src/HOL/Integ/IntDef.thy
 author wenzelm Thu Jan 19 21:22:08 2006 +0100 (2006-01-19 ago) changeset 18708 4b3dadb4fe33 parent 18704 2c86ced392a8 child 18757 f0d901bc0686 permissions -rw-r--r--
setup: theory -> theory;
```     1 (*  Title:      IntDef.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1996  University of Cambridge
```
```     5
```
```     6 *)
```
```     7
```
```     8 header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
```
```     9
```
```    10 theory IntDef
```
```    11 imports Equiv_Relations NatArith
```
```    12 begin
```
```    13
```
```    14 constdefs
```
```    15   intrel :: "((nat * nat) * (nat * nat)) set"
```
```    16     --{*the equivalence relation underlying the integers*}
```
```    17     "intrel == {((x,y),(u,v)) | x y u v. x+v = u+y}"
```
```    18
```
```    19 typedef (Integ)
```
```    20   int = "UNIV//intrel"
```
```    21     by (auto simp add: quotient_def)
```
```    22
```
```    23 instance int :: "{ord, zero, one, plus, times, minus}" ..
```
```    24
```
```    25 constdefs
```
```    26   int :: "nat => int"
```
```    27   "int m == Abs_Integ(intrel `` {(m,0)})"
```
```    28
```
```    29
```
```    30 defs (overloaded)
```
```    31
```
```    32   Zero_int_def:  "0 == int 0"
```
```    33   One_int_def:   "1 == int 1"
```
```    34
```
```    35   minus_int_def:
```
```    36     "- z == Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. intrel``{(y,x)})"
```
```    37
```
```    38   add_int_def:
```
```    39    "z + w ==
```
```    40        Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
```
```    41 		 intrel``{(x+u, y+v)})"
```
```    42
```
```    43   diff_int_def:  "z - (w::int) == z + (-w)"
```
```    44
```
```    45   mult_int_def:
```
```    46    "z * w ==
```
```    47        Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
```
```    48 		  intrel``{(x*u + y*v, x*v + y*u)})"
```
```    49
```
```    50   le_int_def:
```
```    51    "z \<le> (w::int) ==
```
```    52     \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Integ z & (u,v) \<in> Rep_Integ w"
```
```    53
```
```    54   less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
```
```    55
```
```    56
```
```    57 subsection{*Construction of the Integers*}
```
```    58
```
```    59 subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
```
```    60
```
```    61 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    62 by (simp add: intrel_def)
```
```    63
```
```    64 lemma equiv_intrel: "equiv UNIV intrel"
```
```    65 by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
```
```    66
```
```    67 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    68   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    69 lemmas equiv_intrel_iff = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    70
```
```    71 declare equiv_intrel_iff [simp]
```
```    72
```
```    73
```
```    74 text{*All equivalence classes belong to set of representatives*}
```
```    75 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
```
```    76 by (auto simp add: Integ_def intrel_def quotient_def)
```
```    77
```
```    78 text{*Reduces equality on abstractions to equality on representatives:
```
```    79   @{term "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
```
```    80 declare Abs_Integ_inject [simp]  Abs_Integ_inverse [simp]
```
```    81
```
```    82 text{*Case analysis on the representation of an integer as an equivalence
```
```    83       class of pairs of naturals.*}
```
```    84 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    85      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
```
```    86 apply (rule Abs_Integ_cases [of z])
```
```    87 apply (auto simp add: Integ_def quotient_def)
```
```    88 done
```
```    89
```
```    90
```
```    91 subsubsection{*@{term int}: Embedding the Naturals into the Integers*}
```
```    92
```
```    93 lemma inj_int: "inj int"
```
```    94 by (simp add: inj_on_def int_def)
```
```    95
```
```    96 lemma int_int_eq [iff]: "(int m = int n) = (m = n)"
```
```    97 by (fast elim!: inj_int [THEN injD])
```
```    98
```
```    99
```
```   100 subsubsection{*Integer Unary Negation*}
```
```   101
```
```   102 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
```
```   103 proof -
```
```   104   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
```
```   105     by (simp add: congruent_def)
```
```   106   thus ?thesis
```
```   107     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
```
```   108 qed
```
```   109
```
```   110 lemma zminus_zminus: "- (- z) = (z::int)"
```
```   111 by (cases z, simp add: minus)
```
```   112
```
```   113 lemma zminus_0: "- 0 = (0::int)"
```
```   114 by (simp add: int_def Zero_int_def minus)
```
```   115
```
```   116
```
```   117 subsection{*Integer Addition*}
```
```   118
```
```   119 lemma add:
```
```   120      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
```
```   121       Abs_Integ (intrel``{(x+u, y+v)})"
```
```   122 proof -
```
```   123   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
```
```   124         respects2 intrel"
```
```   125     by (simp add: congruent2_def)
```
```   126   thus ?thesis
```
```   127     by (simp add: add_int_def UN_UN_split_split_eq
```
```   128                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   129 qed
```
```   130
```
```   131 lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
```
```   132 by (cases z, cases w, simp add: minus add)
```
```   133
```
```   134 lemma zadd_commute: "(z::int) + w = w + z"
```
```   135 by (cases z, cases w, simp add: add_ac add)
```
```   136
```
```   137 lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
```
```   138 by (cases z1, cases z2, cases z3, simp add: add add_assoc)
```
```   139
```
```   140 (*For AC rewriting*)
```
```   141 lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"
```
```   142   apply (rule mk_left_commute [of "op +"])
```
```   143   apply (rule zadd_assoc)
```
```   144   apply (rule zadd_commute)
```
```   145   done
```
```   146
```
```   147 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
```
```   148
```
```   149 lemmas zmult_ac = OrderedGroup.mult_ac
```
```   150
```
```   151 lemma zadd_int: "(int m) + (int n) = int (m + n)"
```
```   152 by (simp add: int_def add)
```
```   153
```
```   154 lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
```
```   155 by (simp add: zadd_int zadd_assoc [symmetric])
```
```   156
```
```   157 lemma int_Suc: "int (Suc m) = 1 + (int m)"
```
```   158 by (simp add: One_int_def zadd_int)
```
```   159
```
```   160 (*also for the instance declaration int :: comm_monoid_add*)
```
```   161 lemma zadd_0: "(0::int) + z = z"
```
```   162 apply (simp add: Zero_int_def int_def)
```
```   163 apply (cases z, simp add: add)
```
```   164 done
```
```   165
```
```   166 lemma zadd_0_right: "z + (0::int) = z"
```
```   167 by (rule trans [OF zadd_commute zadd_0])
```
```   168
```
```   169 lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
```
```   170 by (cases z, simp add: int_def Zero_int_def minus add)
```
```   171
```
