src/HOL/IntDef.thy
 author huffman Wed Jun 13 03:31:11 2007 +0200 (2007-06-13) changeset 23365 f31794033ae1 parent 23308 95a01ddfb024 child 23372 0035be079bee permissions -rw-r--r--
removed constant int :: nat => int;
int is now an abbreviation for of_nat :: nat => int
```     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 instance int :: zero
```
```    26   Zero_int_def: "0 \<equiv> Abs_Integ (intrel `` {(0, 0)})" ..
```
```    27
```
```    28 instance int :: one
```
```    29   One_int_def: "1 \<equiv> Abs_Integ (intrel `` {(1, 0)})" ..
```
```    30
```
```    31 instance int :: plus
```
```    32   add_int_def: "z + w \<equiv> Abs_Integ
```
```    33     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
```
```    34       intrel `` {(x + u, y + v)})" ..
```
```    35
```
```    36 instance int :: minus
```
```    37   minus_int_def:
```
```    38     "- z \<equiv> Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
```
```    39   diff_int_def:  "z - w \<equiv> z + (-w)" ..
```
```    40
```
```    41 instance int :: times
```
```    42   mult_int_def: "z * w \<equiv>  Abs_Integ
```
```    43     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
```
```    44       intrel `` {(x*u + y*v, x*v + y*u)})" ..
```
```    45
```
```    46 instance int :: ord
```
```    47   le_int_def:
```
```    48    "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"
```
```    49   less_int_def: "z < w \<equiv> z \<le> w \<and> z \<noteq> w" ..
```
```    50
```
```    51 lemmas [code func del] = Zero_int_def One_int_def add_int_def
```
```    52   minus_int_def mult_int_def le_int_def less_int_def
```
```    53
```
```    54
```
```    55 subsection{*Construction of the Integers*}
```
```    56
```
```    57 subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
```
```    58
```
```    59 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    60 by (simp add: intrel_def)
```
```    61
```
```    62 lemma equiv_intrel: "equiv UNIV intrel"
```
```    63 by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
```
```    64
```
```    65 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    66   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    67 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    68
```
```    69 text{*All equivalence classes belong to set of representatives*}
```
```    70 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
```
```    71 by (auto simp add: Integ_def intrel_def quotient_def)
```
```    72
```
```    73 text{*Reduces equality on abstractions to equality on representatives:
```
```    74   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
```
```    75 declare Abs_Integ_inject [simp]  Abs_Integ_inverse [simp]
```
```    76
```
```    77 text{*Case analysis on the representation of an integer as an equivalence
```
```    78       class of pairs of naturals.*}
```
```    79 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    80      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
```
```    81 apply (rule Abs_Integ_cases [of z])
```
```    82 apply (auto simp add: Integ_def quotient_def)
```
```    83 done
```
```    84
```
```    85
```
```    86 subsubsection{*Integer Unary Negation*}
```
```    87
```
```    88 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
```
```    89 proof -
```
```    90   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
```
```    91     by (simp add: congruent_def)
```
```    92   thus ?thesis
```
```    93     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
```
```    94 qed
```
```    95
```
```    96 lemma zminus_zminus: "- (- z) = (z::int)"
```
```    97   by (cases z) (simp add: minus)
```
```    98
```
```    99 lemma zminus_0: "- 0 = (0::int)"
```
```   100   by (simp add: Zero_int_def minus)
```
```   101
```
```   102
```
```   103 subsection{*Integer Addition*}
```
```   104
```
```   105 lemma add:
```
```   106      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
```
```   107       Abs_Integ (intrel``{(x+u, y+v)})"
```
```   108 proof -
```
```   109   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
```
```   110         respects2 intrel"
```
```   111     by (simp add: congruent2_def)
```
```   112   thus ?thesis
```
```   113     by (simp add: add_int_def UN_UN_split_split_eq
```
```   114                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   115 qed
