src/HOL/Int.thy
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simplify theorem references
```     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   "~~/src/Provers/Arith/assoc_fold.ML"
```
```    16   "~~/src/Provers/Arith/cancel_numerals.ML"
```
```    17   "~~/src/Provers/Arith/combine_numerals.ML"
```
```    18   ("Tools/int_arith.ML")
```
```    19 begin
```
```    20
```
```    21 subsection {* The equivalence relation underlying the integers *}
```
```    22
```
```    23 definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where
```
```    24   [code del]: "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
```
```    25
```
```    26 typedef (Integ)
```
```    27   int = "UNIV//intrel"
```
```    28   by (auto simp add: quotient_def)
```
```    29
```
```    30 instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
```
```    31 begin
```
```    32
```
```    33 definition
```
```    34   Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
```
```    35
```
```    36 definition
```
```    37   One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
```
```    38
```
```    39 definition
```
```    40   add_int_def [code del]: "z + w = Abs_Integ
```
```    41     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
```
```    42       intrel `` {(x + u, y + v)})"
```
```    43
```
```    44 definition
```
```    45   minus_int_def [code del]:
```
```    46     "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
```
```    47
```
```    48 definition
```
```    49   diff_int_def [code del]:  "z - w = z + (-w \<Colon> int)"
```
```    50
```
```    51 definition
```
```    52   mult_int_def [code del]: "z * w = Abs_Integ
```
```    53     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
```
```    54       intrel `` {(x*u + y*v, x*v + y*u)})"
```
```    55
```
```    56 definition
```
```    57   le_int_def [code del]:
```
```    58    "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)"
```
```    59
```
```    60 definition
```
```    61   less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```    62
```
```    63 definition
```
```    64   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
```
```    65
```
```    66 definition
```
```    67   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```    68
```
```    69 instance ..
```
```    70
```
```    71 end
```
```    72
```
```    73
```
```    74 subsection{*Construction of the Integers*}
```
```    75
```
```    76 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    77 by (simp add: intrel_def)
```
```    78
```
```    79 lemma equiv_intrel: "equiv UNIV intrel"
```
```    80 by (simp add: intrel_def equiv_def refl_on_def sym_def trans_def)
```
```    81
```
```    82 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    83   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    84 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    85
```
```    86 text{*All equivalence classes belong to set of representatives*}
```
```    87 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
```
```    88 by (auto simp add: Integ_def intrel_def quotient_def)
```
```    89
```
```    90 text{*Reduces equality on abstractions to equality on representatives:
```
```    91   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
```
```    92 declare Abs_Integ_inject [simp,noatp]  Abs_Integ_inverse [simp,noatp]
```
```    93
```
```    94 text{*Case analysis on the representation of an integer as an equivalence
```
```    95       class of pairs of naturals.*}
```
```    96 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```    97      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
```
```    98 apply (rule Abs_Integ_cases [of z])
```
```    99 apply (auto simp add: Integ_def quotient_def)
```
```   100 done
```
```   101
```
```   102
```
```   103 subsection {* Arithmetic Operations *}
```
```   104
```
```   105 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
```
```   106 proof -
```
```   107   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
```
```   108     by (simp add: congruent_def)
```
```   109   thus ?thesis
```
```   110     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
```
```   111 qed
```
```   112
```
```   113 lemma add:
```
```   114      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
```
```   115       Abs_Integ (intrel``{(x+u, y+v)})"
```
```   116 proof -
```
```   117   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
```
```   118         respects2 intrel"
```
```   119     by (simp add: congruent2_def)
```
```   120   thus ?thesis
```
```   121     by (simp add: add_int_def UN_UN_split_split_eq
```
```   122                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   123 qed
```
```   124
```
```   125 text{*Congruence property for multiplication*}
```
```   126 lemma mult_congruent2:
```
```   127      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   128       respects2 intrel"
```
```   129 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   130  apply (force simp add: mult_ac, clarify)
```
```   131 apply (simp add: congruent_def mult_ac)
```
```   132 apply (rename_tac u v w x y z)
```
```   133 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   134 apply (simp add: mult_ac)
```
```   135 apply (simp add: add_mult_distrib [symmetric])
```
```   136 done
```
```   137
```
```   138 lemma mult:
```
```   139      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   140       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   141 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   142               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   143
```
```   144 text{*The integers form a @{text comm_ring_1}*}
```
```   145 instance int :: comm_ring_1
```
```   146 proof
```
```   147   fix i j k :: int
```
```   148   show "(i + j) + k = i + (j + k)"
```
```   149     by (cases i, cases j, cases k) (simp add: add add_assoc)
```
```   150   show "i + j = j + i"
```
```   151     by (cases i, cases j) (simp add: add_ac add)
```
```   152   show "0 + i = i"
```
```   153     by (cases i) (simp add: Zero_int_def add)
```
```   154   show "- i + i = 0"
```
```   155     by (cases i) (simp add: Zero_int_def minus add)
```
```   156   show "i - j = i + - j"
```
```   157     by (simp add: diff_int_def)
```
```   158   show "(i * j) * k = i * (j * k)"
```
```   159     by (cases i, cases j, cases k) (simp add: mult algebra_simps)
```
```   160   show "i * j = j * i"
```
```   161     by (cases i, cases j) (simp add: mult algebra_simps)
```
```   162   show "1 * i = i"
```
```   163     by (cases i) (simp add: One_int_def mult)
```
```   164   show "(i + j) * k = i * k + j * k"
```
```   165     by (cases i, cases j, cases k) (simp add: add mult algebra_simps)
```
```   166   show "0 \<noteq> (1::int)"
```
```   167     by (simp add: Zero_int_def One_int_def)
```
```   168 qed
```
```   169
```
```   170 lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
```
```   171 by (induct m, simp_all add: Zero_int_def One_int_def add)
```
```   172
```
```   173
```
```   174 subsection {* The @{text "\<le>"} Ordering *}
```
```   175
```
```   176 lemma le:
```
```   177   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   178 by (force simp add: le_int_def)
