src/HOL/Int.thy
 author wenzelm Mon Aug 31 21:28:08 2015 +0200 (2015-08-31) changeset 61070 b72a990adfe2 parent 60868 dd18c33c001e child 61076 bdc1e2f0a86a permissions -rw-r--r--
prefer symbols;
```     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" where
```
```    15   "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"
```
```    24     unfolding reflp_def by auto
```
```    25   show "symp intrel"
```
```    26     unfolding symp_def by auto
```
```    27   show "transp intrel"
```
```    28     unfolding transp_def by auto
```
```    29 qed
```
```    30
```
```    31 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    32      "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
```
```    33 by (induct z) auto
```
```    34
```
```    35 subsection \<open>Integers form a commutative ring\<close>
```
```    36
```
```    37 instantiation int :: comm_ring_1
```
```    38 begin
```
```    39
```
```    40 lift_definition zero_int :: "int" is "(0, 0)" .
```
```    41
```
```    42 lift_definition one_int :: "int" is "(1, 0)" .
```
```    43
```
```    44 lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    45   is "\<lambda>(x, y) (u, v). (x + u, y + v)"
```
```    46   by clarsimp
```
```    47
```
```    48 lift_definition uminus_int :: "int \<Rightarrow> int"
```
```    49   is "\<lambda>(x, y). (y, x)"
```
```    50   by clarsimp
```
```    51
```
```    52 lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    53   is "\<lambda>(x, y) (u, v). (x + v, y + u)"
```
```    54   by clarsimp
```
```    55
```
```    56 lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    57   is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
```
```    58 proof (clarsimp)
```
```    59   fix s t u v w x y z :: nat
```
```    60   assume "s + v = u + t" and "w + z = y + x"
```
```    61   hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
```
```    62        = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
```
```    63     by simp
```
```    64   thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
```
```    65     by (simp add: algebra_simps)
```
```    66 qed
```
```    67
```
```    68 instance
```
```    69   by default (transfer, clarsimp simp: algebra_simps)+
```
```    70
```
```    71 end
```
```    72
```
```    73 abbreviation int :: "nat \<Rightarrow> int" where
```
```    74   "int \<equiv> of_nat"
```
```    75
```
```    76 lemma int_def: "int n = Abs_Integ (n, 0)"
```
```    77   by (induct n, simp add: zero_int.abs_eq,
```
```    78     simp add: one_int.abs_eq plus_int.abs_eq)
```
```    79
```
```    80 lemma int_transfer [transfer_rule]:
```
```    81   "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
```
```    82   unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
```
```    83
```
```    84 lemma int_diff_cases:
```
```    85   obtains (diff) m n where "z = int m - int n"
```
```    86   by transfer clarsimp
```
```    87
```
```    88 subsection \<open>Integers are totally ordered\<close>
```
```    89
```
```    90 instantiation int :: linorder
```
```    91 begin
```
```    92
```
```    93 lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    94   is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
```
```    95   by auto
```
```    96
```
```    97 lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    98   is "\<lambda>(x, y) (u, v). x + v < u + y"
```
```    99   by auto
```
```   100
```
```   101 instance
```
```   102   by default (transfer, force)+
```
```   103
```
```   104 end
```
```   105
```
```   106 instantiation int :: distrib_lattice
```
```   107 begin
```
```   108
```
```   109 definition
```
```   110   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   111
```
```   112 definition
```
```   113   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   114
```
```   115 instance
```
```   116   by intro_classes
```
```   117     (auto simp add: inf_int_def sup_int_def max_min_distrib2)
```
```   118
```
```   119 end
```
```   120
```
```   121 subsection \<open>Ordering properties of arithmetic operations\<close>
```
```   122
```
```   123 instance int :: ordered_cancel_ab_semigroup_add
```
```   124 proof
```
```   125   fix i j k :: int
```
```   126   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   127     by transfer clarsimp
```
```   128 qed
```
```   129
```
```   130 text\<open>Strict Monotonicity of Multiplication\<close>
```
```   131
```
```   132 text\<open>strict, in 1st argument; proof is by induction on k>0\<close>
```
```   133 lemma zmult_zless_mono2_lemma:
```
```   134      "(i::int)<j ==> 0<k ==> int k * i < int k * j"
```
```   135 apply (induct k)
```
```   136 apply simp
```
```   137 apply (simp add: distrib_right)
```
```   138 apply (case_tac "k=0")
```
```   139 apply (simp_all add: add_strict_mono)
```
```   140 done
```
```   141
```
```   142 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
```
```   143 apply transfer
```
```   144 apply clarsimp
```
```   145 apply (rule_tac x="a - b" in exI, simp)
```
```   146 done
```
```   147
```
```   148 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
```
```   149 apply transfer
```
```   150 apply clarsimp
```
```   151 apply (rule_tac x="a - b" in exI, simp)
```
```   152 done
```
```   153
```
```   154 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   155 apply (drule zero_less_imp_eq_int)
```
```   156 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   157 done
```
```   158
```
```   159 text\<open>The integers form an ordered integral domain\<close>
```
```   160 instantiation int :: linordered_idom
```
```   161 begin
```
```   162
```
```   163 definition
```
```   164   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
```
```   165
```
```   166 definition
```
```   167   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   168
```
```   169 instance 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\<Colon>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\<Colon>int) \<le> z"
```
```   182   by transfer clarsimp
```
```   183
```
```   184 lemma zless_iff_Suc_zadd:
```
```   185   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
```
```   186 apply transfer
```
```   187 apply auto
```
```   188 apply (rename_tac a b c d)
```
```   189 apply (rule_tac x="c+b - Suc(a+d)" in exI)
```
```   190 apply arith
```
```   191 done
```
```   192
```
```   193 lemmas int_distrib =
```
```   194   distrib_right [of z1 z2 w]
```
```   195   distrib_left [of w z1 z2]
```
```   196   left_diff_distrib [of z1 z2 w]
```
```   197   right_diff_distrib [of w z1 z2]
```
```   198   for z1 z2 w :: int
```
```   199
```
```   200
```
```   201 subsection \<open>Embedding of the Integers into any @{text ring_1}: @{text of_int}\<close>
```
```   202
```
```   203 context ring_1
```
```   204 begin
```
```   205
```
```   206 lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
```
```   207   by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
```
```   208     of_nat_add [symmetric] simp del: of_nat_add)
```
```   209
```
```   210 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   211   by transfer simp
```
```   212
```
```   213 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   214   by transfer simp
```
```   215
```
```   216 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   217   by transfer (clarsimp simp add: algebra_simps)
```
```   218
```
```   219 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   220   by (transfer fixing: uminus) clarsimp
```
```   221
```
```   222 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   223   using of_int_add [of w "- z"] by simp
```
```   224
```
```   225 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   226   by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
```
```   227
```
```   228 text\<open>Collapse nested embeddings\<close>
```
```   229 lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
```
```   230 by (induct n) auto
```
```   231
```
```   232 lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
```
```   233   by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
```
```   234
```
```   235 lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
```
```   236   by simp
```
```   237
```
```   238 lemma of_int_power:
```
```   239   "of_int (z ^ n) = of_int z ^ n"
```
```   240   by (induct n) simp_all
```
```   241
```
```   242 end
```
```   243
```
```   244 context ring_char_0
```
```   245 begin
```
```   246
```
```   247 lemma of_int_eq_iff [simp]:
```
```   248    "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   249   by transfer (clarsimp simp add: algebra_simps
```
```   250     of_nat_add [symmetric] simp del: of_nat_add)
```
```   251
```
```   252 text\<open>Special cases where either operand is zero\<close>
```
```   253 lemma of_int_eq_0_iff [simp]:
```
```   254   "of_int z = 0 \<longleftrightarrow> z = 0"
```
```   255   using of_int_eq_iff [of z 0] by simp
```
```   256
```
```   257 lemma of_int_0_eq_iff [simp]:
```
```   258   "0 = of_int z \<longleftrightarrow> z = 0"
```
```   259   using of_int_eq_iff [of 0 z] by simp
```
```   260
```
```   261 end
```
```   262
```
```   263 context linordered_idom
```
```   264 begin
```
```   265
```
```   266 text\<open>Every @{text linordered_idom} has characteristic zero.\<close>
```
```   267 subclass ring_char_0 ..
