else nat d)"
```
```  1056   by (simp add: Let_def nat_diff_distrib [symmetric])
```
```  1057
```
```  1058 lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
```
```  1059   using diff_nat_numeral [of v Num.One] by simp
```
```  1060
```
```  1061
```
```  1062 subsection \<open>Induction principles for int\<close>
```
```  1063
```
```  1064 text \<open>Well-founded segments of the integers.\<close>
```
```  1065
```
```  1066 definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
```
```  1067   where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
```
```  1068
```
```  1069 lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1070 proof -
```
```  1071   have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
```
```  1072     by (auto simp add: int_ge_less_than_def)
```
```  1073   then show ?thesis
```
```  1074     by (rule wf_subset [OF wf_measure])
```
```  1075 qed
```
```  1076
```
```  1077 text \<open>
```
```  1078   This variant looks odd, but is typical of the relations suggested
```
```  1079   by RankFinder.\<close>
```
```  1080
```
```  1081 definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
```
```  1082   where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
```
```  1083
```
```  1084 lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1085 proof -
```
```  1086   have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
```
```  1087     by (auto simp add: int_ge_less_than2_def)
```
```  1088   then show ?thesis
```
```  1089     by (rule wf_subset [OF wf_measure])
```
```  1090 qed
```
```  1091
```
```  1092 (* `set:int': dummy construction *)
```
```  1093 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1094   fixes i :: int
```
```  1095   assumes ge: "k \<le> i"
```
```  1096     and base: "P k"
```
```  1097     and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1098   shows "P i"
```
```  1099 proof -
```
```  1100   have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
```
```  1101   proof (induct n)
```
```  1102     case 0
```
```  1103     then have "i = k" by arith
```
```  1104     with base show "P i" by simp
```
```  1105   next
```
```  1106     case (Suc n)
```
```  1107     then have "n = nat ((i - 1) - k)" by arith
```
```  1108     moreover have k: "k \<le> i - 1" using Suc.prems by arith
```
```  1109     ultimately have "P (i - 1)" by (rule Suc.hyps)
```
```  1110     from step [OF k this] show ?case by simp
```
```  1111   qed
```
```  1112   with ge show ?thesis by fast
```
```  1113 qed
```
```  1114
```
```  1115 (* `set:int': dummy construction *)
```
```  1116 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1117   fixes i k :: int
```
```  1118   assumes gr: "k < i"
```
```  1119     and base: "P (k + 1)"
```
```  1120     and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1121   shows "P i"
```
```  1122   apply (rule int_ge_induct[of "k + 1"])
```
```  1123   using gr apply arith
```
```  1124    apply (rule base)
```
```  1125   apply (rule step)
```
```  1126    apply simp_all
```
```  1127   done
```
```  1128
```
```  1129 theorem int_le_induct [consumes 1, case_names base step]:
```
```  1130   fixes i k :: int
```
```  1131   assumes le: "i \<le> k"
```
```  1132     and base: "P k"
```
```  1133     and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1134   shows "P i"
```
```  1135 proof -
```
```  1136   have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
```
```  1137   proof (induct n)
```
```  1138     case 0
```
```  1139     then have "i = k" by arith
```
```  1140     with base show "P i" by simp
```
```  1141   next
```
```  1142     case (Suc n)
```
```  1143     then have "n = nat (k - (i + 1))" by arith