```
```   116
```
```   117 lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
```
```   118   by (cases z, cases w) (simp add: minus add)
```
```   119
```
```   120 lemma zadd_commute: "(z::int) + w = w + z"
```
```   121   by (cases z, cases w) (simp add: add_ac add)
```
```   122
```
```   123 lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
```
```   124   by (cases z1, cases z2, cases z3) (simp add: add add_assoc)
```
```   125
```
```   126 (*For AC rewriting*)
```
```   127 lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"
```
```   128   apply (rule mk_left_commute [of "op +"])
```
```   129   apply (rule zadd_assoc)
```
```   130   apply (rule zadd_commute)
```
```   131   done
```
```   132
```
```   133 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
```
```   134
```
```   135 lemmas zmult_ac = OrderedGroup.mult_ac
```
```   136
```
```   137 (*also for the instance declaration int :: comm_monoid_add*)
```
```   138 lemma zadd_0: "(0::int) + z = z"
```
```   139 apply (simp add: Zero_int_def)
```
```   140 apply (cases z, simp add: add)
```
```   141 done
```
```   142
```
```   143 lemma zadd_0_right: "z + (0::int) = z"
```
```   144 by (rule trans [OF zadd_commute zadd_0])
```
```   145
```
```   146 lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
```
```   147 by (cases z, simp add: Zero_int_def minus add)
```
```   148
```
```   149
```
```   150 subsection{*Integer Multiplication*}
```
```   151
```
```   152 text{*Congruence property for multiplication*}
```
```   153 lemma mult_congruent2:
```
```   154      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   155       respects2 intrel"
```
```   156 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   157  apply (force simp add: mult_ac, clarify)
```
```   158 apply (simp add: congruent_def mult_ac)
```
```   159 apply (rename_tac u v w x y z)
```
```   160 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   161 apply (simp add: mult_ac)
```
```   162 apply (simp add: add_mult_distrib [symmetric])
```
```   163 done
```
```   164
```
```   165
```
```   166 lemma mult:
```
```   167      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   168       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   169 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   170               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   171
```
```   172 lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
```
```   173 by (cases z, cases w, simp add: minus mult add_ac)
```
```   174
```
```   175 lemma zmult_commute: "(z::int) * w = w * z"
```
```   176 by (cases z, cases w, simp add: mult add_ac mult_ac)
```
```   177
```
```   178 lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
```
```   179 by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
```
```   180
```
```   181 lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
```
```   182 by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
```
```   183
```
```   184 lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
```
```   185 by (simp add: zmult_commute [of w] zadd_zmult_distrib)
```
```   186
```
```   187 lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
```
```   188 by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)
```
```   189
```
```   190 lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"
```
```   191 by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
```
```   192
```
```   193 lemmas int_distrib =
```
```   194   zadd_zmult_distrib zadd_zmult_distrib2
```
```   195   zdiff_zmult_distrib zdiff_zmult_distrib2
```
```   196
```
```   197
```
```   198 lemma zmult_1: "(1::int) * z = z"
```
```   199 by (cases z, simp add: One_int_def mult)
```
```   200
```
```   201 lemma zmult_1_right: "z * (1::int) = z"
```
```   202 by (rule trans [OF zmult_commute zmult_1])
```
```   203
```
```   204
```
```   205 text{*The integers form a @{text comm_ring_1}*}
```
```   206 instance int :: comm_ring_1
```
```   207 proof
```
```   208   fix i j k :: int
```
```   209   show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
```
```   210   show "i + j = j + i" by (simp add: zadd_commute)
```
```   211   show "0 + i = i" by (rule zadd_0)
```
```   212   show "- i + i = 0" by (rule zadd_zminus_inverse2)
```
