src/HOL/IntDef.thy
 author huffman Wed Jun 06 17:00:09 2007 +0200 (2007-06-06) changeset 23276 a70934b63910 parent 23164 69e55066dbca child 23282 dfc459989d24 permissions -rw-r--r--
generalize of_nat and related constants to class semiring_1
```     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 Nat
```
```    12 begin
```
```    13
```
```    14 text {* the equivalence relation underlying the integers *}
```
```    15
```
```    16 definition
```
```    17   intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
```
```    18 where
```
```    19   "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
```
```    20
```
```    21 typedef (Integ)
```
```    22   int = "UNIV//intrel"
```
```    23   by (auto simp add: quotient_def)
```
```    24
```
```    25 definition
```
```    26   int :: "nat \<Rightarrow> int"
```
```    27 where
```
```    28   [code func del]: "int m = Abs_Integ (intrel `` {(m, 0)})"
```
```    29
```
```    30 instance int :: zero
```
```    31   Zero_int_def: "0 \<equiv> int 0" ..
```
```    32
```
```    33 instance int :: one
```
```    34   One_int_def: "1 \<equiv> int 1" ..
```
```    35
```
```    36 instance int :: plus
```
```    37   add_int_def: "z + w \<equiv> Abs_Integ
```
```    38     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
```
```    39       intrel `` {(x + u, y + v)})" ..
```
```    40
```
```    41 instance int :: minus
```
```    42   minus_int_def:
```
```    43     "- z \<equiv> Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
```
```    44   diff_int_def:  "z - w \<equiv> z + (-w)" ..
```
```    45
```
```    46 instance int :: times
```
```    47   mult_int_def: "z * w \<equiv>  Abs_Integ
```
```    48     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
```
```    49       intrel `` {(x*u + y*v, x*v + y*u)})" ..
```
```    50
```
```    51 instance int :: ord
```
```    52   le_int_def:
```
```    53    "z \<le> w \<equiv> \<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w"
```
```    54   less_int_def: "z < w \<equiv> z \<le> w \<and> z \<noteq> w" ..
```
```    55
```
```    56 lemmas [code func del] = Zero_int_def One_int_def add_int_def
```
```    57   minus_int_def mult_int_def le_int_def less_int_def
```
```    58
```
```    59
```
```    60 subsection{*Construction of the Integers*}
```
```    61
```
```    62 subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
```
```    63
```
```    64 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    65 by (simp add: intrel_def)
```
```    66
```
```    67 lemma equiv_intrel: "equiv UNIV intrel"
```
```    68 by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
```
```    69
```
```    70 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    71   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    72 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    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   @{prop "\<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 (*also for the instance declaration int :: comm_monoid_add*)
```
```   158 lemma zadd_0: "(0::int) + z = z"
```
```   159 apply (simp add: Zero_int_def int_def)
```
```   160 apply (cases z, simp add: add)
```
```   161 done
```
```   162
```
```   163 lemma zadd_0_right: "z + (0::int) = z"
```
```   164 by (rule trans [OF zadd_commute zadd_0])
```
```   165
```
```   166 lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
```
```   167 by (cases z, simp add: int_def Zero_int_def minus add)
```
```   168
```
```   169
```
```   170 subsection{*Integer Multiplication*}
```
```   171
```
```   172 text{*Congruence property for multiplication*}
```
```   173 lemma mult_congruent2:
```
```   174      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   175       respects2 intrel"
```
```   176 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   177  apply (force simp add: mult_ac, clarify)
```
```   178 apply (simp add: congruent_def mult_ac)
```
```   179 apply (rename_tac u v w x y z)
```
```   180 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   181 apply (simp add: mult_ac)
```
```   182 apply (simp add: add_mult_distrib [symmetric])
```
```   183 done
```
```   184
```
```   185
```
```   186 lemma mult:
```
```   187      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   188       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   189 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   190               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   191
```
```   192 lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
```
```   193 by (cases z, cases w, simp add: minus mult add_ac)
```
```   194
```
```   195 lemma zmult_commute: "(z::int) * w = w * z"
```
```   196 by (cases z, cases w, simp add: mult add_ac mult_ac)
```
```   197
```
```   198 lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
```
```   199 by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
```
```   200
```