```
```  1144     moreover have k: "i + 1 \<le> k" using Suc.prems by arith
```
```  1145     ultimately have "P (i + 1)" by (rule Suc.hyps)
```
```  1146     from step[OF k this] show ?case by simp
```
```  1147   qed
```
```  1148   with le show ?thesis by fast
```
```  1149 qed
```
```  1150
```
```  1151 theorem int_less_induct [consumes 1, case_names base step]:
```
```  1152   fixes i k :: int
```
```  1153   assumes less: "i < k"
```
```  1154     and base: "P (k - 1)"
```
```  1155     and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1156   shows "P i"
```
```  1157   apply (rule int_le_induct[of _ "k - 1"])
```
```  1158   using less apply arith
```
```  1159    apply (rule base)
```
```  1160   apply (rule step)
```
```  1161    apply simp_all
```
```  1162   done
```
```  1163
```
```  1164 theorem int_induct [case_names base step1 step2]:
```
```  1165   fixes k :: int
```
```  1166   assumes base: "P k"
```
```  1167     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1168     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1169   shows "P i"
```
```  1170 proof -
```
```  1171   have "i \<le> k \<or> i \<ge> k" by arith
```
```  1172   then show ?thesis
```
```  1173   proof
```
```  1174     assume "i \<ge> k"
```
```  1175     then show ?thesis
```
```  1176       using base by (rule int_ge_induct) (fact step1)
```
```  1177   next
```
```  1178     assume "i \<le> k"
```
```  1179     then show ?thesis
```
```  1180       using base by (rule int_le_induct) (fact step2)
```
```  1181   qed
```
```  1182 qed
```
```  1183
```
```  1184
```
```  1185 subsection \<open>Intermediate value theorems\<close>
```
```  1186
```
```  1187 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)"
```
```  1188   for n :: nat and k :: int
```
```  1189   unfolding One_nat_def
```
```  1190   apply (induct n)
```
```  1191    apply simp
```
```  1192   apply (intro strip)
```
```  1193   apply (erule impE)
```
```  1194    apply simp
```
```  1195   apply (erule_tac x = n in allE)
```
```  1196   apply simp
```
```  1197   apply (case_tac "k = f (Suc n)")
```
```  1198    apply force
```
```  1199   apply (erule impE)
```
```  1200    apply (simp add: abs_if split: if_split_asm)
```
```  1201   apply (blast intro: le_SucI)
```
```  1202   done
```
```  1203
```
```  1204 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1205
```
```  1206 lemma nat_intermed_int_val:
```
```  1207   "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
```
```  1208     f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
```
```  1209     for f :: "nat \<Rightarrow> int" and k :: int
```
```  1210   apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
```
```  1211   unfolding One_nat_def
```
```  1212   apply simp
```
```  1213   apply (erule exE)
```
```  1214   apply (rule_tac x = "i+m" in exI)
```
```  1215   apply arith
```
```  1216   done
```
```  1217
```
```  1218
```
```  1219 subsection \<open>Products and 1, by T. M. Rasmussen\<close>
```
```  1220
```
```  1221 lemma abs_zmult_eq_1:
```
```  1222   fixes m n :: int
```
```  1223   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1224   shows "\<bar>m\<bar> = 1"
```
```  1225 proof -
```
```  1226   from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
```
```  1227   have "\<not> 2 \<le> \<bar>m\<bar>"
```
```  1228   proof
```
```  1229     assume "2 \<le> \<bar>m\<bar>"
```
```  1230     then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
```
```  1231     also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
```
```  1232     also from mn have "\<dots> = 1" by simp
```
```  1233     finally have "2 * \<bar>n\<bar> \<le> 1" .