```   172
```
```   173 subsection{*Integer Multiplication*}
```
```   174
```
```   175 text{*Congruence property for multiplication*}
```
```   176 lemma mult_congruent2:
```
```   177      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   178       respects2 intrel"
```
```   179 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   180  apply (force simp add: mult_ac, clarify)
```
```   181 apply (simp add: congruent_def mult_ac)
```
```   182 apply (rename_tac u v w x y z)
```
```   183 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   184 apply (simp add: mult_ac)
```
```   185 apply (simp add: add_mult_distrib [symmetric])
```
```   186 done
```
```   187
```
```   188
```
```   189 lemma mult:
```
```   190      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   191       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   192 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   193               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   194
```
```   195 lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
```
```   196 by (cases z, cases w, simp add: minus mult add_ac)
```
```   197
```
```   198 lemma zmult_commute: "(z::int) * w = w * z"
```
```   199 by (cases z, cases w, simp add: mult add_ac mult_ac)
```
```   200
```
```   201 lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
```
```   202 by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
```
```   203
```
```   204 lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
```
```   205 by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
```
```   206
```
```   207 lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
```
```   208 by (simp add: zmult_commute [of w] zadd_zmult_distrib)
```
```   209
```
```   210 lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
```
```   211 by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)
```
```   212
```
```   213 lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"
```
```   214 by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
```
```   215
```
```   216 lemmas int_distrib =
```
```   217   zadd_zmult_distrib zadd_zmult_distrib2
```
```   218   zdiff_zmult_distrib zdiff_zmult_distrib2
```
```   219
```
```   220 lemma int_mult: "int (m * n) = (int m) * (int n)"
```
```   221 by (simp add: int_def mult)
```
```   222
```
```   223 text{*Compatibility binding*}
```
```   224 lemmas zmult_int = int_mult [symmetric]
```
```   225
```
```   226 lemma zmult_1: "(1::int) * z = z"
```
```   227 by (cases z, simp add: One_int_def int_def mult)
```
```   228
```
```   229 lemma zmult_1_right: "z * (1::int) = z"
```
```   230 by (rule trans [OF zmult_commute zmult_1])
```
```   231
```
```   232
```
```   233 text{*The integers form a @{text comm_ring_1}*}
```
```   234 instance int :: comm_ring_1
```
```   235 proof
```
```   236   fix i j k :: int
```
```   237   show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
```
```   238   show "i + j = j + i" by (simp add: zadd_commute)
```
```   239   show "0 + i = i" by (rule zadd_0)
```
```   240   show "- i + i = 0" by (rule zadd_zminus_inverse2)
```
```   241   show "i - j = i + (-j)" by (simp add: diff_int_def)
```
```   242   show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
```
```   243   show "i * j = j * i" by (rule zmult_commute)
```
```   244   show "1 * i = i" by (rule zmult_1)
```
```   245   show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
```
```   246   show "0 \<noteq> (1::int)"
```
```   247     by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
```
```   248 qed
```
```   249
```
```   250
```
```   251 subsection{*The @{text "\<le>"} Ordering*}
```
```   252
```
```   253 lemma le:
```
```   254   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   255 by (force simp add: le_int_def)
```
```   256
```
```   257 lemma zle_refl: "w \<le> (w::int)"
```
```   258 by (cases w, simp add: le)
```
```   259
```
```   260 lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
```
```   261 by (cases i, cases j, cases k, simp add: le)
```
```   262
```
```   263 lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
```
```   264 by (cases w, cases z, simp add: le)
```
```   265
```
```   266 (* Axiom 'order_less_le' of class 'order': *)
```
```   267 lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
```
```   268 by (simp add: less_int_def)
```
```   269
```
```   270 instance int :: order
```
```   271   by intro_classes
```
```   272     (assumption |
```
```   273       rule zle_refl zle_trans zle_anti_sym zless_le)+
```
```   274
```
```   275 (* Axiom 'linorder_linear' of class 'linorder': *)
```
```   276 lemma zle_linear: "(z::int) \<le> w | w \<le> z"
```
```   277 by (cases z, cases w) (simp add: le linorder_linear)
```
```   278
```
```   279 instance int :: linorder
```
```   280   by intro_classes (rule zle_linear)
```
```   281
```
```   282
```
```   283 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```   284 lemmas linorder_neqE_int = linorder_neqE[where 'a = int]
```
```   285
```
```   286
```
```   287 lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
```
```   288 by (simp add: Zero_int_def)
```
```   289
```
```   290 lemma zless_int [simp]: "(int m < int n) = (m<n)"
```
```   291 by (simp add: le add int_def linorder_not_le [symmetric])
```
```   292
```
```   293 lemma int_less_0_conv [simp]: "~ (int k < 0)"
```
```   294 by (simp add: Zero_int_def)
```
```   295
```
```   296 lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)"
```
```   297 by (simp add: Zero_int_def)
```
```   298
```
```   299 lemma int_0_less_1: "0 < (1::int)"
```
```   300 by (simp only: Zero_int_def One_int_def One_nat_def zless_int)
```
```   301
```
```   302 lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
```
```   303 by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
```
```   304
```
```   305 lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
```
```   306 by (simp add: linorder_not_less [symmetric])
```
```   307
```
```   308 lemma zero_zle_int [simp]: "(0 \<le> int n)"
```
```   309 by (simp add: Zero_int_def)
```
```   310
```
```   311 lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
```
```   312 by (simp add: Zero_int_def)
```
```   313
```
```   314 lemma int_0 [simp]: "int 0 = (0::int)"
```
```   315 by (simp add: Zero_int_def)
```
```   316
```
```   317 lemma int_1 [simp]: "int 1 = 1"
```
```   318 by (simp add: One_int_def)
```
```   319
```
```   320 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
```
```   321 by (simp add: One_int_def One_nat_def)
```
```   322
```
```   323
```
```   324 subsection{*Monotonicity results*}
```
```   325
```
```   326 lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
```
```   327 by (cases i, cases j, cases k, simp add: le add)
```
```   328
```
```   329 lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
```
```   330 apply (cases i, cases j, cases k)
```
```   331 apply (simp add: linorder_not_le [where 'a = int, symmetric]
```
```   332                  linorder_not_le [where 'a = nat]  le add)
```
```   333 done
```
```   334
```
```   335 lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
