src/HOL/Int.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 49962 a8cc904a6820 child 51112 da97167e03f7 permissions -rw-r--r--
introduce order topology
```     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 header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *}
```
```     7
```
```     8 theory Int
```
```     9 imports Equiv_Relations Wellfounded Quotient
```
```    10 begin
```
```    11
```
```    12 subsection {* Definition of integers as a quotient type *}
```
```    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 {* Integers form a commutative ring *}
```
```    36
```
```    37 instantiation int :: comm_ring_1
```
```    38 begin
```
```    39
```
```    40 lift_definition zero_int :: "int" is "(0, 0)"
```
```    41   by simp
```
```    42
```
```    43 lift_definition one_int :: "int" is "(1, 0)"
```
```    44   by simp
```
```    45
```
```    46 lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    47   is "\<lambda>(x, y) (u, v). (x + u, y + v)"
```
```    48   by clarsimp
```
```    49
```
```    50 lift_definition uminus_int :: "int \<Rightarrow> int"
```
```    51   is "\<lambda>(x, y). (y, x)"
```
```    52   by clarsimp
```
```    53
```
```    54 lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    55   is "\<lambda>(x, y) (u, v). (x + v, y + u)"
```
```    56   by clarsimp
```
```    57
```
```    58 lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    59   is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
```
```    60 proof (clarsimp)
```
```    61   fix s t u v w x y z :: nat
```
```    62   assume "s + v = u + t" and "w + z = y + x"
```
```    63   hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
```
```    64        = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
```
```    65     by simp
```
```    66   thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
```
```    67     by (simp add: algebra_simps)
```
```    68 qed
```
```    69
```
```    70 instance
```
```    71   by default (transfer, clarsimp simp: algebra_simps)+
```
```    72
```
```    73 end
```
```    74
```
```    75 abbreviation int :: "nat \<Rightarrow> int" where
```
```    76   "int \<equiv> of_nat"
```
```    77
```
```    78 lemma int_def: "int n = Abs_Integ (n, 0)"
```
```    79   by (induct n, simp add: zero_int.abs_eq,
```
```    80     simp add: one_int.abs_eq plus_int.abs_eq)
```
```    81
```
```    82 lemma int_transfer [transfer_rule]:
```
```    83   "(fun_rel (op =) cr_int) (\<lambda>n. (n, 0)) int"
```
```    84   unfolding fun_rel_def cr_int_def int_def by simp
```
```    85
```
```    86 lemma int_diff_cases:
```
```    87   obtains (diff) m n where "z = int m - int n"
```
```    88   by transfer clarsimp
```
```    89
```
```    90 subsection {* Integers are totally ordered *}
```
```    91
```
```    92 instantiation int :: linorder
```
```    93 begin
```
```    94
```
```    95 lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    96   is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
```
```    97   by auto
```
```    98
```
```    99 lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```   100   is "\<lambda>(x, y) (u, v). x + v < u + y"
```
```   101   by auto
```
```   102
```
```   103 instance
```
```   104   by default (transfer, force)+
```
```   105
```
```   106 end
```
```   107
```
```   108 instantiation int :: distrib_lattice
```
```   109 begin
```
```   110
```
```   111 definition
```
```   112   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   113
```
```   114 definition
```
```   115   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   116
```
```   117 instance
```
```   118   by intro_classes
```
```   119     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   120
```
```   121 end
```
```   122
```
```   123 subsection {* Ordering properties of arithmetic operations *}
```
```   124
```
```   125 instance int :: ordered_cancel_ab_semigroup_add
```
```   126 proof
```
```   127   fix i j k :: int
```
```   128   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   129     by transfer clarsimp
```
```   130 qed
```
```   131
```
```   132 text{*Strict Monotonicity of Multiplication*}
```
```   133
```
```   134 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   135 lemma zmult_zless_mono2_lemma:
```
```   136      "(i::int)<j ==> 0<k ==> int k * i < int k * j"
```
```   137 apply (induct k)
```
```   138 apply simp
```
```   139 apply (simp add: distrib_right)
```
```   140 apply (case_tac "k=0")
```
```   141 apply (simp_all add: add_strict_mono)
```
```   142 done
```
```   143
```
```   144 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
```
```   145 apply transfer
```
```   146 apply clarsimp
```
```   147 apply (rule_tac x="a - b" in exI, simp)
```
```   148 done
```
```   149
```
```   150 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
```
```   151 apply transfer
```
```   152 apply clarsimp
```
```   153 apply (rule_tac x="a - b" in exI, simp)
```
```   154 done
```
```   155
```
```   156 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   157 apply (drule zero_less_imp_eq_int)
```
```   158 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   159 done
```
```   160
```
```   161 text{*The integers form an ordered integral domain*}
```
```   162 instantiation int :: linordered_idom
```
```   163 begin
```
```   164
```
```   165 definition
```
```   166   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
```
```   167
```
```   168 definition
```
```   169   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   170
```
```   171 instance proof
```
```   172   fix i j k :: int
```
```   173   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   174     by (rule zmult_zless_mono2)
```
```   175   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
```
```   176     by (simp only: zabs_def)
```
```   177   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   178     by (simp only: zsgn_def)
```
```   179 qed
```
```   180
```
```   181 end
```
```   182
```
```   183 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
```
```   184   by transfer clarsimp
```
```   185
```
```   186 lemma zless_iff_Suc_zadd:
```
```   187   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
```
```   188 apply transfer
```
```   189 apply auto
```
```   190 apply (rename_tac a b c d)
```
```   191 apply (rule_tac x="c+b - Suc(a+d)" in exI)
```
```   192 apply arith
```
```   193 done
```
```   194
```
```   195 lemmas int_distrib =
```
```   196   distrib_right [of z1 z2 w]
```
```   197   distrib_left [of w z1 z2]
```
```   198   left_diff_distrib [of z1 z2 w]
```
```   199   right_diff_distrib [of w z1 z2]
```
```   200   for z1 z2 w :: int
```
```   201
```
```   202
```
```   203 subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
```
```   204
```
```   205 context ring_1
```
```   206 begin
```
```   207
```
```   208 lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
```
```   209   by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
```
```   210     of_nat_add [symmetric] simp del: of_nat_add)
```
```   211
```
```   212 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   213   by transfer simp
```
```   214
```
```   215 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   216   by transfer simp
```
```   217
```
```   218 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   219   by transfer (clarsimp simp add: algebra_simps)
```
```   220
```
```   221 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   222   by (transfer fixing: uminus) clarsimp
```
```   223
```
```   224 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   225 by (simp add: diff_minus Groups.diff_minus)
```
```   226
```
```   227 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   228   by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
```
```   229
```
```   230 text{*Collapse nested embeddings*}
```
```   231 lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
```
```   232 by (induct n) auto
```
```   233
```
```   234 lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
```
```   235   by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
```
```   236
```
```   237 lemma of_int_neg_numeral [simp, code_post]: "of_int (neg_numeral k) = neg_numeral k"
```
```   238   unfolding neg_numeral_def neg_numeral_class.neg_numeral_def
```
```   239   by (simp only: of_int_minus of_int_numeral)
```
```   240
```
```   241 lemma of_int_power:
```
```   242   "of_int (z ^ n) = of_int z ^ n"
```
```   243   by (induct n) simp_all
```
```   244
```
```   245 end
```
```   246
```
```   247 context ring_char_0
```
```   248 begin
```
```   249
```
```   250 lemma of_int_eq_iff [simp]:
```
```   251    "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   252   by transfer (clarsimp simp add: algebra_simps
```
```   253     of_nat_add [symmetric] simp del: of_nat_add)
```
```   254
```
```   255 text{*Special cases where either operand is zero*}
```
```   256 lemma of_int_eq_0_iff [simp]:
```
```   257   "of_int z = 0 \<longleftrightarrow> z = 0"
```
```   258   using of_int_eq_iff [of z 0] by simp
```
```   259
```
```   260 lemma of_int_0_eq_iff [simp]:
```
```   261   "0 = of_int z \<longleftrightarrow> z = 0"
```
```   262   using of_int_eq_iff [of 0 z] by simp
```
```   263
```
```   264 end
```
```   265
```
```   266 context linordered_idom
```
```   267 begin
```
```   268
```
```   269 text{*Every @{text linordered_idom} has characteristic zero.*}
```
```   270 subclass ring_char_0 ..