```
```  1234     with 0 show "False" by arith
```
```  1235   qed
```
```  1236   with 0 show ?thesis by auto
```
```  1237 qed
```
```  1238
```
```  1239 lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
```
```  1240   for m n :: int
```
```  1241   using abs_zmult_eq_1 [of m n] by arith
```
```  1242
```
```  1243 lemma pos_zmult_eq_1_iff:
```
```  1244   fixes m n :: int
```
```  1245   assumes "0 < m"
```
```  1246   shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
```
```  1247 proof -
```
```  1248   from assms have "m * n = 1 \<Longrightarrow> m = 1"
```
```  1249     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1250   then show ?thesis
```
```  1251     by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1252 qed
```
```  1253
```
```  1254 lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
```
```  1255   for m n :: int
```
```  1256   apply (rule iffI)
```
```  1257    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1258    apply (simp add: mult.commute [of m])
```
```  1259    apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1260    apply auto
```
```  1261   done
```
```  1262
```
```  1263 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1264 proof
```
```  1265   assume "finite (UNIV::int set)"
```
```  1266   moreover have "inj (\<lambda>i::int. 2 * i)"
```
```  1267     by (rule injI) simp
```
```  1268   ultimately have "surj (\<lambda>i::int. 2 * i)"
```
```  1269     by (rule finite_UNIV_inj_surj)
```
```  1270   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1271   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1272 qed
```
```  1273
```
```  1274
```
```  1275 subsection \<open>Further theorems on numerals\<close>
```
```  1276
```
```  1277 subsubsection \<open>Special Simplification for Constants\<close>
```
```  1278
```
```  1279 text \<open>These distributive laws move literals inside sums and differences.\<close>
```
```  1280
```
```  1281 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
```
```  1282 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
```
```  1283 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
```
```  1284 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
```
```  1285
```
```  1286 text \<open>These are actually for fields, like real: but where else to put them?\<close>
```
```  1287
```
```  1288 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
```
```  1289 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
```
```  1290 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
```
```  1291 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
```
```  1292
```
```  1293
```
```  1294 text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
```
```  1295   strange, but then other simprocs simplify the quotient.\<close>
```
```  1296
```
```  1297 lemmas inverse_eq_divide_numeral [simp] =
```
```  1298   inverse_eq_divide [of "numeral w"] for w
```
```  1299
```
```  1300 lemmas inverse_eq_divide_neg_numeral [simp] =
```
```  1301   inverse_eq_divide [of "- numeral w"] for w
```
```  1302
```
```  1303 text \<open>These laws simplify inequalities, moving unary minus from a term
```
```  1304   into the literal.\<close>
```
```  1305
```
```  1306 lemmas equation_minus_iff_numeral [no_atp] =
```
```  1307   equation_minus_iff [of "numeral v"] for v
```
```  1308
```
```  1309 lemmas minus_equation_iff_numeral [no_atp] =
```
```  1310   minus_equation_iff [of _ "numeral v"] for v
```
```  1311
```
```  1312 lemmas le_minus_iff_numeral [no_atp] =
```
```  1313   le_minus_iff [of "numeral v"] for v
```
```  1314
```
```  1315 lemmas minus_le_iff_numeral [no_atp] =
```
```  1316   minus_le_iff [of _ "numeral v"] for v
```
```  1317
```
```  1318 lemmas less_minus_iff_numeral [no_atp] =
```
```  1319   less_minus_iff [of "numeral v"] for v
```
```  1320
```
```  1321 lemmas minus_less_iff_numeral [no_atp] =
```
```  1322   minus_less_iff [of _ "numeral v"] for v
```
```  1323
```
```  1324 (* FIXME maybe simproc *)
```
```  1325
```
```  1326
```
```  1327 text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
```
```  1328
```
```  1329 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
```
```  1330 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
```
```  1331 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
```
```  1332 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
```
```  1333
```
```  1334
```
```  1335 text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
```
```  1336
```
```  1337 named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
```
```  1338
```
```  1339 lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1340   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1341   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1342
```
```  1343 lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
```
```  1344   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
```
```  1345   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1346
```
```  1347 lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1348   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1349   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1350
```
```  1351 lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
```
```  1352   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
```
```  1353   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1354
```
```  1355 lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
```
```  1356   eq_divide_eq [of _ _ "numeral w"]
```
```  1357   eq_divide_eq [of _ _ "- numeral w"] for w
```
```  1358