```
```   336 by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
```
```   337
```
```   338
```
```   339 subsection{*Strict Monotonicity of Multiplication*}
```
```   340
```
```   341 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   342 lemma zmult_zless_mono2_lemma:
```
```   343      "i<j ==> 0<k ==> int k * i < int k * j"
```
```   344 apply (induct "k", simp)
```
```   345 apply (simp add: int_Suc)
```
```   346 apply (case_tac "k=0")
```
```   347 apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
```
```   348 apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
```
```   349 done
```
```   350
```
```   351 lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
```
```   352 apply (cases k)
```
```   353 apply (auto simp add: le add int_def Zero_int_def)
```
```   354 apply (rule_tac x="x-y" in exI, simp)
```
```   355 done
```
```   356
```
```   357 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   358 apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
```
```   359 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   360 done
```
```   361
```
```   362
```
```   363 defs (overloaded)
```
```   364     zabs_def:  "abs(i::int) == if i < 0 then -i else i"
```
```   365
```
```   366
```
```   367 text{*The integers form an ordered @{text comm_ring_1}*}
```
```   368 instance int :: ordered_idom
```
```   369 proof
```
```   370   fix i j k :: int
```
```   371   show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
```
```   372   show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
```
```   373   show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
```
```   374 qed
```
```   375
```
```   376
```
```   377 lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
```
```   378 apply (cases w, cases z)
```
```   379 apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
```
```   380 done
```
```   381
```
```   382 subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
```
```   383
```
```   384 constdefs
```
```   385    nat  :: "int => nat"
```
```   386     "nat z == contents (\<Union>(x,y) \<in> Rep_Integ z. { x-y })"
```
```   387
```
```   388 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   389 proof -
```
```   390   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   391     by (simp add: congruent_def, arith)
```
```   392   thus ?thesis
```
```   393     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   394 qed
```
```   395
```
```   396 lemma nat_int [simp]: "nat(int n) = n"
```
```   397 by (simp add: nat int_def)
```
```   398
```
```   399 lemma nat_zero [simp]: "nat 0 = 0"
```
```   400 by (simp only: Zero_int_def nat_int)
```
```   401
```
```   402 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   403 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   404
```
```   405 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
```
```   406 by simp
```
```   407
```
```   408 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   409 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   410
```
```   411 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   412 apply (cases w, cases z)
```
```   413 apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
```
```   414 done
```
```   415
```
```   416 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   417 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   418 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   419
```
```   420 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   421 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   422
```
```   423 lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
```
```   424 apply (cases w, cases z)
```
```   425 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   426 done
```
```   427
```
```   428 lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
```
```   429 by (blast dest: nat_0_le sym)
```
```   430
```
```   431 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
```
```   432 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   433
```
```   434 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
```
```   435 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   436
```
```   437 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
```
```   438 apply (cases w)
```
```   439 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   440 done
```
```   441
```
```   442 lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
```
```   443 by (auto simp add: nat_eq_iff2)
```
```   444
```
```   445 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   446 by (insert zless_nat_conj [of 0], auto)
```
```   447
```
```   448 lemma nat_add_distrib:
```
```   449      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   450 by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
```
```   451
```
```   452 lemma nat_diff_distrib:
```
```   453      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   454 by (cases z, cases z',
```
```   455     simp add: nat add minus diff_minus le int_def Zero_int_def)
```
```   456
```
```   457
```
```   458 lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
```
```   459 by (simp add: int_def minus nat Zero_int_def)
```
```   460
```
```   461 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
```
```   462 by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
```
```   463
```
```   464
```
```   465 subsection{*Lemmas about the Function @{term int} and Orderings*}
```
```   466
```
```   467 lemma negative_zless_0: "- (int (Suc n)) < 0"
```
```   468 by (simp add: order_less_le)
```
```   469
```
```   470 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   471 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   472
```
```   473 lemma negative_zle_0: "- int n \<le> 0"
```
```   474 by (simp add: minus_le_iff)
```
```   475
```
```   476 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   477 by (rule order_trans [OF negative_zle_0 zero_zle_int])
```
```   478
```
```   479 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
```
```   480 by (subst le_minus_iff, simp)
```
```   481
```
```   482 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
```
```   483 by (simp add: int_def le minus Zero_int_def)
```
```   484
```
```   485 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
```
```   486 by (simp add: linorder_not_less)
```
```   487
```
```   488 lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
```
```   489 by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
```
```   490
```
```   491 lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
```
```   492 proof (cases w, cases z, simp add: le add int_def)
```
```   493   fix a b c d
```
```   494   assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})"
```
```   495   show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)"
```
```   496   proof
```
```   497     assume "a + d \<le> c + b"
```
```   498     thus "\<exists>n. c + b = a + n + d"
```
```   499       by (auto intro!: exI [where x="c+b - (a+d)"])
```
```   500   next
```
```   501     assume "\<exists>n. c + b = a + n + d"
```
```   502     then obtain n where "c + b = a + n + d" ..
