src/HOL/Int.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 36811 4ab4aa5bee1c child 37767 a2b7a20d6ea3 permissions -rw-r--r--
declare lex_prod_def [code del]
```     1 (*  Title:      Int.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3                 Tobias Nipkow, Florian Haftmann, TU Muenchen
```
```     4     Copyright   1994  University of Cambridge
```
```     5
```
```     6 *)
```
```     7
```
```     8 header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *}
```
```     9
```
```    10 theory Int
```
```    11 imports Equiv_Relations Nat Wellfounded
```
```    12 uses
```
```    13   ("Tools/numeral.ML")
```
```    14   ("Tools/numeral_syntax.ML")
```
```    15   ("Tools/int_arith.ML")
```
```    16 begin
```
```    17
```
```    18 subsection {* The equivalence relation underlying the integers *}
```
```    19
```
```    20 definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where
```
```    21   [code del]: "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
```
```    22
```
```    23 typedef (Integ)
```
```    24   int = "UNIV//intrel"
```
```    25   by (auto simp add: quotient_def)
```
```    26
```
```    27 instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
```
```    28 begin
```
```    29
```
```    30 definition
```
```    31   Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
```
```    32
```
```    33 definition
```
```    34   One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
```
```    35
```
```    36 definition
```
```    37   add_int_def [code del]: "z + w = 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 definition
```
```    42   minus_int_def [code del]:
```
```    43     "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
```
```    44
```
```    45 definition
```
```    46   diff_int_def [code del]:  "z - w = z + (-w \<Colon> int)"
```
```    47
```
```    48 definition
```
```    49   mult_int_def [code del]: "z * w = Abs_Integ
```
```    50     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
```
```    51       intrel `` {(x*u + y*v, x*v + y*u)})"
```
```    52
```
```    53 definition
```
```    54   le_int_def [code del]:
```
```    55    "z \<le> w \<longleftrightarrow> (\<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w)"
```
```    56
```
```    57 definition
```
```    58   less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```    59
```
```    60 definition
```
```    61   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
```
```    62
```
```    63 definition
```
```    64   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```    65
```
```    66 instance ..
```
```    67
```
```    68 end
```
```    69
```
```    70
```
```    71 subsection{*Construction of the Integers*}
```
```    72
```
```    73 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    74 by (simp add: intrel_def)
```
```    75
```
```    76 lemma equiv_intrel: "equiv UNIV intrel"
```
```    77 by (simp add: intrel_def equiv_def refl_on_def sym_def trans_def)
```
```    78
```
```    79 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    80   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    81 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    82
```
```    83 text{*All equivalence classes belong to set of representatives*}
```
```    84 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
```
```    85 by (auto simp add: Integ_def intrel_def quotient_def)
```
```    86
```
```    87 text{*Reduces equality on abstractions to equality on representatives:
```
```    88   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
```
```    89 declare Abs_Integ_inject [simp,no_atp]  Abs_Integ_inverse [simp,no_atp]
```
```    90
```
```    91 text{*Case analysis on the representation of an integer as an equivalence
```
```    92       class of pairs of naturals.*}
```
```    93 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    94      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
```
```    95 apply (rule Abs_Integ_cases [of z])
```
```    96 apply (auto simp add: Integ_def quotient_def)
```
```    97 done
```
```    98
```
```    99
```
```   100 subsection {* Arithmetic Operations *}
```
```   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 add:
```
```   111      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
```
```   112       Abs_Integ (intrel``{(x+u, y+v)})"
```
```   113 proof -
```
```   114   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
```
```   115         respects2 intrel"
```
```   116     by (simp add: congruent2_def)
```
```   117   thus ?thesis
```
```   118     by (simp add: add_int_def UN_UN_split_split_eq
```
```   119                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   120 qed
```
```   121
```
```   122 text{*Congruence property for multiplication*}
```
```   123 lemma mult_congruent2:
```
```   124      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   125       respects2 intrel"
```
```   126 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   127  apply (force simp add: mult_ac, clarify)
```
```   128 apply (simp add: congruent_def mult_ac)
```
```   129 apply (rename_tac u v w x y z)
```
```   130 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   131 apply (simp add: mult_ac)
```
```   132 apply (simp add: add_mult_distrib [symmetric])
```
```   133 done
```
```   134
```
```   135 lemma mult:
```
```   136      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   137       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   138 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   139               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   140
```
```   141 text{*The integers form a @{text comm_ring_1}*}
```
```   142 instance int :: comm_ring_1
```
```   143 proof
```
```   144   fix i j k :: int
```
```   145   show "(i + j) + k = i + (j + k)"
```
```   146     by (cases i, cases j, cases k) (simp add: add add_assoc)
```
```   147   show "i + j = j + i"
```
```   148     by (cases i, cases j) (simp add: add_ac add)
```
```   149   show "0 + i = i"
```
```   150     by (cases i) (simp add: Zero_int_def add)
```
```   151   show "- i + i = 0"
```
```   152     by (cases i) (simp add: Zero_int_def minus add)
```
```   153   show "i - j = i + - j"
```
```   154     by (simp add: diff_int_def)
```
```   155   show "(i * j) * k = i * (j * k)"
```
```   156     by (cases i, cases j, cases k) (simp add: mult algebra_simps)
```
```   157   show "i * j = j * i"
```
```   158     by (cases i, cases j) (simp add: mult algebra_simps)
```
```   159   show "1 * i = i"
```
```   160     by (cases i) (simp add: One_int_def mult)
```
```   161   show "(i + j) * k = i * k + j * k"
```
```   162     by (cases i, cases j, cases k) (simp add: add mult algebra_simps)
```
```   163   show "0 \<noteq> (1::int)"
```
```   164     by (simp add: Zero_int_def One_int_def)
```
```   165 qed
```
```   166
```
```   167 lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
```
```   168 by (induct m, simp_all add: Zero_int_def One_int_def add)
```
```   169
```
```   170
```
```   171 subsection {* The @{text "\<le>"} Ordering *}
```
```   172
```
```   173 lemma le:
```
```   174   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   175 by (force simp add: le_int_def)
```
```   176
```
```   177 lemma less:
```
```   178   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
```
```   179 by (simp add: less_int_def le order_less_le)
```
```   180
```
```   181 instance int :: linorder
```
```   182 proof
```
```   183   fix i j k :: int
```
```   184   show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
```
```   185     by (cases i, cases j) (simp add: le)
```
```   186   show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
```
```   187     by (auto simp add: less_int_def dest: antisym)
```
```   188   show "i \<le> i"
```
```   189     by (cases i) (simp add: le)
```
```   190   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
```
```   191     by (cases i, cases j, cases k) (simp add: le)
```
```   192   show "i \<le> j \<or> j \<le> i"
```
```   193     by (cases i, cases j) (simp add: le linorder_linear)
```
```   194 qed
```
```   195
```
```   196 instantiation int :: distrib_lattice
```
```   197 begin
```
```   198
```
```   199 definition
```
```   200   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   201
```
```   202 definition
```
```   203   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   204
```
```   205 instance
```
```   206   by intro_classes
```
```   207     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   208
```
```   209 end
```
```   210
```
```   211 instance int :: ordered_cancel_ab_semigroup_add
```
```   212 proof
```
```   213   fix i j k :: int
```
```   214   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   215     by (cases i, cases j, cases k) (simp add: le add)
```
```   216 qed
```
```   217
```
```   218
```
```   219 text{*Strict Monotonicity of Multiplication*}
```
```   220
```
```   221 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   222 lemma zmult_zless_mono2_lemma:
```
```   223      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
```
```   224 apply (induct "k", simp)
```
```   225 apply (simp add: left_distrib)
```
```   226 apply (case_tac "k=0")
```
```   227 apply (simp_all add: add_strict_mono)
```
```   228 done
```
```   229
```
```   230 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
```
```   231 apply (cases k)
```
```   232 apply (auto simp add: le add int_def Zero_int_def)
```
```   233 apply (rule_tac x="x-y" in exI, simp)
```
```   234 done
```
```   235
```
```   236 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
```
```   237 apply (cases k)
```
```   238 apply (simp add: less int_def Zero_int_def)
```
```   239 apply (rule_tac x="x-y" in exI, simp)
```
```   240 done
```
```   241
```
```   242 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   243 apply (drule zero_less_imp_eq_int)
```
```   244 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   245 done
```
```   246
```
```   247 text{*The integers form an ordered integral domain*}
```
```   248 instance int :: linordered_idom
```
```   249 proof
```
```   250   fix i j k :: int
```
```   251   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   252     by (rule zmult_zless_mono2)
```
```   253   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
```
```   254     by (simp only: zabs_def)
```
```   255   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   256     by (simp only: zsgn_def)
```
```   257 qed
```
```   258
```
```   259 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
```
```   260 apply (cases w, cases z)
```
```   261 apply (simp add: less le add One_int_def)
```
```   262 done
```
```   263
```
```   264 lemma zless_iff_Suc_zadd:
```
```   265   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
```
```   266 apply (cases z, cases w)
```
```   267 apply (auto simp add: less add int_def)
```
```   268 apply (rename_tac a b c d)
```
```   269 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   270 apply arith
```
```   271 done
```
```   272
```
```   273 lemmas int_distrib =
```
```   274   left_distrib [of "z1::int" "z2" "w", standard]
```
```   275   right_distrib [of "w::int" "z1" "z2", standard]
```
```   276   left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```   277   right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```   278
```
```   279
```
```   280 subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
```
```   281
```
```   282 context ring_1
```
```   283 begin
```
```   284
```
```   285 definition of_int :: "int \<Rightarrow> 'a" where
```
```   286   [code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   287
```
```   288 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   289 proof -
```
```   290   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   291     by (simp add: congruent_def algebra_simps of_nat_add [symmetric]
```
```   292             del: of_nat_add)
```
```   293   thus ?thesis
```
```   294     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   295 qed
```
```   296
```
```   297 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   298 by (simp add: of_int Zero_int_def)
```
```   299
```
```   300 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   301 by (simp add: of_int One_int_def)
```
```   302
```
```   303 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   304 by (cases w, cases z, simp add: algebra_simps of_int add)
```
```   305
```
```   306 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   307 by (cases z, simp add: algebra_simps of_int minus)
```
```   308
```
```   309 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   310 by (simp add: diff_minus Groups.diff_minus)
```
```   311
```
```   312 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   313 apply (cases w, cases z)
```
```   314 apply (simp add: algebra_simps of_int mult of_nat_mult)
```
```   315 done
```
```   316
```
```   317 text{*Collapse nested embeddings*}
```
```   318 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   319 by (induct n) auto
```
```   320
```
```   321 lemma of_int_power:
```
```   322   "of_int (z ^ n) = of_int z ^ n"
```
```   323   by (induct n) simp_all
```
```   324
```
```   325 end
```
```   326
```
```   327 text{*Class for unital rings with characteristic zero.
```
```   328  Includes non-ordered rings like the complex numbers.*}
```
```   329 class ring_char_0 = ring_1 + semiring_char_0
```
```   330 begin
```
```   331
```
```   332 lemma of_int_eq_iff [simp]:
```
```   333    "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   334 apply (cases w, cases z, simp add: of_int)
```
```   335 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
```
```   336 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
```
```   337 done
```
```   338
```
```   339 text{*Special cases where either operand is zero*}
```
```   340 lemma of_int_eq_0_iff [simp]:
```
```   341   "of_int z = 0 \<longleftrightarrow> z = 0"
```
```   342   using of_int_eq_iff [of z 0] by simp
```
```   343
```
```   344 lemma of_int_0_eq_iff [simp]:
```
```   345   "0 = of_int z \<longleftrightarrow> z = 0"
```
```   346   using of_int_eq_iff [of 0 z] by simp
```
```   347
```
```   348 end
```
```   349
```
```   350 context linordered_idom
```
```   351 begin
```
```   352
```
```   353 text{*Every @{text linordered_idom} has characteristic zero.*}
```
```   354 subclass ring_char_0 ..