```   201 lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
```
```   202 by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
```
```   203
```
```   204 lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
```
```   205 by (simp add: zmult_commute [of w] zadd_zmult_distrib)
```
```   206
```
```   207 lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
```
```   208 by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)
```
```   209
```
```   210 lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"
```
```   211 by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
```
```   212
```
```   213 lemmas int_distrib =
```
```   214   zadd_zmult_distrib zadd_zmult_distrib2
```
```   215   zdiff_zmult_distrib zdiff_zmult_distrib2
```
```   216
```
```   217 lemma int_mult: "int (m * n) = (int m) * (int n)"
```
```   218 by (simp add: int_def mult)
```
```   219
```
```   220 text{*Compatibility binding*}
```
```   221 lemmas zmult_int = int_mult [symmetric]
```
```   222
```
```   223 lemma zmult_1: "(1::int) * z = z"
```
```   224 by (cases z, simp add: One_int_def int_def mult)
```
```   225
```
```   226 lemma zmult_1_right: "z * (1::int) = z"
```
```   227 by (rule trans [OF zmult_commute zmult_1])
```
```   228
```
```   229
```
```   230 text{*The integers form a @{text comm_ring_1}*}
```
```   231 instance int :: comm_ring_1
```
```   232 proof
```
```   233   fix i j k :: int
```
```   234   show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
```
```   235   show "i + j = j + i" by (simp add: zadd_commute)
```
```   236   show "0 + i = i" by (rule zadd_0)
```
```   237   show "- i + i = 0" by (rule zadd_zminus_inverse2)
```
```   238   show "i - j = i + (-j)" by (simp add: diff_int_def)
```
```   239   show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
```
```   240   show "i * j = j * i" by (rule zmult_commute)
```
```   241   show "1 * i = i" by (rule zmult_1)
```
```   242   show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
```
```   243   show "0 \<noteq> (1::int)"
```
```   244     by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
```
```   245 qed
```
```   246
```
```   247
```
```   248 subsection{*The @{text "\<le>"} Ordering*}
```
```   249
```
```   250 lemma le:
```
```   251   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   252 by (force simp add: le_int_def)
```
```   253
```
```   254 lemma zle_refl: "w \<le> (w::int)"
```
```   255 by (cases w, simp add: le)
```
```   256
```
```   257 lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
```
```   258 by (cases i, cases j, cases k, simp add: le)
```
```   259
```
```   260 lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
```
```   261 by (cases w, cases z, simp add: le)
```
```   262
```
```   263 instance int :: order
```
```   264   by intro_classes
```
```   265     (assumption |
```
```   266       rule zle_refl zle_trans zle_anti_sym less_int_def [THEN meta_eq_to_obj_eq])+
```
```   267
```
```   268 lemma zle_linear: "(z::int) \<le> w \<or> w \<le> z"
```
```   269 by (cases z, cases w) (simp add: le linorder_linear)
```
```   270
```
```   271 instance int :: linorder
```
```   272   by intro_classes (rule zle_linear)
```
```   273
```
```   274 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```   275
```
```   276
```
```   277 lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
```
```   278 by (simp add: Zero_int_def)
```
```   279
```
```   280 lemma zless_int [simp]: "(int m < int n) = (m<n)"
```
```   281 by (simp add: le add int_def linorder_not_le [symmetric])
```
```   282
```
```   283 lemma int_less_0_conv [simp]: "~ (int k < 0)"
```
```   284 by (simp add: Zero_int_def)
```
```   285
```
```   286 lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)"
```
```   287 by (simp add: Zero_int_def)
```
```   288
```
```   289 lemma int_0_less_1: "0 < (1::int)"
```
```   290 by (simp only: Zero_int_def One_int_def One_nat_def zless_int)
```
```   291
```
```   292 lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
```
```   293 by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
```
```   294
```
```   295 lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
```
```   296 by (simp add: linorder_not_less [symmetric])
```
```   297
```
```   298 lemma zero_zle_int [simp]: "(0 \<le> int n)"
```
```   299 by (simp add: Zero_int_def)
```
```   300
```
```   301 lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
```
```   302 by (simp add: Zero_int_def)
```
```   303
```
```   304 lemma int_0 [simp]: "int 0 = (0::int)"
```
```   305 by (simp add: Zero_int_def)
```
```   306
```
```   307 lemma int_1 [simp]: "int 1 = 1"
```
```   308 by (simp add: One_int_def)
```
```   309
```
```   310 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
```
```   311 by (simp add: One_int_def One_nat_def)
```
```   312
```
```   313 lemma int_Suc: "int (Suc m) = 1 + (int m)"
```
```   314 by (simp add: One_int_def zadd_int)
```
```   315
```