```
```   179
```
```   180 lemma less:
```
```   181   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
```
```   182 by (simp add: less_int_def le order_less_le)
```
```   183
```
```   184 instance int :: linorder
```
```   185 proof
```
```   186   fix i j k :: int
```
```   187   show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
```
```   188     by (cases i, cases j) (simp add: le)
```
```   189   show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
```
```   190     by (auto simp add: less_int_def dest: antisym)
```
```   191   show "i \<le> i"
```
```   192     by (cases i) (simp add: le)
```
```   193   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
```
```   194     by (cases i, cases j, cases k) (simp add: le)
```
```   195   show "i \<le> j \<or> j \<le> i"
```
```   196     by (cases i, cases j) (simp add: le linorder_linear)
```
```   197 qed
```
```   198
```
```   199 instantiation int :: distrib_lattice
```
```   200 begin
```
```   201
```
```   202 definition
```
```   203   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   204
```
```   205 definition
```
```   206   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   207
```
```   208 instance
```
```   209   by intro_classes
```
```   210     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   211
```
```   212 end
```
```   213
```
```   214 instance int :: pordered_cancel_ab_semigroup_add
```
```   215 proof
```
```   216   fix i j k :: int
```
```   217   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   218     by (cases i, cases j, cases k) (simp add: le add)
```
```   219 qed
```
```   220
```
```   221
```
```   222 text{*Strict Monotonicity of Multiplication*}
```
```   223
```
```   224 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   225 lemma zmult_zless_mono2_lemma:
```
```   226      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
```
```   227 apply (induct "k", simp)
```
```   228 apply (simp add: left_distrib)
```
```   229 apply (case_tac "k=0")
```
```   230 apply (simp_all add: add_strict_mono)
```
```   231 done
```
```   232
```
```   233 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
```
```   234 apply (cases k)
```
```   235 apply (auto simp add: le add int_def Zero_int_def)
```
```   236 apply (rule_tac x="x-y" in exI, simp)
```
```   237 done
```
```   238
```
```   239 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
```
```   240 apply (cases k)
```
```   241 apply (simp add: less int_def Zero_int_def)
```
```   242 apply (rule_tac x="x-y" in exI, simp)
```
```   243 done
```
```   244
```
```   245 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   246 apply (drule zero_less_imp_eq_int)
```
```   247 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   248 done
```
```   249
```
```   250 text{*The integers form an ordered integral domain*}
```
```   251 instance int :: ordered_idom
```
```   252 proof
```
```   253   fix i j k :: int
```
```   254   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   255     by (rule zmult_zless_mono2)
```
```   256   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
```
```   257     by (simp only: zabs_def)
```
```   258   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   259     by (simp only: zsgn_def)
```
```   260 qed
```
```   261
```
```   262 instance int :: lordered_ring
```
```   263 proof
```
```   264   fix k :: int
```
```   265   show "abs k = sup k (- k)"
```
```   266     by (auto simp add: sup_int_def zabs_def max_def less_minus_self_iff [symmetric])
```
```   267 qed
```
```   268
```
```   269 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
```
```   270 apply (cases w, cases z)
```
```   271 apply (simp add: less le add One_int_def)
```
```   272 done
```
```   273
```
```   274 lemma zless_iff_Suc_zadd:
```
```   275   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
```
```   276 apply (cases z, cases w)
```
```   277 apply (auto simp add: less add int_def)
```
```   278 apply (rename_tac a b c d)
```
```   279 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   280 apply arith
```
```   281 done
```
```   282
```
```   283 lemmas int_distrib =
```
```   284   left_distrib [of "z1::int" "z2" "w", standard]
```
```   285   right_distrib [of "w::int" "z1" "z2", standard]
```
```   286   left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```   287   right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```   288
```
```   289
```
```   290 subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
```
```   291
```
```   292 context ring_1
```
```   293 begin
```
```   294
```
```   295 definition
```
```   296   of_int :: "int \<Rightarrow> 'a"
```
```   297 where
```
```   298   [code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   299
```
```   300 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   301 proof -
```
```   302   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   303     by (simp add: congruent_def algebra_simps of_nat_add [symmetric]
```
```   304             del: of_nat_add)
```
```   305   thus ?thesis
```
```   306     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   307 qed
```
```   308
```
```   309 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   310 by (simp add: of_int Zero_int_def)
```
```   311
```
```   312 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   313 by (simp add: of_int One_int_def)
```
```   314
```
```   315 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   316 by (cases w, cases z, simp add: algebra_simps of_int add)
```
```   317
```
```   318 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   319 by (cases z, simp add: algebra_simps of_int minus)
```
```   320
```
```   321 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   322 by (simp add: OrderedGroup.diff_minus diff_minus)
```
```   323
```
```   324 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   325 apply (cases w, cases z)
```
```   326 apply (simp add: algebra_simps of_int mult of_nat_mult)
```
```   327 done
```
```   328
```
```   329 text{*Collapse nested embeddings*}
```
```   330 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   331 by (induct n) auto
```
```   332
```
```   333 end
```
```   334
```
```   335 context ordered_idom
```
```   336 begin
```
```   337
```
```   338 lemma of_int_le_iff [simp]:
```
```   339   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   340 by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
```
```   341
```
```   342 text{*Special cases where either operand is zero*}
```
```   343 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
```
```   344 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
```
```   345
```
```   346 lemma of_int_less_iff [simp]:
```
```   347   "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   348   by (simp add: not_le [symmetric] linorder_not_le [symmetric])
```
```   349
```
```   350 text{*Special cases where either operand is zero*}
```
```   351 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
```
```   352 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
```
```   353
```
```   354 end
```
```   355
```
```   356 text{*Class for unital rings with characteristic zero.