```
```   355
```
```   356 lemma of_int_le_iff [simp]:
```
```   357   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   358   by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
```
```   359
```
```   360 lemma of_int_less_iff [simp]:
```
```   361   "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   362   by (simp add: less_le order_less_le)
```
```   363
```
```   364 lemma of_int_0_le_iff [simp]:
```
```   365   "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
```
```   366   using of_int_le_iff [of 0 z] by simp
```
```   367
```
```   368 lemma of_int_le_0_iff [simp]:
```
```   369   "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
```
```   370   using of_int_le_iff [of z 0] by simp
```
```   371
```
```   372 lemma of_int_0_less_iff [simp]:
```
```   373   "0 < of_int z \<longleftrightarrow> 0 < z"
```
```   374   using of_int_less_iff [of 0 z] by simp
```
```   375
```
```   376 lemma of_int_less_0_iff [simp]:
```
```   377   "of_int z < 0 \<longleftrightarrow> z < 0"
```
```   378   using of_int_less_iff [of z 0] by simp
```
```   379
```
```   380 end
```
```   381
```
```   382 lemma of_int_eq_id [simp]: "of_int = id"
```
```   383 proof
```
```   384   fix z show "of_int z = id z"
```
```   385     by (cases z) (simp add: of_int add minus int_def diff_minus)
```
```   386 qed
```
```   387
```
```   388
```
```   389 subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
```
```   390
```
```   391 definition
```
```   392   nat :: "int \<Rightarrow> nat"
```
```   393 where
```
```   394   [code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
```
```   395
```
```   396 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   397 proof -
```
```   398   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   399     by (simp add: congruent_def) arith
```
```   400   thus ?thesis
```
```   401     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   402 qed
```
```   403
```
```   404 lemma nat_int [simp]: "nat (of_nat n) = n"
```
```   405 by (simp add: nat int_def)
```
```   406
```
```   407 (* FIXME: duplicates nat_0 *)
```
```   408 lemma nat_zero [simp]: "nat 0 = 0"
```
```   409 by (simp add: Zero_int_def nat)
```
```   410
```
```   411 lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
```
```   412 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   413
```
```   414 corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
```
```   415 by simp
```
```   416
```
```   417 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   418 by (cases z, simp add: nat le Zero_int_def)
```
```   419
```
```   420 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   421 apply (cases w, cases z)
```
```   422 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
```
```   423 done
```
```   424
```
```   425 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   426 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   427 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   428
```
```   429 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   430 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   431
```
```   432 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   433 apply (cases w, cases z)
```
```   434 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
```
```   435 done
```
```   436
```
```   437 lemma nonneg_eq_int:
```
```   438   fixes z :: int
```
```   439   assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
```
```   440   shows P
```
```   441   using assms by (blast dest: nat_0_le sym)
```
```   442
```
```   443 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   444 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   445
```
```   446 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   447 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   448
```
```   449 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
```
```   450 apply (cases w)
```
```   451 apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)
```
```   452 done
```
```   453
```
```   454 lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
```
```   455 by(simp add: nat_eq_iff) arith
```
```   456
```
```   457 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
```
```   458 by (auto simp add: nat_eq_iff2)
```
```   459
```
```   460 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   461 by (insert zless_nat_conj [of 0], auto)
```
```   462
```
```   463 lemma nat_add_distrib:
```
```   464      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   465 by (cases z, cases z', simp add: nat add le Zero_int_def)
```
```   466
```
```   467 lemma nat_diff_distrib:
```
```   468      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   469 by (cases z, cases z',
```
```   470     simp add: nat add minus diff_minus le Zero_int_def)
```
```   471
```
```   472 lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
```
```   473 by (simp add: int_def minus nat Zero_int_def)
```
```   474
```
```   475 lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
```
```   476 by (cases z, simp add: nat less int_def, arith)
```
```   477
```
```   478 context ring_1
```
```   479 begin
```
```   480
```
```   481 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   482   by (cases z rule: eq_Abs_Integ)
```
```   483    (simp add: nat le of_int Zero_int_def of_nat_diff)
```
```   484
```
```   485 end
```
```   486
```
```   487 text {* For termination proofs: *}
```
```   488 lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
```
```   489
```
```   490
```
```   491 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
```
```   492
```
```   493 lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
```
```   494 by (simp add: order_less_le del: of_nat_Suc)
```
```   495
```
```   496 lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
```
```   497 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   498
```
```   499 lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
```
```   500 by (simp add: minus_le_iff)
```
```   501
```
```   502 lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
```
```   503 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   504
```
```   505 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
```
```   506 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   507
```
```   508 lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
```
```   509 by (simp add: int_def le minus Zero_int_def)
```
```   510
```
```   511 lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
```
```   512 by (simp add: linorder_not_less)
```
```   513
```
```   514 lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
```
```   515 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   516
```
```   517 lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
```
```   518 proof -
```
```   519   have "(w \<le> z) = (0 \<le> z - w)"
```
```   520     by (simp only: le_diff_eq add_0_left)
```
```   521   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
```
```   522     by (auto elim: zero_le_imp_eq_int)
```
```   523   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
```
```   524     by (simp only: algebra_simps)
```
```   525   finally show ?thesis .
```
```   526 qed
```
```   527
```
```   528 lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
```
```   529 by simp
```
```   530
```
```   531 lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
```
```   532 by simp
```
```   533
```
```   534 text{*This version is proved for all ordered rings, not just integers!
```
```   535       It is proved here because attribute @{text arith_split} is not available
```
```   536       in theory @{text Rings}.
```
```   537       But is it really better than just rewriting with @{text abs_if}?*}
```
```   538 lemma abs_split [arith_split,no_atp]:
```
```   539      "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   540 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   541
```
```   542 lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
```
```   543 apply (cases x)
```
```   544 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   545 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   546 done
```
```   547
```
```   548
```
```   549 subsection {* Cases and induction *}
```
```   550
```
```   551 text{*Now we replace the case analysis rule by a more conventional one:
```
```   552 whether an integer is negative or not.*}
```
```   553
```
```   554 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   555   "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
```
```   556 apply (cases "z < 0", blast dest!: negD)
```
```   557 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   558 apply auto
```
```   559 apply (blast dest: nat_0_le [THEN sym])
```
```   560 done
```
```   561
```
```   562 theorem int_of_nat_induct [induct type: int, case_names nonneg neg]:
```
```   563      "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
```
```   564   by (cases z rule: int_cases) auto
```
```   565
```
```   566 text{*Contributed by Brian Huffman*}
```
```   567 theorem int_diff_cases:
```
```   568   obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
```
```   569 apply (cases z rule: eq_Abs_Integ)
```
```   570 apply (rule_tac m=x and n=y in diff)
```
```   571 apply (simp add: int_def diff_def minus add)
```
```   572 done
```
```   573
```
```   574
```
```   575 subsection {* Binary representation *}
```
```   576
```
```   577 text {*
```
```   578   This formalization defines binary arithmetic in terms of the integers
```
```   579   rather than using a datatype. This avoids multiple representations (leading
```
```   580   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
```
```   581   int_of_binary}, for the numerical interpretation.
```
```   582
```
```   583   The representation expects that @{text "(m mod 2)"} is 0 or 1,
```
```   584   even if m is negative;
```
```   585   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
```
```   586   @{text "-5 = (-3)*2 + 1"}.
```
```   587
```
```   588   This two's complement binary representation derives from the paper
```
```   589   "An Efficient Representation of Arithmetic for Term Rewriting" by
```
```   590   Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
```
```   591   Springer LNCS 488 (240-251), 1991.
```
```   592 *}
```
```   593
```
```   594 subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
```
```   595
```
```   596 definition
```
```   597   Pls :: int where
```
```   598   [code del]: "Pls = 0"
```
```   599
```
```   600 definition
```
```   601   Min :: int where
```
```   602   [code del]: "Min = - 1"
```
```   603
```
```   604 definition
```
```   605   Bit0 :: "int \<Rightarrow> int" where
```
```   606   [code del]: "Bit0 k = k + k"
```
```   607
```
```   608 definition
```
```   609   Bit1 :: "int \<Rightarrow> int" where
```
```   610   [code del]: "Bit1 k = 1 + k + k"
```
```   611
```
```   612 class number = -- {* for numeric types: nat, int, real, \dots *}
```
```   613   fixes number_of :: "int \<Rightarrow> 'a"
```
```   614
```
```   615 use "Tools/numeral.ML"
```
```   616
```
```   617 syntax
```
```   618   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```   619
```
```   620 use "Tools/numeral_syntax.ML"
```
```   621 setup Numeral_Syntax.setup
```
```   622
```
```   623 abbreviation
```
```   624   "Numeral0 \<equiv> number_of Pls"
```
```   625
```
```   626 abbreviation
```
```   627   "Numeral1 \<equiv> number_of (Bit1 Pls)"
```
```   628
```
```   629 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
```
```   630   -- {* Unfold all @{text let}s involving constants *}
```
```   631   unfolding Let_def ..