```   316
```
```   317 subsection{*Monotonicity results*}
```
```   318
```
```   319 lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
```
```   320 by (cases i, cases j, cases k, simp add: le add)
```
```   321
```
```   322 lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
```
```   323 apply (cases i, cases j, cases k)
```
```   324 apply (simp add: linorder_not_le [where 'a = int, symmetric]
```
```   325                  linorder_not_le [where 'a = nat]  le add)
```
```   326 done
```
```   327
```
```   328 lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
```
```   329 by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
```
```   330
```
```   331
```
```   332 subsection{*Strict Monotonicity of Multiplication*}
```
```   333
```
```   334 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   335 lemma zmult_zless_mono2_lemma:
```
```   336      "i<j ==> 0<k ==> int k * i < int k * j"
```
```   337 apply (induct "k", simp)
```
```   338 apply (simp add: int_Suc)
```
```   339 apply (case_tac "k=0")
```
```   340 apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
```
```   341 apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
```
```   342 done
```
```   343
```
```   344 lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
```
```   345 apply (cases k)
```
```   346 apply (auto simp add: le add int_def Zero_int_def)
```
```   347 apply (rule_tac x="x-y" in exI, simp)
```
```   348 done
```
```   349
```
```   350 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   351 apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
```
```   352 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   353 done
```
```   354
```
```   355 instance int :: minus
```
```   356   zabs_def: "\<bar>i\<Colon>int\<bar> \<equiv> if i < 0 then - i else i" ..
```
```   357
```
```   358 instance int :: distrib_lattice
```
```   359   "inf \<equiv> min"
```
```   360   "sup \<equiv> max"
```
```   361   by intro_classes
```
```   362     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   363
```
```   364 text{*The integers form an ordered @{text comm_ring_1}*}
```
```   365 instance int :: ordered_idom
```
```   366 proof
```
```   367   fix i j k :: int
```
```   368   show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
```
```   369   show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
```
```   370   show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
```
```   371 qed
```
```   372
```
```   373 lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
```
```   374 apply (cases w, cases z)
```
```   375 apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
```
```   376 done
```
```   377
```
```   378 subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
```
```   379
```
```   380 definition
```
```   381   nat :: "int \<Rightarrow> nat"
```
```   382 where
```
```   383   [code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
```
```   384
```
```   385 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   386 proof -
```
```   387   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   388     by (simp add: congruent_def) arith
```
```   389   thus ?thesis
```
```   390     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   391 qed
```
```   392
```
```   393 lemma nat_int [simp]: "nat(int n) = n"
```
```   394 by (simp add: nat int_def)
```
```   395
```
```   396 lemma nat_zero [simp]: "nat 0 = 0"
```
```   397 by (simp only: Zero_int_def nat_int)
```
```   398
```
```   399 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   400 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   401
```
```   402 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
```
```   403 by simp
```
```   404
```
```   405 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   406 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   407
```
```   408 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   409 apply (cases w, cases z)
```
```   410 apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
```
```   411 done
```
```   412
```
```   413 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   414 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   415 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   416
```
```   417 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   418 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   419
```
```   420 lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
```
```   421 apply (cases w, cases z)
```
```   422 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   423 done
```
```   424
```
```   425 lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
```
```   426 by (blast dest: nat_0_le sym)
```
```   427
```
```   428 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
```
```   429 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   430
```
```   431 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
```