```   213   show "i - j = i + (-j)" by (simp add: diff_int_def)
```
```   214   show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
```
```   215   show "i * j = j * i" by (rule zmult_commute)
```
```   216   show "1 * i = i" by (rule zmult_1)
```
```   217   show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
```
```   218   show "0 \<noteq> (1::int)" by (simp add: Zero_int_def One_int_def)
```
```   219 qed
```
```   220
```
```   221 abbreviation
```
```   222   int :: "nat \<Rightarrow> int"
```
```   223 where
```
```   224   "int \<equiv> of_nat"
```
```   225
```
```   226 lemma int_def: "int m = Abs_Integ (intrel `` {(m, 0)})"
```
```   227 by (induct m, simp_all add: Zero_int_def One_int_def add)
```
```   228
```
```   229
```
```   230 subsection{*The @{text "\<le>"} Ordering*}
```
```   231
```
```   232 lemma le:
```
```   233   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   234 by (force simp add: le_int_def)
```
```   235
```
```   236 lemma less:
```
```   237   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
```
```   238 by (simp add: less_int_def le order_less_le)
```
```   239
```
```   240 lemma zle_refl: "w \<le> (w::int)"
```
```   241 by (cases w, simp add: le)
```
```   242
```
```   243 lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
```
```   244 by (cases i, cases j, cases k, simp add: le)
```
```   245
```
```   246 lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
```
```   247 by (cases w, cases z, simp add: le)
```
```   248
```
```   249 instance int :: order
```
```   250   by intro_classes
```
```   251     (assumption |
```
```   252       rule zle_refl zle_trans zle_anti_sym less_int_def [THEN meta_eq_to_obj_eq])+
```
```   253
```
```   254 lemma zle_linear: "(z::int) \<le> w \<or> w \<le> z"
```
```   255 by (cases z, cases w) (simp add: le linorder_linear)
```
```   256
```
```   257 instance int :: linorder
```
```   258   by intro_classes (rule zle_linear)
```
```   259
```
```   260 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```   261
```
```   262
```
```   263 lemma int_0_less_1: "0 < (1::int)"
```
```   264 by (simp add: Zero_int_def One_int_def less)
```
```   265
```
```   266 lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
```
```   267 by (rule int_0_less_1 [THEN less_imp_neq])
```
```   268
```
```   269
```
```   270 subsection{*Monotonicity results*}
```
```   271
```
```   272 instance int :: pordered_cancel_ab_semigroup_add
```
```   273 proof
```
```   274   fix a b c :: int
```
```   275   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
```
```   276     by (cases a, cases b, cases c, simp add: le add)
```
```   277 qed
```
```   278
```
```   279 lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
```
```   280 by (rule add_left_mono)
```
```   281
```
```   282 lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
```
```   283 by (rule add_strict_right_mono)
```
```   284
```
```   285 lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
```
```   286 by (rule add_less_le_mono)
```
```   287
```
```   288
```
```   289 subsection{*Strict Monotonicity of Multiplication*}
```
```   290
```
```   291 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   292 lemma zmult_zless_mono2_lemma:
```
```   293      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
```
```   294 apply (induct "k", simp)
```
```   295 apply (simp add: left_distrib)
```
```   296 apply (case_tac "k=0")
```
```   297 apply (simp_all add: add_strict_mono)
```
```   298 done
```
```   299
```
```   300 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
```
```   301 apply (cases k)
```
```   302 apply (auto simp add: le add int_def Zero_int_def)
```
```   303 apply (rule_tac x="x-y" in exI, simp)
```
```   304 done
```
```   305
```
```   306 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
```
```   307 apply (cases k)
```
```   308 apply (simp add: less int_def Zero_int_def)
```
```   309 apply (rule_tac x="x-y" in exI, simp)
```
```   310 done
```
```   311
```
```   312 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   313 apply (drule zero_less_imp_eq_int)
```
```   314 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   315 done
```
```   316
```
```   317 instance int :: minus
```
```   318   zabs_def: "\<bar>i\<Colon>int\<bar> \<equiv> if i < 0 then - i else i" ..