```
```   268
```
```   269 lemma of_int_le_iff [simp]:
```
```   270   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   271   by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
```
```   272     of_nat_add [symmetric] simp del: of_nat_add)
```
```   273
```
```   274 lemma of_int_less_iff [simp]:
```
```   275   "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   276   by (simp add: less_le order_less_le)
```
```   277
```
```   278 lemma of_int_0_le_iff [simp]:
```
```   279   "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
```
```   280   using of_int_le_iff [of 0 z] by simp
```
```   281
```
```   282 lemma of_int_le_0_iff [simp]:
```
```   283   "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
```
```   284   using of_int_le_iff [of z 0] by simp
```
```   285
```
```   286 lemma of_int_0_less_iff [simp]:
```
```   287   "0 < of_int z \<longleftrightarrow> 0 < z"
```
```   288   using of_int_less_iff [of 0 z] by simp
```
```   289
```
```   290 lemma of_int_less_0_iff [simp]:
```
```   291   "of_int z < 0 \<longleftrightarrow> z < 0"
```
```   292   using of_int_less_iff [of z 0] by simp
```
```   293
```
```   294 end
```
```   295
```
```   296 lemma of_nat_less_of_int_iff:
```
```   297   "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
```
```   298   by (metis of_int_of_nat_eq of_int_less_iff)
```
```   299
```
```   300 lemma of_int_eq_id [simp]: "of_int = id"
```
```   301 proof
```
```   302   fix z show "of_int z = id z"
```
```   303     by (cases z rule: int_diff_cases, simp)
```
```   304 qed
```
```   305
```
```   306
```
```   307 instance int :: no_top
```
```   308   apply default
```
```   309   apply (rule_tac x="x + 1" in exI)
```
```   310   apply simp
```
```   311   done
```
```   312
```
```   313 instance int :: no_bot
```
```   314   apply default
```
```   315   apply (rule_tac x="x - 1" in exI)
```
```   316   apply simp
```
```   317   done
```
```   318
```
```   319 subsection \<open>Magnitude of an Integer, as a Natural Number: @{text nat}\<close>
```
```   320
```
```   321 lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
```
```   322   by auto
```
```   323
```
```   324 lemma nat_int [simp]: "nat (int n) = n"
```
```   325   by transfer simp
```
```   326
```
```   327 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   328   by transfer clarsimp
```
```   329
```
```   330 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
```
```   331 by simp
```
```   332
```
```   333 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   334   by transfer clarsimp
```
```   335
```
```   336 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   337   by transfer (clarsimp, arith)
```
```   338
```
```   339 text\<open>An alternative condition is @{term "0 \<le> w"}\<close>
```
```   340 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   341 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   342
```
```   343 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   344 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   345
```
```   346 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   347   by transfer (clarsimp, arith)
```
```   348
```
```   349 lemma nonneg_eq_int:
```
```   350   fixes z :: int
```
```   351   assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
```
```   352   shows P
```
```   353   using assms by (blast dest: nat_0_le sym)
```
```   354
```
```   355 lemma nat_eq_iff:
```
```   356   "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   357   by transfer (clarsimp simp add: le_imp_diff_is_add)
```
```   358
```
```   359 corollary nat_eq_iff2:
```
```   360   "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
```
```   361   using nat_eq_iff [of w m] by auto
```
```   362
```
```   363 lemma nat_0 [simp]:
```
```   364   "nat 0 = 0"
```
```   365   by (simp add: nat_eq_iff)
```
```   366
```
```   367 lemma nat_1 [simp]:
```
```   368   "nat 1 = Suc 0"
```
```   369   by (simp add: nat_eq_iff)
```
```   370
```
```   371 lemma nat_numeral [simp]:
```
```   372   "nat (numeral k) = numeral k"
```
```   373   by (simp add: nat_eq_iff)
```
```   374
```
```   375 lemma nat_neg_numeral [simp]:
```
```   376   "nat (- numeral k) = 0"
```
```   377   by simp
```
```   378
```
```   379 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   380   by simp
```
```   381
```
```   382 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
```
```   383   by transfer (clarsimp, arith)
```
```   384
```
```   385 lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
```
```   386   by transfer (clarsimp simp add: le_diff_conv)
```
```   387
```
```   388 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
```
```   389   by transfer auto
```
```   390
```
```   391 lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
```
```   392   by transfer clarsimp
```
```   393
```
```   394 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
```
```   395 by (auto simp add: nat_eq_iff2)
```
```   396
```
```   397 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   398 by (insert zless_nat_conj [of 0], auto)
```
```   399
```
```   400 lemma nat_add_distrib:
```
```   401   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
```
```   402   by transfer clarsimp
```
```   403
```
```   404 lemma nat_diff_distrib':
```
```   405   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
```
```   406   by transfer clarsimp
```
```   407
```
```   408 lemma nat_diff_distrib:
```
```   409   "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
```
```   410   by (rule nat_diff_distrib') auto
```
```   411
```
```   412 lemma nat_zminus_int [simp]: "nat (- int n) = 0"
```
```   413   by transfer simp
```
```   414
```
```   415 lemma le_nat_iff:
```
```   416   "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
```
```   417   by transfer auto
```
```   418
```
```   419 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
```
```   420   by transfer (clarsimp simp add: less_diff_conv)
```
```   421
```
```   422 context ring_1
```
```   423 begin
```
```   424
```
```   425 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   426   by transfer (clarsimp simp add: of_nat_diff)
```
```   427
```
```   428 end
```
```   429
```
```   430 lemma diff_nat_numeral [simp]:
```
```   431   "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
```
```   432   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
```
```   433
```
```   434
```
```   435 text \<open>For termination proofs:\<close>
```
```   436 lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
```
```   437
```
```   438
```
```   439 subsection\<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
```
```   440
```
```   441 lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
```
```   442 by (simp add: order_less_le del: of_nat_Suc)
```
```   443
```
```   444 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   445 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   446
```
```   447 lemma negative_zle_0: "- int n \<le> 0"
```
```   448 by (simp add: minus_le_iff)
```
```   449
```
```   450 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   451 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   452
```
```   453 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
```
```   454 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   455
```
```   456 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
```
```   457   by transfer simp
```
```   458
```
```   459 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
```
```   460 by (simp add: linorder_not_less)
```
```   461
```
```   462 lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
```
```   463 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   464
```
```   465 lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
```
```   466 proof -
```
```   467   have "(w \<le> z) = (0 \<le> z - w)"
```
```   468     by (simp only: le_diff_eq add_0_left)
```
```   469   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
```
```   470     by (auto elim: zero_le_imp_eq_int)
```
```   471   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
```
```   472     by (simp only: algebra_simps)
```
```   473   finally show ?thesis .