```
```   503     thus "a + d \<le> c + b" by arith
```
```   504   qed
```
```   505 qed
```
```   506
```
```   507 lemma abs_int_eq [simp]: "abs (int m) = int m"
```
```   508 by (simp add: abs_if)
```
```   509
```
```   510 text{*This version is proved for all ordered rings, not just integers!
```
```   511       It is proved here because attribute @{text arith_split} is not available
```
```   512       in theory @{text Ring_and_Field}.
```
```   513       But is it really better than just rewriting with @{text abs_if}?*}
```
```   514 lemma abs_split [arith_split]:
```
```   515      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   516 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   517
```
```   518
```
```   519
```
```   520 subsection{*The Constants @{term neg} and @{term iszero}*}
```
```   521
```
```   522 constdefs
```
```   523
```
```   524   neg   :: "'a::ordered_idom => bool"
```
```   525   "neg(Z) == Z < 0"
```
```   526
```
```   527   (*For simplifying equalities*)
```
```   528   iszero :: "'a::comm_semiring_1_cancel => bool"
```
```   529   "iszero z == z = (0)"
```
```   530
```
```   531
```
```   532 lemma not_neg_int [simp]: "~ neg(int n)"
```
```   533 by (simp add: neg_def)
```
```   534
```
```   535 lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
```
```   536 by (simp add: neg_def neg_less_0_iff_less)
```
```   537
```
```   538 lemmas neg_eq_less_0 = neg_def
```
```   539
```
```   540 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   541 by (simp add: neg_def linorder_not_less)
```
```   542
```
```   543
```
```   544 subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   545
```
```   546 lemma not_neg_0: "~ neg 0"
```
```   547 by (simp add: One_int_def neg_def)
```
```   548
```
```   549 lemma not_neg_1: "~ neg 1"
```
```   550 by (simp add: neg_def linorder_not_less zero_le_one)
```
```   551
```
```   552 lemma iszero_0: "iszero 0"
```
```   553 by (simp add: iszero_def)
```
```   554
```
```   555 lemma not_iszero_1: "~ iszero 1"
```
```   556 by (simp add: iszero_def eq_commute)
```
```   557
```
```   558 lemma neg_nat: "neg z ==> nat z = 0"
```
```   559 by (simp add: neg_def order_less_imp_le)
```
```   560
```
```   561 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
```
```   562 by (simp add: linorder_not_less neg_def)
```
```   563
```
```   564
```
```   565 subsection{*The Set of Natural Numbers*}
```
```   566
```
```   567 constdefs
```
```   568    Nats  :: "'a::comm_semiring_1_cancel set"
```
```   569     "Nats == range of_nat"
```
```   570
```
```   571 syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
```
```   572
```
```   573 lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
```
```   574 by (simp add: Nats_def)
```
```   575
```
```   576 lemma Nats_0 [simp]: "0 \<in> Nats"
```
```   577 apply (simp add: Nats_def)
```
```   578 apply (rule range_eqI)
```
```   579 apply (rule of_nat_0 [symmetric])
```
```   580 done
```
```   581
```
```   582 lemma Nats_1 [simp]: "1 \<in> Nats"
```
```   583 apply (simp add: Nats_def)
```
```   584 apply (rule range_eqI)
```
```   585 apply (rule of_nat_1 [symmetric])
```
```   586 done
```
```   587
```
```   588 lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
```
```   589 apply (auto simp add: Nats_def)
```
```   590 apply (rule range_eqI)
```
```   591 apply (rule of_nat_add [symmetric])
```
```   592 done
```
```   593
```
```   594 lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
```
```   595 apply (auto simp add: Nats_def)
```
```   596 apply (rule range_eqI)
```
```   597 apply (rule of_nat_mult [symmetric])
```
```   598 done
```
```   599
```
```   600 text{*Agreement with the specific embedding for the integers*}
```
```   601 lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
```
```   602 proof
```
```   603   fix n
```
```   604   show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
```
```   605 qed
```
```   606
```
```   607 lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
```
```   608 proof
```
```   609   fix n
```
```   610   show "of_nat n = id n"  by (induct n, simp_all)
```
```   611 qed
```
```   612
```
```   613
```
```   614 subsection{*Embedding of the Integers into any @{text comm_ring_1}:
```
```   615 @{term of_int}*}
```
```   616
```
```   617 constdefs
```
```   618    of_int :: "int => 'a::comm_ring_1"
```
```   619    "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   620
```
```   621
```
```   622 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   623 proof -
```
```   624   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   625     by (simp add: congruent_def compare_rls of_nat_add [symmetric]
```
```   626             del: of_nat_add)
```
```   627   thus ?thesis
```
```   628     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   629 qed
```
```   630
```
```   631 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   632 by (simp add: of_int Zero_int_def int_def)
```
```   633
```