```
```  1359 lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
```
```  1360   divide_eq_eq [of _ "numeral w"]
```
```  1361   divide_eq_eq [of _ "- numeral w"] for w
```
```  1362
```
```  1363
```
```  1364 subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
```
```  1365
```
```  1366 text \<open>Simplify quotients that are compared with a literal constant.\<close>
```
```  1367
```
```  1368 lemmas le_divide_eq_numeral [divide_const_simps] =
```
```  1369   le_divide_eq [of "numeral w"]
```
```  1370   le_divide_eq [of "- numeral w"] for w
```
```  1371
```
```  1372 lemmas divide_le_eq_numeral [divide_const_simps] =
```
```  1373   divide_le_eq [of _ _ "numeral w"]
```
```  1374   divide_le_eq [of _ _ "- numeral w"] for w
```
```  1375
```
```  1376 lemmas less_divide_eq_numeral [divide_const_simps] =
```
```  1377   less_divide_eq [of "numeral w"]
```
```  1378   less_divide_eq [of "- numeral w"] for w
```
```  1379
```
```  1380 lemmas divide_less_eq_numeral [divide_const_simps] =
```
```  1381   divide_less_eq [of _ _ "numeral w"]
```
```  1382   divide_less_eq [of _ _ "- numeral w"] for w
```
```  1383
```
```  1384 lemmas eq_divide_eq_numeral [divide_const_simps] =
```
```  1385   eq_divide_eq [of "numeral w"]
```
```  1386   eq_divide_eq [of "- numeral w"] for w
```
```  1387
```
```  1388 lemmas divide_eq_eq_numeral [divide_const_simps] =
```
```  1389   divide_eq_eq [of _ _ "numeral w"]
```
```  1390   divide_eq_eq [of _ _ "- numeral w"] for w
```
```  1391
```
```  1392
```
```  1393 text \<open>Not good as automatic simprules because they cause case splits.\<close>
```
```  1394 lemmas [divide_const_simps] =
```
```  1395   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  1396
```
```  1397
```
```  1398 subsection \<open>The divides relation\<close>
```
```  1399
```
```  1400 lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
```
```  1401   for m n :: int
```
```  1402   by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  1403
```
```  1404 lemma zdvd_antisym_abs:
```
```  1405   fixes a b :: int
```
```  1406   assumes "a dvd b" and "b dvd a"
```
```  1407   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  1408 proof (cases "a = 0")
```
```  1409   case True
```
```  1410   with assms show ?thesis by simp
```
```  1411 next
```
```  1412   case False
```
```  1413   from \<open>a dvd b\<close> obtain k where k: "b = a * k"
```
```  1414     unfolding dvd_def by blast
```
```  1415   from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
```
```  1416     unfolding dvd_def by blast
```
```  1417   from k k' have "a = a * k * k'" by simp
```
```  1418   with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
```
```  1419     using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
```
```  1420   then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
```
```  1421     by (simp add: zmult_eq_1_iff)
```
```  1422   with k k' show ?thesis by auto
```
```  1423 qed
```
```  1424
```
```  1425 lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
```
```  1426   for k m n :: int
```
```  1427   using dvd_add_right_iff [of k "- n" m] by simp
```
```  1428
```
```  1429 lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
```
```  1430   for k m n :: int
```
```  1431   using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
```
```  1432
```
```  1433 lemma dvd_imp_le_int:
```
```  1434   fixes d i :: int
```
```  1435   assumes "i \<noteq> 0" and "d dvd i"
```
```  1436   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  1437 proof -
```
```  1438   from \<open>d dvd i\<close> obtain k where "i = d * k" ..
```
```  1439   with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
```
```  1440   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  1441   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  1442   with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
```
```  1443 qed
```
```  1444
```
```  1445 lemma zdvd_not_zless:
```
```  1446   fixes m n :: int
```
```  1447   assumes "0 < m" and "m < n"
```
```  1448   shows "\<not> n dvd m"
```
```  1449 proof
```
```  1450   from assms have "0 < n" by auto
```
```  1451   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  1452   with \<open>0 < m\<close> have "0 < n * k" by auto
```
```  1453   with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
```
```  1454   with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
```
```  1455   with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
```
```  1456 qed
```
```  1457
```
```  1458 lemma zdvd_mult_cancel:
```
```  1459   fixes k m n :: int
```
```  1460   assumes d: "k * m dvd k * n"
```
```  1461     and "k \<noteq> 0"
```
```  1462   shows "m dvd n"
```
```  1463 proof -
```
```  1464   from d obtain h where h: "k * n = k * m * h"
```
```  1465     unfolding dvd_def by blast
```
```  1466   have "n = m * h"
```
```  1467   proof (rule ccontr)
```
```  1468     assume "\<not> ?thesis"
```
```  1469     with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
```
```  1470     with h show False
```
```  1471       by (simp add: mult.assoc)
```
```  1472   qed
```
```  1473   then show ?thesis by simp
```
```  1474 qed
```
```  1475
```
```  1476 theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
```
```  1477 proof -
```
```  1478   have "x dvd y" if "int y = int x * k" for k
```
```  1479   proof (cases k)
```
```  1480     case (nonneg n)
```
```  1481     with that have "y = x * n"
```
```  1482       by (simp del: of_nat_mult add: of_nat_mult [symmetric])
```
```  1483     then show ?thesis ..