```   432 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   433
```
```   434 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
```
```   435 apply (cases w)
```
```   436 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   437 done
```
```   438
```
```   439 lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
```
```   440 by (auto simp add: nat_eq_iff2)
```
```   441
```
```   442 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   443 by (insert zless_nat_conj [of 0], auto)
```
```   444
```
```   445 lemma nat_add_distrib:
```
```   446      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   447 by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
```
```   448
```
```   449 lemma nat_diff_distrib:
```
```   450      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   451 by (cases z, cases z',
```
```   452     simp add: nat add minus diff_minus le int_def Zero_int_def)
```
```   453
```
```   454
```
```   455 lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
```
```   456 by (simp add: int_def minus nat Zero_int_def)
```
```   457
```
```   458 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
```
```   459 by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
```
```   460
```
```   461
```
```   462 subsection{*Lemmas about the Function @{term int} and Orderings*}
```
```   463
```
```   464 lemma negative_zless_0: "- (int (Suc n)) < 0"
```
```   465 by (simp add: order_less_le)
```
```   466
```
```   467 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   468 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   469
```
```   470 lemma negative_zle_0: "- int n \<le> 0"
```
```   471 by (simp add: minus_le_iff)
```
```   472
```
```   473 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   474 by (rule order_trans [OF negative_zle_0 zero_zle_int])
```
```   475
```
```   476 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
```
```   477 by (subst le_minus_iff, simp)
```
```   478
```
```   479 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
```
```   480 by (simp add: int_def le minus Zero_int_def)
```
```   481
```
```   482 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
```
```   483 by (simp add: linorder_not_less)
```
```   484
```
```   485 lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
```
```   486 by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
```
```   487
```
```   488 lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
```
```   489 proof (cases w, cases z, simp add: le add int_def)
```
```   490   fix a b c d
```
```   491   assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})"
```
```   492   show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)"
```
```   493   proof
```
```   494     assume "a + d \<le> c + b"
```
```   495     thus "\<exists>n. c + b = a + n + d"
```
```   496       by (auto intro!: exI [where x="c+b - (a+d)"])
```
```   497   next
```
```   498     assume "\<exists>n. c + b = a + n + d"
```
```   499     then obtain n where "c + b = a + n + d" ..
```
```   500     thus "a + d \<le> c + b" by arith
```
```   501   qed
```
```   502 qed
```
```   503
```
```   504 lemma abs_int_eq [simp]: "abs (int m) = int m"
```
```   505 by (simp add: abs_if)
```
```   506
```
```   507 text{*This version is proved for all ordered rings, not just integers!
```
```   508       It is proved here because attribute @{text arith_split} is not available
```
```   509       in theory @{text Ring_and_Field}.
```
```   510       But is it really better than just rewriting with @{text abs_if}?*}
```
```   511 lemma abs_split [arith_split]:
```
```   512      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   513 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   514
```
```   515
```
```   516 subsection {* Constants @{term neg} and @{term iszero} *}
```
```   517
```
```   518 definition
```
```   519   neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
```
```   520 where
```
```   521   [code inline]: "neg Z \<longleftrightarrow> Z < 0"
```
```   522
```
```   523 definition (*for simplifying equalities*)
```
```   524   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
```
```   525 where
```
```   526   "iszero z \<longleftrightarrow> z = 0"
```
```   527
```
```   528 lemma not_neg_int [simp]: "~ neg(int n)"
```
```   529 by (simp add: neg_def)
```
```   530
```
```   531 lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
```
```   532 by (simp add: neg_def neg_less_0_iff_less)
```
```   533
```
```   534 lemmas neg_eq_less_0 = neg_def
```
```   535
```
```   536 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   537 by (simp add: neg_def linorder_not_less)
```
```   538
```
```   539
```
```   540 subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   541
```
```   542 lemma not_neg_0: "~ neg 0"
```
```   543 by (simp add: One_int_def neg_def)
```
```   544
```
```   545 lemma not_neg_1: "~ neg 1"
```
```   546 by (simp add: neg_def linorder_not_less zero_le_one)