```
```   357  Includes non-ordered rings like the complex numbers.*}
```
```   358 class ring_char_0 = ring_1 + semiring_char_0
```
```   359 begin
```
```   360
```
```   361 lemma of_int_eq_iff [simp]:
```
```   362    "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   363 apply (cases w, cases z, simp add: of_int)
```
```   364 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
```
```   365 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
```
```   366 done
```
```   367
```
```   368 text{*Special cases where either operand is zero*}
```
```   369 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
```
```   370 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
```
```   371
```
```   372 end
```
```   373
```
```   374 text{*Every @{text ordered_idom} has characteristic zero.*}
```
```   375 subclass (in ordered_idom) ring_char_0 by intro_locales
```
```   376
```
```   377 lemma of_int_eq_id [simp]: "of_int = id"
```
```   378 proof
```
```   379   fix z show "of_int z = id z"
```
```   380     by (cases z) (simp add: of_int add minus int_def diff_minus)
```
```   381 qed
```
```   382
```
```   383
```
```   384 subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
```
```   385
```
```   386 definition
```
```   387   nat :: "int \<Rightarrow> nat"
```
```   388 where
```
```   389   [code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
```
```   390
```
```   391 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   392 proof -
```
```   393   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   394     by (simp add: congruent_def) arith
```
```   395   thus ?thesis
```
```   396     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   397 qed
```
```   398
```
```   399 lemma nat_int [simp]: "nat (of_nat n) = n"
```
```   400 by (simp add: nat int_def)
```
```   401
```
```   402 lemma nat_zero [simp]: "nat 0 = 0"
```
```   403 by (simp add: Zero_int_def nat)
```
```   404
```
```   405 lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
```
```   406 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   407
```
```   408 corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
```
```   409 by simp
```
```   410
```
```   411 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   412 by (cases z, simp add: nat le Zero_int_def)
```
```   413
```
```   414 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   415 apply (cases w, cases z)
```
```   416 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
```
```   417 done
```
```   418
```
```   419 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   420 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   421 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   422
```
```   423 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   424 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   425
```
```   426 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   427 apply (cases w, cases z)
```
```   428 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
```
```   429 done
```
```   430
```
```   431 lemma nonneg_eq_int:
```
```   432   fixes z :: int
```
```   433   assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
```
```   434   shows P
```
```   435   using assms by (blast dest: nat_0_le sym)
```
```   436
```
```   437 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   438 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   439
```
```   440 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   441 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   442
```
```   443 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
```
```   444 apply (cases w)
```
```   445 apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)
```
```   446 done
```
```   447
```
```   448 lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
```
```   449 by(simp add: nat_eq_iff) arith
```
```   450
```
```   451 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
```
```   452 by (auto simp add: nat_eq_iff2)
```
```   453
```
```   454 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   455 by (insert zless_nat_conj [of 0], auto)
```
```   456
```
```   457 lemma nat_add_distrib:
```
```   458      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   459 by (cases z, cases z', simp add: nat add le Zero_int_def)
```
```   460
```
```   461 lemma nat_diff_distrib:
```
```   462      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   463 by (cases z, cases z',
```
```   464     simp add: nat add minus diff_minus le Zero_int_def)
```
```   465
```
```   466 lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
```
```   467 by (simp add: int_def minus nat Zero_int_def)
```
```   468
```
```   469 lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
```
```   470 by (cases z, simp add: nat less int_def, arith)
```
```   471
```
```   472 context ring_1
```
```   473 begin
```
```   474
```
```   475 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   476   by (cases z rule: eq_Abs_Integ)
```
```   477    (simp add: nat le of_int Zero_int_def of_nat_diff)
```
```   478
```
```   479 end
```
```   480
```
```   481 text {* For termination proofs: *}
```
```   482 lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
```
```   483
```
```   484
```
```   485 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
```
```   486
```
```   487 lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
```
```   488 by (simp add: order_less_le del: of_nat_Suc)
```
```   489
```
```   490 lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
```
```   491 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   492
```
```   493 lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
```
```   494 by (simp add: minus_le_iff)
```
```   495
```
```   496 lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
```
```   497 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   498
```
```   499 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
```
```   500 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   501
```
```   502 lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
```
```   503 by (simp add: int_def le minus Zero_int_def)
```
```   504
```
```   505 lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
```
```   506 by (simp add: linorder_not_less)
```
```   507
```
```   508 lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
```
```   509 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   510
```
```   511 lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
```
```   512 proof -
```
```   513   have "(w \<le> z) = (0 \<le> z - w)"
```
```   514     by (simp only: le_diff_eq add_0_left)
```
```   515   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
```
```   516     by (auto elim: zero_le_imp_eq_int)
```
```   517   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
```
```   518     by (simp only: algebra_simps)
```
```   519   finally show ?thesis .