```
```   271
```
```   272 lemma of_int_le_iff [simp]:
```
```   273   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   274   by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
```
```   275     of_nat_add [symmetric] simp del: of_nat_add)
```
```   276
```
```   277 lemma of_int_less_iff [simp]:
```
```   278   "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   279   by (simp add: less_le order_less_le)
```
```   280
```
```   281 lemma of_int_0_le_iff [simp]:
```
```   282   "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
```
```   283   using of_int_le_iff [of 0 z] by simp
```
```   284
```
```   285 lemma of_int_le_0_iff [simp]:
```
```   286   "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
```
```   287   using of_int_le_iff [of z 0] by simp
```
```   288
```
```   289 lemma of_int_0_less_iff [simp]:
```
```   290   "0 < of_int z \<longleftrightarrow> 0 < z"
```
```   291   using of_int_less_iff [of 0 z] by simp
```
```   292
```
```   293 lemma of_int_less_0_iff [simp]:
```
```   294   "of_int z < 0 \<longleftrightarrow> z < 0"
```
```   295   using of_int_less_iff [of z 0] by simp
```
```   296
```
```   297 end
```
```   298
```
```   299 lemma of_int_eq_id [simp]: "of_int = id"
```
```   300 proof
```
```   301   fix z show "of_int z = id z"
```
```   302     by (cases z rule: int_diff_cases, simp)
```
```   303 qed
```
```   304
```
```   305
```
```   306 subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
```
```   307
```
```   308 lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
```
```   309   by auto
```
```   310
```
```   311 lemma nat_int [simp]: "nat (int n) = n"
```
```   312   by transfer simp
```
```   313
```
```   314 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   315   by transfer clarsimp
```
```   316
```
```   317 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
```
```   318 by simp
```
```   319
```
```   320 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   321   by transfer clarsimp
```
```   322
```
```   323 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   324   by transfer (clarsimp, arith)
```
```   325
```
```   326 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   327 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   328 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   329
```
```   330 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   331 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   332
```
```   333 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   334   by transfer (clarsimp, arith)
```
```   335
```
```   336 lemma nonneg_eq_int:
```
```   337   fixes z :: int
```
```   338   assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
```
```   339   shows P
```
```   340   using assms by (blast dest: nat_0_le sym)
```
```   341
```
```   342 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
```
```   343   by transfer (clarsimp simp add: le_imp_diff_is_add)
```
```   344
```
```   345 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
```
```   346 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   347
```
```   348 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
```
```   349   by transfer (clarsimp, arith)
```
```   350
```
```   351 lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
```
```   352   by transfer (clarsimp simp add: le_diff_conv)
```
```   353
```
```   354 lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
```
```   355   by transfer auto
```
```   356
```
```   357 lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
```
```   358   by transfer clarsimp
```
```   359
```
```   360 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
```
```   361 by (auto simp add: nat_eq_iff2)
```
```   362
```
```   363 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   364 by (insert zless_nat_conj [of 0], auto)
```
```   365
```
```   366 lemma nat_add_distrib:
```
```   367      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   368   by transfer clarsimp
```
```   369
```
```   370 lemma nat_diff_distrib:
```
```   371      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   372   by transfer clarsimp
```
```   373
```
```   374 lemma nat_zminus_int [simp]: "nat (- int n) = 0"
```
```   375   by transfer simp
```
```   376
```
```   377 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
```
```   378   by transfer (clarsimp simp add: less_diff_conv)
```
```   379
```
```   380 context ring_1
```
```   381 begin
```
```   382
```
```   383 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   384   by transfer (clarsimp simp add: of_nat_diff)
```
```   385
```
```   386 end
```
```   387
```
```   388 text {* For termination proofs: *}
```
```   389 lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
```
```   390
```
```   391
```
```   392 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
```
```   393
```
```   394 lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
```
```   395 by (simp add: order_less_le del: of_nat_Suc)
```
```   396
```
```   397 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   398 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   399
```
```   400 lemma negative_zle_0: "- int n \<le> 0"
```
```   401 by (simp add: minus_le_iff)
```
```   402
```
```   403 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   404 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   405
```
```   406 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
```
```   407 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   408
```
```   409 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
```
```   410   by transfer simp
```
```   411
```
```   412 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
```
```   413 by (simp add: linorder_not_less)
```
```   414
```
```   415 lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
```
```   416 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   417
```
```   418 lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
```
```   419 proof -
```
```   420   have "(w \<le> z) = (0 \<le> z - w)"
```
```   421     by (simp only: le_diff_eq add_0_left)
```
```   422   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
```
```   423     by (auto elim: zero_le_imp_eq_int)
```
```   424   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
```
```   425     by (simp only: algebra_simps)
```
```   426   finally show ?thesis .
```
```   427 qed
```
```   428
```
```   429 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
```
```   430 by simp
```
```   431
```
```   432 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
```
```   433 by simp
```
```   434
```
```   435 text{*This version is proved for all ordered rings, not just integers!
```
```   436       It is proved here because attribute @{text arith_split} is not available
```
```   437       in theory @{text Rings}.
```
```   438       But is it really better than just rewriting with @{text abs_if}?*}
```
```   439 lemma abs_split [arith_split,no_atp]:
```
```   440      "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   441 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   442
```
```   443 lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
```
```   444 apply transfer
```
```   445 apply clarsimp
```
```   446 apply (rule_tac x="b - Suc a" in exI, arith)
```
```   447 done
```
```   448
```
```   449
```
```   450 subsection {* Cases and induction *}
```
```   451
```
```   452 text{*Now we replace the case analysis rule by a more conventional one:
```
```   453 whether an integer is negative or not.*}
```
```   454
```
```   455 theorem int_cases [case_names nonneg neg, cases type: int]:
```
```   456   "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
```
```   457 apply (cases "z < 0")
```
```   458 apply (blast dest!: negD)
```
```   459 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   460 apply auto
```
```   461 apply (blast dest: nat_0_le [THEN sym])
```
```   462 done
```
```   463
```
```   464 theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
```
```   465      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
```
```   466   by (cases z) auto
```
```   467
```
```   468 lemma nonneg_int_cases:
```
```   469   assumes "0 \<le> k" obtains n where "k = int n"
```
```   470   using assms by (cases k, simp, simp del: of_nat_Suc)
```
```   471
```
```   472 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```   473   -- {* Unfold all @{text let}s involving constants *}
```
```   474   unfolding Let_def ..
```
```   475
```
```   476 lemma Let_neg_numeral [simp]: "Let (neg_numeral v) f = f (neg_numeral v)"
```
```   477   -- {* Unfold all @{text let}s involving constants *}
```
```   478   unfolding Let_def ..