```   634 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   635 by (simp add: of_int One_int_def int_def)
```
```   636
```
```   637 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   638 by (cases w, cases z, simp add: compare_rls of_int add)
```
```   639
```
```   640 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   641 by (cases z, simp add: compare_rls of_int minus)
```
```   642
```
```   643 lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
```
```   644 by (simp add: diff_minus)
```
```   645
```
```   646 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   647 apply (cases w, cases z)
```
```   648 apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
```
```   649                  mult add_ac)
```
```   650 done
```
```   651
```
```   652 lemma of_int_le_iff [simp]:
```
```   653      "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
```
```   654 apply (cases w)
```
```   655 apply (cases z)
```
```   656 apply (simp add: compare_rls of_int le diff_int_def add minus
```
```   657                  of_nat_add [symmetric]   del: of_nat_add)
```
```   658 done
```
```   659
```
```   660 text{*Special cases where either operand is zero*}
```
```   661 lemmas of_int_0_le_iff = of_int_le_iff [of 0, simplified]
```
```   662 lemmas of_int_le_0_iff = of_int_le_iff [of _ 0, simplified]
```
```   663 declare of_int_0_le_iff [simp]
```
```   664 declare of_int_le_0_iff [simp]
```
```   665
```
```   666 lemma of_int_less_iff [simp]:
```
```   667      "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
```
```   668 by (simp add: linorder_not_le [symmetric])
```
```   669
```
```   670 text{*Special cases where either operand is zero*}
```
```   671 lemmas of_int_0_less_iff = of_int_less_iff [of 0, simplified]
```
```   672 lemmas of_int_less_0_iff = of_int_less_iff [of _ 0, simplified]
```
```   673 declare of_int_0_less_iff [simp]
```
```   674 declare of_int_less_0_iff [simp]
```
```   675
```
```   676 text{*The ordering on the @{text comm_ring_1} is necessary.
```
```   677  See @{text of_nat_eq_iff} above.*}
```
```   678 lemma of_int_eq_iff [simp]:
```
```   679      "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)"
```
```   680 by (simp add: order_eq_iff)
```
```   681
```
```   682 text{*Special cases where either operand is zero*}
```
```   683 lemmas of_int_0_eq_iff = of_int_eq_iff [of 0, simplified]
```
```   684 lemmas of_int_eq_0_iff = of_int_eq_iff [of _ 0, simplified]
```
```   685 declare of_int_0_eq_iff [simp]
```
```   686 declare of_int_eq_0_iff [simp]
```
```   687
```
```   688 lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
```
```   689 proof
```
```   690  fix z
```
```   691  show "of_int z = id z"
```
```   692   by (cases z,
```
```   693       simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)
```
```   694 qed
```
```   695
```
```   696
```
```   697 subsection{*The Set of Integers*}
```
```   698
```
```   699 constdefs
```
```   700    Ints  :: "'a::comm_ring_1 set"
```
```   701     "Ints == range of_int"
```
```   702
```
```   703
```
```   704 syntax (xsymbols)
```
```   705   Ints      :: "'a set"                   ("\<int>")
```
```   706
```
```   707 lemma Ints_0 [simp]: "0 \<in> Ints"
```
```   708 apply (simp add: Ints_def)
```
```   709 apply (rule range_eqI)
```
```   710 apply (rule of_int_0 [symmetric])
```
```   711 done
```
```   712
```
```   713 lemma Ints_1 [simp]: "1 \<in> Ints"
```
```   714 apply (simp add: Ints_def)
```
```   715 apply (rule range_eqI)
```
```   716 apply (rule of_int_1 [symmetric])
```
```   717 done
```
```   718
```
```   719 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
```
```   720 apply (auto simp add: Ints_def)
```
```   721 apply (rule range_eqI)
```
```   722 apply (rule of_int_add [symmetric])
```
```   723 done
```
```   724
```
```   725 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
```
```   726 apply (auto simp add: Ints_def)
```
```   727 apply (rule range_eqI)
```
```   728 apply (rule of_int_minus [symmetric])
```
```   729 done
```
```   730
```
```   731 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
```
```   732 apply (auto simp add: Ints_def)
```
```   733 apply (rule range_eqI)
```
```   734 apply (rule of_int_diff [symmetric])
```
```   735 done
```
```   736
```
```   737 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
```
```   738 apply (auto simp add: Ints_def)
```
```   739 apply (rule range_eqI)
```
```   740 apply (rule of_int_mult [symmetric])
```
```   741 done
```
```   742
```
```   743 text{*Collapse nested embeddings*}
```
```   744 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   745 by (induct n, auto)
```
```   746
```
```   747 lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n"
```
```   748 by (simp add: int_eq_of_nat)
```
```   749
```
```   750 lemma Ints_cases [case_names of_int, cases set: Ints]:
```
```   751   "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
```
```   752 proof (simp add: Ints_def)
```
```   753   assume "!!z. q = of_int z ==> C"
```
```   754   assume "q \<in> range of_int" thus C ..