```
```   520 qed
```
```   521
```
```   522 lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
```
```   523 by simp
```
```   524
```
```   525 lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
```
```   526 by simp
```
```   527
```
```   528 text{*This version is proved for all ordered rings, not just integers!
```
```   529       It is proved here because attribute @{text arith_split} is not available
```
```   530       in theory @{text Ring_and_Field}.
```
```   531       But is it really better than just rewriting with @{text abs_if}?*}
```
```   532 lemma abs_split [arith_split,noatp]:
```
```   533      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   534 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   535
```
```   536 lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
```
```   537 apply (cases x)
```
```   538 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   539 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   540 done
```
```   541
```
```   542
```
```   543 subsection {* Cases and induction *}
```
```   544
```
```   545 text{*Now we replace the case analysis rule by a more conventional one:
```
```   546 whether an integer is negative or not.*}
```
```   547
```
```   548 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   549   "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
```
```   550 apply (cases "z < 0", blast dest!: negD)
```
```   551 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   552 apply auto
```
```   553 apply (blast dest: nat_0_le [THEN sym])
```
```   554 done
```
```   555
```
```   556 theorem int_induct [induct type: int, case_names nonneg neg]:
```
```   557      "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
```
```   558   by (cases z rule: int_cases) auto
```
```   559
```
```   560 text{*Contributed by Brian Huffman*}
```
```   561 theorem int_diff_cases:
```
```   562   obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
```
```   563 apply (cases z rule: eq_Abs_Integ)
```
```   564 apply (rule_tac m=x and n=y in diff)
```
```   565 apply (simp add: int_def diff_def minus add)
```
```   566 done
```
```   567
```
```   568
```
```   569 subsection {* Binary representation *}
```
```   570
```
```   571 text {*
```
```   572   This formalization defines binary arithmetic in terms of the integers
```
```   573   rather than using a datatype. This avoids multiple representations (leading
```
```   574   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
```
```   575   int_of_binary}, for the numerical interpretation.
```
```   576
```
```   577   The representation expects that @{text "(m mod 2)"} is 0 or 1,
```
```   578   even if m is negative;
```
```   579   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
```
```   580   @{text "-5 = (-3)*2 + 1"}.
```
```   581
```
```   582   This two's complement binary representation derives from the paper
```
```   583   "An Efficient Representation of Arithmetic for Term Rewriting" by
```
```   584   Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
```
```   585   Springer LNCS 488 (240-251), 1991.
```
```   586 *}
```
```   587
```
```   588 subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
```
```   589
```
```   590 definition
```
```   591   Pls :: int where
```
```   592   [code del]: "Pls = 0"
```
```   593
```
```   594 definition
```
```   595   Min :: int where
```
```   596   [code del]: "Min = - 1"
```
```   597
```
```   598 definition
```
```   599   Bit0 :: "int \<Rightarrow> int" where
```
```   600   [code del]: "Bit0 k = k + k"
```
```   601
```
```   602 definition
```
```   603   Bit1 :: "int \<Rightarrow> int" where
```
```   604   [code del]: "Bit1 k = 1 + k + k"
```
```   605
```
```   606 class number = -- {* for numeric types: nat, int, real, \dots *}
```
```   607   fixes number_of :: "int \<Rightarrow> 'a"
```
```   608
```
```   609 use "Tools/numeral.ML"
```
```   610
```
```   611 syntax
```
```   612   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```   613
```
```   614 use "Tools/numeral_syntax.ML"
```
```   615 setup NumeralSyntax.setup
```
```   616
```
```   617 abbreviation
```
```   618   "Numeral0 \<equiv> number_of Pls"
```
```   619
```
```   620 abbreviation
```
```   621   "Numeral1 \<equiv> number_of (Bit1 Pls)"
```
```   622
```
```   623 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
```
```   624   -- {* Unfold all @{text let}s involving constants *}
```
```   625   unfolding Let_def ..