```
```  1484   next
```
```  1485     case (neg n)
```
```  1486     with that have "int y = int x * (- int (Suc n))"
```
```  1487       by simp
```
```  1488     also have "\<dots> = - (int x * int (Suc n))"
```
```  1489       by (simp only: mult_minus_right)
```
```  1490     also have "\<dots> = - int (x * Suc n)"
```
```  1491       by (simp only: of_nat_mult [symmetric])
```
```  1492     finally have "- int (x * Suc n) = int y" ..
```
```  1493     then show ?thesis
```
```  1494       by (simp only: negative_eq_positive) auto
```
```  1495   qed
```
```  1496   then show ?thesis
```
```  1497     by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  1498 qed
```
```  1499
```
```  1500 lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
```
```  1501   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1502   for x :: int
```
```  1503 proof
```
```  1504   assume ?lhs
```
```  1505   then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  1506   then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  1507   then have "nat \<bar>x\<bar> = 1" by simp
```
```  1508   then show ?rhs by (cases "x < 0") auto
```
```  1509 next
```
```  1510   assume ?rhs
```
```  1511   then have "x = 1 \<or> x = - 1" by auto
```
```  1512   then show ?lhs by (auto intro: dvdI)
```
```  1513 qed
```
```  1514
```
```  1515 lemma zdvd_mult_cancel1:
```
```  1516   fixes m :: int
```
```  1517   assumes mp: "m \<noteq> 0"
```
```  1518   shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
```
```  1519     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1520 proof
```
```  1521   assume ?rhs
```
```  1522   then show ?lhs
```
```  1523     by (cases "n > 0") (auto simp add: minus_equation_iff)
```
```  1524 next
```
```  1525   assume ?lhs
```
```  1526   then have "m * n dvd m * 1" by simp
```
```  1527   from zdvd_mult_cancel[OF this mp] show ?rhs
```
```  1528     by (simp only: zdvd1_eq)
```
```  1529 qed
```
```  1530
```
```  1531 lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
```
```  1532   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1533
```
```  1534 lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
```
```  1535   by (cases "z \<ge> 0") (simp_all add: zdvd_int)
```
```  1536
```
```  1537 lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
```
```  1538   by (simp add: dvd_int_iff [symmetric])
```
```  1539
```
```  1540 lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
```
```  1541   by (auto simp add: dvd_int_iff)
```
```  1542
```
```  1543 lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  1544   by (auto elim!: nonneg_eq_int)
```
```  1545
```
```  1546 lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  1547   by (induct n) (simp_all add: nat_mult_distrib)
```
```  1548
```
```  1549 lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
```
```  1550   for n z :: int
```
```  1551   apply (cases n)
```
```  1552    apply (auto simp add: dvd_int_iff)
```
```  1553   apply (cases z)
```
```  1554    apply (auto simp add: dvd_imp_le)
```
```  1555   done
```
```  1556
```
```  1557 lemma zdvd_period:
```
```  1558   fixes a d :: int
```
```  1559   assumes "a dvd d"
```
```  1560   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  1561     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1562 proof -
```
```  1563   from assms obtain k where "d = a * k" by (rule dvdE)
```
```  1564   show ?thesis
```
```  1565   proof
```
```  1566     assume ?lhs
```
```  1567     then obtain l where "x + t = a * l" by (rule dvdE)
```
```  1568     then have "x = a * l - t" by simp
```
```  1569     with \<open>d = a * k\<close> show ?rhs by simp
```
```  1570   next
```
```  1571     assume ?rhs
```
```  1572     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
```
```  1573     then have "x = a * l - c * d - t" by simp
```
```  1574     with \<open>d = a * k\<close> show ?lhs by simp
```
```  1575   qed
```
```  1576 qed
```
```  1577
```
```  1578
```
```  1579 subsection \<open>Finiteness of intervals\<close>