```
```   632
```
```   633 definition
```
```   634   succ :: "int \<Rightarrow> int" where
```
```   635   [code del]: "succ k = k + 1"
```
```   636
```
```   637 definition
```
```   638   pred :: "int \<Rightarrow> int" where
```
```   639   [code del]: "pred k = k - 1"
```
```   640
```
```   641 lemmas
```
```   642   max_number_of [simp] = max_def
```
```   643     [of "number_of u" "number_of v", standard]
```
```   644 and
```
```   645   min_number_of [simp] = min_def
```
```   646     [of "number_of u" "number_of v", standard]
```
```   647   -- {* unfolding @{text minx} and @{text max} on numerals *}
```
```   648
```
```   649 lemmas numeral_simps =
```
```   650   succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
```
```   651
```
```   652 text {* Removal of leading zeroes *}
```
```   653
```
```   654 lemma Bit0_Pls [simp, code_post]:
```
```   655   "Bit0 Pls = Pls"
```
```   656   unfolding numeral_simps by simp
```
```   657
```
```   658 lemma Bit1_Min [simp, code_post]:
```
```   659   "Bit1 Min = Min"
```
```   660   unfolding numeral_simps by simp
```
```   661
```
```   662 lemmas normalize_bin_simps =
```
```   663   Bit0_Pls Bit1_Min
```
```   664
```
```   665
```
```   666 subsubsection {* Successor and predecessor functions *}
```
```   667
```
```   668 text {* Successor *}
```
```   669
```
```   670 lemma succ_Pls:
```
```   671   "succ Pls = Bit1 Pls"
```
```   672   unfolding numeral_simps by simp
```
```   673
```
```   674 lemma succ_Min:
```
```   675   "succ Min = Pls"
```
```   676   unfolding numeral_simps by simp
```
```   677
```
```   678 lemma succ_Bit0:
```
```   679   "succ (Bit0 k) = Bit1 k"
```
```   680   unfolding numeral_simps by simp
```
```   681
```
```   682 lemma succ_Bit1:
```
```   683   "succ (Bit1 k) = Bit0 (succ k)"
```
```   684   unfolding numeral_simps by simp
```
```   685
```
```   686 lemmas succ_bin_simps [simp] =
```
```   687   succ_Pls succ_Min succ_Bit0 succ_Bit1
```
```   688
```
```   689 text {* Predecessor *}
```
```   690
```
```   691 lemma pred_Pls:
```
```   692   "pred Pls = Min"
```
```   693   unfolding numeral_simps by simp
```
```   694
```
```   695 lemma pred_Min:
```
```   696   "pred Min = Bit0 Min"
```
```   697   unfolding numeral_simps by simp
```
```   698
```
```   699 lemma pred_Bit0:
```
```   700   "pred (Bit0 k) = Bit1 (pred k)"
```
```   701   unfolding numeral_simps by simp
```
```   702
```
```   703 lemma pred_Bit1:
```
```   704   "pred (Bit1 k) = Bit0 k"
```
```   705   unfolding numeral_simps by simp
```
```   706
```
```   707 lemmas pred_bin_simps [simp] =
```
```   708   pred_Pls pred_Min pred_Bit0 pred_Bit1
```
```   709
```
```   710
```
```   711 subsubsection {* Binary arithmetic *}
```
```   712
```
```   713 text {* Addition *}
```
```   714
```
```   715 lemma add_Pls:
```
```   716   "Pls + k = k"
```
```   717   unfolding numeral_simps by simp
```
```   718
```
```   719 lemma add_Min:
```
```   720   "Min + k = pred k"
```
```   721   unfolding numeral_simps by simp
```
```   722
```
```   723 lemma add_Bit0_Bit0:
```
```   724   "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
```
```   725   unfolding numeral_simps by simp
```
```   726
```
```   727 lemma add_Bit0_Bit1:
```
```   728   "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
```
```   729   unfolding numeral_simps by simp
```
```   730
```
```   731 lemma add_Bit1_Bit0:
```
```   732   "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
```
```   733   unfolding numeral_simps by simp
```
```   734
```
```   735 lemma add_Bit1_Bit1:
```
```   736   "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
```
```   737   unfolding numeral_simps by simp
```
```   738
```
```   739 lemma add_Pls_right:
```
```   740   "k + Pls = k"
```
```   741   unfolding numeral_simps by simp
```
```   742
```
```   743 lemma add_Min_right:
```
```   744   "k + Min = pred k"
```
```   745   unfolding numeral_simps by simp
```
```   746
```
```   747 lemmas add_bin_simps [simp] =
```
```   748   add_Pls add_Min add_Pls_right add_Min_right
```
```   749   add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
```
```   750
```
```   751 text {* Negation *}
```
```   752
```
```   753 lemma minus_Pls:
```
```   754   "- Pls = Pls"
```
```   755   unfolding numeral_simps by simp
```
```   756
```
```   757 lemma minus_Min:
```
```   758   "- Min = Bit1 Pls"
```
```   759   unfolding numeral_simps by simp
```
```   760
```
```   761 lemma minus_Bit0:
```
```   762   "- (Bit0 k) = Bit0 (- k)"
```
```   763   unfolding numeral_simps by simp
```
```   764
```
```   765 lemma minus_Bit1:
```
```   766   "- (Bit1 k) = Bit1 (pred (- k))"
```
```   767   unfolding numeral_simps by simp
```
```   768
```
```   769 lemmas minus_bin_simps [simp] =
```
```   770   minus_Pls minus_Min minus_Bit0 minus_Bit1
```
```   771
```
```   772 text {* Subtraction *}
```
```   773
```
```   774 lemma diff_bin_simps [simp]:
```
```   775   "k - Pls = k"
```
```   776   "k - Min = succ k"
```
```   777   "Pls - (Bit0 l) = Bit0 (Pls - l)"
```
```   778   "Pls - (Bit1 l) = Bit1 (Min - l)"
```
```   779   "Min - (Bit0 l) = Bit1 (Min - l)"
```
```   780   "Min - (Bit1 l) = Bit0 (Min - l)"
```
```   781   "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
```
```   782   "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
```
```   783   "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
```
```   784   "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
```
```   785   unfolding numeral_simps by simp_all
```
```   786
```
```   787 text {* Multiplication *}
```
```   788
```
```   789 lemma mult_Pls:
```
```   790   "Pls * w = Pls"
```
```   791   unfolding numeral_simps by simp
```
```   792
```
```   793 lemma mult_Min:
```
```   794   "Min * k = - k"
```
```   795   unfolding numeral_simps by simp
```
```   796
```
```   797 lemma mult_Bit0:
```
```   798   "(Bit0 k) * l = Bit0 (k * l)"
```
```   799   unfolding numeral_simps int_distrib by simp
```
```   800
```
```   801 lemma mult_Bit1:
```
```   802   "(Bit1 k) * l = (Bit0 (k * l)) + l"
```
```   803   unfolding numeral_simps int_distrib by simp
```
```   804
```
```   805 lemmas mult_bin_simps [simp] =
```
```   806   mult_Pls mult_Min mult_Bit0 mult_Bit1
```
```   807
```
```   808
```
```   809 subsubsection {* Binary comparisons *}
```
```   810
```
```   811 text {* Preliminaries *}
```
```   812
```
```   813 lemma even_less_0_iff:
```
```   814   "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
```
```   815 proof -
```
```   816   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
```
```   817   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
```
```   818     by (simp add: mult_less_0_iff zero_less_two
```
```   819                   order_less_not_sym [OF zero_less_two])
```
```   820   finally show ?thesis .
```
```   821 qed
```
```   822
```
```   823 lemma le_imp_0_less:
```
```   824   assumes le: "0 \<le> z"
```
```   825   shows "(0::int) < 1 + z"
```
```   826 proof -
```
```   827   have "0 \<le> z" by fact
```
```   828   also have "... < z + 1" by (rule less_add_one)
```
```   829   also have "... = 1 + z" by (simp add: add_ac)
```
```   830   finally show "0 < 1 + z" .
```
```   831 qed
```
```   832
```
```   833 lemma odd_less_0_iff:
```
```   834   "(1 + z + z < 0) = (z < (0::int))"
```
```   835 proof (cases z rule: int_cases)
```
```   836   case (nonneg n)
```
```   837   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```   838                              le_imp_0_less [THEN order_less_imp_le])
```
```   839 next
```
```   840   case (neg n)
```
```   841   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   842     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   843 qed
```
```   844
```
```   845 lemma bin_less_0_simps:
```
```   846   "Pls < 0 \<longleftrightarrow> False"
```
```   847   "Min < 0 \<longleftrightarrow> True"
```
```   848   "Bit0 w < 0 \<longleftrightarrow> w < 0"
```
```   849   "Bit1 w < 0 \<longleftrightarrow> w < 0"
```
```   850   unfolding numeral_simps
```
```   851   by (simp_all add: even_less_0_iff odd_less_0_iff)
```
```   852
```
```   853 lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
```
```   854   by simp
```
```   855
```
```   856 lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
```
```   857   unfolding numeral_simps
```
```   858   proof
```
```   859     have "k - 1 < k" by simp
```
```   860     also assume "k \<le> l"
```
```   861     finally show "k - 1 < l" .
```
```   862   next
```
```   863     assume "k - 1 < l"
```
```   864     hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
```
```   865     thus "k \<le> l" by simp
```
```   866   qed
```
```   867
```
```   868 lemma succ_pred: "succ (pred x) = x"
```
```   869   unfolding numeral_simps by simp
```
```   870
```
```   871 text {* Less-than *}
```
```   872
```
```   873 lemma less_bin_simps [simp]:
```
```   874   "Pls < Pls \<longleftrightarrow> False"
```
```   875   "Pls < Min \<longleftrightarrow> False"
```
```   876   "Pls < Bit0 k \<longleftrightarrow> Pls < k"
```
```   877   "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
```
```   878   "Min < Pls \<longleftrightarrow> True"
```
```   879   "Min < Min \<longleftrightarrow> False"
```
```   880   "Min < Bit0 k \<longleftrightarrow> Min < k"
```
```   881   "Min < Bit1 k \<longleftrightarrow> Min < k"
```
```   882   "Bit0 k < Pls \<longleftrightarrow> k < Pls"
```
```   883   "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
```
```   884   "Bit1 k < Pls \<longleftrightarrow> k < Pls"
```
```   885   "Bit1 k < Min \<longleftrightarrow> k < Min"
```
```   886   "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
```
```   887   "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
```
```   888   "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
```
```   889   "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
```
```   890   unfolding le_iff_pred_less
```
```   891     less_bin_lemma [of Pls]
```
```   892     less_bin_lemma [of Min]
```
```   893     less_bin_lemma [of "k"]
```
```   894     less_bin_lemma [of "Bit0 k"]
```
```   895     less_bin_lemma [of "Bit1 k"]
```
```   896     less_bin_lemma [of "pred Pls"]
```
```   897     less_bin_lemma [of "pred k"]
```
```   898   by (simp_all add: bin_less_0_simps succ_pred)
```
```   899
```
```   900 text {* Less-than-or-equal *}
```
```   901
```
```   902 lemma le_bin_simps [simp]:
```
```   903   "Pls \<le> Pls \<longleftrightarrow> True"
```
```   904   "Pls \<le> Min \<longleftrightarrow> False"
```
```   905   "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
```
```   906   "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
```
```   907   "Min \<le> Pls \<longleftrightarrow> True"
```
```   908   "Min \<le> Min \<longleftrightarrow> True"
```
```   909   "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
```
```   910   "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
```
```   911   "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
```
```   912   "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
```
```   913   "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
```
```   914   "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
```
```   915   "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
```
```   916   "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
```
```   917   "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
```
```   918   "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
```
```   919   unfolding not_less [symmetric]
```
```   920   by (simp_all add: not_le)
```
```   921
```
```   922 text {* Equality *}
```
```   923
```
```   924 lemma eq_bin_simps [simp]:
```
```   925   "Pls = Pls \<longleftrightarrow> True"
```
```   926   "Pls = Min \<longleftrightarrow> False"
```
```   927   "Pls = Bit0 l \<longleftrightarrow> Pls = l"
```
```   928   "Pls = Bit1 l \<longleftrightarrow> False"
```
```   929   "Min = Pls \<longleftrightarrow> False"
```
```   930   "Min = Min \<longleftrightarrow> True"
```
```   931   "Min = Bit0 l \<longleftrightarrow> False"
```
```   932   "Min = Bit1 l \<longleftrightarrow> Min = l"
```
```   933   "Bit0 k = Pls \<longleftrightarrow> k = Pls"
```
```   934   "Bit0 k = Min \<longleftrightarrow> False"
```
```   935   "Bit1 k = Pls \<longleftrightarrow> False"
```
```   936   "Bit1 k = Min \<longleftrightarrow> k = Min"
```
```   937   "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
```
```   938   "Bit0 k = Bit1 l \<longleftrightarrow> False"
```
```   939   "Bit1 k = Bit0 l \<longleftrightarrow> False"
```
```   940   "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
```
```   941   unfolding order_eq_iff [where 'a=int]
```
```   942   by (simp_all add: not_less)
```
```   943
```
```   944
```
```   945 subsection {* Converting Numerals to Rings: @{term number_of} *}
```
```   946
```
```   947 class number_ring = number + comm_ring_1 +
```
```   948   assumes number_of_eq: "number_of k = of_int k"
```
```   949
```
```   950 text {* self-embedding of the integers *}
```
```   951
```
```   952 instantiation int :: number_ring
```
```   953 begin
```
```   954
```
```   955 definition int_number_of_def [code del]:
```
```   956   "number_of w = (of_int w \<Colon> int)"
```
```   957
```
```   958 instance proof
```
```   959 qed (simp only: int_number_of_def)
```
```   960
```
```   961 end
```
```   962
```
```   963 lemma number_of_is_id:
```
```   964   "number_of (k::int) = k"
```
```   965   unfolding int_number_of_def by simp
```
```   966
```
```   967 lemma number_of_succ:
```
```   968   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
```
```   969   unfolding number_of_eq numeral_simps by simp
```
```   970
```
```   971 lemma number_of_pred:
```
```   972   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
```
```   973   unfolding number_of_eq numeral_simps by simp
```
```   974
```
```   975 lemma number_of_minus:
```
```   976   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
```
```   977   unfolding number_of_eq by (rule of_int_minus)
```
```   978
```
```   979 lemma number_of_add:
```
```   980   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
```
```   981   unfolding number_of_eq by (rule of_int_add)
```
```   982
```
```   983 lemma number_of_diff:
```
```   984   "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
```
```   985   unfolding number_of_eq by (rule of_int_diff)
```
```   986
```
```   987 lemma number_of_mult:
```
```   988   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
```
```   989   unfolding number_of_eq by (rule of_int_mult)
```
```   990
```
```   991 text {*
```
```   992   The correctness of shifting.