```
```   547
```
```   548 lemma iszero_0: "iszero 0"
```
```   549 by (simp add: iszero_def)
```
```   550
```
```   551 lemma not_iszero_1: "~ iszero 1"
```
```   552 by (simp add: iszero_def eq_commute)
```
```   553
```
```   554 lemma neg_nat: "neg z ==> nat z = 0"
```
```   555 by (simp add: neg_def order_less_imp_le)
```
```   556
```
```   557 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
```
```   558 by (simp add: linorder_not_less neg_def)
```
```   559
```
```   560
```
```   561 subsection{*The Set of Natural Numbers*}
```
```   562
```
```   563 constdefs
```
```   564   Nats  :: "'a::semiring_1 set"
```
```   565   "Nats == range of_nat"
```
```   566
```
```   567 notation (xsymbols)
```
```   568   Nats  ("\<nat>")
```
```   569
```
```   570 lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
```
```   571 by (simp add: Nats_def)
```
```   572
```
```   573 lemma Nats_0 [simp]: "0 \<in> Nats"
```
```   574 apply (simp add: Nats_def)
```
```   575 apply (rule range_eqI)
```
```   576 apply (rule of_nat_0 [symmetric])
```
```   577 done
```
```   578
```
```   579 lemma Nats_1 [simp]: "1 \<in> Nats"
```
```   580 apply (simp add: Nats_def)
```
```   581 apply (rule range_eqI)
```
```   582 apply (rule of_nat_1 [symmetric])
```
```   583 done
```
```   584
```
```   585 lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
```
```   586 apply (auto simp add: Nats_def)
```
```   587 apply (rule range_eqI)
```
```   588 apply (rule of_nat_add [symmetric])
```
```   589 done
```
```   590
```
```   591 lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
```
```   592 apply (auto simp add: Nats_def)
```
```   593 apply (rule range_eqI)
```
```   594 apply (rule of_nat_mult [symmetric])
```
```   595 done
```
```   596
```
```   597 text{*Agreement with the specific embedding for the integers*}
```
```   598 lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
```
```   599 proof
```
```   600   fix n
```
```   601   show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
```
```   602 qed
```
```   603
```
```   604 lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
```
```   605 proof
```
```   606   fix n
```
```   607   show "of_nat n = id n"  by (induct n, simp_all)
```
```   608 qed
```
```   609
```
```   610
```
```   611 subsection{*Embedding of the Integers into any @{text ring_1}:
```
```   612 @{term of_int}*}
```
```   613
```
```   614 constdefs
```
```   615    of_int :: "int => 'a::ring_1"
```
```   616    "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   617
```
```   618
```
```   619 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   620 proof -
```
```   621   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   622     by (simp add: congruent_def compare_rls of_nat_add [symmetric]
```
```   623             del: of_nat_add)
```
```   624   thus ?thesis
```
```   625     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   626 qed
```
```   627
```
```   628 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   629 by (simp add: of_int Zero_int_def int_def)
```
```   630
```
```   631 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   632 by (simp add: of_int One_int_def int_def)
```
```   633
```
```   634 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   635 by (cases w, cases z, simp add: compare_rls of_int add)
```
```   636
```
```   637 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   638 by (cases z, simp add: compare_rls of_int minus)
```
```   639
```
```   640 lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
```
```   641 by (simp add: diff_minus)
```
```   642
```
```   643 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   644 apply (cases w, cases z)
```
```   645 apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
```
```   646                  mult add_ac)
```
```   647 done
```
```   648
```
```   649 lemma of_int_le_iff [simp]:
```
```   650      "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
```
```   651 apply (cases w)
```
```   652 apply (cases z)
```
```   653 apply (simp add: compare_rls of_int le diff_int_def add minus
```
```   654                  of_nat_add [symmetric]   del: of_nat_add)
```
```   655 done
```
```   656
```
```   657 text{*Special cases where either operand is zero*}
```
```   658 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
```
```   659 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
```
```   660
```
```   661
```
```   662 lemma of_int_less_iff [simp]:
```
```   663      "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
```
```   664 by (simp add: linorder_not_le [symmetric])
```
```   665
```
```   666 text{*Special cases where either operand is zero*}
```
```   667 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
```
```   668 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
```
```   669
```
```   670 text{*Class for unital rings with characteristic zero.