```
```   474 qed
```
```   475
```
```   476 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
```
```   477 by simp
```
```   478
```
```   479 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
```
```   480 by simp
```
```   481
```
```   482 text\<open>This version is proved for all ordered rings, not just integers!
```
```   483       It is proved here because attribute @{text arith_split} is not available
```
```   484       in theory @{text Rings}.
```
```   485       But is it really better than just rewriting with @{text abs_if}?\<close>
```
```   486 lemma abs_split [arith_split, no_atp]:
```
```   487      "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   488 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   489
```
```   490 lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
```
```   491 apply transfer
```
```   492 apply clarsimp
```
```   493 apply (rule_tac x="b - Suc a" in exI, arith)
```
```   494 done
```
```   495
```
```   496 subsection \<open>Cases and induction\<close>
```
```   497
```
```   498 text\<open>Now we replace the case analysis rule by a more conventional one:
```
```   499 whether an integer is negative or not.\<close>
```
```   500
```
```   501 text\<open>This version is symmetric in the two subgoals.\<close>
```
```   502 theorem int_cases2 [case_names nonneg nonpos, cases type: int]:
```
```   503   "\<lbrakk>!! n. z = int n \<Longrightarrow> P;  !! n. z = - (int n) \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```   504 apply (cases "z < 0")
```
```   505 apply (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
```
```   506 done
```
```   507
```
```   508 text\<open>This is the default, with a negative case.\<close>
```
```   509 theorem int_cases [case_names nonneg neg, cases type: int]:
```
```   510   "[|!! n. z = int n ==> P;  !! n. z = - (int (Suc n)) ==> P |] ==> P"
```
```   511 apply (cases "z < 0")
```
```   512 apply (blast dest!: negD)
```
```   513 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   514 apply auto
```
```   515 apply (blast dest: nat_0_le [THEN sym])
```
```   516 done
```
```   517
```
```   518 lemma int_cases3 [case_names zero pos neg]:
```
```   519   fixes k :: int
```
```   520   assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   521     and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
```
```   522   shows "P"
```
```   523 proof (cases k "0::int" rule: linorder_cases)
```
```   524   case equal with assms(1) show P by simp
```
```   525 next
```
```   526   case greater
```
```   527   then have "nat k > 0" by simp
```
```   528   moreover from this have "k = int (nat k)" by auto
```
```   529   ultimately show P using assms(2) by blast
```
```   530 next
```
```   531   case less
```
```   532   then have "nat (- k) > 0" by simp
```
```   533   moreover from this have "k = - int (nat (- k))" by auto
```
```   534   ultimately show P using assms(3) by blast
```
```   535 qed
```
```   536
```
```   537 theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
```
```   538      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
```
```   539   by (cases z) auto
```
```   540
```
```   541 lemma nonneg_int_cases:
```
```   542   assumes "0 \<le> k" obtains n where "k = int n"
```
```   543   using assms by (rule nonneg_eq_int)
```
```   544
```
```   545 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```   546   -- \<open>Unfold all @{text let}s involving constants\<close>
```
```   547   by (fact Let_numeral) -- \<open>FIXME drop\<close>
```
```   548
```
```   549 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
```
```   550   -- \<open>Unfold all @{text let}s involving constants\<close>
```
```   551   by (fact Let_neg_numeral) -- \<open>FIXME drop\<close>
```
```   552
```
```   553 text \<open>Unfold @{text min} and @{text max} on numerals.\<close>
```
```   554
```
```   555 lemmas max_number_of [simp] =
```
```   556   max_def [of "numeral u" "numeral v"]
```
```   557   max_def [of "numeral u" "- numeral v"]
```
```   558   max_def [of "- numeral u" "numeral v"]
```
```   559   max_def [of "- numeral u" "- numeral v"] for u v
```
```   560
```
```   561 lemmas min_number_of [simp] =
```
```   562   min_def [of "numeral u" "numeral v"]
```
```   563   min_def [of "numeral u" "- numeral v"]
```
```   564   min_def [of "- numeral u" "numeral v"]
```
```   565   min_def [of "- numeral u" "- numeral v"] for u v
```
```   566
```
```   567
```
```   568 subsubsection \<open>Binary comparisons\<close>
```
```   569
```
```   570 text \<open>Preliminaries\<close>
```
```   571
```
```   572 lemma le_imp_0_less:
```
```   573   assumes le: "0 \<le> z"
```
```   574   shows "(0::int) < 1 + z"
```
```   575 proof -
```
```   576   have "0 \<le> z" by fact
```
```   577   also have "... < z + 1" by (rule less_add_one)
```
```   578   also have "... = 1 + z" by (simp add: ac_simps)
```
```   579   finally show "0 < 1 + z" .
```
```   580 qed
```
```   581
```
```   582 lemma odd_less_0_iff:
```
```   583   "(1 + z + z < 0) = (z < (0::int))"
```
```   584 proof (cases z)
```
```   585   case (nonneg n)
```
```   586   thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
```
```   587                              le_imp_0_less [THEN order_less_imp_le])
```
```   588 next
```
```   589   case (neg n)
```
```   590   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   591     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   592 qed
```
```   593
```
```   594 subsubsection \<open>Comparisons, for Ordered Rings\<close>
```
```   595
```
```   596 lemmas double_eq_0_iff = double_zero
```
```   597
```
```   598 lemma odd_nonzero:
```
```   599   "1 + z + z \<noteq> (0::int)"
```
```   600 proof (cases z)
```
```   601   case (nonneg n)
```
```   602   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```   603   thus ?thesis using  le_imp_0_less [OF le]
```
```   604     by (auto simp add: add.assoc)
```
```   605 next
```
```   606   case (neg n)
```
```   607   show ?thesis
```
```   608   proof
```
```   609     assume eq: "1 + z + z = 0"
```
```   610     have "(0::int) < 1 + (int n + int n)"
```
```   611       by (simp add: le_imp_0_less add_increasing)
```
```   612     also have "... = - (1 + z + z)"
```
```   613       by (simp add: neg add.assoc [symmetric])
```
```   614     also have "... = 0" by (simp add: eq)
```
```   615     finally have "0<0" ..
```
```   616     thus False by blast
```
```   617   qed
```
```   618 qed
```
```   619
```
```   620
```
```   621 subsection \<open>The Set of Integers\<close>
```
```   622
```
```   623 context ring_1
```
```   624 begin
```
```   625
```
```   626 definition Ints :: "'a set"  ("\<int>")
```
```   627   where "\<int> = range of_int"
```
```   628
```
```   629 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```   630   by (simp add: Ints_def)
```
```   631
```
```   632 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```   633   using Ints_of_int [of "of_nat n"] by simp
```
```   634
```
```   635 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```   636   using Ints_of_int [of "0"] by simp
```
```   637
```
```   638 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```   639   using Ints_of_int [of "1"] by simp
```
```   640
```
```   641 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```   642 apply (auto simp add: Ints_def)
```
```   643 apply (rule range_eqI)
```
```   644 apply (rule of_int_add [symmetric])
```
```   645 done
```
```   646
```
```   647 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```   648 apply (auto simp add: Ints_def)
```
```   649 apply (rule range_eqI)
```
```   650 apply (rule of_int_minus [symmetric])
```
```   651 done
```
```   652
```
```   653 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```   654 apply (auto simp add: Ints_def)
```
```   655 apply (rule range_eqI)
```
```   656 apply (rule of_int_diff [symmetric])
```
```   657 done
```
```   658
```
```   659 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```   660 apply (auto simp add: Ints_def)
```
```   661 apply (rule range_eqI)
```
```   662 apply (rule of_int_mult [symmetric])
```
```   663 done
```
```   664
```
```   665 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```   666 by (induct n) simp_all
```
```   667
```
```   668 lemma Ints_cases [cases set: Ints]:
```
```   669   assumes "q \<in> \<int>"
```
```   670   obtains (of_int) z where "q = of_int z"
```
```   671   unfolding Ints_def
```
```   672 proof -
```
```   673   from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
```
```   674   then obtain z where "q = of_int z" ..