```
```   479
```
```   480 text {* Unfold @{text min} and @{text max} on numerals. *}
```
```   481
```
```   482 lemmas max_number_of [simp] =
```
```   483   max_def [of "numeral u" "numeral v"]
```
```   484   max_def [of "numeral u" "neg_numeral v"]
```
```   485   max_def [of "neg_numeral u" "numeral v"]
```
```   486   max_def [of "neg_numeral u" "neg_numeral v"] for u v
```
```   487
```
```   488 lemmas min_number_of [simp] =
```
```   489   min_def [of "numeral u" "numeral v"]
```
```   490   min_def [of "numeral u" "neg_numeral v"]
```
```   491   min_def [of "neg_numeral u" "numeral v"]
```
```   492   min_def [of "neg_numeral u" "neg_numeral v"] for u v
```
```   493
```
```   494
```
```   495 subsubsection {* Binary comparisons *}
```
```   496
```
```   497 text {* Preliminaries *}
```
```   498
```
```   499 lemma even_less_0_iff:
```
```   500   "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
```
```   501 proof -
```
```   502   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one)
```
```   503   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
```
```   504     by (simp add: mult_less_0_iff zero_less_two
```
```   505                   order_less_not_sym [OF zero_less_two])
```
```   506   finally show ?thesis .
```
```   507 qed
```
```   508
```
```   509 lemma le_imp_0_less:
```
```   510   assumes le: "0 \<le> z"
```
```   511   shows "(0::int) < 1 + z"
```
```   512 proof -
```
```   513   have "0 \<le> z" by fact
```
```   514   also have "... < z + 1" by (rule less_add_one)
```
```   515   also have "... = 1 + z" by (simp add: add_ac)
```
```   516   finally show "0 < 1 + z" .
```
```   517 qed
```
```   518
```
```   519 lemma odd_less_0_iff:
```
```   520   "(1 + z + z < 0) = (z < (0::int))"
```
```   521 proof (cases z)
```
```   522   case (nonneg n)
```
```   523   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```   524                              le_imp_0_less [THEN order_less_imp_le])
```
```   525 next
```
```   526   case (neg n)
```
```   527   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   528     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   529 qed
```
```   530
```
```   531 subsubsection {* Comparisons, for Ordered Rings *}
```
```   532
```
```   533 lemmas double_eq_0_iff = double_zero
```
```   534
```
```   535 lemma odd_nonzero:
```
```   536   "1 + z + z \<noteq> (0::int)"
```
```   537 proof (cases z)
```
```   538   case (nonneg n)
```
```   539   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```   540   thus ?thesis using  le_imp_0_less [OF le]
```
```   541     by (auto simp add: add_assoc)
```
```   542 next
```
```   543   case (neg n)
```
```   544   show ?thesis
```
```   545   proof
```
```   546     assume eq: "1 + z + z = 0"
```
```   547     have "(0::int) < 1 + (int n + int n)"
```
```   548       by (simp add: le_imp_0_less add_increasing)
```
```   549     also have "... = - (1 + z + z)"
```
```   550       by (simp add: neg add_assoc [symmetric])
```
```   551     also have "... = 0" by (simp add: eq)
```
```   552     finally have "0<0" ..
```
```   553     thus False by blast
```
```   554   qed
```
```   555 qed
```
```   556
```
```   557
```
```   558 subsection {* The Set of Integers *}
```
```   559
```
```   560 context ring_1
```
```   561 begin
```
```   562
```
```   563 definition Ints  :: "'a set" where
```
```   564   "Ints = range of_int"
```
```   565
```
```   566 notation (xsymbols)
```
```   567   Ints  ("\<int>")
```
```   568
```
```   569 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```   570   by (simp add: Ints_def)
```
```   571
```
```   572 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```   573   using Ints_of_int [of "of_nat n"] by simp
```
```   574
```
```   575 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```   576   using Ints_of_int [of "0"] by simp
```
```   577
```
```   578 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```   579   using Ints_of_int [of "1"] by simp
```
```   580
```
```   581 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```   582 apply (auto simp add: Ints_def)
```
```   583 apply (rule range_eqI)
```
```   584 apply (rule of_int_add [symmetric])
```
```   585 done
```
```   586
```
```   587 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```   588 apply (auto simp add: Ints_def)
```
```   589 apply (rule range_eqI)
```
```   590 apply (rule of_int_minus [symmetric])
```
```   591 done
```
```   592
```
```   593 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```   594 apply (auto simp add: Ints_def)
```
```   595 apply (rule range_eqI)
```
```   596 apply (rule of_int_diff [symmetric])
```
```   597 done
```
```   598
```
```   599 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```   600 apply (auto simp add: Ints_def)
```
```   601 apply (rule range_eqI)
```
```   602 apply (rule of_int_mult [symmetric])
```
```   603 done
```
```   604
```
```   605 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```   606 by (induct n) simp_all
```
```   607
```
```   608 lemma Ints_cases [cases set: Ints]:
```
```   609   assumes "q \<in> \<int>"
```
```   610   obtains (of_int) z where "q = of_int z"
```
```   611   unfolding Ints_def
```
```   612 proof -
```
```   613   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```   614   then obtain z where "q = of_int z" ..
```
```   615   then show thesis ..
```
```   616 qed
```
```   617
```
```   618 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   619   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```   620   by (rule Ints_cases) auto
```
```   621
```
```   622 end
```
```   623
```
```   624 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```   625
```
```   626 lemma Ints_double_eq_0_iff:
```
```   627   assumes in_Ints: "a \<in> Ints"
```
```   628   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```   629 proof -
```
```   630   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   631   then obtain z where a: "a = of_int z" ..
```
```   632   show ?thesis
```
```   633   proof
```
```   634     assume "a = 0"
```
```   635     thus "a + a = 0" by simp
```
```   636   next
```
```   637     assume eq: "a + a = 0"
```
```   638     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   639     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```   640     hence "z = 0" by (simp only: double_eq_0_iff)
```
```   641     thus "a = 0" by (simp add: a)
```
```   642   qed
```
```   643 qed
```
```   644
```
```   645 lemma Ints_odd_nonzero:
```
```   646   assumes in_Ints: "a \<in> Ints"
```
```   647   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```   648 proof -
```
```   649   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   650   then obtain z where a: "a = of_int z" ..
```
```   651   show ?thesis
```
```   652   proof
```
```   653     assume eq: "1 + a + a = 0"
```
```   654     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   655     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   656     with odd_nonzero show False by blast
```
```   657   qed
```
```   658 qed
```
```   659
```
```   660 lemma Nats_numeral [simp]: "numeral w \<in> Nats"
```
```   661   using of_nat_in_Nats [of "numeral w"] by simp
```
```   662
```
```   663 lemma Ints_odd_less_0:
```
```   664   assumes in_Ints: "a \<in> Ints"
```
```   665   shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
```
```   666 proof -
```
```   667   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   668   then obtain z where a: "a = of_int z" ..
```
```   669   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```   670     by (simp add: a)
```
```   671   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
```
```   672   also have "... = (a < 0)" by (simp add: a)
```
```   673   finally show ?thesis .