```
```   319
```
```   320 instance int :: distrib_lattice
```
```   321   "inf \<equiv> min"
```
```   322   "sup \<equiv> max"
```
```   323   by intro_classes
```
```   324     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   325
```
```   326 text{*The integers form an ordered integral domain*}
```
```   327 instance int :: ordered_idom
```
```   328 proof
```
```   329   fix i j k :: int
```
```   330   show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
```
```   331   show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
```
```   332 qed
```
```   333
```
```   334 lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
```
```   335 apply (cases w, cases z)
```
```   336 apply (simp add: less le add One_int_def)
```
```   337 done
```
```   338
```
```   339
```
```   340 subsection{*Magnitude of an Integer, as a Natural Number: @{term nat}*}
```
```   341
```
```   342 definition
```
```   343   nat :: "int \<Rightarrow> nat"
```
```   344 where
```
```   345   [code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
```
```   346
```
```   347 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   348 proof -
```
```   349   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   350     by (simp add: congruent_def) arith
```
```   351   thus ?thesis
```
```   352     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   353 qed
```
```   354
```
```   355 lemma nat_int [simp]: "nat (int n) = n"
```
```   356 by (simp add: nat int_def)
```
```   357
```
```   358 lemma nat_zero [simp]: "nat 0 = 0"
```
```   359 by (simp add: Zero_int_def nat)
```
```   360
```
```   361 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
```
```   362 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   363
```
```   364 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
```
```   365 by simp
```
```   366
```
```   367 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   368 by (cases z, simp add: nat le Zero_int_def)
```
```   369
```
```   370 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   371 apply (cases w, cases z)
```
```   372 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
```
```   373 done
```
```   374
```
```   375 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   376 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   377 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   378
```
```   379 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   380 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   381
```
```   382 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   383 apply (cases w, cases z)
```
```   384 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
```
```   385 done
```
```   386
```
```   387 lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
```
```   388 by (blast dest: nat_0_le sym)
```
```   389
```
```   390 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
```
```   391 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   392
```
```   393 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
```
```   394 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   395
```
```   396 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
```
```   397 apply (cases w)
```
```   398 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   399 done
```
```   400
```
```   401 lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
```
```   402 by (auto simp add: nat_eq_iff2)
```
```   403
```
```   404 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   405 by (insert zless_nat_conj [of 0], auto)
```
```   406
```
```   407 lemma nat_add_distrib:
```
```   408      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   409 by (cases z, cases z', simp add: nat add le Zero_int_def)
```
```   410
```
```   411 lemma nat_diff_distrib:
```
```   412      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   413 by (cases z, cases z',
```
```   414     simp add: nat add minus diff_minus le Zero_int_def)
```
```   415
```
```   416 lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
```
```   417 by (simp add: int_def minus nat Zero_int_def)
```
```   418
```
```   419 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
```
```   420 by (cases z, simp add: nat less int_def, arith)
```
```   421
```
```   422
```
```   423 subsection{*Lemmas about the Function @{term int} and Orderings*}
```
```   424
```
```   425 lemma negative_zless_0: "- (int (Suc n)) < 0"
```
```   426 by (simp add: order_less_le del: of_nat_Suc)
```
```   427
```
```   428 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
```
```   429 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   430
```
```   431 lemma negative_zle_0: "- int n \<le> 0"
```
```   432 by (simp add: minus_le_iff)
```
```   433
```
```   434 lemma negative_zle [iff]: "- int n \<le> int m"
```
```   435 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   436
```
```   437 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
```
```   438 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   439
```
```   440 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
```
```   441 by (simp add: int_def le minus Zero_int_def)
```
```   442
```
```   443 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
```
```   444 by (simp add: linorder_not_less)
```
```   445
```
```   446 lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
```
```   447 by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
```
```   448
```
```   449 lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
```
```   450 proof (cases w, cases z, simp add: le add int_def)
```
```   451   fix a b c d
```
```   452   assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})"
```
```   453   show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)"
```
```   454   proof
```
```   455     assume "a + d \<le> c + b"
```
```   456     thus "\<exists>n. c + b = a + n + d"
```
```   457       by (auto intro!: exI [where x="c+b - (a+d)"])
```
```   458   next
```
```   459     assume "\<exists>n. c + b = a + n + d"
```
```   460     then obtain n where "c + b = a + n + d" ..
```
```   461     thus "a + d \<le> c + b" by arith
```
```   462   qed
```
```   463 qed
```
```   464
```
```   465 lemma abs_of_nat [simp]: "\<bar>of_nat n::'a::ordered_idom\<bar> = of_nat n"
```
```   466 by (rule  of_nat_0_le_iff [THEN abs_of_nonneg]) (* belongs in Nat.thy *)
```
```   467
```
```   468 text{*This version is proved for all ordered rings, not just integers!
```
```   469       It is proved here because attribute @{text arith_split} is not available
```
```   470       in theory @{text Ring_and_Field}.