```
```   993   But it doesn't seem to give a measurable speed-up.
```
```   994 *}
```
```   995
```
```   996 lemma double_number_of_Bit0:
```
```   997   "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
```
```   998   unfolding number_of_eq numeral_simps left_distrib by simp
```
```   999
```
```  1000 text {*
```
```  1001   Converting numerals 0 and 1 to their abstract versions.
```
```  1002 *}
```
```  1003
```
```  1004 lemma numeral_0_eq_0 [simp, code_post]:
```
```  1005   "Numeral0 = (0::'a::number_ring)"
```
```  1006   unfolding number_of_eq numeral_simps by simp
```
```  1007
```
```  1008 lemma numeral_1_eq_1 [simp, code_post]:
```
```  1009   "Numeral1 = (1::'a::number_ring)"
```
```  1010   unfolding number_of_eq numeral_simps by simp
```
```  1011
```
```  1012 text {*
```
```  1013   Special-case simplification for small constants.
```
```  1014 *}
```
```  1015
```
```  1016 text{*
```
```  1017   Unary minus for the abstract constant 1. Cannot be inserted
```
```  1018   as a simprule until later: it is @{text number_of_Min} re-oriented!
```
```  1019 *}
```
```  1020
```
```  1021 lemma numeral_m1_eq_minus_1:
```
```  1022   "(-1::'a::number_ring) = - 1"
```
```  1023   unfolding number_of_eq numeral_simps by simp
```
```  1024
```
```  1025 lemma mult_minus1 [simp]:
```
```  1026   "-1 * z = -(z::'a::number_ring)"
```
```  1027   unfolding number_of_eq numeral_simps by simp
```
```  1028
```
```  1029 lemma mult_minus1_right [simp]:
```
```  1030   "z * -1 = -(z::'a::number_ring)"
```
```  1031   unfolding number_of_eq numeral_simps by simp
```
```  1032
```
```  1033 (*Negation of a coefficient*)
```
```  1034 lemma minus_number_of_mult [simp]:
```
```  1035    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
```
```  1036    unfolding number_of_eq by simp
```
```  1037
```
```  1038 text {* Subtraction *}
```
```  1039
```
```  1040 lemma diff_number_of_eq:
```
```  1041   "number_of v - number_of w =
```
```  1042     (number_of (v + uminus w)::'a::number_ring)"
```
```  1043   unfolding number_of_eq by simp
```
```  1044
```
```  1045 lemma number_of_Pls:
```
```  1046   "number_of Pls = (0::'a::number_ring)"
```
```  1047   unfolding number_of_eq numeral_simps by simp
```
```  1048
```
```  1049 lemma number_of_Min:
```
```  1050   "number_of Min = (- 1::'a::number_ring)"
```
```  1051   unfolding number_of_eq numeral_simps by simp
```
```  1052
```
```  1053 lemma number_of_Bit0:
```
```  1054   "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
```
```  1055   unfolding number_of_eq numeral_simps by simp
```
```  1056
```
```  1057 lemma number_of_Bit1:
```
```  1058   "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
```
```  1059   unfolding number_of_eq numeral_simps by simp
```
```  1060
```
```  1061
```
```  1062 subsubsection {* Equality of Binary Numbers *}
```
```  1063
```
```  1064 text {* First version by Norbert Voelker *}
```
```  1065
```
```  1066 definition (*for simplifying equalities*) iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" where
```
```  1067   "iszero z \<longleftrightarrow> z = 0"
```
```  1068
```
```  1069 lemma iszero_0: "iszero 0"
```
```  1070   by (simp add: iszero_def)
```
```  1071
```
```  1072 lemma iszero_Numeral0: "iszero (Numeral0 :: 'a::number_ring)"
```
```  1073   by (simp add: iszero_0)
```
```  1074
```
```  1075 lemma not_iszero_1: "\<not> iszero 1"
```
```  1076   by (simp add: iszero_def)
```
```  1077
```
```  1078 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1 :: 'a::number_ring)"
```
```  1079   by (simp add: not_iszero_1)
```
```  1080
```
```  1081 lemma eq_number_of_eq [simp]:
```
```  1082   "((number_of x::'a::number_ring) = number_of y) =
```
```  1083      iszero (number_of (x + uminus y) :: 'a)"
```
```  1084 unfolding iszero_def number_of_add number_of_minus
```
```  1085 by (simp add: algebra_simps)
```
```  1086
```
```  1087 lemma iszero_number_of_Pls:
```
```  1088   "iszero ((number_of Pls)::'a::number_ring)"
```
```  1089 unfolding iszero_def numeral_0_eq_0 ..
```
```  1090
```
```  1091 lemma nonzero_number_of_Min:
```
```  1092   "~ iszero ((number_of Min)::'a::number_ring)"
```
```  1093 unfolding iszero_def numeral_m1_eq_minus_1 by simp
```
```  1094
```
```  1095
```
```  1096 subsubsection {* Comparisons, for Ordered Rings *}
```
```  1097
```
```  1098 lemmas double_eq_0_iff = double_zero
```
```  1099
```
```  1100 lemma odd_nonzero:
```
```  1101   "1 + z + z \<noteq> (0::int)"
```
```  1102 proof (cases z rule: int_cases)
```
```  1103   case (nonneg n)
```
```  1104   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```  1105   thus ?thesis using  le_imp_0_less [OF le]
```
```  1106     by (auto simp add: add_assoc)
```
```  1107 next
```
```  1108   case (neg n)
```
```  1109   show ?thesis
```
```  1110   proof
```
```  1111     assume eq: "1 + z + z = 0"
```
```  1112     have "(0::int) < 1 + (of_nat n + of_nat n)"
```
```  1113       by (simp add: le_imp_0_less add_increasing)
```
```  1114     also have "... = - (1 + z + z)"
```
```  1115       by (simp add: neg add_assoc [symmetric])
```
```  1116     also have "... = 0" by (simp add: eq)
```
```  1117     finally have "0<0" ..
```
```  1118     thus False by blast
```
```  1119   qed
```
```  1120 qed
```
```  1121
```
```  1122 lemma iszero_number_of_Bit0:
```
```  1123   "iszero (number_of (Bit0 w)::'a) =
```
```  1124    iszero (number_of w::'a::{ring_char_0,number_ring})"
```
```  1125 proof -
```
```  1126   have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
```
```  1127   proof -
```
```  1128     assume eq: "of_int w + of_int w = (0::'a)"
```
```  1129     then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
```
```  1130     then have "w + w = 0" by (simp only: of_int_eq_iff)
```
```  1131     then show "w = 0" by (simp only: double_eq_0_iff)
```
```  1132   qed
```
```  1133   thus ?thesis
```
```  1134     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```  1135 qed
```
```  1136
```
```  1137 lemma iszero_number_of_Bit1:
```
```  1138   "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
```
```  1139 proof -
```
```  1140   have "1 + of_int w + of_int w \<noteq> (0::'a)"
```
```  1141   proof
```
```  1142     assume eq: "1 + of_int w + of_int w = (0::'a)"
```
```  1143     hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp
```
```  1144     hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
```
```  1145     with odd_nonzero show False by blast
```
```  1146   qed
```
```  1147   thus ?thesis
```
```  1148     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```  1149 qed
```
```  1150
```
```  1151 lemmas iszero_simps [simp] =
```
```  1152   iszero_0 not_iszero_1
```
```  1153   iszero_number_of_Pls nonzero_number_of_Min
```
```  1154   iszero_number_of_Bit0 iszero_number_of_Bit1
```
```  1155 (* iszero_number_of_Pls would never normally be used
```
```  1156    because its lhs simplifies to "iszero 0" *)
```
```  1157
```
```  1158 subsubsection {* The Less-Than Relation *}
```
```  1159
```
```  1160 lemma double_less_0_iff:
```
```  1161   "(a + a < 0) = (a < (0::'a::linordered_idom))"
```
```  1162 proof -
```
```  1163   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
```
```  1164   also have "... = (a < 0)"
```
```  1165     by (simp add: mult_less_0_iff zero_less_two
```
```  1166                   order_less_not_sym [OF zero_less_two])
```
```  1167   finally show ?thesis .
```
```  1168 qed
```
```  1169
```
```  1170 lemma odd_less_0:
```
```  1171   "(1 + z + z < 0) = (z < (0::int))"
```
```  1172 proof (cases z rule: int_cases)
```
```  1173   case (nonneg n)
```
```  1174   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```  1175                              le_imp_0_less [THEN order_less_imp_le])
```
```  1176 next
```
```  1177   case (neg n)
```
```  1178   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```  1179     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```  1180 qed
```
```  1181
```
```  1182 text {* Less-Than or Equals *}
```
```  1183
```
```  1184 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
```
```  1185
```
```  1186 lemmas le_number_of_eq_not_less =
```
```  1187   linorder_not_less [of "number_of w" "number_of v", symmetric,
```
```  1188   standard]
```
```  1189
```
```  1190
```
```  1191 text {* Absolute value (@{term abs}) *}
```
```  1192
```
```  1193 lemma abs_number_of:
```
```  1194   "abs(number_of x::'a::{linordered_idom,number_ring}) =
```
```  1195    (if number_of x < (0::'a) then -number_of x else number_of x)"
```
```  1196   by (simp add: abs_if)
```
```  1197
```
```  1198
```
```  1199 text {* Re-orientation of the equation nnn=x *}
```
```  1200
```
```  1201 lemma number_of_reorient:
```
```  1202   "(number_of w = x) = (x = number_of w)"
```
```  1203   by auto
```
```  1204
```
```  1205
```
```  1206 subsubsection {* Simplification of arithmetic operations on integer constants. *}
```
```  1207
```
```  1208 lemmas arith_extra_simps [standard, simp] =
```
```  1209   number_of_add [symmetric]
```
```  1210   number_of_minus [symmetric]
```
```  1211   numeral_m1_eq_minus_1 [symmetric]
```
```  1212   number_of_mult [symmetric]
```
```  1213   diff_number_of_eq abs_number_of
```
```  1214
```
```  1215 text {*
```
```  1216   For making a minimal simpset, one must include these default simprules.
```
```  1217   Also include @{text simp_thms}.