```
```   671  Includes non-ordered rings like the complex numbers.*}
```
```   672 axclass ring_char_0 < ring_1
```
```   673   of_int_inject: "of_int w = of_int z ==> w = z"
```
```   674
```
```   675 lemma of_int_eq_iff [simp]:
```
```   676      "(of_int w = (of_int z::'a::ring_char_0)) = (w = z)"
```
```   677 by (safe elim!: of_int_inject)
```
```   678
```
```   679 text{*Every @{text ordered_idom} has characteristic zero.*}
```
```   680 instance ordered_idom < ring_char_0
```
```   681 by intro_classes (simp add: order_eq_iff)
```
```   682
```
```   683 text{*Special cases where either operand is zero*}
```
```   684 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
```
```   685 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
```
```   686
```
```   687 lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
```
```   688 proof
```
```   689   fix z
```
```   690   show "of_int z = id z"
```
```   691     by (cases z)
```
```   692       (simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)
```
```   693 qed
```
```   694
```
```   695
```
```   696 subsection{*The Set of Integers*}
```
```   697
```
```   698 constdefs
```
```   699   Ints  :: "'a::ring_1 set"
```
```   700   "Ints == range of_int"
```
```   701
```
```   702 notation (xsymbols)
```
```   703   Ints  ("\<int>")
```
```   704
```
```   705 lemma Ints_0 [simp]: "0 \<in> Ints"
```
```   706 apply (simp add: Ints_def)
```
```   707 apply (rule range_eqI)
```
```   708 apply (rule of_int_0 [symmetric])
```
```   709 done
```
```   710
```
```   711 lemma Ints_1 [simp]: "1 \<in> Ints"
```
```   712 apply (simp add: Ints_def)
```
```   713 apply (rule range_eqI)
```
```   714 apply (rule of_int_1 [symmetric])
```
```   715 done
```
```   716
```
```   717 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
```
```   718 apply (auto simp add: Ints_def)
```
```   719 apply (rule range_eqI)
```
```   720 apply (rule of_int_add [symmetric])
```
```   721 done
```
```   722
```
```   723 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
```
```   724 apply (auto simp add: Ints_def)
```
```   725 apply (rule range_eqI)
```
```   726 apply (rule of_int_minus [symmetric])
```
```   727 done
```
```   728
```
```   729 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
```
```   730 apply (auto simp add: Ints_def)
```
```   731 apply (rule range_eqI)
```
```   732 apply (rule of_int_diff [symmetric])
```
```   733 done
```
```   734
```
```   735 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
```
```   736 apply (auto simp add: Ints_def)
```
```   737 apply (rule range_eqI)
```
```   738 apply (rule of_int_mult [symmetric])
```
```   739 done
```
```   740
```
```   741 text{*Collapse nested embeddings*}
```
```   742 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   743 by (induct n, auto)
```
```   744
```
```   745 lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n"
```
```   746 by (simp add: int_eq_of_nat)
```
```   747
```
```   748 lemma Ints_cases [cases set: Ints]:
```
```   749   assumes "q \<in> \<int>"
```
```   750   obtains (of_int) z where "q = of_int z"
```
```   751   unfolding Ints_def
```
```   752 proof -
```
```   753   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```   754   then obtain z where "q = of_int z" ..
```
```   755   then show thesis ..