```
```   674 qed
```
```   675
```
```   676
```
```   677 subsection {* @{term setsum} and @{term setprod} *}
```
```   678
```
```   679 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   680   apply (cases "finite A")
```
```   681   apply (erule finite_induct, auto)
```
```   682   done
```
```   683
```
```   684 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   685   apply (cases "finite A")
```
```   686   apply (erule finite_induct, auto)
```
```   687   done
```
```   688
```
```   689 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   690   apply (cases "finite A")
```
```   691   apply (erule finite_induct, auto simp add: of_nat_mult)
```
```   692   done
```
```   693
```
```   694 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   695   apply (cases "finite A")
```
```   696   apply (erule finite_induct, auto)
```
```   697   done
```
```   698
```
```   699 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```   700 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```   701
```
```   702
```
```   703 text {* Legacy theorems *}
```
```   704
```
```   705 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```   706 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```   707 lemmas numeral_1_eq_1 = numeral_One
```
```   708
```
```   709 subsection {* Setting up simplification procedures *}
```
```   710
```
```   711 lemmas int_arith_rules =
```
```   712   neg_le_iff_le numeral_One
```
```   713   minus_zero diff_minus left_minus right_minus
```
```   714   mult_zero_left mult_zero_right mult_1_left mult_1_right
```
```   715   mult_minus_left mult_minus_right
```
```   716   minus_add_distrib minus_minus mult_assoc
```
```   717   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
```
```   718   of_int_0 of_int_1 of_int_add of_int_mult
```
```   719
```
```   720 ML_file "Tools/int_arith.ML"
```
```   721 declaration {* K Int_Arith.setup *}
```
```   722
```
```   723 simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
```
```   724   "(m::'a::linordered_idom) <= n" |
```
```   725   "(m::'a::linordered_idom) = n") =
```
```   726   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
```
```   727
```
```   728
```
```   729 subsection{*Lemmas About Small Numerals*}
```
```   730
```
```   731 lemma abs_power_minus_one [simp]:
```
```   732   "abs(-1 ^ n) = (1::'a::linordered_idom)"
```
```   733 by (simp add: power_abs)
```
```   734
```
```   735
```
```   736 subsection{*More Inequality Reasoning*}
```
```   737
```
```   738 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```   739 by arith
```
```   740
```
```   741 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```   742 by arith
```
```   743
```
```   744 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```   745 by arith
```
```   746
```
```   747 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```   748 by arith
```
```   749
```
```   750 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```   751 by arith
```
```   752
```
```   753
```
```   754 subsection{*The functions @{term nat} and @{term int}*}
```
```   755
```
```   756 text{*Simplify the term @{term "w + - z"}*}
```
```   757 lemmas diff_int_def_symmetric = diff_def [where 'a=int, symmetric, simp]
```
```   758
```
```   759 lemma nat_0 [simp]: "nat 0 = 0"
```
```   760 by (simp add: nat_eq_iff)
```
```   761
```
```   762 lemma nat_1 [simp]: "nat 1 = Suc 0"
```
```   763 by (subst nat_eq_iff, simp)
```
```   764
```
```   765 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   766 by (subst nat_eq_iff, simp)
```
```   767
```
```   768 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```   769 apply (insert zless_nat_conj [of 1 z])
```
```   770 apply auto
```
```   771 done
```
```   772
```
```   773 text{*This simplifies expressions of the form @{term "int n = z"} where
```
```   774       z is an integer literal.*}
```
```   775 lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
```
```   776
```
```   777 lemma split_nat [arith_split]:
```
```   778   "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```   779   (is "?P = (?L & ?R)")
```
```   780 proof (cases "i < 0")
```
```   781   case True thus ?thesis by auto
```
```   782 next
```
```   783   case False
```
```   784   have "?P = ?L"
```
```   785   proof
```
```   786     assume ?P thus ?L using False by clarsimp
```
```   787   next
```
```   788     assume ?L thus ?P using False by simp
```
```   789   qed
```
```   790   with False show ?thesis by simp
```
```   791 qed
```
```   792
```
```   793 context ring_1
```
```   794 begin
```
```   795
```
```   796 lemma of_int_of_nat [nitpick_simp]:
```
```   797   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```   798 proof (cases "k < 0")
```
```   799   case True then have "0 \<le> - k" by simp
```
```   800   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```   801   with True show ?thesis by simp
```
```   802 next
```
```   803   case False then show ?thesis by (simp add: not_less of_nat_nat)
```
```   804 qed
```
```   805
```
```   806 end
```
```   807
```
```   808 lemma nat_mult_distrib:
```
```   809   fixes z z' :: int
```
```   810   assumes "0 \<le> z"
```
```   811   shows "nat (z * z') = nat z * nat z'"
```
```   812 proof (cases "0 \<le> z'")
```
```   813   case False with assms have "z * z' \<le> 0"
```
```   814     by (simp add: not_le mult_le_0_iff)
```
```   815   then have "nat (z * z') = 0" by simp
```
```   816   moreover from False have "nat z' = 0" by simp
```
```   817   ultimately show ?thesis by simp
```
```   818 next
```
```   819   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```   820   show ?thesis
```
```   821     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```   822       (simp only: of_nat_mult of_nat_nat [OF True]
```
```   823          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```   824 qed
```
```   825
```
```   826 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```   827 apply (rule trans)
```
```   828 apply (rule_tac  nat_mult_distrib, auto)
```
```   829 done
```
```   830
```
```   831 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```   832 apply (cases "z=0 | w=0")
```
```   833 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```   834                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```   835 done
```
```   836
```
```   837 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
```
```   838 apply (rule sym)
```
```   839 apply (simp add: nat_eq_iff)
```
```   840 done
```
```   841
```
```   842 lemma diff_nat_eq_if:
```
```   843      "nat z - nat z' =
```
```   844         (if z' < 0 then nat z
```
```   845          else let d = z-z' in
```
```   846               if d < 0 then 0 else nat d)"
```
```   847 by (simp add: Let_def nat_diff_distrib [symmetric])
```
```   848
```
```   849 (* nat_diff_distrib has too-strong premises *)
```
```   850 lemma nat_diff_distrib': "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x - y) = nat x - nat y"
```
```   851 apply (rule int_int_eq [THEN iffD1], clarsimp)
```
```   852 apply (subst of_nat_diff)
```
```   853 apply (rule nat_mono, simp_all)
```
```   854 done
```
```   855
```
```   856 lemma nat_numeral [simp, code_abbrev]:
```
```   857   "nat (numeral k) = numeral k"
```
```   858   by (simp add: nat_eq_iff)
```
```   859
```
```   860 lemma nat_neg_numeral [simp]:
```
```   861   "nat (neg_numeral k) = 0"
```