```
```  1218 *}
```
```  1219
```
```  1220 lemmas arith_simps =
```
```  1221   normalize_bin_simps pred_bin_simps succ_bin_simps
```
```  1222   add_bin_simps minus_bin_simps mult_bin_simps
```
```  1223   abs_zero abs_one arith_extra_simps
```
```  1224
```
```  1225 text {* Simplification of relational operations *}
```
```  1226
```
```  1227 lemma less_number_of [simp]:
```
```  1228   "(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
```
```  1229   unfolding number_of_eq by (rule of_int_less_iff)
```
```  1230
```
```  1231 lemma le_number_of [simp]:
```
```  1232   "(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
```
```  1233   unfolding number_of_eq by (rule of_int_le_iff)
```
```  1234
```
```  1235 lemma eq_number_of [simp]:
```
```  1236   "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
```
```  1237   unfolding number_of_eq by (rule of_int_eq_iff)
```
```  1238
```
```  1239 lemmas rel_simps =
```
```  1240   less_number_of less_bin_simps
```
```  1241   le_number_of le_bin_simps
```
```  1242   eq_number_of_eq eq_bin_simps
```
```  1243   iszero_simps
```
```  1244
```
```  1245
```
```  1246 subsubsection {* Simplification of arithmetic when nested to the right. *}
```
```  1247
```
```  1248 lemma add_number_of_left [simp]:
```
```  1249   "number_of v + (number_of w + z) =
```
```  1250    (number_of(v + w) + z::'a::number_ring)"
```
```  1251   by (simp add: add_assoc [symmetric])
```
```  1252
```
```  1253 lemma mult_number_of_left [simp]:
```
```  1254   "number_of v * (number_of w * z) =
```
```  1255    (number_of(v * w) * z::'a::number_ring)"
```
```  1256   by (simp add: mult_assoc [symmetric])
```
```  1257
```
```  1258 lemma add_number_of_diff1:
```
```  1259   "number_of v + (number_of w - c) =
```
```  1260   number_of(v + w) - (c::'a::number_ring)"
```
```  1261   by (simp add: diff_minus)
```
```  1262
```
```  1263 lemma add_number_of_diff2 [simp]:
```
```  1264   "number_of v + (c - number_of w) =
```
```  1265    number_of (v + uminus w) + (c::'a::number_ring)"
```
```  1266 by (simp add: algebra_simps diff_number_of_eq [symmetric])
```
```  1267
```
```  1268
```
```  1269
```
```  1270
```
```  1271 subsection {* The Set of Integers *}
```
```  1272
```
```  1273 context ring_1
```
```  1274 begin
```
```  1275
```
```  1276 definition Ints  :: "'a set" where
```
```  1277   [code del]: "Ints = range of_int"
```
```  1278
```
```  1279 notation (xsymbols)
```
```  1280   Ints  ("\<int>")
```
```  1281
```
```  1282 lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
```
```  1283   by (simp add: Ints_def)
```
```  1284
```
```  1285 lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
```
```  1286 apply (simp add: Ints_def)
```
```  1287 apply (rule range_eqI)
```
```  1288 apply (rule of_int_of_nat_eq [symmetric])
```
```  1289 done
```
```  1290
```
```  1291 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```  1292 apply (simp add: Ints_def)
```
```  1293 apply (rule range_eqI)
```
```  1294 apply (rule of_int_0 [symmetric])
```
```  1295 done
```
```  1296
```
```  1297 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```  1298 apply (simp add: Ints_def)
```
```  1299 apply (rule range_eqI)
```
```  1300 apply (rule of_int_1 [symmetric])
```
```  1301 done
```
```  1302
```
```  1303 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```  1304 apply (auto simp add: Ints_def)
```
```  1305 apply (rule range_eqI)
```
```  1306 apply (rule of_int_add [symmetric])
```
```  1307 done
```
```  1308
```
```  1309 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```  1310 apply (auto simp add: Ints_def)
```
```  1311 apply (rule range_eqI)
```
```  1312 apply (rule of_int_minus [symmetric])
```
```  1313 done
```
```  1314
```
```  1315 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
```
```  1316 apply (auto simp add: Ints_def)
```
```  1317 apply (rule range_eqI)
```
```  1318 apply (rule of_int_diff [symmetric])
```
```  1319 done
```
```  1320
```
```  1321 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```  1322 apply (auto simp add: Ints_def)
```
```  1323 apply (rule range_eqI)
```
```  1324 apply (rule of_int_mult [symmetric])
```
```  1325 done
```
```  1326
```
```  1327 lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
```
```  1328 by (induct n) simp_all
```
```  1329
```
```  1330 lemma Ints_cases [cases set: Ints]:
```
```  1331   assumes "q \<in> \<int>"
```
```  1332   obtains (of_int) z where "q = of_int z"
```
```  1333   unfolding Ints_def
```
```  1334 proof -
```
```  1335   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```  1336   then obtain z where "q = of_int z" ..
```
```  1337   then show thesis ..
```
```  1338 qed
```
```  1339
```
```  1340 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```  1341   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```  1342   by (rule Ints_cases) auto
```
```  1343
```
```  1344 end
```
```  1345
```
```  1346 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```  1347
```
```  1348 lemma Ints_double_eq_0_iff:
```
```  1349   assumes in_Ints: "a \<in> Ints"
```
```  1350   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```  1351 proof -
```
```  1352   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1353   then obtain z where a: "a = of_int z" ..
```
```  1354   show ?thesis
```
```  1355   proof
```
```  1356     assume "a = 0"
```
```  1357     thus "a + a = 0" by simp
```
```  1358   next
```
```  1359     assume eq: "a + a = 0"
```
```  1360     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1361     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```  1362     hence "z = 0" by (simp only: double_eq_0_iff)
```
```  1363     thus "a = 0" by (simp add: a)
```
```  1364   qed
```
```  1365 qed
```
```  1366
```
```  1367 lemma Ints_odd_nonzero:
```
```  1368   assumes in_Ints: "a \<in> Ints"
```
```  1369   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```  1370 proof -
```
```  1371   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1372   then obtain z where a: "a = of_int z" ..
```
```  1373   show ?thesis
```
```  1374   proof
```
```  1375     assume eq: "1 + a + a = 0"
```
```  1376     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1377     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```  1378     with odd_nonzero show False by blast
```
```  1379   qed
```
```  1380 qed
```
```  1381
```
```  1382 lemma Ints_number_of [simp]:
```
```  1383   "(number_of w :: 'a::number_ring) \<in> Ints"
```
```  1384   unfolding number_of_eq Ints_def by simp
```
```  1385
```
```  1386 lemma Nats_number_of [simp]:
```
```  1387   "Int.Pls \<le> w \<Longrightarrow> (number_of w :: 'a::number_ring) \<in> Nats"
```
```  1388 unfolding Int.Pls_def number_of_eq
```
```  1389 by (simp only: of_nat_nat [symmetric] of_nat_in_Nats)
```
```  1390
```
```  1391 lemma Ints_odd_less_0:
```
```  1392   assumes in_Ints: "a \<in> Ints"
```
```  1393   shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
```
```  1394 proof -
```
```  1395   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1396   then obtain z where a: "a = of_int z" ..
```
```  1397   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```  1398     by (simp add: a)
```
```  1399   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```  1400   also have "... = (a < 0)" by (simp add: a)
```
```  1401   finally show ?thesis .
```
```  1402 qed
```
```  1403
```
```  1404
```
```  1405 subsection {* @{term setsum} and @{term setprod} *}
```
```  1406
```
```  1407 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```  1408   apply (cases "finite A")
```
```  1409   apply (erule finite_induct, auto)
```
```  1410   done
```
```  1411
```
```  1412 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```  1413   apply (cases "finite A")
```
```  1414   apply (erule finite_induct, auto)
```
```  1415   done
```
```  1416
```
```  1417 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```  1418   apply (cases "finite A")
```
```  1419   apply (erule finite_induct, auto simp add: of_nat_mult)
```
```  1420   done
```
```  1421
```
```  1422 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```  1423   apply (cases "finite A")
```
```  1424   apply (erule finite_induct, auto)
```
```  1425   done
```
```  1426
```
```  1427 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```  1428 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```  1429
```
```  1430
```
```  1431 subsection{*Inequality Reasoning for the Arithmetic Simproc*}
```
```  1432
```
```  1433 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
```
```  1434 by simp
```
```  1435
```
```  1436 lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
```
```  1437 by simp
```
```  1438
```
```  1439 lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
```
```  1440 by simp
```
```  1441
```
```  1442 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
```
```  1443 by simp
```
```  1444
```
```  1445 lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
```
```  1446 by simp
```
```  1447
```
```  1448 lemma inverse_numeral_1:
```
```  1449   "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
```
```  1450 by simp
```
```  1451
```
```  1452 text{*Theorem lists for the cancellation simprocs. The use of binary numerals
```
```  1453 for 0 and 1 reduces the number of special cases.*}
```
```  1454
```
```  1455 lemmas add_0s = add_numeral_0 add_numeral_0_right
```
```  1456 lemmas mult_1s = mult_numeral_1 mult_numeral_1_right
```
```  1457                  mult_minus1 mult_minus1_right
```
```  1458
```
```  1459
```
```  1460 subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
```
```  1461
```
```  1462 text{*Arithmetic computations are defined for binary literals, which leaves 0
```
```  1463 and 1 as special cases. Addition already has rules for 0, but not 1.
```
```  1464 Multiplication and unary minus already have rules for both 0 and 1.*}
```
```  1465
```
```  1466
```
```  1467 lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
```
```  1468 by simp
```
```  1469
```
```  1470
```
```  1471 lemmas add_number_of_eq = number_of_add [symmetric]
```
```  1472
```
```  1473 text{*Allow 1 on either or both sides*}
```
```  1474 lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
```
```  1475 by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric])
```
```  1476
```
```  1477 lemmas add_special =
```
```  1478     one_add_one_is_two
```
```  1479     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1480     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
```
```  1481
```
```  1482 text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
```
```  1483 lemmas diff_special =
```
```  1484     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1485     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
```
```  1486
```
```  1487 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1488 lemmas eq_special =
```
```  1489     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
```
```  1490     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1491     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
```
```  1492     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
```
```  1493
```
```  1494 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1495 lemmas less_special =
```
```  1496   binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]
```
```  1497   binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]
```
```  1498   binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]
```
```  1499   binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]
```
```  1500
```
```  1501 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1502 lemmas le_special =
```
```  1503     binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]
```
```  1504     binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]
```
```  1505     binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]
```
```  1506     binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]
```
```  1507
```
```  1508 lemmas arith_special[simp] =
```
```  1509        add_special diff_special eq_special less_special le_special
```
```  1510
```
```  1511
```
```  1512 text {* Legacy theorems *}
```
```  1513
```
```  1514 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```  1515 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```  1516
```
```  1517 subsection {* Setting up simplification procedures *}
```
```  1518
```
```  1519 lemmas int_arith_rules =
```
```  1520   neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1
```
```  1521   minus_zero diff_minus left_minus right_minus
```
```  1522   mult_zero_left mult_zero_right mult_Bit1 mult_1_left mult_1_right
```
```  1523   mult_minus_left mult_minus_right
```
```  1524   minus_add_distrib minus_minus mult_assoc
```
```  1525   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
```
```  1526   of_int_0 of_int_1 of_int_add of_int_mult
```
```  1527
```
```  1528 use "Tools/int_arith.ML"
```
```  1529 setup {* Int_Arith.global_setup *}
```
```  1530 declaration {* K Int_Arith.setup *}
```
```  1531
```
```  1532 setup {*
```
```  1533   Reorient_Proc.add
```
```  1534     (fn Const (@{const_name number_of}, _) \$ _ => true | _ => false)
```
```  1535 *}
```
```  1536
```
```  1537 simproc_setup reorient_numeral ("number_of w = x") = Reorient_Proc.proc
```
```  1538
```
```  1539
```
```  1540 subsection{*Lemmas About Small Numerals*}
```
```  1541
```
```  1542 lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
```
```  1543 proof -
```
```  1544   have "(of_int -1 :: 'a) = of_int (- 1)" by simp
```
```  1545   also have "... = - of_int 1" by (simp only: of_int_minus)
```
```  1546   also have "... = -1" by simp
```
```  1547   finally show ?thesis .