```
```   471       But is it really better than just rewriting with @{text abs_if}?*}
```
```   472 lemma abs_split [arith_split]:
```
```   473      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   474 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   475
```
```   476
```
```   477 subsection {* Constants @{term neg} and @{term iszero} *}
```
```   478
```
```   479 definition
```
```   480   neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
```
```   481 where
```
```   482   [code inline]: "neg Z \<longleftrightarrow> Z < 0"
```
```   483
```
```   484 definition (*for simplifying equalities*)
```
```   485   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
```
```   486 where
```
```   487   "iszero z \<longleftrightarrow> z = 0"
```
```   488
```
```   489 lemma not_neg_int [simp]: "~ neg (int n)"
```
```   490 by (simp add: neg_def)
```
```   491
```
```   492 lemma neg_zminus_int [simp]: "neg (- (int (Suc n)))"
```
```   493 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
```
```   494
```
```   495 lemmas neg_eq_less_0 = neg_def
```
```   496
```
```   497 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   498 by (simp add: neg_def linorder_not_less)
```
```   499
```
```   500
```
```   501 subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   502
```
```   503 lemma not_neg_0: "~ neg 0"
```
```   504 by (simp add: One_int_def neg_def)
```
```   505
```
```   506 lemma not_neg_1: "~ neg 1"
```
```   507 by (simp add: neg_def linorder_not_less zero_le_one)
```
```   508
```
```   509 lemma iszero_0: "iszero 0"
```
```   510 by (simp add: iszero_def)
```
```   511
```
```   512 lemma not_iszero_1: "~ iszero 1"
```
```   513 by (simp add: iszero_def eq_commute)
```
```   514
```
```   515 lemma neg_nat: "neg z ==> nat z = 0"
```
```   516 by (simp add: neg_def order_less_imp_le)
```
```   517
```
```   518 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
```
```   519 by (simp add: linorder_not_less neg_def)
```
```   520
```
```   521
```
```   522 subsection{*The Set of Natural Numbers*}
```
```   523
```
```   524 constdefs
```
```   525   Nats  :: "'a::semiring_1 set"
```
```   526   "Nats == range of_nat"
```
```   527
```
```   528 notation (xsymbols)
```
```   529   Nats  ("\<nat>")
```
```   530
```
```   531 lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
```
```   532 by (simp add: Nats_def)
```
```   533
```
```   534 lemma Nats_0 [simp]: "0 \<in> Nats"
```
```   535 apply (simp add: Nats_def)
```
```   536 apply (rule range_eqI)
```
```   537 apply (rule of_nat_0 [symmetric])
```
```   538 done
```
```   539
```
```   540 lemma Nats_1 [simp]: "1 \<in> Nats"
```
```   541 apply (simp add: Nats_def)
```
```   542 apply (rule range_eqI)
```
```   543 apply (rule of_nat_1 [symmetric])
```
```   544 done
```
```   545
```
```   546 lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
```
```   547 apply (auto simp add: Nats_def)
```
```   548 apply (rule range_eqI)
```
```   549 apply (rule of_nat_add [symmetric])
```
```   550 done
```
```   551
```
```   552 lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
```
```   553 apply (auto simp add: Nats_def)
```
```   554 apply (rule range_eqI)
```
```   555 apply (rule of_nat_mult [symmetric])
```
```   556 done
```
```   557
```
```   558 lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
```
```   559 proof
```
```   560   fix n
```
```   561   show "of_nat n = id n"  by (induct n, simp_all)
```
```   562 qed (* belongs in Nat.thy *)
```
```   563
```
```   564
```
```   565 subsection{*Embedding of the Integers into any @{text ring_1}:
```
```   566 @{term of_int}*}
```
```   567
```
```   568 constdefs
```
```   569    of_int :: "int => 'a::ring_1"
```
```   570    "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   571
```
```   572
```
```   573 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   574 proof -
```
```   575   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   576     by (simp add: congruent_def compare_rls of_nat_add [symmetric]
```
```   577             del: of_nat_add)
```
```   578   thus ?thesis
```
```   579     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   580 qed
```
```   581
```
```   582 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   583 by (simp add: of_int Zero_int_def)
```
```   584
```
```   585 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   586 by (simp add: of_int One_int_def)
```
```   587
```
```   588 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   589 by (cases w, cases z, simp add: compare_rls of_int add)
```
```   590
```
```   591 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   592 by (cases z, simp add: compare_rls of_int minus)
```
```   593
```
```   594 lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
```
```   595 by (simp add: diff_minus)
```
```   596
```
```   597 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   598 apply (cases w, cases z)
```
```   599 apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
```
```   600                  mult add_ac)
```
```   601 done
```
```   602
```
```   603 lemma of_int_le_iff [simp]:
```
```   604      "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
```
```   605 apply (cases w)
```
```   606 apply (cases z)
```
```   607 apply (simp add: compare_rls of_int le diff_int_def add minus
```
```   608                  of_nat_add [symmetric]   del: of_nat_add)
```
```   609 done
```
```   610
```
```   611 text{*Special cases where either operand is zero*}
```
```   612 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
```
```   613 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
```
```   614
```
```   615
```
```   616 lemma of_int_less_iff [simp]:
```
```   617      "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
```
```   618 by (simp add: linorder_not_le [symmetric])
```
```   619
```
```   620 text{*Special cases where either operand is zero*}
```
```   621 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
```
```   622 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
```
```   623
```
```   624 text{*Class for unital rings with characteristic zero.