```
```   755 qed
```
```   756
```
```   757 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   758   "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
```
```   759   by (rule Ints_cases) auto
```
```   760
```
```   761
```
```   762 (* int (Suc n) = 1 + int n *)
```
```   763 declare int_Suc [simp]
```
```   764
```
```   765
```
```   766 subsection{*More Properties of @{term setsum} and  @{term setprod}*}
```
```   767
```
```   768 text{*By Jeremy Avigad*}
```
```   769
```
```   770
```
```   771 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   772   apply (case_tac "finite A")
```
```   773   apply (erule finite_induct, auto)
```
```   774   done
```
```   775
```
```   776 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   777   apply (case_tac "finite A")
```
```   778   apply (erule finite_induct, auto)
```
```   779   done
```
```   780
```
```   781 lemma int_setsum: "int (setsum f A) = (\<Sum>x\<in>A. int(f x))"
```
```   782   by (simp add: int_eq_of_nat of_nat_setsum)
```
```   783
```
```   784 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   785   apply (case_tac "finite A")
```
```   786   apply (erule finite_induct, auto)
```
```   787   done
```
```   788
```
```   789 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   790   apply (case_tac "finite A")
```
```   791   apply (erule finite_induct, auto)
```
```   792   done
```
```   793
```
```   794 lemma int_setprod: "int (setprod f A) = (\<Prod>x\<in>A. int(f x))"
```
```   795   by (simp add: int_eq_of_nat of_nat_setprod)
```
```   796
```
```   797 lemma setprod_nonzero_nat:
```
```   798     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
```
```   799   by (rule setprod_nonzero, auto)
```
```   800
```
```   801 lemma setprod_zero_eq_nat:
```
```   802     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
```
```   803   by (rule setprod_zero_eq, auto)
```
```   804
```
```   805 lemma setprod_nonzero_int:
```
```   806     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
```
```   807   by (rule setprod_nonzero, auto)
```
```   808
```
```   809 lemma setprod_zero_eq_int:
```
```   810     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
```
```   811   by (rule setprod_zero_eq, auto)
```
```   812
```
```   813
```
```   814 text{*Now we replace the case analysis rule by a more conventional one:
```
```   815 whether an integer is negative or not.*}
```
```   816
```
```   817 lemma zless_iff_Suc_zadd:
```
```   818     "(w < z) = (\<exists>n. z = w + int(Suc n))"
```
```   819 apply (cases z, cases w)
```
```   820 apply (auto simp add: le add int_def linorder_not_le [symmetric])
```
```   821 apply (rename_tac a b c d)
```
```   822 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   823 apply arith
```
```   824 done
```
```   825
```
```   826 lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
```
```   827 apply (cases x)
```
```   828 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   829 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   830 done
```
```   831
```
```   832 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   833      "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
```
```   834 apply (case_tac "z < 0", blast dest!: negD)
```
```   835 apply (simp add: linorder_not_less)
```
```   836 apply (blast dest: nat_0_le [THEN sym])
```
```   837 done
```
```   838
```
```   839 theorem int_induct [induct type: int, case_names nonneg neg]:
```
```   840      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
```
```   841   by (cases z) auto
```
```   842
```
```   843 text{*Contributed by Brian Huffman*}
```
```   844 theorem int_diff_cases [case_names diff]:
```
```   845 assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
```
```   846  apply (rule_tac z=z in int_cases)
```
```   847   apply (rule_tac m=n and n=0 in prem, simp)
```
```   848  apply (rule_tac m=0 and n="Suc n" in prem, simp)
```
```   849 done
```
```   850
```
```   851 lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
```
```   852 apply (cases z)
```
```   853 apply (simp_all add: not_zle_0_negative del: int_Suc)
```
```   854 done
```
```   855
```
```   856
```
```   857 subsection {* Configuration of the code generator *}
```
```   858
```
```   859 (*FIXME: the IntInf.fromInt below hides a dependence on fixed-precision ints!*)
```
```   860
```
```   861 types_code
```
```   862   "int" ("int")
```
```   863 attach (term_of) {*
```
```   864 val term_of_int = HOLogic.mk_int o IntInf.fromInt;
```
```   865 *}
```
```   866 attach (test) {*
```
```   867 fun gen_int i = one_of [~1, 1] * random_range 0 i;
```
```   868 *}
```
```   869
```
```   870 constdefs
```
```   871   int_aux :: "int \<Rightarrow> nat \<Rightarrow> int"
```
```   872   "int_aux i n == (i + int n)"
```
```   873   nat_aux :: "nat \<Rightarrow> int \<Rightarrow> nat"
```
```   874   "nat_aux n i == (n + nat i)"
```