```
```   675   then show thesis ..
```
```   676 qed
```
```   677
```
```   678 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   679   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```   680   by (rule Ints_cases) auto
```
```   681
```
```   682 end
```
```   683
```
```   684 text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
```
```   685
```
```   686 lemma Ints_double_eq_0_iff:
```
```   687   assumes in_Ints: "a \<in> \<int>"
```
```   688   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```   689 proof -
```
```   690   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   691   then obtain z where a: "a = of_int z" ..
```
```   692   show ?thesis
```
```   693   proof
```
```   694     assume "a = 0"
```
```   695     thus "a + a = 0" by simp
```
```   696   next
```
```   697     assume eq: "a + a = 0"
```
```   698     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   699     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```   700     hence "z = 0" by (simp only: double_eq_0_iff)
```
```   701     thus "a = 0" by (simp add: a)
```
```   702   qed
```
```   703 qed
```
```   704
```
```   705 lemma Ints_odd_nonzero:
```
```   706   assumes in_Ints: "a \<in> \<int>"
```
```   707   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```   708 proof -
```
```   709   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   710   then obtain z where a: "a = of_int z" ..
```
```   711   show ?thesis
```
```   712   proof
```
```   713     assume eq: "1 + a + a = 0"
```
```   714     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   715     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   716     with odd_nonzero show False by blast
```
```   717   qed
```
```   718 qed
```
```   719
```
```   720 lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
```
```   721   using of_nat_in_Nats [of "numeral w"] by simp
```
```   722
```
```   723 lemma Ints_odd_less_0:
```
```   724   assumes in_Ints: "a \<in> \<int>"
```
```   725   shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
```
```   726 proof -
```
```   727   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   728   then obtain z where a: "a = of_int z" ..
```
```   729   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```   730     by (simp add: a)
```
```   731   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
```
```   732   also have "... = (a < 0)" by (simp add: a)
```
```   733   finally show ?thesis .
```
```   734 qed
```
```   735
```
```   736
```
```   737 subsection \<open>@{term setsum} and @{term setprod}\<close>
```
```   738
```
```   739 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   740   apply (cases "finite A")
```
```   741   apply (erule finite_induct, auto)
```
```   742   done
```
```   743
```
```   744 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   745   apply (cases "finite A")
```
```   746   apply (erule finite_induct, auto)
```
```   747   done
```
```   748
```
```   749 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   750   apply (cases "finite A")
```
```   751   apply (erule finite_induct, auto simp add: of_nat_mult)
```
```   752   done
```
```   753
```
```   754 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   755   apply (cases "finite A")
```
```   756   apply (erule finite_induct, auto)
```
```   757   done
```
```   758
```
```   759 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```   760 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```   761
```
```   762
```
```   763 text \<open>Legacy theorems\<close>
```
```   764
```
```   765 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```   766 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```   767 lemmas numeral_1_eq_1 = numeral_One
```
```   768
```
```   769 subsection \<open>Setting up simplification procedures\<close>
```
```   770
```
```   771 lemmas of_int_simps =
```
```   772   of_int_0 of_int_1 of_int_add of_int_mult
```
```   773
```
```   774 ML_file "Tools/int_arith.ML"
```
```   775 declaration \<open>K Int_Arith.setup\<close>
```
```   776
```
```   777 simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
```
```   778   "(m::'a::linordered_idom) <= n" |
```
```   779   "(m::'a::linordered_idom) = n") =
```
```   780   \<open>fn _ => fn ss => fn ct => Lin_Arith.simproc ss (Thm.term_of ct)\<close>
```
```   781
```
```   782
```
```   783 subsection\<open>More Inequality Reasoning\<close>
```
```   784
```
```   785 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```   786 by arith
```
```   787
```
```   788 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```   789 by arith
```
```   790
```
```   791 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```   792 by arith
```
```   793
```
```   794 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```   795 by arith
```
```   796
```
```   797 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```   798 by arith
```
```   799
```
```   800
```
```   801 subsection\<open>The functions @{term nat} and @{term int}\<close>
```
```   802
```
```   803 text\<open>Simplify the term @{term "w + - z"}\<close>
```
```   804
```
```   805 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```   806   using zless_nat_conj [of 1 z] by auto
```
```   807
```
```   808 text\<open>This simplifies expressions of the form @{term "int n = z"} where
```
```   809       z is an integer literal.\<close>
```
```   810 lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
```
```   811
```
```   812 lemma split_nat [arith_split]:
```
```   813   "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```   814   (is "?P = (?L & ?R)")
```
```   815 proof (cases "i < 0")
```
```   816   case True thus ?thesis by auto
```
```   817 next
```
```   818   case False
```
```   819   have "?P = ?L"
```
```   820   proof
```
```   821     assume ?P thus ?L using False by clarsimp
```
```   822   next
```
```   823     assume ?L thus ?P using False by simp
```
```   824   qed
```
```   825   with False show ?thesis by simp
```
```   826 qed
```
```   827
```
```   828 lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
```
```   829   by auto
```
```   830
```
```   831 lemma nat_int_add: "nat (int a + int b) = a + b"
```
```   832   by auto
```
```   833
```
```   834 context ring_1
```
```   835 begin
```
```   836
```
```   837 lemma of_int_of_nat [nitpick_simp]:
```
```   838   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```   839 proof (cases "k < 0")
```
```   840   case True then have "0 \<le> - k" by simp
```
```   841   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```   842   with True show ?thesis by simp
```
```   843 next
```
```   844   case False then show ?thesis by (simp add: not_less of_nat_nat)
```
```   845 qed
```
```   846
```
```   847 end
```
```   848
```
```   849 lemma nat_mult_distrib:
```
```   850   fixes z z' :: int
```
```   851   assumes "0 \<le> z"
```
```   852   shows "nat (z * z') = nat z * nat z'"
```
```   853 proof (cases "0 \<le> z'")
```
```   854   case False with assms have "z * z' \<le> 0"
```
```   855     by (simp add: not_le mult_le_0_iff)
```
```   856   then have "nat (z * z') = 0" by simp
```
```   857   moreover from False have "nat z' = 0" by simp
```
```   858   ultimately show ?thesis by simp
```
```   859 next
```
```   860   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```   861   show ?thesis
```
```   862     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```   863       (simp only: of_nat_mult of_nat_nat [OF True]
```
```   864          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```   865 qed
```
```   866
```
```   867 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```   868 apply (rule trans)
```
```   869 apply (rule_tac [2] nat_mult_distrib, auto)
```
```   870 done
```
```   871
```
```   872 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```   873 apply (cases "z=0 | w=0")
```
```   874 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```   875                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```   876 done
```
```   877
```
```   878 lemma int_in_range_abs [simp]:
```
```   879   "int n \<in> range abs"
```
```   880 proof (rule range_eqI)
```
```   881   show "int n = \<bar>int n\<bar>"
```
```   882     by simp
```
```   883 qed
```
```   884
```
```   885 lemma range_abs_Nats [simp]:
```
```   886   "range abs = (\<nat> :: int set)"
```
```   887 proof -
```
```   888   have "\<bar>k\<bar> \<in> \<nat>" for k :: int
```
```   889     by (cases k) simp_all
```
```   890   moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
```
```   891     using that by induct simp
```
```   892   ultimately show ?thesis by blast
```
```   893 qed
```
```   894
```
```   895 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   896 apply (rule sym)
```
```   897 apply (simp add: nat_eq_iff)
```
```   898 done
```
```   899
```
```   900 lemma diff_nat_eq_if:
```
```   901      "nat z - nat z' =
```
```   902         (if z' < 0 then nat z
```
```   903          else let d = z-z' in
```
```   904               if d < 0 then 0 else nat d)"
```
```   905 by (simp add: Let_def nat_diff_distrib [symmetric])
```
```   906
```
```   907 lemma nat_numeral_diff_1 [simp]:
```
```   908   "numeral v - (1::nat) = nat (numeral v - 1)"
```
```   909   using diff_nat_numeral [of v Num.One] by simp
```
```   910
```
```   911
```
```   912 subsection "Induction principles for int"
```
```   913
```
```   914 text\<open>Well-founded segments of the integers\<close>
```
```   915
```
```   916 definition
```
```   917   int_ge_less_than  ::  "int => (int * int) set"
```
```   918 where
```
```   919   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
```
```   920
```
```   921 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```   922 proof -
```
```   923   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
```
```   924     by (auto simp add: int_ge_less_than_def)
```
```   925   thus ?thesis
```
```   926     by (rule wf_subset [OF wf_measure])
```
```   927 qed
```
```   928
```
```   929 text\<open>This variant looks odd, but is typical of the relations suggested
```
```   930 by RankFinder.\<close>
```
```   931
```
```   932 definition
```
```   933   int_ge_less_than2 ::  "int => (int * int) set"
```
```   934 where
```
```   935   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
```
```   936
```
```   937 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```   938 proof -
```
```   939   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
```
```   940     by (auto simp add: int_ge_less_than2_def)
```
```   941   thus ?thesis
```
```   942     by (rule wf_subset [OF wf_measure])
```
```   943 qed
```
```   944
```
```   945 (* `set:int': dummy construction *)
```
```   946 theorem int_ge_induct [case_names base step, induct set: int]:
```
```   947   fixes i :: int
```
```   948   assumes ge: "k \<le> i" and
```
```   949     base: "P k" and
```
```   950     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```   951   shows "P i"
```
```   952 proof -
```
```   953   { fix n
```
```   954     have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```   955     proof (induct n)
```
```   956       case 0
```
```   957       hence "i = k" by arith
```
```   958       thus "P i" using base by simp
```
```   959     next
```
```   960       case (Suc n)
```
```   961       then have "n = nat((i - 1) - k)" by arith
```
```   962       moreover
```
```   963       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```   964       ultimately
```
```   965       have "P (i - 1)" by (rule Suc.hyps)
```
```   966       from step [OF ki1 this] show ?case by simp
```
```   967     qed
```
```   968   }
```
```   969   with ge show ?thesis by fast
```
```   970 qed
```
```   971
```
```   972 (* `set:int': dummy construction *)
```
```   973 theorem int_gr_induct [case_names base step, induct set: int]:
```
```   974   assumes gr: "k < (i::int)" and
```
```   975         base: "P(k+1)" and
```
```   976         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   977   shows "P i"
```
```   978 apply(rule int_ge_induct[of "k + 1"])
```
```   979   using gr apply arith
```
```   980  apply(rule base)
```
```   981 apply (rule step, simp+)
```
```   982 done
```
```   983
```
```   984 theorem int_le_induct [consumes 1, case_names base step]:
```
```   985   assumes le: "i \<le> (k::int)" and
```
```   986         base: "P(k)" and
```
```   987         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   988   shows "P i"
```
```   989 proof -
```
```   990   { fix n
```
```   991     have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
```
```   992     proof (induct n)
```
```   993       case 0
```
```   994       hence "i = k" by arith
```
```   995       thus "P i" using base by simp
```
```   996     next
```
```   997       case (Suc n)
```
```   998       hence "n = nat (k - (i + 1))" by arith
```
```   999       moreover
```
```  1000       have ki1: "i + 1 \<le> k" using Suc.prems by arith
```
```  1001       ultimately
```
```  1002       have "P (i + 1)" by(rule Suc.hyps)
```
```  1003       from step[OF ki1 this] show ?case by simp
```
```  1004     qed
```
```  1005   }
```
```  1006   with le show ?thesis by fast
```
```  1007 qed
```
```  1008
```
```  1009 theorem int_less_induct [consumes 1, case_names base step]:
```
```  1010   assumes less: "(i::int) < k" and
```
```  1011         base: "P(k - 1)" and
```
```  1012         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1013   shows "P i"
```
```  1014 apply(rule int_le_induct[of _ "k - 1"])
```
```  1015   using less apply arith
```
```  1016  apply(rule base)
```
```  1017 apply (rule step, simp+)
```
```  1018 done
```
```  1019
```
```  1020 theorem int_induct [case_names base step1 step2]:
```
```  1021   fixes k :: int
```
```  1022   assumes base: "P k"
```
```  1023     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1024     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1025   shows "P i"
```
```  1026 proof -
```
```  1027   have "i \<le> k \<or> i \<ge> k" by arith
```
```  1028   then show ?thesis
```
```  1029   proof
```
```  1030     assume "i \<ge> k"
```
```  1031     then show ?thesis using base
```
```  1032       by (rule int_ge_induct) (fact step1)
```
```  1033   next
```
```  1034     assume "i \<le> k"
```
```  1035     then show ?thesis using base
```
```  1036       by (rule int_le_induct) (fact step2)
```
```  1037   qed
```
```  1038 qed
```
```  1039
```
```  1040 subsection\<open>Intermediate value theorems\<close>
```
```  1041
```
```  1042 lemma int_val_lemma:
```
```  1043      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```  1044       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```  1045 unfolding One_nat_def
```
```  1046 apply (induct n)
```
```  1047 apply simp
```
```  1048 apply (intro strip)
```
```  1049 apply (erule impE, simp)
```
```  1050 apply (erule_tac x = n in allE, simp)
```
```  1051 apply (case_tac "k = f (Suc n)")
```
```  1052 apply force
```
```  1053 apply (erule impE)
```
```  1054  apply (simp add: abs_if split add: split_if_asm)
```
```  1055 apply (blast intro: le_SucI)
```
```  1056 done
```
```  1057
```
```  1058 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1059
```
```  1060 lemma nat_intermed_int_val:
```
```  1061      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```  1062          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```  1063 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```  1064        in int_val_lemma)
```
```  1065 unfolding One_nat_def
```
```  1066 apply simp
```
```  1067 apply (erule exE)
```
```  1068 apply (rule_tac x = "i+m" in exI, arith)
```
```  1069 done
```
```  1070
```
```  1071
```
```  1072 subsection\<open>Products and 1, by T. M. Rasmussen\<close>
```
```  1073
```
```  1074 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```  1075 by arith
```
```  1076
```
```  1077 lemma abs_zmult_eq_1:
```
```  1078   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1079   shows "\<bar>m\<bar> = (1::int)"
```
```  1080 proof -
```
```  1081   have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
```
```  1082     by auto
```
```  1083   have "~ (2 \<le> \<bar>m\<bar>)"
```
```  1084   proof
```
```  1085     assume "2 \<le> \<bar>m\<bar>"
```
```  1086     hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
```
```  1087       by (simp add: mult_mono 0)
```
```  1088     also have "... = \<bar>m*n\<bar>"
```
```  1089       by (simp add: abs_mult)
```
```  1090     also have "... = 1"
```
```  1091       by (simp add: mn)
```
```  1092     finally have "2*\<bar>n\<bar> \<le> 1" .