```
```   625  Includes non-ordered rings like the complex numbers.*}
```
```   626 axclass ring_char_0 < ring_1, semiring_char_0
```
```   627
```
```   628 lemma of_int_eq_iff [simp]:
```
```   629      "(of_int w = (of_int z::'a::ring_char_0)) = (w = z)"
```
```   630 apply (cases w, cases z, simp add: of_int)
```
```   631 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
```
```   632 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
```
```   633 done
```
```   634
```
```   635 text{*Every @{text ordered_idom} has characteristic zero.*}
```
```   636 instance ordered_idom < ring_char_0 ..
```
```   637
```
```   638 text{*Special cases where either operand is zero*}
```
```   639 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
```
```   640 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
```
```   641
```
```   642 lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
```
```   643 proof
```
```   644   fix z
```
```   645   show "of_int z = id z"
```
```   646     by (cases z)
```
```   647       (simp add: of_int add minus int_def diff_minus)
```
```   648 qed
```
```   649
```
```   650
```
```   651 subsection{*The Set of Integers*}
```
```   652
```
```   653 constdefs
```
```   654   Ints  :: "'a::ring_1 set"
```
```   655   "Ints == range of_int"
```
```   656
```
```   657 notation (xsymbols)
```
```   658   Ints  ("\<int>")
```
```   659
```
```   660 lemma Ints_0 [simp]: "0 \<in> Ints"
```
```   661 apply (simp add: Ints_def)
```
```   662 apply (rule range_eqI)
```
```   663 apply (rule of_int_0 [symmetric])
```
```   664 done
```
```   665
```
```   666 lemma Ints_1 [simp]: "1 \<in> Ints"
```
```   667 apply (simp add: Ints_def)
```
```   668 apply (rule range_eqI)
```
```   669 apply (rule of_int_1 [symmetric])
```
```   670 done
```
```   671
```
```   672 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
```
```   673 apply (auto simp add: Ints_def)
```
```   674 apply (rule range_eqI)
```
```   675 apply (rule of_int_add [symmetric])
```
```   676 done
```
```   677
```
```   678 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
```
```   679 apply (auto simp add: Ints_def)
```
```   680 apply (rule range_eqI)
```
```   681 apply (rule of_int_minus [symmetric])
```
```   682 done
```
```   683
```
```   684 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
```
```   685 apply (auto simp add: Ints_def)
```
```   686 apply (rule range_eqI)
```
```   687 apply (rule of_int_diff [symmetric])
```
```   688 done
```
```   689
```
```   690 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
```
```   691 apply (auto simp add: Ints_def)
```
```   692 apply (rule range_eqI)
```
```   693 apply (rule of_int_mult [symmetric])
```
```   694 done
```
```   695
```
```   696 text{*Collapse nested embeddings*}
```
```   697 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   698 by (induct n, auto)
```
```   699
```
```   700 lemma Ints_cases [cases set: Ints]:
```
```   701   assumes "q \<in> \<int>"
```
```   702   obtains (of_int) z where "q = of_int z"
```
```   703   unfolding Ints_def
```
```   704 proof -
```
```   705   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```   706   then obtain z where "q = of_int z" ..
```
```   707   then show thesis ..