```
```   626
```
```   627 definition
```
```   628   succ :: "int \<Rightarrow> int" where
```
```   629   [code del]: "succ k = k + 1"
```
```   630
```
```   631 definition
```
```   632   pred :: "int \<Rightarrow> int" where
```
```   633   [code del]: "pred k = k - 1"
```
```   634
```
```   635 lemmas
```
```   636   max_number_of [simp] = max_def
```
```   637     [of "number_of u" "number_of v", standard, simp]
```
```   638 and
```
```   639   min_number_of [simp] = min_def
```
```   640     [of "number_of u" "number_of v", standard, simp]
```
```   641   -- {* unfolding @{text minx} and @{text max} on numerals *}
```
```   642
```
```   643 lemmas numeral_simps =
```
```   644   succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
```
```   645
```
```   646 text {* Removal of leading zeroes *}
```
```   647
```
```   648 lemma Bit0_Pls [simp, code post]:
```
```   649   "Bit0 Pls = Pls"
```
```   650   unfolding numeral_simps by simp
```
```   651
```
```   652 lemma Bit1_Min [simp, code post]:
```
```   653   "Bit1 Min = Min"
```
```   654   unfolding numeral_simps by simp
```
```   655
```
```   656 lemmas normalize_bin_simps =
```
```   657   Bit0_Pls Bit1_Min
```
```   658
```
```   659
```
```   660 subsubsection {* Successor and predecessor functions *}
```
```   661
```
```   662 text {* Successor *}
```
```   663
```
```   664 lemma succ_Pls:
```
```   665   "succ Pls = Bit1 Pls"
```
```   666   unfolding numeral_simps by simp
```
```   667
```
```   668 lemma succ_Min:
```
```   669   "succ Min = Pls"
```
```   670   unfolding numeral_simps by simp
```
```   671
```
```   672 lemma succ_Bit0:
```
```   673   "succ (Bit0 k) = Bit1 k"
```
```   674   unfolding numeral_simps by simp
```
```   675
```
```   676 lemma succ_Bit1:
```
```   677   "succ (Bit1 k) = Bit0 (succ k)"
```
```   678   unfolding numeral_simps by simp
```
```   679
```
```   680 lemmas succ_bin_simps [simp] =
```
```   681   succ_Pls succ_Min succ_Bit0 succ_Bit1
```
```   682
```
```   683 text {* Predecessor *}
```
```   684
```
```   685 lemma pred_Pls:
```
```   686   "pred Pls = Min"
```
```   687   unfolding numeral_simps by simp
```
```   688
```
```   689 lemma pred_Min:
```
```   690   "pred Min = Bit0 Min"
```
```   691   unfolding numeral_simps by simp
```
```   692
```
```   693 lemma pred_Bit0:
```
```   694   "pred (Bit0 k) = Bit1 (pred k)"
```
```   695   unfolding numeral_simps by simp
```
```   696
```
```   697 lemma pred_Bit1:
```
```   698   "pred (Bit1 k) = Bit0 k"
```
```   699   unfolding numeral_simps by simp
```
```   700
```
```   701 lemmas pred_bin_simps [simp] =
```
```   702   pred_Pls pred_Min pred_Bit0 pred_Bit1
```
```   703
```
```   704
```
```   705 subsubsection {* Binary arithmetic *}
```
```   706
```
```   707 text {* Addition *}
```
```   708
```
```   709 lemma add_Pls:
```
```   710   "Pls + k = k"
```
```   711   unfolding numeral_simps by simp
```
```   712
```
```   713 lemma add_Min:
```
```   714   "Min + k = pred k"
```
```   715   unfolding numeral_simps by simp
```
```   716
```
```   717 lemma add_Bit0_Bit0:
```
```   718   "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
```
```   719   unfolding numeral_simps by simp
```
```   720
```
```   721 lemma add_Bit0_Bit1:
```
```   722   "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
```
```   723   unfolding numeral_simps by simp
```
```   724
```
```   725 lemma add_Bit1_Bit0:
```
```   726   "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
```
```   727   unfolding numeral_simps by simp
```
```   728
```
```   729 lemma add_Bit1_Bit1:
```
```   730   "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
```
```   731   unfolding numeral_simps by simp
```
```   732
```
```   733 lemma add_Pls_right:
```
```   734   "k + Pls = k"
```
```   735   unfolding numeral_simps by simp
```
```   736
```
```   737 lemma add_Min_right:
```
```   738   "k + Min = pred k"
```
```   739   unfolding numeral_simps by simp
```
```   740
```
```   741 lemmas add_bin_simps [simp] =
```
```   742   add_Pls add_Min add_Pls_right add_Min_right
```
```   743   add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
```
```   744
```
```   745 text {* Negation *}
```
```   746
```
```   747 lemma minus_Pls:
```
```   748   "- Pls = Pls"
```
```   749   unfolding numeral_simps by simp
```
```   750
```
```   751 lemma minus_Min:
```
```   752   "- Min = Bit1 Pls"
```
```   753   unfolding numeral_simps by simp
```
```   754
```
```   755 lemma minus_Bit0:
```
```   756   "- (Bit0 k) = Bit0 (- k)"
```
```   757   unfolding numeral_simps by simp
```
```   758
```
```   759 lemma minus_Bit1:
```
```   760   "- (Bit1 k) = Bit1 (pred (- k))"
```
```   761   unfolding numeral_simps by simp
```
```   762
```
```   763 lemmas minus_bin_simps [simp] =
```
```   764   minus_Pls minus_Min minus_Bit0 minus_Bit1
```
```   765
```
```   766 text {* Subtraction *}
```
```   767
```
```   768 lemma diff_bin_simps [simp]:
```
```   769   "k - Pls = k"
```
```   770   "k - Min = succ k"
```
```   771   "Pls - (Bit0 l) = Bit0 (Pls - l)"
```
```   772   "Pls - (Bit1 l) = Bit1 (Min - l)"
```
```   773   "Min - (Bit0 l) = Bit1 (Min - l)"
```
```   774   "Min - (Bit1 l) = Bit0 (Min - l)"
```
```   775   "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
```
```   776   "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
```
```   777   "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
```
```   778   "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
```
```   779   unfolding numeral_simps by simp_all
```
```   780
```
```   781 text {* Multiplication *}
```
```   782
```
```   783 lemma mult_Pls:
```
```   784   "Pls * w = Pls"
```
```   785   unfolding numeral_simps by simp
```
```   786
```
```   787 lemma mult_Min:
```
```   788   "Min * k = - k"
```
```   789   unfolding numeral_simps by simp
```
```   790
```
```   791 lemma mult_Bit0:
```
```   792   "(Bit0 k) * l = Bit0 (k * l)"
```
```   793   unfolding numeral_simps int_distrib by simp
```
```   794
```
```   795 lemma mult_Bit1:
```
```   796   "(Bit1 k) * l = (Bit0 (k * l)) + l"
```
```   797   unfolding numeral_simps int_distrib by simp
```
```   798
```
```   799 lemmas mult_bin_simps [simp] =
```
```   800   mult_Pls mult_Min mult_Bit0 mult_Bit1
```
```   801
```
```   802
```
```   803 subsubsection {* Binary comparisons *}
```
```   804
```
```   805 text {* Preliminaries *}
```
```   806
```
```   807 lemma even_less_0_iff:
```
```   808   "a + a < 0 \<longleftrightarrow> a < (0::'a::ordered_idom)"
```
```   809 proof -
```
```   810   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
```
```   811   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
```
```   812     by (simp add: mult_less_0_iff zero_less_two
```
```   813                   order_less_not_sym [OF zero_less_two])
```
```   814   finally show ?thesis .