```
```  1548 qed
```
```  1549
```
```  1550 lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})"
```
```  1551 by (simp add: abs_if)
```
```  1552
```
```  1553 lemma abs_power_minus_one [simp]:
```
```  1554   "abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})"
```
```  1555 by (simp add: power_abs)
```
```  1556
```
```  1557 lemma of_int_number_of_eq [simp]:
```
```  1558      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
```
```  1559 by (simp add: number_of_eq)
```
```  1560
```
```  1561 text{*Lemmas for specialist use, NOT as default simprules*}
```
```  1562 lemma mult_2: "2 * z = (z+z::'a::number_ring)"
```
```  1563 unfolding one_add_one_is_two [symmetric] left_distrib by simp
```
```  1564
```
```  1565 lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
```
```  1566 by (subst mult_commute, rule mult_2)
```
```  1567
```
```  1568
```
```  1569 subsection{*More Inequality Reasoning*}
```
```  1570
```
```  1571 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```  1572 by arith
```
```  1573
```
```  1574 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```  1575 by arith
```
```  1576
```
```  1577 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```  1578 by arith
```
```  1579
```
```  1580 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```  1581 by arith
```
```  1582
```
```  1583 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```  1584 by arith
```
```  1585
```
```  1586
```
```  1587 subsection{*The functions @{term nat} and @{term int}*}
```
```  1588
```
```  1589 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
```
```  1590   @{term "w + - z"}*}
```
```  1591 declare Zero_int_def [symmetric, simp]
```
```  1592 declare One_int_def [symmetric, simp]
```
```  1593
```
```  1594 lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
```
```  1595
```
```  1596 (* FIXME: duplicates nat_zero *)
```
```  1597 lemma nat_0: "nat 0 = 0"
```
```  1598 by (simp add: nat_eq_iff)
```
```  1599
```
```  1600 lemma nat_1: "nat 1 = Suc 0"
```
```  1601 by (subst nat_eq_iff, simp)
```
```  1602
```
```  1603 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```  1604 by (subst nat_eq_iff, simp)
```
```  1605
```
```  1606 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```  1607 apply (insert zless_nat_conj [of 1 z])
```
```  1608 apply (auto simp add: nat_1)
```
```  1609 done
```
```  1610
```
```  1611 text{*This simplifies expressions of the form @{term "int n = z"} where
```
```  1612       z is an integer literal.*}
```
```  1613 lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
```
```  1614
```
```  1615 lemma split_nat [arith_split]:
```
```  1616   "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```  1617   (is "?P = (?L & ?R)")
```
```  1618 proof (cases "i < 0")
```
```  1619   case True thus ?thesis by auto
```
```  1620 next
```
```  1621   case False
```
```  1622   have "?P = ?L"
```
```  1623   proof
```
```  1624     assume ?P thus ?L using False by clarsimp
```
```  1625   next
```
```  1626     assume ?L thus ?P using False by simp
```
```  1627   qed
```
```  1628   with False show ?thesis by simp
```
```  1629 qed
```
```  1630
```
```  1631 context ring_1
```
```  1632 begin
```
```  1633
```
```  1634 lemma of_int_of_nat [nitpick_simp]:
```
```  1635   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```  1636 proof (cases "k < 0")
```
```  1637   case True then have "0 \<le> - k" by simp
```
```  1638   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```  1639   with True show ?thesis by simp
```
```  1640 next
```
```  1641   case False then show ?thesis by (simp add: not_less of_nat_nat)
```
```  1642 qed
```
```  1643
```
```  1644 end
```
```  1645
```
```  1646 lemma nat_mult_distrib:
```
```  1647   fixes z z' :: int
```
```  1648   assumes "0 \<le> z"
```
```  1649   shows "nat (z * z') = nat z * nat z'"
```
```  1650 proof (cases "0 \<le> z'")
```
```  1651   case False with assms have "z * z' \<le> 0"
```
```  1652     by (simp add: not_le mult_le_0_iff)
```
```  1653   then have "nat (z * z') = 0" by simp
```
```  1654   moreover from False have "nat z' = 0" by simp
```
```  1655   ultimately show ?thesis by simp
```
```  1656 next
```
```  1657   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1658   show ?thesis
```
```  1659     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1660       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1661          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1662 qed
```
```  1663
```
```  1664 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```  1665 apply (rule trans)
```
```  1666 apply (rule_tac [2] nat_mult_distrib, auto)
```
```  1667 done
```
```  1668
```
```  1669 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```  1670 apply (cases "z=0 | w=0")
```
```  1671 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1672                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1673 done
```
```  1674
```
```  1675
```
```  1676 subsection "Induction principles for int"
```
```  1677
```
```  1678 text{*Well-founded segments of the integers*}
```
```  1679
```
```  1680 definition
```
```  1681   int_ge_less_than  ::  "int => (int * int) set"
```
```  1682 where
```
```  1683   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
```
```  1684
```
```  1685 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1686 proof -
```
```  1687   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
```
```  1688     by (auto simp add: int_ge_less_than_def)
```
```  1689   thus ?thesis
```
```  1690     by (rule wf_subset [OF wf_measure])
```
```  1691 qed
```
```  1692
```
```  1693 text{*This variant looks odd, but is typical of the relations suggested
```
```  1694 by RankFinder.*}
```
```  1695
```
```  1696 definition
```
```  1697   int_ge_less_than2 ::  "int => (int * int) set"
```
```  1698 where
```
```  1699   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
```
```  1700
```
```  1701 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1702 proof -
```
```  1703   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
```
```  1704     by (auto simp add: int_ge_less_than2_def)
```
```  1705   thus ?thesis
```
```  1706     by (rule wf_subset [OF wf_measure])
```
```  1707 qed
```
```  1708
```
```  1709 abbreviation
```
```  1710   int :: "nat \<Rightarrow> int"
```
```  1711 where
```
```  1712   "int \<equiv> of_nat"
```
```  1713
```
```  1714 (* `set:int': dummy construction *)
```
```  1715 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1716   fixes i :: int
```
```  1717   assumes ge: "k \<le> i" and
```
```  1718     base: "P k" and
```
```  1719     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1720   shows "P i"
```
```  1721 proof -
```
```  1722   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```  1723     proof (induct n)
```
```  1724       case 0
```
```  1725       hence "i = k" by arith
```
```  1726       thus "P i" using base by simp
```
```  1727     next
```
```  1728       case (Suc n)
```
```  1729       then have "n = nat((i - 1) - k)" by arith
```
```  1730       moreover
```
```  1731       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```  1732       ultimately
```
```  1733       have "P(i - 1)" by(rule Suc.hyps)
```
```  1734       from step[OF ki1 this] show ?case by simp
```
```  1735     qed
```
```  1736   }
```
```  1737   with ge show ?thesis by fast
```
```  1738 qed
```
```  1739
```
```  1740 (* `set:int': dummy construction *)
```
```  1741 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1742   assumes gr: "k < (i::int)" and
```
```  1743         base: "P(k+1)" and
```
```  1744         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```  1745   shows "P i"
```
```  1746 apply(rule int_ge_induct[of "k + 1"])
```
```  1747   using gr apply arith
```
```  1748  apply(rule base)
```
```  1749 apply (rule step, simp+)
```
```  1750 done
```
```  1751
```
```  1752 theorem int_le_induct[consumes 1,case_names base step]:
```
```  1753   assumes le: "i \<le> (k::int)" and
```
```  1754         base: "P(k)" and
```
```  1755         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1756   shows "P i"
```
```  1757 proof -
```
```  1758   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
```
```  1759     proof (induct n)
```
```  1760       case 0
```
```  1761       hence "i = k" by arith
```
```  1762       thus "P i" using base by simp
```
```  1763     next
```
```  1764       case (Suc n)
```
```  1765       hence "n = nat(k - (i+1))" by arith
```
```  1766       moreover
```
```  1767       have ki1: "i + 1 \<le> k" using Suc.prems by arith
```
```  1768       ultimately
```
```  1769       have "P(i+1)" by(rule Suc.hyps)
```
```  1770       from step[OF ki1 this] show ?case by simp
```
```  1771     qed
```
```  1772   }
```
```  1773   with le show ?thesis by fast
```
```  1774 qed
```
```  1775
```
```  1776 theorem int_less_induct [consumes 1,case_names base step]:
```
```  1777   assumes less: "(i::int) < k" and
```
```  1778         base: "P(k - 1)" and
```
```  1779         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1780   shows "P i"
```
```  1781 apply(rule int_le_induct[of _ "k - 1"])
```
```  1782   using less apply arith
```
```  1783  apply(rule base)
```
```  1784 apply (rule step, simp+)
```
```  1785 done
```
```  1786
```
```  1787 theorem int_induct [case_names base step1 step2]:
```
```  1788   fixes k :: int
```
```  1789   assumes base: "P k"
```
```  1790     and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1791     and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
```
```  1792   shows "P i"
```
```  1793 proof -
```
```  1794   have "i \<le> k \<or> i \<ge> k" by arith
```
```  1795   then show ?thesis proof
```
```  1796     assume "i \<ge> k" then show ?thesis using base
```
```  1797       by (rule int_ge_induct) (fact step1)
```
```  1798   next
```
```  1799     assume "i \<le> k" then show ?thesis using base
```
```  1800       by (rule int_le_induct) (fact step2)
```
```  1801   qed
```
```  1802 qed
```
```  1803
```
```  1804 subsection{*Intermediate value theorems*}
```
```  1805
```
```  1806 lemma int_val_lemma:
```
```  1807      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```  1808       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```  1809 unfolding One_nat_def
```
```  1810 apply (induct n, simp)
```
```  1811 apply (intro strip)
```
```  1812 apply (erule impE, simp)
```
```  1813 apply (erule_tac x = n in allE, simp)
```
```  1814 apply (case_tac "k = f (Suc n)")
```
```  1815 apply force
```
```  1816 apply (erule impE)
```
```  1817  apply (simp add: abs_if split add: split_if_asm)
```
```  1818 apply (blast intro: le_SucI)
```
```  1819 done
```
```  1820
```
```  1821 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1822
```
```  1823 lemma nat_intermed_int_val:
```
```  1824      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```  1825          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```  1826 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```  1827        in int_val_lemma)
```
```  1828 unfolding One_nat_def
```
```  1829 apply simp
```
```  1830 apply (erule exE)
```
```  1831 apply (rule_tac x = "i+m" in exI, arith)
```
```  1832 done
```
```  1833
```
```  1834
```
```  1835 subsection{*Products and 1, by T. M. Rasmussen*}
```
```  1836
```
```  1837 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```  1838 by arith
```
```  1839
```
```  1840 lemma abs_zmult_eq_1:
```
```  1841   assumes mn: "\<bar>m * n\<bar> = 1"
```
```  1842   shows "\<bar>m\<bar> = (1::int)"
```
```  1843 proof -
```
```  1844   have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
```
```  1845     by auto
```
```  1846   have "~ (2 \<le> \<bar>m\<bar>)"
```
```  1847   proof
```
```  1848     assume "2 \<le> \<bar>m\<bar>"
```
```  1849     hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
```
```  1850       by (simp add: mult_mono 0)
```
```  1851     also have "... = \<bar>m*n\<bar>"
```
```  1852       by (simp add: abs_mult)
```
```  1853     also have "... = 1"
```
```  1854       by (simp add: mn)
```
```  1855     finally have "2*\<bar>n\<bar> \<le> 1" .