```
```  1580
```
```  1581 lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
```
```  1582 proof (cases "a \<le> b")
```
```  1583   case True
```
```  1584   then show ?thesis
```
```  1585   proof (induct b rule: int_ge_induct)
```
```  1586     case base
```
```  1587     have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
```
```  1588     then show ?case by simp
```
```  1589   next
```
```  1590     case (step b)
```
```  1591     then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
```
```  1592     with step show ?case by simp
```
```  1593   qed
```
```  1594 next
```
```  1595   case False
```
```  1596   then show ?thesis
```
```  1597     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
```
```  1598 qed
```
```  1599
```
```  1600 lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
```
```  1601   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1602
```
```  1603 lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
```
```  1604   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1605
```
```  1606 lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
```
```  1607   by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1608
```
```  1609
```
```  1610 subsection \<open>Configuration of the code generator\<close>
```
```  1611
```
```  1612 text \<open>Constructors\<close>
```
```  1613
```
```  1614 definition Pos :: "num \<Rightarrow> int"
```
```  1615   where [simp, code_abbrev]: "Pos = numeral"
```
```  1616
```
```  1617 definition Neg :: "num \<Rightarrow> int"
```
```  1618   where [simp, code_abbrev]: "Neg n = - (Pos n)"
```
```  1619
```
```  1620 code_datatype "0::int" Pos Neg
```
```  1621
```
```  1622
```
```  1623 text \<open>Auxiliary operations.\<close>
```
```  1624
```
```  1625 definition dup :: "int \<Rightarrow> int"
```
```  1626   where [simp]: "dup k = k + k"
```
```  1627
```
```  1628 lemma dup_code [code]:
```
```  1629   "dup 0 = 0"
```
```  1630   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```  1631   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```  1632   by (simp_all add: numeral_Bit0)
```
```  1633
```
```  1634 definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
```
```  1635   where [simp]: "sub m n = numeral m - numeral n"
```
```  1636
```
```  1637 lemma sub_code [code]:
```
```  1638   "sub Num.One Num.One = 0"
```
```  1639   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```  1640   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```  1641   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```  1642   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```  1643   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```  1644   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```  1645   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```  1646   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```  1647           apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
```
```  1648         apply (simp_all only: algebra_simps minus_diff_eq)
```
```  1649   apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
```
```  1650   apply (simp_all only: minus_add add.assoc left_minus)
```
```  1651   done
```
```  1652
```
```  1653 text \<open>Implementations.\<close>
```
```  1654
```
```  1655 lemma one_int_code [code, code_unfold]: "1 = Pos Num.One"
```
```  1656   by simp
```
```  1657
```
```  1658 lemma plus_int_code [code]:
```
```  1659   "k + 0 = k"
```
```  1660   "0 + l = l"
```
```  1661   "Pos m + Pos n = Pos (m + n)"
```
```  1662   "Pos m + Neg n = sub m n"
```
```  1663   "Neg m + Pos n = sub n m"
```
```  1664   "Neg m + Neg n = Neg (m + n)"
```
```  1665   for k l :: int
```
```  1666   by simp_all
```
```  1667
```
```  1668 lemma uminus_int_code [code]:
```
```  1669   "uminus 0 = (0::int)"
```
```  1670   "uminus (Pos m) = Neg m"
```
```  1671   "uminus (Neg m) = Pos m"
```
```  1672   by simp_all
```
```  1673
```
```  1674 lemma minus_int_code [code]:
```
```  1675   "k - 0 = k"