```
```   708 qed
```
```   709
```
```   710 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```   711   "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
```
```   712   by (rule Ints_cases) auto
```
```   713
```
```   714
```
```   715 (* int (Suc n) = 1 + int n *)
```
```   716
```
```   717
```
```   718
```
```   719 subsection{*More Properties of @{term setsum} and  @{term setprod}*}
```
```   720
```
```   721 text{*By Jeremy Avigad*}
```
```   722
```
```   723
```
```   724 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```   725   apply (cases "finite A")
```
```   726   apply (erule finite_induct, auto)
```
```   727   done
```
```   728
```
```   729 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```   730   apply (cases "finite A")
```
```   731   apply (erule finite_induct, auto)
```
```   732   done
```
```   733
```
```   734 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```   735   apply (cases "finite A")
```
```   736   apply (erule finite_induct, auto)
```
```   737   done
```
```   738
```
```   739 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```   740   apply (cases "finite A")
```
```   741   apply (erule finite_induct, auto)
```
```   742   done
```
```   743
```
```   744 lemma setprod_nonzero_nat:
```
```   745     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
```
```   746   by (rule setprod_nonzero, auto)
```
```   747
```
```   748 lemma setprod_zero_eq_nat:
```
```   749     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
```
```   750   by (rule setprod_zero_eq, auto)
```
```   751
```
```   752 lemma setprod_nonzero_int:
```
```   753     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
```
```   754   by (rule setprod_nonzero, auto)
```
```   755
```
```   756 lemma setprod_zero_eq_int:
```
```   757     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
```
```   758   by (rule setprod_zero_eq, auto)
```
```   759
```
```   760
```
```   761 subsection {* Further properties *}
```
```   762
```
```   763 text{*Now we replace the case analysis rule by a more conventional one:
```
```   764 whether an integer is negative or not.*}
```
```   765
```
```   766 lemma zless_iff_Suc_zadd:
```
```   767     "(w < z) = (\<exists>n. z = w + int (Suc n))"
```
```   768 apply (cases z, cases w)
```
```   769 apply (auto simp add: le add int_def linorder_not_le [symmetric])
```
```   770 apply (rename_tac a b c d)
```
```   771 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   772 apply arith
```
```   773 done
```
```   774
```
```   775 lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
```
```   776 apply (cases x)
```
```   777 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   778 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   779 done
```
```   780
```
```   781 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   782      "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
```
```   783 apply (cases "z < 0", blast dest!: negD)
```
```   784 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   785 apply (blast dest: nat_0_le [THEN sym])
```
```   786 done
```
```   787
```
```   788 theorem int_induct'[induct type: int, case_names nonneg neg]:
```
```   789      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
```
```   790   by (cases z rule: int_cases) auto
```
```   791
```
```   792 text{*Contributed by Brian Huffman*}
```
```   793 theorem int_diff_cases [case_names diff]:
```
```   794 assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
```
```   795 apply (cases z rule: eq_Abs_Integ)
```
```   796 apply (rule_tac m=x and n=y in prem)
```
```   797 apply (simp add: int_def diff_def minus add)
```
```   798 done
```
```   799
```
```   800 lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
```
```   801 by (cases z rule: eq_Abs_Integ, simp add: nat le of_int Zero_int_def)
```
```   802
```
```   803
```
```   804 subsection {* Legacy theorems *}
```
```   805
```
```   806 lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
```
```   807 by simp
```
```   808
```
```   809 lemma int_Suc: "int (Suc m) = 1 + (int m)"
```
```   810 by simp
```
```   811
```
```   812 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
```
```   813 by simp
```
```   814
```
```   815 lemmas inj_int = inj_of_nat [where 'a=int]
```
```   816 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```   817 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```   818 lemmas int_mult = of_nat_mult [where 'a=int]
```
```   819 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```   820 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int]
```
```   821 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```   822 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int]
```
```   823 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```   824 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```   825 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```   826 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int]
```
```   827 lemmas int_0 = of_nat_0 [where ?'a_1.0=int]
```
```   828 lemmas int_1 = of_nat_1 [where 'a=int]
```
```   829 lemmas abs_int_eq = abs_of_nat [where 'a=int]
```
```   830 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```   831 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```   832 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```   833 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```   834 lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq]
```
```   835 lemmas int_eq_of_nat = TrueI
```
```   836
```
```   837 abbreviation
```
```   838   int_of_nat :: "nat \<Rightarrow> int"
```
```   839 where
```
```   840   "int_of_nat \<equiv> of_nat"
```
```   841
```
```   842 end
```