```
```  1856     thus "False" using 0
```
```  1857       by auto
```
```  1858   qed
```
```  1859   thus ?thesis using 0
```
```  1860     by auto
```
```  1861 qed
```
```  1862
```
```  1863 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```  1864 by (insert abs_zmult_eq_1 [of m n], arith)
```
```  1865
```
```  1866 lemma pos_zmult_eq_1_iff:
```
```  1867   assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
```
```  1868 proof -
```
```  1869   from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1870   thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1871 qed
```
```  1872
```
```  1873 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```  1874 apply (rule iffI)
```
```  1875  apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1876  apply (simp add: mult_commute [of m])
```
```  1877  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1878 done
```
```  1879
```
```  1880 lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
```
```  1881 proof
```
```  1882   assume "finite (UNIV::int set)"
```
```  1883   moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1884     by (rule injI) simp
```
```  1885   ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
```
```  1886     by (rule finite_UNIV_inj_surj)
```
```  1887   then obtain i :: int where "1 = 2 * i" by (rule surjE)
```
```  1888   then show False by (simp add: pos_zmult_eq_1_iff)
```
```  1889 qed
```
```  1890
```
```  1891
```
```  1892 subsection {* Further theorems on numerals *}
```
```  1893
```
```  1894 subsubsection{*Special Simplification for Constants*}
```
```  1895
```
```  1896 text{*These distributive laws move literals inside sums and differences.*}
```
```  1897
```
```  1898 lemmas left_distrib_number_of [simp] =
```
```  1899   left_distrib [of _ _ "number_of v", standard]
```
```  1900
```
```  1901 lemmas right_distrib_number_of [simp] =
```
```  1902   right_distrib [of "number_of v", standard]
```
```  1903
```
```  1904 lemmas left_diff_distrib_number_of [simp] =
```
```  1905   left_diff_distrib [of _ _ "number_of v", standard]
```
```  1906
```
```  1907 lemmas right_diff_distrib_number_of [simp] =
```
```  1908   right_diff_distrib [of "number_of v", standard]
```
```  1909
```
```  1910 text{*These are actually for fields, like real: but where else to put them?*}
```
```  1911
```
```  1912 lemmas zero_less_divide_iff_number_of [simp, no_atp] =
```
```  1913   zero_less_divide_iff [of "number_of w", standard]
```
```  1914
```
```  1915 lemmas divide_less_0_iff_number_of [simp, no_atp] =
```
```  1916   divide_less_0_iff [of "number_of w", standard]
```
```  1917
```
```  1918 lemmas zero_le_divide_iff_number_of [simp, no_atp] =
```
```  1919   zero_le_divide_iff [of "number_of w", standard]
```
```  1920
```
```  1921 lemmas divide_le_0_iff_number_of [simp, no_atp] =
```
```  1922   divide_le_0_iff [of "number_of w", standard]
```
```  1923
```
```  1924
```
```  1925 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```  1926   strange, but then other simprocs simplify the quotient.*}
```
```  1927
```
```  1928 lemmas inverse_eq_divide_number_of [simp] =
```
```  1929   inverse_eq_divide [of "number_of w", standard]
```
```  1930
```
```  1931 text {*These laws simplify inequalities, moving unary minus from a term
```
```  1932 into the literal.*}
```
```  1933
```
```  1934 lemmas less_minus_iff_number_of [simp, no_atp] =
```
```  1935   less_minus_iff [of "number_of v", standard]
```
```  1936
```
```  1937 lemmas le_minus_iff_number_of [simp, no_atp] =
```
```  1938   le_minus_iff [of "number_of v", standard]
```
```  1939
```
```  1940 lemmas equation_minus_iff_number_of [simp, no_atp] =
```
```  1941   equation_minus_iff [of "number_of v", standard]
```
```  1942
```
```  1943 lemmas minus_less_iff_number_of [simp, no_atp] =
```
```  1944   minus_less_iff [of _ "number_of v", standard]
```
```  1945
```
```  1946 lemmas minus_le_iff_number_of [simp, no_atp] =
```
```  1947   minus_le_iff [of _ "number_of v", standard]
```
```  1948
```
```  1949 lemmas minus_equation_iff_number_of [simp, no_atp] =
```
```  1950   minus_equation_iff [of _ "number_of v", standard]
```
```  1951
```
```  1952
```
```  1953 text{*To Simplify Inequalities Where One Side is the Constant 1*}
```
```  1954
```
```  1955 lemma less_minus_iff_1 [simp,no_atp]:
```
```  1956   fixes b::"'b::{linordered_idom,number_ring}"
```
```  1957   shows "(1 < - b) = (b < -1)"
```
```  1958 by auto
```
```  1959
```
```  1960 lemma le_minus_iff_1 [simp,no_atp]:
```
```  1961   fixes b::"'b::{linordered_idom,number_ring}"
```
```  1962   shows "(1 \<le> - b) = (b \<le> -1)"
```
```  1963 by auto
```
```  1964
```
```  1965 lemma equation_minus_iff_1 [simp,no_atp]:
```
```  1966   fixes b::"'b::number_ring"
```
```  1967   shows "(1 = - b) = (b = -1)"
```
```  1968 by (subst equation_minus_iff, auto)
```
```  1969
```
```  1970 lemma minus_less_iff_1 [simp,no_atp]:
```
```  1971   fixes a::"'b::{linordered_idom,number_ring}"
```
```  1972   shows "(- a < 1) = (-1 < a)"
```
```  1973 by auto
```
```  1974
```
```  1975 lemma minus_le_iff_1 [simp,no_atp]:
```
```  1976   fixes a::"'b::{linordered_idom,number_ring}"
```
```  1977   shows "(- a \<le> 1) = (-1 \<le> a)"
```
```  1978 by auto
```
```  1979
```
```  1980 lemma minus_equation_iff_1 [simp,no_atp]:
```
```  1981   fixes a::"'b::number_ring"
```
```  1982   shows "(- a = 1) = (a = -1)"
```
```  1983 by (subst minus_equation_iff, auto)
```
```  1984
```
```  1985
```
```  1986 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```  1987
```
```  1988 lemmas mult_less_cancel_left_number_of [simp, no_atp] =
```
```  1989   mult_less_cancel_left [of "number_of v", standard]
```
```  1990
```
```  1991 lemmas mult_less_cancel_right_number_of [simp, no_atp] =
```
```  1992   mult_less_cancel_right [of _ "number_of v", standard]
```
```  1993
```
```  1994 lemmas mult_le_cancel_left_number_of [simp, no_atp] =
```
```  1995   mult_le_cancel_left [of "number_of v", standard]
```
```  1996
```
```  1997 lemmas mult_le_cancel_right_number_of [simp, no_atp] =
```
```  1998   mult_le_cancel_right [of _ "number_of v", standard]
```
```  1999
```
```  2000
```
```  2001 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```  2002
```
```  2003 lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
```
```  2004 lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
```
```  2005 lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
```
```  2006 lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
```
```  2007 lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
```
```  2008 lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
```
```  2009
```
```  2010
```
```  2011 subsubsection{*Optional Simplification Rules Involving Constants*}
```
```  2012
```
```  2013 text{*Simplify quotients that are compared with a literal constant.*}
```
```  2014
```
```  2015 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
```
```  2016 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
```
```  2017 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
```
```  2018 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
```
```  2019 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
```
```  2020 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
```
```  2021
```
```  2022
```
```  2023 text{*Not good as automatic simprules because they cause case splits.*}
```
```  2024 lemmas divide_const_simps =
```
```  2025   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
```
```  2026   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
```
```  2027   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  2028
```
```  2029 text{*Division By @{text "-1"}*}
```
```  2030
```
```  2031 lemma divide_minus1 [simp]:
```
```  2032      "x/-1 = -(x::'a::{field_inverse_zero, number_ring})"
```
```  2033 by simp
```
```  2034
```
```  2035 lemma minus1_divide [simp]:
```
```  2036      "-1 / (x::'a::{field_inverse_zero, number_ring}) = - (1/x)"
```
```  2037 by (simp add: divide_inverse)
```
```  2038
```
```  2039 lemma half_gt_zero_iff:
```
```  2040      "(0 < r/2) = (0 < (r::'a::{linordered_field_inverse_zero,number_ring}))"
```
```  2041 by auto
```
```  2042
```
```  2043 lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
```
```  2044
```
```  2045 lemma divide_Numeral1:
```
```  2046   "(x::'a::{field, number_ring}) / Numeral1 = x"
```
```  2047   by simp
```
```  2048
```
```  2049 lemma divide_Numeral0:
```
```  2050   "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
```
```  2051   by simp
```
```  2052
```
```  2053
```
```  2054 subsection {* The divides relation *}
```
```  2055
```
```  2056 lemma zdvd_antisym_nonneg:
```
```  2057     "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
```
```  2058   apply (simp add: dvd_def, auto)
```
```  2059   apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
```
```  2060   done
```
```  2061
```
```  2062 lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
```
```  2063   shows "\<bar>a\<bar> = \<bar>b\<bar>"
```
```  2064 proof cases
```
```  2065   assume "a = 0" with assms show ?thesis by simp
```
```  2066 next
```
```  2067   assume "a \<noteq> 0"
```
```  2068   from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast
```
```  2069   from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast
```
```  2070   from k k' have "a = a*k*k'" by simp
```
```  2071   with mult_cancel_left1[where c="a" and b="k*k'"]
```
```  2072   have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
```
```  2073   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
```
```  2074   thus ?thesis using k k' by auto
```
```  2075 qed
```
```  2076
```
```  2077 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
```
```  2078   apply (subgoal_tac "m = n + (m - n)")
```
```  2079    apply (erule ssubst)
```
```  2080    apply (blast intro: dvd_add, simp)
```
```  2081   done
```
```  2082
```
```  2083 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
```
```  2084 apply (rule iffI)
```
```  2085  apply (erule_tac [2] dvd_add)
```
```  2086  apply (subgoal_tac "n = (n + k * m) - k * m")
```
```  2087   apply (erule ssubst)
```
```  2088   apply (erule dvd_diff)
```
```  2089   apply(simp_all)
```
```  2090 done
```
```  2091
```
```  2092 lemma dvd_imp_le_int:
```
```  2093   fixes d i :: int
```
```  2094   assumes "i \<noteq> 0" and "d dvd i"
```
```  2095   shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
```
```  2096 proof -
```
```  2097   from `d dvd i` obtain k where "i = d * k" ..
```
```  2098   with `i \<noteq> 0` have "k \<noteq> 0" by auto
```
```  2099   then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
```
```  2100   then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
```
```  2101   with `i = d * k` show ?thesis by (simp add: abs_mult)
```
```  2102 qed
```
```  2103
```
```  2104 lemma zdvd_not_zless:
```
```  2105   fixes m n :: int
```
```  2106   assumes "0 < m" and "m < n"
```
```  2107   shows "\<not> n dvd m"
```
```  2108 proof
```
```  2109   from assms have "0 < n" by auto
```
```  2110   assume "n dvd m" then obtain k where k: "m = n * k" ..
```
```  2111   with `0 < m` have "0 < n * k" by auto
```
```  2112   with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
```
```  2113   with k `0 < n` `m < n` have "n * k < n * 1" by simp
```
```  2114   with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
```
```  2115 qed
```
```  2116
```
```  2117 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
```
```  2118   shows "m dvd n"
```
```  2119 proof-
```
```  2120   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
```
```  2121   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
```
```  2122     with h have False by (simp add: mult_assoc)}
```
```  2123   hence "n = m * h" by blast
```
```  2124   thus ?thesis by simp
```
```  2125 qed
```
```  2126
```
```  2127 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
```
```  2128 proof -
```
```  2129   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
```
```  2130   proof -
```
```  2131     fix k
```
```  2132     assume A: "int y = int x * k"
```
```  2133     then show "x dvd y" proof (cases k)
```
```  2134       case (1 n) with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
```
```  2135       then show ?thesis ..
```
```  2136     next
```
```  2137       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
```
```  2138       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
```
```  2139       also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
```
```  2140       finally have "- int (x * Suc n) = int y" ..