```
```  1093     thus "False" using 0
```
```  1094       by arith
```
```  1095   qed
```
```  1096   thus ?thesis using 0
```
```  1097     by auto
```
```  1098 qed
```
```  1099
```
```  1100 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```  1101 by (insert abs_zmult_eq_1 [of m n], arith)
```
```  1102
```
```  1103 lemma pos_zmult_eq_1_iff:
```
```  1104   assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
```
```  1105 proof -
```
```  1106   from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1107   thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1108 qed
```
```  1109
```
```  1110 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```  1111 apply (rule iffI)
```
```  1112  apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1113  apply (simp add: mult.commute [of m])
```
```  1114  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1115 done
```
```  1116
```
```  1117 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1118 proof
```
```  1119   assume "finite (UNIV::int set)"
```
```  1120   moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1121     by (rule injI) simp
```
```  1122   ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1123     by (rule finite_UNIV_inj_surj)
```
```  1124   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1125   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1126 qed
```
```  1127
```
```  1128
```
```  1129 subsection \<open>Further theorems on numerals\<close>
```
```  1130
```
```  1131 subsubsection\<open>Special Simplification for Constants\<close>
```
```  1132
```
```  1133 text\<open>These distributive laws move literals inside sums and differences.\<close>
```
```  1134
```
```  1135 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
```
```  1136 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
```
```  1137 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
```
```  1138 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
```
```  1139
```
```  1140 text\<open>These are actually for fields, like real: but where else to put them?\<close>
```
```  1141
```
```  1142 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
```
```  1143 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
```
```  1144 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
```
```  1145 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
```
```  1146
```
```  1147
```
```  1148 text \<open>Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```  1149   strange, but then other simprocs simplify the quotient.\<close>
```
```  1150
```
```  1151 lemmas inverse_eq_divide_numeral [simp] =
```
```  1152   inverse_eq_divide [of "numeral w"] for w
```
```  1153
```
```  1154 lemmas inverse_eq_divide_neg_numeral [simp] =
```
```  1155   inverse_eq_divide [of "- numeral w"] for w
```
```  1156
```
```  1157 text \<open>These laws simplify inequalities, moving unary minus from a term
```
```  1158 into the literal.\<close>
```
```  1159
```
```  1160 lemmas equation_minus_iff_numeral [no_atp] =
```
```  1161   equation_minus_iff [of "numeral v"] for v
```
```  1162
```
```  1163 lemmas minus_equation_iff_numeral [no_atp] =
```
```  1164   minus_equation_iff [of _ "numeral v"] for v
```
```  1165
```
```  1166 lemmas le_minus_iff_numeral [no_atp] =
```
```  1167   le_minus_iff [of "numeral v"] for v
```
```  1168
```
```  1169 lemmas minus_le_iff_numeral [no_atp] =
```
```  1170   minus_le_iff [of _ "numeral v"] for v
```
```  1171
```
```  1172 lemmas less_minus_iff_numeral [no_atp] =
```
```  1173   less_minus_iff [of "numeral v"] for v
```
```  1174
```
```  1175 lemmas minus_less_iff_numeral [no_atp] =
```
```  1176   minus_less_iff [of _ "numeral v"] for v
```
```  1177
```
```  1178 -- \<open>FIXME maybe simproc\<close>
```
```  1179
```
```  1180
```
```  1181 text \<open>Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"})\<close>
```
```  1182
```
```  1183 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
```
```  1184 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
```
```  1185 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
```
```  1186 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
```
```  1187
```
```  1188
```
```  1189 text \<open>Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="})\<close>
```
```  1190
```
```  1191 lemmas le_divide_eq_numeral1 [simp] =
```
```  1192   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1193   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1194
```
```  1195 lemmas divide_le_eq_numeral1 [simp] =
```
```  1196   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
```
```  1197   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1198
```
```  1199 lemmas less_divide_eq_numeral1 [simp] =
```
```  1200   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1201   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1202
```
```  1203 lemmas divide_less_eq_numeral1 [simp] =
```
```  1204   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
```
```  1205   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
```
```  1206
```
```  1207 lemmas eq_divide_eq_numeral1 [simp] =
```
```  1208   eq_divide_eq [of _ _ "numeral w"]
```
```  1209   eq_divide_eq [of _ _ "- numeral w"] for w
```
```  1210
```
```  1211 lemmas divide_eq_eq_numeral1 [simp] =
```
```  1212   divide_eq_eq [of _ "numeral w"]
```
```  1213   divide_eq_eq [of _ "- numeral w"] for w
```
```  1214
```
```  1215
```
```  1216 subsubsection\<open>Optional Simplification Rules Involving Constants\<close>
```
```  1217
```
```  1218 text\<open>Simplify quotients that are compared with a literal constant.\<close>
```
```  1219
```
```  1220 lemmas le_divide_eq_numeral =
```
```  1221   le_divide_eq [of "numeral w"]
```
```  1222   le_divide_eq [of "- numeral w"] for w
```
```  1223
```
```  1224 lemmas divide_le_eq_numeral =
```
```  1225   divide_le_eq [of _ _ "numeral w"]
```
```  1226   divide_le_eq [of _ _ "- numeral w"] for w
```
```  1227
```
```  1228 lemmas less_divide_eq_numeral =
```
```  1229   less_divide_eq [of "numeral w"]
```
```  1230   less_divide_eq [of "- numeral w"] for w
```
```  1231
```
```  1232 lemmas divide_less_eq_numeral =
```
```  1233   divide_less_eq [of _ _ "numeral w"]
```
```  1234   divide_less_eq [of _ _ "- numeral w"] for w
```
```  1235
```
```  1236 lemmas eq_divide_eq_numeral =
```
```  1237   eq_divide_eq [of "numeral w"]
```
```  1238   eq_divide_eq [of "- numeral w"] for w
```
```  1239
```
```  1240 lemmas divide_eq_eq_numeral =
```
```  1241   divide_eq_eq [of _ _ "numeral w"]
```
```  1242   divide_eq_eq [of _ _ "- numeral w"] for w
```
```  1243
```
```  1244
```
```  1245 text\<open>Not good as automatic simprules because they cause case splits.\<close>
```
```  1246 lemmas divide_const_simps =
```
```  1247   le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
```
```  1248   divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
```
```  1249   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  1250
```
```  1251
```
```  1252 subsection \<open>The divides relation\<close>
```
```  1253
```
```  1254 lemma zdvd_antisym_nonneg:
```
```  1255     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
```
```  1256   apply (simp add: dvd_def, auto)
```
```  1257   apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  1258   done
```
```  1259
```
```  1260 lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
```
```  1261   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  1262 proof cases
```
```  1263   assume "a = 0" with assms show ?thesis by simp
```
```  1264 next
```
```  1265   assume "a \<noteq> 0"
```
```  1266   from \<open>a dvd b\<close> obtain k where k:"b = a*k" unfolding dvd_def by blast
```
```  1267   from \<open>b dvd a\<close> obtain k' where k':"a = b*k'" unfolding dvd_def by blast
```
```  1268   from k k' have "a = a*k*k'" by simp
```
```  1269   with mult_cancel_left1[where c="a" and b="k*k'"]
```
```  1270   have kk':"k*k' = 1" using \<open>a\<noteq>0\<close> by (simp add: mult.assoc)
```
```  1271   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
```
```  1272   thus ?thesis using k k' by auto
```
```  1273 qed
```
```  1274
```
```  1275 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
```
```  1276   using dvd_add_right_iff [of k "- n" m] by simp
```
```  1277
```
```  1278 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
```
```  1279   using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
```
```  1280
```
```  1281 lemma dvd_imp_le_int:
```
```  1282   fixes d i :: int
```
```  1283   assumes "i \<noteq> 0" and "d dvd i"
```
```  1284   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  1285 proof -
```
```  1286   from \<open>d dvd i\<close> obtain k where "i = d * k" ..