```
```   815 qed
```
```   816
```
```   817 lemma le_imp_0_less:
```
```   818   assumes le: "0 \<le> z"
```
```   819   shows "(0::int) < 1 + z"
```
```   820 proof -
```
```   821   have "0 \<le> z" by fact
```
```   822   also have "... < z + 1" by (rule less_add_one)
```
```   823   also have "... = 1 + z" by (simp add: add_ac)
```
```   824   finally show "0 < 1 + z" .
```
```   825 qed
```
```   826
```
```   827 lemma odd_less_0_iff:
```
```   828   "(1 + z + z < 0) = (z < (0::int))"
```
```   829 proof (cases z rule: int_cases)
```
```   830   case (nonneg n)
```
```   831   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```   832                              le_imp_0_less [THEN order_less_imp_le])
```
```   833 next
```
```   834   case (neg n)
```
```   835   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```   836     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```   837 qed
```
```   838
```
```   839 lemma bin_less_0_simps:
```
```   840   "Pls < 0 \<longleftrightarrow> False"
```
```   841   "Min < 0 \<longleftrightarrow> True"
```
```   842   "Bit0 w < 0 \<longleftrightarrow> w < 0"
```
```   843   "Bit1 w < 0 \<longleftrightarrow> w < 0"
```
```   844   unfolding numeral_simps
```
```   845   by (simp_all add: even_less_0_iff odd_less_0_iff)
```
```   846
```
```   847 lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
```
```   848   by simp
```
```   849
```
```   850 lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
```
```   851   unfolding numeral_simps
```
```   852   proof
```
```   853     have "k - 1 < k" by simp
```
```   854     also assume "k \<le> l"
```
```   855     finally show "k - 1 < l" .
```
```   856   next
```
```   857     assume "k - 1 < l"
```
```   858     hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
```
```   859     thus "k \<le> l" by simp
```
```   860   qed
```
```   861
```
```   862 lemma succ_pred: "succ (pred x) = x"
```
```   863   unfolding numeral_simps by simp
```
```   864
```
```   865 text {* Less-than *}
```
```   866
```
```   867 lemma less_bin_simps [simp]:
```
```   868   "Pls < Pls \<longleftrightarrow> False"
```
```   869   "Pls < Min \<longleftrightarrow> False"
```
```   870   "Pls < Bit0 k \<longleftrightarrow> Pls < k"
```
```   871   "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
```
```   872   "Min < Pls \<longleftrightarrow> True"
```
```   873   "Min < Min \<longleftrightarrow> False"
```
```   874   "Min < Bit0 k \<longleftrightarrow> Min < k"
```
```   875   "Min < Bit1 k \<longleftrightarrow> Min < k"
```
```   876   "Bit0 k < Pls \<longleftrightarrow> k < Pls"
```
```   877   "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
```
```   878   "Bit1 k < Pls \<longleftrightarrow> k < Pls"
```
```   879   "Bit1 k < Min \<longleftrightarrow> k < Min"
```
```   880   "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
```
```   881   "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
```
```   882   "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
```
```   883   "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
```
```   884   unfolding le_iff_pred_less
```
```   885     less_bin_lemma [of Pls]
```
```   886     less_bin_lemma [of Min]
```
```   887     less_bin_lemma [of "k"]
```
```   888     less_bin_lemma [of "Bit0 k"]
```
```   889     less_bin_lemma [of "Bit1 k"]
```
```   890     less_bin_lemma [of "pred Pls"]
```
```   891     less_bin_lemma [of "pred k"]
```
```   892   by (simp_all add: bin_less_0_simps succ_pred)
```
```   893
```
```   894 text {* Less-than-or-equal *}
```
```   895
```
```   896 lemma le_bin_simps [simp]:
```
```   897   "Pls \<le> Pls \<longleftrightarrow> True"
```
```   898   "Pls \<le> Min \<longleftrightarrow> False"
```
```   899   "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
```
```   900   "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
```
```   901   "Min \<le> Pls \<longleftrightarrow> True"
```
```   902   "Min \<le> Min \<longleftrightarrow> True"
```
```   903   "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
```
```   904   "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
```
```   905   "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
```
```   906   "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
```
```   907   "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
```
```   908   "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
```
```   909   "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