```   862   by simp
```
```   863
```
```   864 lemma diff_nat_numeral [simp]:
```
```   865   "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
```
```   866   by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
```
```   867
```
```   868 lemma nat_numeral_diff_1 [simp]:
```
```   869   "numeral v - (1::nat) = nat (numeral v - 1)"
```
```   870   using diff_nat_numeral [of v Num.One] by simp
```
```   871
```
```   872 lemmas nat_arith = diff_nat_numeral
```
```   873
```
```   874
```
```   875 subsection "Induction principles for int"
```
```   876
```
```   877 text{*Well-founded segments of the integers*}
```
```   878
```
```   879 definition
```
```   880   int_ge_less_than  ::  "int => (int * int) set"
```
```   881 where
```
```   882   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
```
```   883
```
```   884 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```   885 proof -
```
```   886   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
```
```   887     by (auto simp add: int_ge_less_than_def)
```
```   888   thus ?thesis
```
```   889     by (rule wf_subset [OF wf_measure])
```
```   890 qed
```
```   891
```
```   892 text{*This variant looks odd, but is typical of the relations suggested
```
```   893 by RankFinder.*}
```
```   894
```
```   895 definition
```
```   896   int_ge_less_than2 ::  "int => (int * int) set"
```
```   897 where
```
```   898   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
```
```   899
```
```   900 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```   901 proof -
```
```   902   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
```
```   903     by (auto simp add: int_ge_less_than2_def)
```
```   904   thus ?thesis
```
```   905     by (rule wf_subset [OF wf_measure])
```
```   906 qed
```
```   907
```
```   908 (* `set:int': dummy construction *)
```
```   909 theorem int_ge_induct [case_names base step, induct set: int]:
```
```   910   fixes i :: int
```
```   911   assumes ge: "k \<le> i" and
```
```   912     base: "P k" and
```
```   913     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```   914   shows "P i"
```
```   915 proof -
```
```   916   { fix n
```
```   917     have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```   918     proof (induct n)
```
```   919       case 0
```
```   920       hence "i = k" by arith
```
```   921       thus "P i" using base by simp
```
```   922     next
```
```   923       case (Suc n)
```
```   924       then have "n = nat((i - 1) - k)" by arith
```
```   925       moreover
```
```   926       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```   927       ultimately
```
```   928       have "P (i - 1)" by (rule Suc.hyps)
```
```   929       from step [OF ki1 this] show ?case by simp
```
```   930     qed
```
```   931   }
```
```   932   with ge show ?thesis by fast
```
```   933 qed
```
```   934
```
```   935 (* `set:int': dummy construction *)
```
```   936 theorem int_gr_induct [case_names base step, induct set: int]:
```
```   937   assumes gr: "k < (i::int)" and
```
```   938         base: "P(k+1)" and
```
```   939         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   940   shows "P i"
```
```   941 apply(rule int_ge_induct[of "k + 1"])
```
```   942   using gr apply arith
```
```   943  apply(rule base)
```
```   944 apply (rule step, simp+)
```
```   945 done
```
```   946
```
```   947 theorem int_le_induct [consumes 1, case_names base step]:
```
```   948   assumes le: "i \<le> (k::int)" and
```
```   949         base: "P(k)" and
```
```   950         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   951   shows "P i"
```
```   952 proof -
```
```   953   { fix n
```
```   954     have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<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       hence "n = nat (k - (i + 1))" by arith
```
```   962       moreover
```
```   963       have ki1: "i + 1 \<le> k" 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 le show ?thesis by fast
```
```   970 qed
```
```   971
```
```   972 theorem int_less_induct [consumes 1, case_names base step]:
```
```   973   assumes less: "(i::int) < k" and
```
```   974         base: "P(k - 1)" and
```
```   975         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   976   shows "P i"
```
```   977 apply(rule int_le_induct[of _ "k - 1"])
```
```   978   using less apply arith
```
```   979  apply(rule base)
```
```   980 apply (rule step, simp+)
```
```   981 done
```
```   982
```
```   983 theorem int_induct [case_names base step1 step2]:
```
```   984   fixes k :: int
```
```   985   assumes base: "P k"
```
```   986     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```   987     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```   988   shows "P i"
```
```   989 proof -
```
```   990   have "i \<le> k \<or> i \<ge> k" by arith
```
```   991   then show ?thesis
```
```   992   proof
```
```   993     assume "i \<ge> k"
```
```   994     then show ?thesis using base
```
```   995       by (rule int_ge_induct) (fact step1)
```
```   996   next
```
```   997     assume "i \<le> k"
```
```   998     then show ?thesis using base
```
```   999       by (rule int_le_induct) (fact step2)
```
```  1000   qed
```
```  1001 qed
```
```  1002
```
```  1003 subsection{*Intermediate value theorems*}
```
```  1004
```
```  1005 lemma int_val_lemma:
```
```  1006      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```  1007       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```  1008 unfolding One_nat_def
```
```  1009 apply (induct n)
```
```  1010 apply simp
```
```  1011 apply (intro strip)
```
```  1012 apply (erule impE, simp)
```
```  1013 apply (erule_tac x = n in allE, simp)
```
```  1014 apply (case_tac "k = f (Suc n)")
```
```  1015 apply force
```
```  1016 apply (erule impE)
```
```  1017  apply (simp add: abs_if split add: split_if_asm)
```
```  1018 apply (blast intro: le_SucI)
```
```  1019 done
```
```  1020
```
```  1021 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1022
```
```  1023 lemma nat_intermed_int_val:
```
```  1024      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```  1025          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```  1026 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```  1027        in int_val_lemma)
```
```  1028 unfolding One_nat_def
```
```  1029 apply simp
```
```  1030 apply (erule exE)
```
```  1031 apply (rule_tac x = "i+m" in exI, arith)
```
```  1032 done
```
```  1033
```
```  1034
```
```  1035 subsection{*Products and 1, by T. M. Rasmussen*}
```
```  1036
```
```  1037 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```  1038 by arith
```
```  1039
```
```  1040 lemma abs_zmult_eq_1:
```
```  1041   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1042   shows "\<bar>m\<bar> = (1::int)"
```
```  1043 proof -
```
```  1044   have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
```
```  1045     by auto
```
```  1046   have "~ (2 \<le> \<bar>m\<bar>)"
```
```  1047   proof
```
```  1048     assume "2 \<le> \<bar>m\<bar>"
```
```  1049     hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
```
```  1050       by (simp add: mult_mono 0)
```
```  1051     also have "... = \<bar>m*n\<bar>"
```
```  1052       by (simp add: abs_mult)
```
```  1053     also have "... = 1"
```
```  1054       by (simp add: mn)
```
```  1055     finally have "2*\<bar>n\<bar> \<le> 1" .