```
```   756 qed
```
```   757
```
```   758 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   759   "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
```
```   760   by (rule Ints_cases) auto
```
```   761
```
```   762
```
```   763 (* int (Suc n) = 1 + int n *)
```
```   764
```
```   765
```
```   766
```
```   767 subsection{*More Properties of @{term setsum} and  @{term setprod}*}
```
```   768
```
```   769 text{*By Jeremy Avigad*}
```
```   770
```
```   771
```
```   772 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   773   apply (cases "finite A")
```
```   774   apply (erule finite_induct, auto)
```
```   775   done
```
```   776
```
```   777 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   778   apply (cases "finite A")
```
```   779   apply (erule finite_induct, auto)
```
```   780   done
```
```   781
```
```   782 lemma int_setsum: "int (setsum f A) = (\<Sum>x\<in>A. int(f x))"
```
```   783   by (simp add: int_eq_of_nat of_nat_setsum)
```
```   784
```
```   785 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   786   apply (cases "finite A")
```
```   787   apply (erule finite_induct, auto)
```
```   788   done
```
```   789
```
```   790 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   791   apply (cases "finite A")
```
```   792   apply (erule finite_induct, auto)
```
```   793   done
```
```   794
```
```   795 lemma int_setprod: "int (setprod f A) = (\<Prod>x\<in>A. int(f x))"
```
```   796   by (simp add: int_eq_of_nat of_nat_setprod)
```
```   797
```
```   798 lemma setprod_nonzero_nat:
```
```   799     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
```
```   800   by (rule setprod_nonzero, auto)
```
```   801
```
```   802 lemma setprod_zero_eq_nat:
```
```   803     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
```
```   804   by (rule setprod_zero_eq, auto)
```
```   805
```
```   806 lemma setprod_nonzero_int:
```
```   807     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
```
```   808   by (rule setprod_nonzero, auto)
```
```   809
```
```   810 lemma setprod_zero_eq_int:
```
```   811     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
```
```   812   by (rule setprod_zero_eq, auto)
```
```   813
```
```   814
```
```   815 subsection {* Further properties *}
```
```   816
```
```   817 text{*Now we replace the case analysis rule by a more conventional one:
```
```   818 whether an integer is negative or not.*}
```
```   819
```
```   820 lemma zless_iff_Suc_zadd:
```
```   821     "(w < z) = (\<exists>n. z = w + int(Suc n))"
```
```   822 apply (cases z, cases w)
```
```   823 apply (auto simp add: le add int_def linorder_not_le [symmetric])
```
```   824 apply (rename_tac a b c d)
```
```   825 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   826 apply arith
```
```   827 done
```
```   828
```
```   829 lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
```
```   830 apply (cases x)
```
```   831 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   832 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   833 done
```
```   834
```
```   835 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   836      "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
```
```   837 apply (cases "z < 0", blast dest!: negD)
```
```   838 apply (simp add: linorder_not_less)
```
```   839 apply (blast dest: nat_0_le [THEN sym])
```
```   840 done
```
```   841
```
```   842 theorem int_induct [induct type: int, case_names nonneg neg]:
```
```   843      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
```
```   844   by (cases z) auto
```
```   845
```
```   846 text{*Contributed by Brian Huffman*}
```
```   847 theorem int_diff_cases [case_names diff]:
```
```   848 assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
```
```   849  apply (rule_tac z=z in int_cases)
```
```   850   apply (rule_tac m=n and n=0 in prem, simp)
```
```   851  apply (rule_tac m=0 and n="Suc n" in prem, simp)
```
```   852 done
```
```   853
```
```   854 lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
```
```   855 apply (cases z)
```
```   856 apply (simp_all add: not_zle_0_negative del: int_Suc)
```
```   857 done
```
```   858
```
```   859 lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq]
```
```   860
```
```   861 lemmas [simp] = int_Suc
```
```   862
```
```   863
```
```   864 subsection {* Legacy ML bindings *}
```
```   865
```
```   866 ML {*
```
```   867 val of_nat_0 = @{thm of_nat_0};
```
```   868 val of_nat_1 = @{thm of_nat_1};
```
```   869 val of_nat_Suc = @{thm of_nat_Suc};
```
```   870 val of_nat_add = @{thm of_nat_add};
```
```   871 val of_nat_mult = @{thm of_nat_mult};
```
```   872 val of_int_0 = @{thm of_int_0};
```
```   873 val of_int_1 = @{thm of_int_1};
```
```   874 val of_int_add = @{thm of_int_add};
```
```   875 val of_int_mult = @{thm of_int_mult};
```
```   876 val int_eq_of_nat = @{thm int_eq_of_nat};
```
```   877 val zle_int = @{thm zle_int};
```
```   878 val int_int_eq = @{thm int_int_eq};
```
```   879 val diff_int_def = @{thm diff_int_def};
```
```   880 val zadd_ac = @{thms zadd_ac};
```
```   881 val zless_int = @{thm zless_int};
```
```   882 val zadd_int = @{thm zadd_int};
```
```   883 val zmult_int = @{thm zmult_int};
```
```   884 val nat_0_le = @{thm nat_0_le};
```
```   885 val int_0 = @{thm int_0};
```
```   886 val int_1 = @{thm int_1};
```
```   887 val abs_split = @{thm abs_split};
```
```   888 *}
```
```   889
```
```   890 end
```