```
```  1287   with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
```
```  1288   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  1289   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  1290   with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
```
```  1291 qed
```
```  1292
```
```  1293 lemma zdvd_not_zless:
```
```  1294   fixes m n :: int
```
```  1295   assumes "0 < m" and "m < n"
```
```  1296   shows "\<not> n dvd m"
```
```  1297 proof
```
```  1298   from assms have "0 < n" by auto
```
```  1299   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  1300   with \<open>0 < m\<close> have "0 < n * k" by auto
```
```  1301   with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
```
```  1302   with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
```
```  1303   with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
```
```  1304 qed
```
```  1305
```
```  1306 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
```
```  1307   shows "m dvd n"
```
```  1308 proof-
```
```  1309   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
```
```  1310   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
```
```  1311     with h have False by (simp add: mult.assoc)}
```
```  1312   hence "n = m * h" by blast
```
```  1313   thus ?thesis by simp
```
```  1314 qed
```
```  1315
```
```  1316 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
```
```  1317 proof -
```
```  1318   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
```
```  1319   proof -
```
```  1320     fix k
```
```  1321     assume A: "int y = int x * k"
```
```  1322     then show "x dvd y"
```
```  1323     proof (cases k)
```
```  1324       case (nonneg n)
```
```  1325       with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
```
```  1326       then show ?thesis ..
```
```  1327     next
```
```  1328       case (neg n)
```
```  1329       with A have "int y = int x * (- int (Suc n))" by simp
```
```  1330       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
```
```  1331       also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
```
```  1332       finally have "- int (x * Suc n) = int y" ..
```
```  1333       then show ?thesis by (simp only: negative_eq_positive) auto
```
```  1334     qed
```
```  1335   qed
```
```  1336   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  1337 qed
```
```  1338
```
```  1339 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
```
```  1340 proof
```
```  1341   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  1342   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  1343   hence "nat \<bar>x\<bar> = 1"  by simp
```
```  1344   thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
```
```  1345 next
```
```  1346   assume "\<bar>x\<bar>=1"
```
```  1347   then have "x = 1 \<or> x = -1" by auto
```
```  1348   then show "x dvd 1" by (auto intro: dvdI)
```
```  1349 qed
```
```  1350
```
```  1351 lemma zdvd_mult_cancel1:
```
```  1352   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
```
```  1353 proof
```
```  1354   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
```
```  1355     by (cases "n >0") (auto simp add: minus_equation_iff)
```
```  1356 next
```
```  1357   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
```
```  1358   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
```
```  1359 qed
```
```  1360
```
```  1361 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
```
```  1362   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  1363
```
```  1364 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
```
```  1365   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  1366
```
```  1367 lemma dvd_int_unfold_dvd_nat:
```
```  1368   "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
```
```  1369   unfolding dvd_int_iff [symmetric] by simp
```
```  1370
```
```  1371 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
```
```  1372   by (auto simp add: dvd_int_iff)
```
```  1373
```
```  1374 lemma eq_nat_nat_iff:
```
```  1375   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  1376   by (auto elim!: nonneg_eq_int)
```
```  1377
```
```  1378 lemma nat_power_eq:
```
```  1379   "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  1380   by (induct n) (simp_all add: nat_mult_distrib)
```
```  1381
```
```  1382 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
```
```  1383   apply (cases n)
```
```  1384   apply (auto simp add: dvd_int_iff)
```
```  1385   apply (cases z)
```
```  1386   apply (auto simp add: dvd_imp_le)
```
```  1387   done
```
```  1388
```
```  1389 lemma zdvd_period:
```
```  1390   fixes a d :: int
```
```  1391   assumes "a dvd d"
```
```  1392   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  1393 proof -
```
```  1394   from assms obtain k where "d = a * k" by (rule dvdE)
```
```  1395   show ?thesis
```
```  1396   proof
```
```  1397     assume "a dvd (x + t)"
```
```  1398     then obtain l where "x + t = a * l" by (rule dvdE)
```
```  1399     then have "x = a * l - t" by simp
```
```  1400     with \<open>d = a * k\<close> show "a dvd x + c * d + t" by simp
```
```  1401   next
```
```  1402     assume "a dvd x + c * d + t"
```
```  1403     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
```
```  1404     then have "x = a * l - c * d - t" by simp
```
```  1405     with \<open>d = a * k\<close> show "a dvd (x + t)" by simp
```
```  1406   qed
```
```  1407 qed
```
```  1408
```
```  1409
```
```  1410 subsection \<open>Finiteness of intervals\<close>
```
```  1411
```
```  1412 lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
```
```  1413 proof (cases "a <= b")
```
```  1414   case True
```
```  1415   from this show ?thesis
```
```  1416   proof (induct b rule: int_ge_induct)
```
```  1417     case base
```
```  1418     have "{i. a <= i & i <= a} = {a}" by auto
```
```  1419     from this show ?case by simp
```
```  1420   next
```
```  1421     case (step b)
```
```  1422     from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
```
```  1423     from this step show ?case by simp
```
```  1424   qed
```
```  1425 next
```
```  1426   case False from this show ?thesis
```
```  1427     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
```
```  1428 qed
```
```  1429
```
```  1430 lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
```
```  1431 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1432
```
```  1433 lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
```
```  1434 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1435
```
```  1436 lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
```
```  1437 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1438
```
```  1439
```
```  1440 subsection \<open>Configuration of the code generator\<close>
```
```  1441
```
```  1442 text \<open>Constructors\<close>
```
```  1443
```
```  1444 definition Pos :: "num \<Rightarrow> int" where
```
```  1445   [simp, code_abbrev]: "Pos = numeral"
```
```  1446
```
```  1447 definition Neg :: "num \<Rightarrow> int" where
```
```  1448   [simp, code_abbrev]: "Neg n = - (Pos n)"
```
```  1449
```
```  1450 code_datatype "0::int" Pos Neg
```
```  1451
```
```  1452
```
```  1453 text \<open>Auxiliary operations\<close>
```
```  1454
```
```  1455 definition dup :: "int \<Rightarrow> int" where
```
```  1456   [simp]: "dup k = k + k"
```
```  1457
```
```  1458 lemma dup_code [code]:
```
```  1459   "dup 0 = 0"
```
```  1460   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```  1461   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```  1462   unfolding Pos_def Neg_def
```
```  1463   by (simp_all add: numeral_Bit0)
```