```
```  1056     thus "False" using 0
```
```  1057       by arith
```
```  1058   qed
```
```  1059   thus ?thesis using 0
```
```  1060     by auto
```
```  1061 qed
```
```  1062
```
```  1063 ML_val {* @{const_name neg_numeral} *}
```
```  1064
```
```  1065 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```  1066 by (insert abs_zmult_eq_1 [of m n], arith)
```
```  1067
```
```  1068 lemma pos_zmult_eq_1_iff:
```
```  1069   assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
```
```  1070 proof -
```
```  1071   from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1072   thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1073 qed
```
```  1074
```
```  1075 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```  1076 apply (rule iffI)
```
```  1077  apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1078  apply (simp add: mult_commute [of m])
```
```  1079  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1080 done
```
```  1081
```
```  1082 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1083 proof
```
```  1084   assume "finite (UNIV::int set)"
```
```  1085   moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1086     by (rule injI) simp
```
```  1087   ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1088     by (rule finite_UNIV_inj_surj)
```
```  1089   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1090   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1091 qed
```
```  1092
```
```  1093
```
```  1094 subsection {* Further theorems on numerals *}
```
```  1095
```
```  1096 subsubsection{*Special Simplification for Constants*}
```
```  1097
```
```  1098 text{*These distributive laws move literals inside sums and differences.*}
```
```  1099
```
```  1100 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
```
```  1101 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
```
```  1102 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
```
```  1103 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
```
```  1104
```
```  1105 text{*These are actually for fields, like real: but where else to put them?*}
```
```  1106
```
```  1107 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
```
```  1108 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
```
```  1109 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
```
```  1110 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
```
```  1111
```
```  1112
```
```  1113 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```  1114   strange, but then other simprocs simplify the quotient.*}
```
```  1115
```
```  1116 lemmas inverse_eq_divide_numeral [simp] =
```
```  1117   inverse_eq_divide [of "numeral w"] for w
```
```  1118
```
```  1119 lemmas inverse_eq_divide_neg_numeral [simp] =
```
```  1120   inverse_eq_divide [of "neg_numeral w"] for w
```
```  1121
```
```  1122 text {*These laws simplify inequalities, moving unary minus from a term
```
```  1123 into the literal.*}
```
```  1124
```
```  1125 lemmas le_minus_iff_numeral [simp, no_atp] =
```
```  1126   le_minus_iff [of "numeral v"]
```
```  1127   le_minus_iff [of "neg_numeral v"] for v
```
```  1128
```
```  1129 lemmas equation_minus_iff_numeral [simp, no_atp] =
```
```  1130   equation_minus_iff [of "numeral v"]
```
```  1131   equation_minus_iff [of "neg_numeral v"] for v
```
```  1132
```
```  1133 lemmas minus_less_iff_numeral [simp, no_atp] =
```
```  1134   minus_less_iff [of _ "numeral v"]
```
```  1135   minus_less_iff [of _ "neg_numeral v"] for v
```
```  1136
```
```  1137 lemmas minus_le_iff_numeral [simp, no_atp] =
```
```  1138   minus_le_iff [of _ "numeral v"]
```
```  1139   minus_le_iff [of _ "neg_numeral v"] for v
```
```  1140
```
```  1141 lemmas minus_equation_iff_numeral [simp, no_atp] =
```
```  1142   minus_equation_iff [of _ "numeral v"]
```
```  1143   minus_equation_iff [of _ "neg_numeral v"] for v
```
```  1144
```
```  1145 text{*To Simplify Inequalities Where One Side is the Constant 1*}
```
```  1146
```
```  1147 lemma less_minus_iff_1 [simp,no_atp]:
```
```  1148   fixes b::"'b::linordered_idom"
```
```  1149   shows "(1 < - b) = (b < -1)"
```
```  1150 by auto
```
```  1151
```
```  1152 lemma le_minus_iff_1 [simp,no_atp]:
```
```  1153   fixes b::"'b::linordered_idom"
```
```  1154   shows "(1 \<le> - b) = (b \<le> -1)"
```
```  1155 by auto
```
```  1156
```
```  1157 lemma equation_minus_iff_1 [simp,no_atp]:
```
```  1158   fixes b::"'b::ring_1"
```
```  1159   shows "(1 = - b) = (b = -1)"
```
```  1160 by (subst equation_minus_iff, auto)
```
```  1161
```
```  1162 lemma minus_less_iff_1 [simp,no_atp]:
```
```  1163   fixes a::"'b::linordered_idom"
```
```  1164   shows "(- a < 1) = (-1 < a)"
```
```  1165 by auto
```
```  1166
```
```  1167 lemma minus_le_iff_1 [simp,no_atp]:
```
```  1168   fixes a::"'b::linordered_idom"
```
```  1169   shows "(- a \<le> 1) = (-1 \<le> a)"
```
```  1170 by auto
```
```  1171
```
```  1172 lemma minus_equation_iff_1 [simp,no_atp]:
```
```  1173   fixes a::"'b::ring_1"
```
```  1174   shows "(- a = 1) = (a = -1)"
```
```  1175 by (subst minus_equation_iff, auto)
```
```  1176
```
```  1177
```
```  1178 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```  1179
```
```  1180 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
```
```  1181 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
```
```  1182 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
```
```  1183 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
```
```  1184
```
```  1185
```
```  1186 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```  1187
```
```  1188 lemmas le_divide_eq_numeral1 [simp] =
```
```  1189   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1190   neg_le_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
```
```  1191
```
```  1192 lemmas divide_le_eq_numeral1 [simp] =
```
```  1193   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
```
```  1194   neg_divide_le_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
```
```  1195
```
```  1196 lemmas less_divide_eq_numeral1 [simp] =
```
```  1197   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
```
```  1198   neg_less_divide_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
```
```  1199
```
```  1200 lemmas divide_less_eq_numeral1 [simp] =
```
```  1201   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
```
```  1202   neg_divide_less_eq [of "neg_numeral w", OF neg_numeral_less_zero] for w
```
```  1203
```
```  1204 lemmas eq_divide_eq_numeral1 [simp] =
```
```  1205   eq_divide_eq [of _ _ "numeral w"]
```
```  1206   eq_divide_eq [of _ _ "neg_numeral w"] for w
```
```  1207
```
```  1208 lemmas divide_eq_eq_numeral1 [simp] =
```
```  1209   divide_eq_eq [of _ "numeral w"]
```
```  1210   divide_eq_eq [of _ "neg_numeral w"] for w
```
```  1211
```
```  1212 subsubsection{*Optional Simplification Rules Involving Constants*}
```
```  1213
```
```  1214 text{*Simplify quotients that are compared with a literal constant.*}
```
```  1215
```
```  1216 lemmas le_divide_eq_numeral =
```
```  1217   le_divide_eq [of "numeral w"]
```
```  1218   le_divide_eq [of "neg_numeral w"] for w
```
```  1219
```
```  1220 lemmas divide_le_eq_numeral =
```
```  1221   divide_le_eq [of _ _ "numeral w"]
```
```  1222   divide_le_eq [of _ _ "neg_numeral w"] for w
```
```  1223
```
```  1224 lemmas less_divide_eq_numeral =
```
```  1225   less_divide_eq [of "numeral w"]
```
```  1226   less_divide_eq [of "neg_numeral w"] for w
```
```  1227
```
```  1228 lemmas divide_less_eq_numeral =
```
```  1229   divide_less_eq [of _ _ "numeral w"]
```
```  1230   divide_less_eq [of _ _ "neg_numeral w"] for w
```
```  1231
```
```  1232 lemmas eq_divide_eq_numeral =
```
```  1233   eq_divide_eq [of "numeral w"]
```
```  1234   eq_divide_eq [of "neg_numeral w"] for w
```
```  1235
```
```  1236 lemmas divide_eq_eq_numeral =
```
```  1237   divide_eq_eq [of _ _ "numeral w"]
```
```  1238   divide_eq_eq [of _ _ "neg_numeral w"] for w
```
```  1239
```
```  1240
```
```  1241 text{*Not good as automatic simprules because they cause case splits.*}
```
```  1242 lemmas divide_const_simps =
```
```  1243   le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
```
```  1244   divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
```
```  1245   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  1246
```
```  1247 text{*Division By @{text "-1"}*}
```
```  1248
```
```  1249 lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x"
```
```  1250   unfolding minus_one [symmetric]
```
```  1251   unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric]
```
```  1252   by simp
```
```  1253
```
```  1254 lemma minus1_divide [simp]: "-1 / (x::'a::field) = - (1 / x)"
```
```  1255   unfolding minus_one [symmetric] by (rule divide_minus_left)
```
```  1256
```
```  1257 lemma half_gt_zero_iff:
```
```  1258      "(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))"
```
```  1259 by auto
```
```  1260
```
```  1261 lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2]
```
```  1262
```
```  1263 lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x"
```
```  1264   by simp
```
```  1265
```
```  1266
```
```  1267 subsection {* The divides relation *}
```
```  1268
```
```  1269 lemma zdvd_antisym_nonneg:
```
```  1270     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
```
```  1271   apply (simp add: dvd_def, auto)
```
```  1272   apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  1273   done
```
```  1274
```
```  1275 lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
```
```  1276   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  1277 proof cases
```
```  1278   assume "a = 0" with assms show ?thesis by simp
```
```  1279 next
```
```  1280   assume "a \<noteq> 0"
```
```  1281   from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast
```
```  1282   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast
```
```  1283   from k k' have "a = a*k*k'" by simp
```
```  1284   with mult_cancel_left1[where c="a" and b="k*k'"]
```
```  1285   have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
```
```  1286   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
```
```  1287   thus ?thesis using k k' by auto
```
```  1288 qed
```
```  1289
```
```  1290 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
```
```  1291   apply (subgoal_tac "m = n + (m - n)")
```
```  1292    apply (erule ssubst)
```
```  1293    apply (blast intro: dvd_add, simp)
```
```  1294   done
```
```  1295
```
```  1296 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
```
```  1297 apply (rule iffI)
```
```  1298  apply (erule_tac  dvd_add)
```
```  1299  apply (subgoal_tac "n = (n + k * m) - k * m")
```
```  1300   apply (erule ssubst)
```
```  1301   apply (erule dvd_diff)
```
```  1302   apply(simp_all)
```
```  1303 done
```
```  1304
```
```  1305 lemma dvd_imp_le_int:
```
```  1306   fixes d i :: int
```
```  1307   assumes "i \<noteq> 0" and "d dvd i"
```
```  1308   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  1309 proof -
```
```  1310   from `d dvd i` obtain k where "i = d * k" ..