```   875
```
```   876 lemma [code]:
```
```   877   "int_aux i 0 = i"
```
```   878   "int_aux i (Suc n) = int_aux (i + 1) n" -- {* tail recursive *}
```
```   879   "int n = int_aux 0 n"
```
```   880   by (simp add: int_aux_def)+
```
```   881
```
```   882 lemma [code]: "nat_aux n i = (if i <= 0 then n else nat_aux (Suc n) (i - 1))"
```
```   883   -- {* tail recursive *}
```
```   884   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
```
```   885     dest: zless_imp_add1_zle)
```
```   886 lemma [code]: "nat i = nat_aux 0 i"
```
```   887   by (simp add: nat_aux_def)
```
```   888
```
```   889 consts_code
```
```   890   "0" :: "int"                  ("0")
```
```   891   "1" :: "int"                  ("1")
```
```   892   "uminus" :: "int => int"      ("~")
```
```   893   "op +" :: "int => int => int" ("(_ +/ _)")
```
```   894   "op *" :: "int => int => int" ("(_ */ _)")
```
```   895   "op <" :: "int => int => bool" ("(_ </ _)")
```
```   896   "op <=" :: "int => int => bool" ("(_ <=/ _)")
```
```   897   "neg"                         ("(_ < 0)")
```
```   898
```
```   899 code_syntax_tyco int
```
```   900   ml (atom "IntInf.int")
```
```   901   haskell (atom "Integer")
```
```   902
```
```   903 code_syntax_const
```
```   904   0 :: "int"
```
```   905     ml (atom "(0:IntInf.int)")
```
```   906     haskell (atom "0")
```
```   907   1 :: "int"
```
```   908     ml (atom "(1:IntInf.int)")
```
```   909     haskell (atom "1")
```
```   910   "op +" :: "int \<Rightarrow> int \<Rightarrow> int"
```
```   911     ml (infixl 8 "+")
```
```   912     haskell (infixl 6 "+")
```
```   913   "op *" :: "int \<Rightarrow> int \<Rightarrow> int"
```
```   914     ml (infixl 9 "*")
```
```   915     haskell (infixl 7 "*")
```
```   916   uminus :: "int \<Rightarrow> int"
```
```   917     ml (atom "~")
```
```   918     haskell (atom "negate")
```
```   919   "op <" :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```   920     ml (infix 6 "<")
```
```   921     haskell (infix 4 "<")
```
```   922   "op <=" :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```   923     ml (infix 6 "<=")
```
```   924     haskell (infix 4 "<=")
```
```   925   "neg" :: "int \<Rightarrow> bool"
```
```   926     ml ("_/ </ 0")
```
```   927     haskell ("_/ </ 0")
```
```   928
```
```   929 ML {*
```
```   930 fun mk_int_to_nat bin =
```
```   931   Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT)
```
```   932   \$ (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) \$ bin);
```
```   933
```
```   934 fun bin_to_int bin = HOLogic.dest_binum bin
```
```   935   handle TERM _
```
```   936     => error ("not a number: " ^ Sign.string_of_term thy bin);
```
```   937
```
```   938 fun number_of_codegen thy defs gr dep module b (Const ("Numeral.number_of",
```
```   939       Type ("fun", [_, T as Type ("IntDef.int", [])])) \$ bin) =
```
```   940         (SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, T)),
```
```   941            Pretty.str (IntInf.toString (HOLogic.dest_binum bin))) handle TERM _ => NONE)
```
```   942   | number_of_codegen thy defs gr s thyname b (Const ("Numeral.number_of",
```
```   943       Type ("fun", [_, Type ("nat", [])])) \$ bin) =
```
```   944         SOME (Codegen.invoke_codegen thy defs s thyname b (gr,
```
```   945           Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT) \$
```
```   946             (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) \$ bin)))
```
```   947   | number_of_codegen _ _ _ _ _ _ _ = NONE;
```
```   948
```
```   949 *}
```
```   950
```
```   951 setup {*
```
```   952   Codegen.add_codegen "number_of_codegen" number_of_codegen #>
```
```   953   CodegenPackage.add_appconst
```
```   954     ("Numeral.number_of", ((1, 1), CodegenPackage.appgen_number_of mk_int_to_nat bin_to_int "IntDef.int" "nat")) #>
```
```   955   CodegenPackage.set_int_tyco "IntDef.int"
```
```   956 *}
```
```   957
```
```   958 quickcheck_params [default_type = int]
```
```   959
```
```   960
```
```   961 (*Legacy ML bindings, but no longer the structure Int.*)
```
```   962 ML
```
```   963 {*
```
```   964 val zabs_def = thm "zabs_def"
```
```   965
```
```   966 val int_0 = thm "int_0";
```
```   967 val int_1 = thm "int_1";
```
```   968 val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
```
```   969 val neg_eq_less_0 = thm "neg_eq_less_0";
```
```   970 val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
```
```   971 val not_neg_0 = thm "not_neg_0";
```
```   972 val not_neg_1 = thm "not_neg_1";
```
```   973 val iszero_0 = thm "iszero_0";
```
```   974 val not_iszero_1 = thm "not_iszero_1";
```
```   975 val int_0_less_1 = thm "int_0_less_1";
```
```   976 val int_0_neq_1 = thm "int_0_neq_1";
```
```   977 val negative_zless = thm "negative_zless";
```
```   978 val negative_zle = thm "negative_zle";
```
```   979 val not_zle_0_negative = thm "not_zle_0_negative";