```  1464
```
```  1465 definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
```
```  1466   [simp]: "sub m n = numeral m - numeral n"
```
```  1467
```
```  1468 lemma sub_code [code]:
```
```  1469   "sub Num.One Num.One = 0"
```
```  1470   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```  1471   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```  1472   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```  1473   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```  1474   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```  1475   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```  1476   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```  1477   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```  1478   apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
```
```  1479   apply (simp_all only: algebra_simps minus_diff_eq)
```
```  1480   apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
```
```  1481   apply (simp_all only: minus_add add.assoc left_minus)
```
```  1482   done
```
```  1483
```
```  1484 text \<open>Implementations\<close>
```
```  1485
```
```  1486 lemma one_int_code [code, code_unfold]:
```
```  1487   "1 = Pos Num.One"
```
```  1488   by simp
```
```  1489
```
```  1490 lemma plus_int_code [code]:
```
```  1491   "k + 0 = (k::int)"
```
```  1492   "0 + l = (l::int)"
```
```  1493   "Pos m + Pos n = Pos (m + n)"
```
```  1494   "Pos m + Neg n = sub m n"
```
```  1495   "Neg m + Pos n = sub n m"
```
```  1496   "Neg m + Neg n = Neg (m + n)"
```
```  1497   by simp_all
```
```  1498
```
```  1499 lemma uminus_int_code [code]:
```
```  1500   "uminus 0 = (0::int)"
```
```  1501   "uminus (Pos m) = Neg m"
```
```  1502   "uminus (Neg m) = Pos m"
```
```  1503   by simp_all
```
```  1504
```
```  1505 lemma minus_int_code [code]:
```
```  1506   "k - 0 = (k::int)"
```
```  1507   "0 - l = uminus (l::int)"
```
```  1508   "Pos m - Pos n = sub m n"
```
```  1509   "Pos m - Neg n = Pos (m + n)"
```
```  1510   "Neg m - Pos n = Neg (m + n)"
```
```  1511   "Neg m - Neg n = sub n m"
```
```  1512   by simp_all
```
```  1513
```
```  1514 lemma times_int_code [code]:
```
```  1515   "k * 0 = (0::int)"
```
```  1516   "0 * l = (0::int)"
```
```  1517   "Pos m * Pos n = Pos (m * n)"
```
```  1518   "Pos m * Neg n = Neg (m * n)"
```
```  1519   "Neg m * Pos n = Neg (m * n)"
```
```  1520   "Neg m * Neg n = Pos (m * n)"
```
```  1521   by simp_all
```
```  1522
```
```  1523 instantiation int :: equal
```
```  1524 begin
```
```  1525
```
```  1526 definition
```
```  1527   "HOL.equal k l \<longleftrightarrow> k = (l::int)"
```
```  1528
```
```  1529 instance by default (rule equal_int_def)
```
```  1530
```
```  1531 end
```
```  1532
```
```  1533 lemma equal_int_code [code]:
```
```  1534   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
```
```  1535   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```  1536   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```  1537   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```  1538   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```  1539   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```  1540   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```  1541   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```  1542   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```  1543   by (auto simp add: equal)
```
```  1544
```
```  1545 lemma equal_int_refl [code nbe]:
```
```  1546   "HOL.equal (k::int) k \<longleftrightarrow> True"
```
```  1547   by (fact equal_refl)
```
```  1548
```
```  1549 lemma less_eq_int_code [code]:
```
```  1550   "0 \<le> (0::int) \<longleftrightarrow> True"
```
```  1551   "0 \<le> Pos l \<longleftrightarrow> True"
```
```  1552   "0 \<le> Neg l \<longleftrightarrow> False"
```
```  1553   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```  1554   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```  1555   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```  1556   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```  1557   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```  1558   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```  1559   by simp_all
```
```  1560
```
```  1561 lemma less_int_code [code]:
```
```  1562   "0 < (0::int) \<longleftrightarrow> False"
```
```  1563   "0 < Pos l \<longleftrightarrow> True"
```
```  1564   "0 < Neg l \<longleftrightarrow> False"
```
```  1565   "Pos k < 0 \<longleftrightarrow> False"
```
```  1566   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```  1567   "Pos k < Neg l \<longleftrightarrow> False"
```
```  1568   "Neg k < 0 \<longleftrightarrow> True"
```
```  1569   "Neg k < Pos l \<longleftrightarrow> True"
```
```  1570   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```  1571   by simp_all
```
```  1572
```
```  1573 lemma nat_code [code]:
```
```  1574   "nat (Int.Neg k) = 0"
```
```  1575   "nat 0 = 0"
```
```  1576   "nat (Int.Pos k) = nat_of_num k"
```
```  1577   by (simp_all add: nat_of_num_numeral)
```
```  1578
```
```  1579 lemma (in ring_1) of_int_code [code]:
```
```  1580   "of_int (Int.Neg k) = - numeral k"
```
```  1581   "of_int 0 = 0"
```
```  1582   "of_int (Int.Pos k) = numeral k"
```
```  1583   by simp_all
```
```  1584
```
```  1585
```
```  1586 text \<open>Serializer setup\<close>
```
```  1587
```
```  1588 code_identifier
```
```  1589   code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1590
```
```  1591 quickcheck_params [default_type = int]
```
```  1592
```
```  1593 hide_const (open) Pos Neg sub dup
```
```  1594
```
```  1595
```
```  1596 subsection \<open>Legacy theorems\<close>
```
```  1597
```
```  1598 lemmas inj_int = inj_of_nat [where 'a=int]
```
```  1599 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```  1600 lemmas int_mult = of_nat_mult [where 'a=int]
```
```  1601 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```  1602 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
```
```  1603 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```  1604 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
```
```  1605 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```  1606 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```  1607 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
```
```  1608 lemmas int_0 = of_nat_0 [where 'a=int]
```
```  1609 lemmas int_1 = of_nat_1 [where 'a=int]
```
```  1610 lemmas int_Suc = of_nat_Suc [where 'a=int]
```
```  1611 lemmas int_numeral = of_nat_numeral [where 'a=int]
```
```  1612 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
```
```  1613 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```  1614 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```  1615 lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
```
```  1616 lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
```
```  1617
```
```  1618 lemma zpower_zpower:
```
```  1619   "(x ^ y) ^ z = (x ^ (y * z)::int)"
```
```  1620   by (rule power_mult [symmetric])
```
```  1621
```
```  1622 lemma int_power:
```
```  1623   "int (m ^ n) = int m ^ n"
```
```  1624   by (fact of_nat_power)
```
```  1625
```
```  1626 lemmas zpower_int = int_power [symmetric]
```
```  1627
```
```  1628 text \<open>De-register @{text "int"} as a quotient type:\<close>
```
```  1629
```
```  1630 lifting_update int.lifting
```
```  1631 lifting_forget int.lifting
```
```  1632
```
```  1633 text\<open>Also the class for fields with characteristic zero.\<close>
```
```  1634 class field_char_0 = field + ring_char_0
```
```  1635 subclass (in linordered_field) field_char_0 ..
```
```  1636
```
```  1637 end
```