```
```  1311   with `i \<noteq> 0` have "k \<noteq> 0" by auto
```
```  1312   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  1313   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  1314   with `i = d * k` show ?thesis by (simp add: abs_mult)
```
```  1315 qed
```
```  1316
```
```  1317 lemma zdvd_not_zless:
```
```  1318   fixes m n :: int
```
```  1319   assumes "0 < m" and "m < n"
```
```  1320   shows "\<not> n dvd m"
```
```  1321 proof
```
```  1322   from assms have "0 < n" by auto
```
```  1323   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  1324   with `0 < m` have "0 < n * k" by auto
```
```  1325   with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
```
```  1326   with k `0 < n` `m < n` have "n * k < n * 1" by simp
```
```  1327   with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
```
```  1328 qed
```
```  1329
```
```  1330 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
```
```  1331   shows "m dvd n"
```
```  1332 proof-
```
```  1333   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
```
```  1334   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
```
```  1335     with h have False by (simp add: mult_assoc)}
```
```  1336   hence "n = m * h" by blast
```
```  1337   thus ?thesis by simp
```
```  1338 qed
```
```  1339
```
```  1340 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
```
```  1341 proof -
```
```  1342   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
```
```  1343   proof -
```
```  1344     fix k
```
```  1345     assume A: "int y = int x * k"
```
```  1346     then show "x dvd y"
```
```  1347     proof (cases k)
```
```  1348       case (nonneg n)
```
```  1349       with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
```
```  1350       then show ?thesis ..
```
```  1351     next
```
```  1352       case (neg n)
```
```  1353       with A have "int y = int x * (- int (Suc n))" by simp
```
```  1354       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
```
```  1355       also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
```
```  1356       finally have "- int (x * Suc n) = int y" ..
```
```  1357       then show ?thesis by (simp only: negative_eq_positive) auto
```
```  1358     qed
```
```  1359   qed
```
```  1360   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  1361 qed
```
```  1362
```
```  1363 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
```
```  1364 proof
```
```  1365   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  1366   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  1367   hence "nat \<bar>x\<bar> = 1"  by simp
```
```  1368   thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
```
```  1369 next
```
```  1370   assume "\<bar>x\<bar>=1"
```
```  1371   then have "x = 1 \<or> x = -1" by auto
```
```  1372   then show "x dvd 1" by (auto intro: dvdI)
```
```  1373 qed
```
```  1374
```
```  1375 lemma zdvd_mult_cancel1:
```
```  1376   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
```
```  1377 proof
```
```  1378   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
```
```  1379     by (cases "n >0") (auto simp add: minus_equation_iff)
```
```  1380 next
```
```  1381   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
```
```  1382   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
```
```  1383 qed
```
```  1384
```
```  1385 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
```
```  1386   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  1387
```
```  1388 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
```
```  1389   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  1390
```
```  1391 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
```
```  1392   by (auto simp add: dvd_int_iff)
```
```  1393
```
```  1394 lemma eq_nat_nat_iff:
```
```  1395   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  1396   by (auto elim!: nonneg_eq_int)
```
```  1397
```
```  1398 lemma nat_power_eq:
```
```  1399   "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  1400   by (induct n) (simp_all add: nat_mult_distrib)
```
```  1401
```
```  1402 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
```
```  1403   apply (cases n)
```
```  1404   apply (auto simp add: dvd_int_iff)
```
```  1405   apply (cases z)
```
```  1406   apply (auto simp add: dvd_imp_le)
```
```  1407   done
```
```  1408
```
```  1409 lemma zdvd_period:
```
```  1410   fixes a d :: int
```
```  1411   assumes "a dvd d"
```
```  1412   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  1413 proof -
```
```  1414   from assms obtain k where "d = a * k" by (rule dvdE)
```
```  1415   show ?thesis
```
```  1416   proof
```
```  1417     assume "a dvd (x + t)"
```
```  1418     then obtain l where "x + t = a * l" by (rule dvdE)
```
```  1419     then have "x = a * l - t" by simp
```
```  1420     with `d = a * k` show "a dvd x + c * d + t" by simp
```
```  1421   next
```
```  1422     assume "a dvd x + c * d + t"
```
```  1423     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
```
```  1424     then have "x = a * l - c * d - t" by simp
```
```  1425     with `d = a * k` show "a dvd (x + t)" by simp
```
```  1426   qed
```
```  1427 qed
```
```  1428
```
```  1429
```
```  1430 subsection {* Finiteness of intervals *}
```
```  1431
```
```  1432 lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
```
```  1433 proof (cases "a <= b")
```
```  1434   case True
```
```  1435   from this show ?thesis
```
```  1436   proof (induct b rule: int_ge_induct)
```
```  1437     case base
```
```  1438     have "{i. a <= i & i <= a} = {a}" by auto
```
```  1439     from this show ?case by simp
```
```  1440   next
```
```  1441     case (step b)
```
```  1442     from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
```
```  1443     from this step show ?case by simp
```
```  1444   qed
```
```  1445 next
```
```  1446   case False from this show ?thesis
```
```  1447     by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
```
```  1448 qed
```
```  1449
```
```  1450 lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
```
```  1451 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1452
```
```  1453 lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
```
```  1454 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1455
```
```  1456 lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
```
```  1457 by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
```
```  1458
```
```  1459
```
```  1460 subsection {* Configuration of the code generator *}
```
```  1461
```
```  1462 text {* Constructors *}
```
```  1463
```
```  1464 definition Pos :: "num \<Rightarrow> int" where
```
```  1465   [simp, code_abbrev]: "Pos = numeral"
```
```  1466
```
```  1467 definition Neg :: "num \<Rightarrow> int" where
```
```  1468   [simp, code_abbrev]: "Neg = neg_numeral"
```
```  1469
```
```  1470 code_datatype "0::int" Pos Neg
```
```  1471
```
```  1472
```
```  1473 text {* Auxiliary operations *}
```
```  1474
```
```  1475 definition dup :: "int \<Rightarrow> int" where
```
```  1476   [simp]: "dup k = k + k"
```
```  1477
```
```  1478 lemma dup_code [code]:
```
```  1479   "dup 0 = 0"
```
```  1480   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```  1481   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```  1482   unfolding Pos_def Neg_def neg_numeral_def
```
```  1483   by (simp_all add: numeral_Bit0)
```
```  1484
```
```  1485 definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
```
```  1486   [simp]: "sub m n = numeral m - numeral n"
```
```  1487
```