```
```   980 val not_int_zless_negative = thm "not_int_zless_negative";
```
```   981 val negative_eq_positive = thm "negative_eq_positive";
```
```   982 val zle_iff_zadd = thm "zle_iff_zadd";
```
```   983 val abs_int_eq = thm "abs_int_eq";
```
```   984 val abs_split = thm"abs_split";
```
```   985 val nat_int = thm "nat_int";
```
```   986 val nat_zminus_int = thm "nat_zminus_int";
```
```   987 val nat_zero = thm "nat_zero";
```
```   988 val not_neg_nat = thm "not_neg_nat";
```
```   989 val neg_nat = thm "neg_nat";
```
```   990 val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
```
```   991 val nat_0_le = thm "nat_0_le";
```
```   992 val nat_le_0 = thm "nat_le_0";
```
```   993 val zless_nat_conj = thm "zless_nat_conj";
```
```   994 val int_cases = thm "int_cases";
```
```   995
```
```   996 val int_def = thm "int_def";
```
```   997 val Zero_int_def = thm "Zero_int_def";
```
```   998 val One_int_def = thm "One_int_def";
```
```   999 val diff_int_def = thm "diff_int_def";
```
```  1000
```
```  1001 val inj_int = thm "inj_int";
```
```  1002 val zminus_zminus = thm "zminus_zminus";
```
```  1003 val zminus_0 = thm "zminus_0";
```
```  1004 val zminus_zadd_distrib = thm "zminus_zadd_distrib";
```
```  1005 val zadd_commute = thm "zadd_commute";
```
```  1006 val zadd_assoc = thm "zadd_assoc";
```
```  1007 val zadd_left_commute = thm "zadd_left_commute";
```
```  1008 val zadd_ac = thms "zadd_ac";
```
```  1009 val zmult_ac = thms "zmult_ac";
```
```  1010 val zadd_int = thm "zadd_int";
```
```  1011 val zadd_int_left = thm "zadd_int_left";
```
```  1012 val int_Suc = thm "int_Suc";
```
```  1013 val zadd_0 = thm "zadd_0";
```
```  1014 val zadd_0_right = thm "zadd_0_right";
```
```  1015 val zmult_zminus = thm "zmult_zminus";
```
```  1016 val zmult_commute = thm "zmult_commute";
```
```  1017 val zmult_assoc = thm "zmult_assoc";
```
```  1018 val zadd_zmult_distrib = thm "zadd_zmult_distrib";
```
```  1019 val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
```
```  1020 val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
```
```  1021 val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
```
```  1022 val int_distrib = thms "int_distrib";
```
```  1023 val zmult_int = thm "zmult_int";
```
```  1024 val zmult_1 = thm "zmult_1";
```
```  1025 val zmult_1_right = thm "zmult_1_right";
```
```  1026 val int_int_eq = thm "int_int_eq";
```
```  1027 val int_eq_0_conv = thm "int_eq_0_conv";
```
```  1028 val zless_int = thm "zless_int";
```
```  1029 val int_less_0_conv = thm "int_less_0_conv";
```
```  1030 val zero_less_int_conv = thm "zero_less_int_conv";
```
```  1031 val zle_int = thm "zle_int";
```
```  1032 val zero_zle_int = thm "zero_zle_int";
```
```  1033 val int_le_0_conv = thm "int_le_0_conv";
```
```  1034 val zle_refl = thm "zle_refl";
```
```  1035 val zle_linear = thm "zle_linear";
```
```  1036 val zle_trans = thm "zle_trans";
```
```  1037 val zle_anti_sym = thm "zle_anti_sym";
```
```  1038
```
```  1039 val Ints_def = thm "Ints_def";
```
```  1040 val Nats_def = thm "Nats_def";
```
```  1041
```
```  1042 val of_nat_0 = thm "of_nat_0";
```
```  1043 val of_nat_Suc = thm "of_nat_Suc";
```
```  1044 val of_nat_1 = thm "of_nat_1";
```
```  1045 val of_nat_add = thm "of_nat_add";
```
```  1046 val of_nat_mult = thm "of_nat_mult";
```
```  1047 val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
```
```  1048 val less_imp_of_nat_less = thm "less_imp_of_nat_less";
```
```  1049 val of_nat_less_imp_less = thm "of_nat_less_imp_less";
```
```  1050 val of_nat_less_iff = thm "of_nat_less_iff";
```
```  1051 val of_nat_le_iff = thm "of_nat_le_iff";
```
```  1052 val of_nat_eq_iff = thm "of_nat_eq_iff";
```
```  1053 val Nats_0 = thm "Nats_0";
```
```  1054 val Nats_1 = thm "Nats_1";
```
```  1055 val Nats_add = thm "Nats_add";
```
```  1056 val Nats_mult = thm "Nats_mult";
```
```  1057 val int_eq_of_nat = thm"int_eq_of_nat";
```
```  1058 val of_int = thm "of_int";
```
```  1059 val of_int_0 = thm "of_int_0";
```
```  1060 val of_int_1 = thm "of_int_1";
```
```  1061 val of_int_add = thm "of_int_add";
```
```  1062 val of_int_minus = thm "of_int_minus";
```
```  1063 val of_int_diff = thm "of_int_diff";
```
```  1064 val of_int_mult = thm "of_int_mult";
```
```  1065 val of_int_le_iff = thm "of_int_le_iff";
```
```  1066 val of_int_less_iff = thm "of_int_less_iff";
```
```  1067 val of_int_eq_iff = thm "of_int_eq_iff";
```
```  1068 val Ints_0 = thm "Ints_0";
```
```  1069 val Ints_1 = thm "Ints_1";
```
```  1070 val Ints_add = thm "Ints_add";
```
```  1071 val Ints_minus = thm "Ints_minus";
```
```  1072 val Ints_diff = thm "Ints_diff";
```
```  1073 val Ints_mult = thm "Ints_mult";
```
```  1074 val of_int_of_nat_eq = thm"of_int_of_nat_eq";
```
```  1075 val Ints_cases = thm "Ints_cases";
```
```  1076 val Ints_induct = thm "Ints_induct";
```
```  1077 *}
```
```  1078
```
```  1079 end
```