```
```   910   "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
```
```   911   "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
```
```   912   "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
```
```   913   unfolding not_less [symmetric]
```
```   914   by (simp_all add: not_le)
```
```   915
```
```   916 text {* Equality *}
```
```   917
```
```   918 lemma eq_bin_simps [simp]:
```
```   919   "Pls = Pls \<longleftrightarrow> True"
```
```   920   "Pls = Min \<longleftrightarrow> False"
```
```   921   "Pls = Bit0 l \<longleftrightarrow> Pls = l"
```
```   922   "Pls = Bit1 l \<longleftrightarrow> False"
```
```   923   "Min = Pls \<longleftrightarrow> False"
```
```   924   "Min = Min \<longleftrightarrow> True"
```
```   925   "Min = Bit0 l \<longleftrightarrow> False"
```
```   926   "Min = Bit1 l \<longleftrightarrow> Min = l"
```
```   927   "Bit0 k = Pls \<longleftrightarrow> k = Pls"
```
```   928   "Bit0 k = Min \<longleftrightarrow> False"
```
```   929   "Bit1 k = Pls \<longleftrightarrow> False"
```
```   930   "Bit1 k = Min \<longleftrightarrow> k = Min"
```
```   931   "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
```
```   932   "Bit0 k = Bit1 l \<longleftrightarrow> False"
```
```   933   "Bit1 k = Bit0 l \<longleftrightarrow> False"
```
```   934   "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
```
```   935   unfolding order_eq_iff [where 'a=int]
```
```   936   by (simp_all add: not_less)
```
```   937
```
```   938
```
```   939 subsection {* Converting Numerals to Rings: @{term number_of} *}
```
```   940
```
```   941 class number_ring = number + comm_ring_1 +
```
```   942   assumes number_of_eq: "number_of k = of_int k"
```
```   943
```
```   944 text {* self-embedding of the integers *}
```
```   945
```
```   946 instantiation int :: number_ring
```
```   947 begin
```
```   948
```
```   949 definition int_number_of_def [code del]:
```
```   950   "number_of w = (of_int w \<Colon> int)"
```
```   951
```
```   952 instance proof
```
```   953 qed (simp only: int_number_of_def)
```
```   954
```
```   955 end
```
```   956
```
```   957 lemma number_of_is_id:
```
```   958   "number_of (k::int) = k"
```
```   959   unfolding int_number_of_def by simp
```
```   960
```
```   961 lemma number_of_succ:
```
```   962   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
```
```   963   unfolding number_of_eq numeral_simps by simp
```
```   964
```
```   965 lemma number_of_pred:
```
```   966   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
```
```   967   unfolding number_of_eq numeral_simps by simp
```
```   968
```
```   969 lemma number_of_minus:
```
```   970   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
```
```   971   unfolding number_of_eq by (rule of_int_minus)
```
```   972
```
```   973 lemma number_of_add:
```
```   974   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
```
```   975   unfolding number_of_eq by (rule of_int_add)
```
```   976
```
```   977 lemma number_of_diff:
```
```   978   "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
```
```   979   unfolding number_of_eq by (rule of_int_diff)
```
```   980
```
```   981 lemma number_of_mult:
```
```   982   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
```
```   983   unfolding number_of_eq by (rule of_int_mult)
```
```   984
```
```   985 text {*
```
```   986   The correctness of shifting.
```
```   987   But it doesn't seem to give a measurable speed-up.
```
```   988 *}
```
```   989
```
```   990 lemma double_number_of_Bit0:
```
```   991   "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
```
```   992   unfolding number_of_eq numeral_simps left_distrib by simp
```
```   993
```
```   994 text {*
```
```   995   Converting numerals 0 and 1 to their abstract versions.
```
```   996 *}
```
```   997
```
```   998 lemma numeral_0_eq_0 [simp]:
```
```   999   "Numeral0 = (0::'a::number_ring)"
```
```  1000   unfolding number_of_eq numeral_simps by simp
```
```  1001
```
```  1002 lemma numeral_1_eq_1 [simp]:
```
```  1003   "Numeral1 = (1::'a::number_ring)"
```
```  1004   unfolding number_of_eq numeral_simps by simp
```
```  1005
```
```  1006 text {*
```
```  1007   Special-case simplification for small constants.
```
```  1008 *}
```
```  1009
```
```  1010 text{*
```
```  1011   Unary minus for the abstract constant 1. Cannot be inserted
```
```  1012   as a simprule until later: it is @{text number_of_Min} re-oriented!