```  1488 lemma sub_code [code]:
```
```  1489   "sub Num.One Num.One = 0"
```
```  1490   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```  1491   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```  1492   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```  1493   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```  1494   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```  1495   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```  1496   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```  1497   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```  1498   unfolding sub_def dup_def numeral.simps Pos_def Neg_def
```
```  1499     neg_numeral_def numeral_BitM
```
```  1500   by (simp_all only: algebra_simps)
```
```  1501
```
```  1502
```
```  1503 text {* Implementations *}
```
```  1504
```
```  1505 lemma one_int_code [code, code_unfold]:
```
```  1506   "1 = Pos Num.One"
```
```  1507   by simp
```
```  1508
```
```  1509 lemma plus_int_code [code]:
```
```  1510   "k + 0 = (k::int)"
```
```  1511   "0 + l = (l::int)"
```
```  1512   "Pos m + Pos n = Pos (m + n)"
```
```  1513   "Pos m + Neg n = sub m n"
```
```  1514   "Neg m + Pos n = sub n m"
```
```  1515   "Neg m + Neg n = Neg (m + n)"
```
```  1516   by simp_all
```
```  1517
```
```  1518 lemma uminus_int_code [code]:
```
```  1519   "uminus 0 = (0::int)"
```
```  1520   "uminus (Pos m) = Neg m"
```
```  1521   "uminus (Neg m) = Pos m"
```
```  1522   by simp_all
```
```  1523
```
```  1524 lemma minus_int_code [code]:
```
```  1525   "k - 0 = (k::int)"
```
```  1526   "0 - l = uminus (l::int)"
```
```  1527   "Pos m - Pos n = sub m n"
```
```  1528   "Pos m - Neg n = Pos (m + n)"
```
```  1529   "Neg m - Pos n = Neg (m + n)"
```
```  1530   "Neg m - Neg n = sub n m"
```
```  1531   by simp_all
```
```  1532
```
```  1533 lemma times_int_code [code]:
```
```  1534   "k * 0 = (0::int)"
```
```  1535   "0 * l = (0::int)"
```
```  1536   "Pos m * Pos n = Pos (m * n)"
```
```  1537   "Pos m * Neg n = Neg (m * n)"
```
```  1538   "Neg m * Pos n = Neg (m * n)"
```
```  1539   "Neg m * Neg n = Pos (m * n)"
```
```  1540   by simp_all
```
```  1541
```
```  1542 instantiation int :: equal
```
```  1543 begin
```
```  1544
```
```  1545 definition
```
```  1546   "HOL.equal k l \<longleftrightarrow> k = (l::int)"
```
```  1547
```
```  1548 instance by default (rule equal_int_def)
```
```  1549
```
```  1550 end
```
```  1551
```
```  1552 lemma equal_int_code [code]:
```
```  1553   "HOL.equal 0 (0::int) \<longleftrightarrow> True"
```
```  1554   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```  1555   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```  1556   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```  1557   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```  1558   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```  1559   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```  1560   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```  1561   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```  1562   by (auto simp add: equal)
```
```  1563
```
```  1564 lemma equal_int_refl [code nbe]:
```
```  1565   "HOL.equal (k::int) k \<longleftrightarrow> True"
```
```  1566   by (fact equal_refl)
```
```  1567
```
```  1568 lemma less_eq_int_code [code]:
```
```  1569   "0 \<le> (0::int) \<longleftrightarrow> True"
```
```  1570   "0 \<le> Pos l \<longleftrightarrow> True"
```
```  1571   "0 \<le> Neg l \<longleftrightarrow> False"
```
```  1572   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```  1573   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```  1574   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```  1575   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```  1576   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```  1577   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```  1578   by simp_all
```
```  1579
```
```  1580 lemma less_int_code [code]:
```
```  1581   "0 < (0::int) \<longleftrightarrow> False"
```
```  1582   "0 < Pos l \<longleftrightarrow> True"
```
```  1583   "0 < Neg l \<longleftrightarrow> False"
```
```  1584   "Pos k < 0 \<longleftrightarrow> False"
```
```  1585   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```  1586   "Pos k < Neg l \<longleftrightarrow> False"
```
```  1587   "Neg k < 0 \<longleftrightarrow> True"
```
```  1588   "Neg k < Pos l \<longleftrightarrow> True"
```
```  1589   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```  1590   by simp_all
```
```  1591
```
```  1592 lemma nat_code [code]:
```
```  1593   "nat (Int.Neg k) = 0"
```
```  1594   "nat 0 = 0"
```
```  1595   "nat (Int.Pos k) = nat_of_num k"
```
```  1596   by (simp_all add: nat_of_num_numeral nat_numeral)
```
```  1597
```
```  1598 lemma (in ring_1) of_int_code [code]:
```
```  1599   "of_int (Int.Neg k) = neg_numeral k"
```
```  1600   "of_int 0 = 0"
```
```  1601   "of_int (Int.Pos k) = numeral k"
```
```  1602   by simp_all
```
```  1603
```
```  1604
```
```  1605 text {* Serializer setup *}
```
```  1606
```
```  1607 code_modulename SML
```
```  1608   Int Arith
```
```  1609
```
```  1610 code_modulename OCaml
```
```  1611   Int Arith
```
```  1612
```
```  1613 code_modulename Haskell
```
```  1614   Int Arith
```
```  1615
```
```  1616 quickcheck_params [default_type = int]
```
```  1617
```
```  1618 hide_const (open) Pos Neg sub dup
```
```  1619
```
```  1620
```
```  1621 subsection {* Legacy theorems *}
```
```  1622
```
```  1623 lemmas inj_int = inj_of_nat [where 'a=int]
```
```  1624 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```  1625 lemmas int_mult = of_nat_mult [where 'a=int]
```
```  1626 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```  1627 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
```
```  1628 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```  1629 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
```
```  1630 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```  1631 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```  1632 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n
```
```  1633 lemmas int_0 = of_nat_0 [where 'a=int]
```
```  1634 lemmas int_1 = of_nat_1 [where 'a=int]
```
```  1635 lemmas int_Suc = of_nat_Suc [where 'a=int]
```
```  1636 lemmas int_numeral = of_nat_numeral [where 'a=int]
```
```  1637 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m
```
```  1638 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```  1639 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```  1640 lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
```
```  1641 lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
```
```  1642
```
```  1643 lemma zpower_zpower:
```
```  1644   "(x ^ y) ^ z = (x ^ (y * z)::int)"
```
```  1645   by (rule power_mult [symmetric])
```
```  1646
```
```  1647 lemma int_power:
```
```  1648   "int (m ^ n) = int m ^ n"
```
```  1649   by (rule of_nat_power)
```
```  1650
```
```  1651 lemmas zpower_int = int_power [symmetric]
```
```  1652
```
```  1653 text {* De-register @{text "int"} as a quotient type: *}
```
```  1654
```
```  1655 lemmas [transfer_rule del] =
```
```  1656   int.id_abs_transfer int.rel_eq_transfer zero_int.transfer one_int.transfer
```
```  1657   plus_int.transfer uminus_int.transfer minus_int.transfer times_int.transfer
```
```  1658   int_transfer less_eq_int.transfer less_int.transfer of_int.transfer
```
```  1659   nat.transfer
```
```  1660
```
```  1661 declare Quotient_int [quot_del]
```
```  1662
```
```  1663 end
```