```
```  2141       then show ?thesis by (simp only: negative_eq_positive) auto
```
```  2142     qed
```
```  2143   qed
```
```  2144   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
```
```  2145 qed
```
```  2146
```
```  2147 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
```
```  2148 proof
```
```  2149   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
```
```  2150   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
```
```  2151   hence "nat \<bar>x\<bar> = 1"  by simp
```
```  2152   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
```
```  2153 next
```
```  2154   assume "\<bar>x\<bar>=1"
```
```  2155   then have "x = 1 \<or> x = -1" by auto
```
```  2156   then show "x dvd 1" by (auto intro: dvdI)
```
```  2157 qed
```
```  2158
```
```  2159 lemma zdvd_mult_cancel1:
```
```  2160   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
```
```  2161 proof
```
```  2162   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
```
```  2163     by (cases "n >0", auto simp add: minus_equation_iff)
```
```  2164 next
```
```  2165   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
```
```  2166   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
```
```  2167 qed
```
```  2168
```
```  2169 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
```
```  2170   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  2171
```
```  2172 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
```
```  2173   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
```
```  2174
```
```  2175 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
```
```  2176   by (auto simp add: dvd_int_iff)
```
```  2177
```
```  2178 lemma eq_nat_nat_iff:
```
```  2179   "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
```
```  2180   by (auto elim!: nonneg_eq_int)
```
```  2181
```
```  2182 lemma nat_power_eq:
```
```  2183   "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
```
```  2184   by (induct n) (simp_all add: nat_mult_distrib)
```
```  2185
```
```  2186 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
```
```  2187   apply (rule_tac z=n in int_cases)
```
```  2188   apply (auto simp add: dvd_int_iff)
```
```  2189   apply (rule_tac z=z in int_cases)
```
```  2190   apply (auto simp add: dvd_imp_le)
```
```  2191   done
```
```  2192
```
```  2193 lemma zdvd_period:
```
```  2194   fixes a d :: int
```
```  2195   assumes "a dvd d"
```
```  2196   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```  2197 proof -
```
```  2198   from assms obtain k where "d = a * k" by (rule dvdE)
```
```  2199   show ?thesis proof
```
```  2200     assume "a dvd (x + t)"
```
```  2201     then obtain l where "x + t = a * l" by (rule dvdE)
```
```  2202     then have "x = a * l - t" by simp
```
```  2203     with `d = a * k` show "a dvd x + c * d + t" by simp
```
```  2204   next
```
```  2205     assume "a dvd x + c * d + t"
```
```  2206     then obtain l where "x + c * d + t = a * l" by (rule dvdE)
```
```  2207     then have "x = a * l - c * d - t" by simp
```
```  2208     with `d = a * k` show "a dvd (x + t)" by simp
```
```  2209   qed
```
```  2210 qed
```
```  2211
```
```  2212
```
```  2213 subsection {* Configuration of the code generator *}
```
```  2214
```
```  2215 code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
```
```  2216
```
```  2217 lemmas pred_succ_numeral_code [code] =
```
```  2218   pred_bin_simps succ_bin_simps
```
```  2219
```
```  2220 lemmas plus_numeral_code [code] =
```
```  2221   add_bin_simps
```
```  2222   arith_extra_simps(1) [where 'a = int]
```
```  2223
```
```  2224 lemmas minus_numeral_code [code] =
```
```  2225   minus_bin_simps
```
```  2226   arith_extra_simps(2) [where 'a = int]
```
```  2227   arith_extra_simps(5) [where 'a = int]
```
```  2228
```
```  2229 lemmas times_numeral_code [code] =
```
```  2230   mult_bin_simps
```
```  2231   arith_extra_simps(4) [where 'a = int]
```
```  2232
```
```  2233 instantiation int :: eq
```
```  2234 begin
```
```  2235
```
```  2236 definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)"
```
```  2237
```
```  2238 instance by default (simp add: eq_int_def)
```
```  2239
```
```  2240 end
```
```  2241
```
```  2242 lemma eq_number_of_int_code [code]:
```
```  2243   "eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l"
```
```  2244   unfolding eq_int_def number_of_is_id ..
```
```  2245
```
```  2246 lemma eq_int_code [code]:
```
```  2247   "eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True"
```
```  2248   "eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False"
```
```  2249   "eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2"
```
```  2250   "eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
```
```  2251   "eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False"
```
```  2252   "eq_class.eq Int.Min Int.Min \<longleftrightarrow> True"
```
```  2253   "eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
```
```  2254   "eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2"
```
```  2255   "eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls"
```
```  2256   "eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
```
```  2257   "eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
```
```  2258   "eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min"
```
```  2259   "eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2"
```
```  2260   "eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
```
```  2261   "eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
```
```  2262   "eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2"
```
```  2263   unfolding eq_equals by simp_all
```
```  2264
```
```  2265 lemma eq_int_refl [code nbe]:
```
```  2266   "eq_class.eq (k::int) k \<longleftrightarrow> True"
```
```  2267   by (rule HOL.eq_refl)
```
```  2268
```
```  2269 lemma less_eq_number_of_int_code [code]:
```
```  2270   "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
```
```  2271   unfolding number_of_is_id ..
```
```  2272
```
```  2273 lemma less_eq_int_code [code]:
```
```  2274   "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
```
```  2275   "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
```
```  2276   "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2277   "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2278   "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
```
```  2279   "Int.Min \<le> Int.Min \<longleftrightarrow> True"
```
```  2280   "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
```
```  2281   "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
```
```  2282   "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
```
```  2283   "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2284   "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2285   "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2286   "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2287   "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2288   "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2289   "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2290   by simp_all
```
```  2291
```
```  2292 lemma less_number_of_int_code [code]:
```
```  2293   "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
```
```  2294   unfolding number_of_is_id ..
```
```  2295
```
```  2296 lemma less_int_code [code]:
```
```  2297   "Int.Pls < Int.Pls \<longleftrightarrow> False"
```
```  2298   "Int.Pls < Int.Min \<longleftrightarrow> False"
```
```  2299   "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
```
```  2300   "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2301   "Int.Min < Int.Pls \<longleftrightarrow> True"
```
```  2302   "Int.Min < Int.Min \<longleftrightarrow> False"
```
```  2303   "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
```
```  2304   "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
```
```  2305   "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2306   "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2307   "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2308   "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
```
```  2309   "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2310   "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2311   "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2312   "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
```
```  2313   by simp_all
```
```  2314
```
```  2315 definition
```
```  2316   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
```
```  2317   "nat_aux i n = nat i + n"
```
```  2318
```
```  2319 lemma [code]:
```
```  2320   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
```
```  2321   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
```
```  2322     dest: zless_imp_add1_zle)
```
```  2323
```
```  2324 lemma [code]: "nat i = nat_aux i 0"
```
```  2325   by (simp add: nat_aux_def)
```
```  2326
```
```  2327 hide_const (open) nat_aux
```
```  2328
```
```  2329 lemma zero_is_num_zero [code, code_unfold_post]:
```
```  2330   "(0\<Colon>int) = Numeral0"
```
```  2331   by simp
```
```  2332
```
```  2333 lemma one_is_num_one [code, code_unfold_post]:
```
```  2334   "(1\<Colon>int) = Numeral1"
```
```  2335   by simp
```
```  2336
```
```  2337 code_modulename SML
```
```  2338   Int Arith
```
```  2339
```
```  2340 code_modulename OCaml
```
```  2341   Int Arith
```
```  2342
```
```  2343 code_modulename Haskell
```
```  2344   Int Arith
```
```  2345
```
```  2346 types_code
```
```  2347   "int" ("int")
```
```  2348 attach (term_of) {*
```
```  2349 val term_of_int = HOLogic.mk_number HOLogic.intT;
```
```  2350 *}
```
```  2351 attach (test) {*
```
```  2352 fun gen_int i =
```
```  2353   let val j = one_of [~1, 1] * random_range 0 i
```
```  2354   in (j, fn () => term_of_int j) end;
```
```  2355 *}
```
```  2356
```
```  2357 setup {*
```
```  2358 let
```
```  2359
```
```  2360 fun strip_number_of (@{term "Int.number_of :: int => int"} \$ t) = t
```
```  2361   | strip_number_of t = t;
```
```  2362
```
```  2363 fun numeral_codegen thy defs dep module b t gr =
```
```  2364   let val i = HOLogic.dest_numeral (strip_number_of t)
```
```  2365   in
```
```  2366     SOME (Codegen.str (string_of_int i),
```
```  2367       snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr))
```
```  2368   end handle TERM _ => NONE;
```
```  2369
```
```  2370 in
```
```  2371
```
```  2372 Codegen.add_codegen "numeral_codegen" numeral_codegen
```
```  2373
```
```  2374 end
```
```  2375 *}
```
```  2376
```
```  2377 consts_code
```
```  2378   "number_of :: int \<Rightarrow> int"    ("(_)")
```
```  2379   "0 :: int"                   ("0")
```
```  2380   "1 :: int"                   ("1")
```
```  2381   "uminus :: int => int"       ("~")
```
```  2382   "op + :: int => int => int"  ("(_ +/ _)")
```
```  2383   "op * :: int => int => int"  ("(_ */ _)")
```
```  2384   "op \<le> :: int => int => bool" ("(_ <=/ _)")
```
```  2385   "op < :: int => int => bool" ("(_ </ _)")
```
```  2386
```
```  2387 quickcheck_params [default_type = int]
```
```  2388
```
```  2389 hide_const (open) Pls Min Bit0 Bit1 succ pred
```
```  2390
```
```  2391
```
```  2392 subsection {* Legacy theorems *}
```
```  2393
```
```  2394 lemmas zminus_zminus = minus_minus [of "z::int", standard]
```
```  2395 lemmas zminus_0 = minus_zero [where 'a=int]
```
```  2396 lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
```
```  2397 lemmas zadd_commute = add_commute [of "z::int" "w", standard]
```
```  2398 lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
```
```  2399 lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
```
```  2400 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
```
```  2401 lemmas zmult_ac = mult_ac
```
```  2402 lemmas zadd_0 = add_0_left [of "z::int", standard]
```
```  2403 lemmas zadd_0_right = add_0_right [of "z::int", standard]
```
```  2404 lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
```
```  2405 lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
```
```  2406 lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
```
```  2407 lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
```
```  2408 lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
```
```  2409 lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
```
```  2410 lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```  2411 lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```  2412
```
```  2413 lemmas zmult_1 = mult_1_left [of "z::int", standard]
```
```  2414 lemmas zmult_1_right = mult_1_right [of "z::int", standard]
```
```  2415
```
```  2416 lemmas zle_refl = order_refl [of "w::int", standard]
```
```  2417 lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
```
```  2418 lemmas zle_antisym = order_antisym [of "z::int" "w", standard]
```
```  2419 lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
```
```  2420 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```  2421
```
```  2422 lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
```
```  2423 lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
```
```  2424 lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
```
```  2425
```
```  2426 lemmas int_0_less_1 = zero_less_one [where 'a=int]
```
```  2427 lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
```
```  2428
```
```  2429 lemmas inj_int = inj_of_nat [where 'a=int]
```
```  2430 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```  2431 lemmas int_mult = of_nat_mult [where 'a=int]
```
```  2432 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```  2433 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
```
```  2434 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```  2435 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
```
```  2436 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```  2437 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```  2438 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
```
```  2439 lemmas int_0 = of_nat_0 [where 'a=int]
```
```  2440 lemmas int_1 = of_nat_1 [where 'a=int]
```
```  2441 lemmas int_Suc = of_nat_Suc [where 'a=int]
```
```  2442 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
```
```  2443 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```  2444 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```  2445 lemmas zless_le = less_int_def
```
```  2446 lemmas int_eq_of_nat = TrueI
```
```  2447
```
```  2448 lemma zpower_zadd_distrib:
```
```  2449   "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
```
```  2450   by (rule power_add)
```
```  2451
```
```  2452 lemma zero_less_zpower_abs_iff:
```
```  2453   "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
```
```  2454   by (rule zero_less_power_abs_iff)
```
```  2455
```
```  2456 lemma zero_le_zpower_abs: "(0::int) \<le> abs x ^ n"
```
```  2457   by (rule zero_le_power_abs)
```
```  2458
```
```  2459 lemma zpower_zpower:
```
```  2460   "(x ^ y) ^ z = (x ^ (y * z)::int)"
```
```  2461   by (rule power_mult [symmetric])
```
```  2462
```
```  2463 lemma int_power:
```
```  2464   "int (m ^ n) = int m ^ n"
```
```  2465   by (rule of_nat_power)
```
```  2466
```
```  2467 lemmas zpower_int = int_power [symmetric]
```
```  2468
```
```  2469 end
```