```
```  1676   "0 - l = uminus l"
```
```  1677   "Pos m - Pos n = sub m n"
```
```  1678   "Pos m - Neg n = Pos (m + n)"
```
```  1679   "Neg m - Pos n = Neg (m + n)"
```
```  1680   "Neg m - Neg n = sub n m"
```
```  1681   for k l :: int
```
```  1682   by simp_all
```
```  1683
```
```  1684 lemma times_int_code [code]:
```
```  1685   "k * 0 = 0"
```
```  1686   "0 * l = 0"
```
```  1687   "Pos m * Pos n = Pos (m * n)"
```
```  1688   "Pos m * Neg n = Neg (m * n)"
```
```  1689   "Neg m * Pos n = Neg (m * n)"
```
```  1690   "Neg m * Neg n = Pos (m * n)"
```
```  1691   for k l :: int
```
```  1692   by simp_all
```
```  1693
```
```  1694 instantiation int :: equal
```
```  1695 begin
```
```  1696
```
```  1697 definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
```
```  1698
```
```  1699 instance
```
```  1700   by standard (rule equal_int_def)
```
```  1701
```
```  1702 end
```
```  1703
```
```  1704 lemma equal_int_code [code]:
```
```  1705   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
```
```  1706   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```  1707   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```  1708   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```  1709   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```  1710   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```  1711   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```  1712   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```  1713   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```  1714   by (auto simp add: equal)
```
```  1715
```
```  1716 lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
```
```  1717   for k :: int
```
```  1718   by (fact equal_refl)
```
```  1719
```
```  1720 lemma less_eq_int_code [code]:
```
```  1721   "0 \<le> (0::int) \<longleftrightarrow> True"
```
```  1722   "0 \<le> Pos l \<longleftrightarrow> True"
```
```  1723   "0 \<le> Neg l \<longleftrightarrow> False"
```
```  1724   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```  1725   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```  1726   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```  1727   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```  1728   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```  1729   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```  1730   by simp_all
```
```  1731
```
```  1732 lemma less_int_code [code]:
```
```  1733   "0 < (0::int) \<longleftrightarrow> False"
```
```  1734   "0 < Pos l \<longleftrightarrow> True"
```
```  1735   "0 < Neg l \<longleftrightarrow> False"
```
```  1736   "Pos k < 0 \<longleftrightarrow> False"
```
```  1737   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```  1738   "Pos k < Neg l \<longleftrightarrow> False"
```
```  1739   "Neg k < 0 \<longleftrightarrow> True"
```
```  1740   "Neg k < Pos l \<longleftrightarrow> True"
```
```  1741   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```  1742   by simp_all
```
```  1743
```
```  1744 lemma nat_code [code]:
```
```  1745   "nat (Int.Neg k) = 0"
```
```  1746   "nat 0 = 0"
```
```  1747   "nat (Int.Pos k) = nat_of_num k"
```
```  1748   by (simp_all add: nat_of_num_numeral)
```
```  1749
```
```  1750 lemma (in ring_1) of_int_code [code]:
```
```  1751   "of_int (Int.Neg k) = - numeral k"
```
```  1752   "of_int 0 = 0"
```
```  1753   "of_int (Int.Pos k) = numeral k"
```
```  1754   by simp_all
```
```  1755
```
```  1756
```
```  1757 text \<open>Serializer setup.\<close>
```
```  1758
```
```  1759 code_identifier
```
```  1760   code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1761
```
```  1762 quickcheck_params [default_type = int]
```
```  1763
```
```  1764 hide_const (open) Pos Neg sub dup
```
```  1765
```
```  1766
```
```  1767 text \<open>De-register \<open>int\<close> as a quotient type:\<close>
```
```  1768
```
```  1769 lifting_update int.lifting
```
```  1770 lifting_forget int.lifting
```
```  1771
```
```  1772 end
```