```
```  1013 *}
```
```  1014
```
```  1015 lemma numeral_m1_eq_minus_1:
```
```  1016   "(-1::'a::number_ring) = - 1"
```
```  1017   unfolding number_of_eq numeral_simps by simp
```
```  1018
```
```  1019 lemma mult_minus1 [simp]:
```
```  1020   "-1 * z = -(z::'a::number_ring)"
```
```  1021   unfolding number_of_eq numeral_simps by simp
```
```  1022
```
```  1023 lemma mult_minus1_right [simp]:
```
```  1024   "z * -1 = -(z::'a::number_ring)"
```
```  1025   unfolding number_of_eq numeral_simps by simp
```
```  1026
```
```  1027 (*Negation of a coefficient*)
```
```  1028 lemma minus_number_of_mult [simp]:
```
```  1029    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
```
```  1030    unfolding number_of_eq by simp
```
```  1031
```
```  1032 text {* Subtraction *}
```
```  1033
```
```  1034 lemma diff_number_of_eq:
```
```  1035   "number_of v - number_of w =
```
```  1036     (number_of (v + uminus w)::'a::number_ring)"
```
```  1037   unfolding number_of_eq by simp
```
```  1038
```
```  1039 lemma number_of_Pls:
```
```  1040   "number_of Pls = (0::'a::number_ring)"
```
```  1041   unfolding number_of_eq numeral_simps by simp
```
```  1042
```
```  1043 lemma number_of_Min:
```
```  1044   "number_of Min = (- 1::'a::number_ring)"
```
```  1045   unfolding number_of_eq numeral_simps by simp
```
```  1046
```
```  1047 lemma number_of_Bit0:
```
```  1048   "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
```
```  1049   unfolding number_of_eq numeral_simps by simp
```
```  1050
```
```  1051 lemma number_of_Bit1:
```
```  1052   "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
```
```  1053   unfolding number_of_eq numeral_simps by simp
```
```  1054
```
```  1055
```
```  1056 subsubsection {* Equality of Binary Numbers *}
```
```  1057
```
```  1058 text {* First version by Norbert Voelker *}
```
```  1059
```
```  1060 definition (*for simplifying equalities*)
```
```  1061   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
```
```  1062 where
```
```  1063   "iszero z \<longleftrightarrow> z = 0"
```
```  1064
```
```  1065 lemma iszero_0: "iszero 0"
```
```  1066 by (simp add: iszero_def)
```
```  1067
```
```  1068 lemma not_iszero_1: "~ iszero 1"
```
```  1069 by (simp add: iszero_def eq_commute)
```
```  1070
```
```  1071 lemma eq_number_of_eq:
```
```  1072   "((number_of x::'a::number_ring) = number_of y) =
```
```  1073    iszero (number_of (x + uminus y) :: 'a)"
```
```  1074 unfolding iszero_def number_of_add number_of_minus
```
```  1075 by (simp add: algebra_simps)
```
```  1076
```
```  1077 lemma iszero_number_of_Pls:
```
```  1078   "iszero ((number_of Pls)::'a::number_ring)"
```
```  1079 unfolding iszero_def numeral_0_eq_0 ..
```
```  1080
```
```  1081 lemma nonzero_number_of_Min:
```
```  1082   "~ iszero ((number_of Min)::'a::number_ring)"
```
```  1083 unfolding iszero_def numeral_m1_eq_minus_1 by simp
```
```  1084
```
```  1085
```
```  1086 subsubsection {* Comparisons, for Ordered Rings *}
```
```  1087
```
```  1088 lemmas double_eq_0_iff = double_zero
```
```  1089
```
```  1090 lemma odd_nonzero:
```
```  1091   "1 + z + z \<noteq> (0::int)";
```
```  1092 proof (cases z rule: int_cases)
```
```  1093   case (nonneg n)
```
```  1094   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```  1095   thus ?thesis using  le_imp_0_less [OF le]
```
```  1096     by (auto simp add: add_assoc)
```
```  1097 next
```
```  1098   case (neg n)
```
```  1099   show ?thesis
```
```  1100   proof
```
```  1101     assume eq: "1 + z + z = 0"
```
```  1102     have "(0::int) < 1 + (of_nat n + of_nat n)"
```
```  1103       by (simp add: le_imp_0_less add_increasing)
```
```  1104     also have "... = - (1 + z + z)"
```
```  1105       by (simp add: neg add_assoc [symmetric])
```
```  1106     also have "... = 0" by (simp add: eq)
```
```  1107     finally have "0<0" ..
```
```  1108     thus False by blast
```
```  1109   qed
```
```  1110 qed
```
```  1111
```
```  1112 lemma iszero_number_of_Bit0:
```
```  1113   "iszero (number_of (Bit0 w)::'a) =
```
```  1114    iszero (number_of w::'a::{ring_char_0,number_ring})"
```
```  1115 proof -
```
```  1116   have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
```
```  1117   proof -
```
```  1118     assume eq: "of_int w + of_int w = (0::'a)"
```
```  1119     then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
```
```  1120     then have "w + w = 0" by (simp only: of_int_eq_iff)
```
```  1121     then show "w = 0" by (simp only: double_eq_0_iff)
```
```  1122   qed
```
```  1123   thus ?thesis
```
```  1124     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```  1125 qed
```
```  1126
```
```  1127 lemma iszero_number_of_Bit1:
```
```  1128   "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
```
```  1129 proof -
```
```  1130   have "1 + of_int w + of_int w \<noteq> (0::'a)"
```
```  1131   proof
```
```  1132     assume eq: "1 + of_int w + of_int w = (0::'a)"
```
```  1133     hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp
```
```  1134     hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
```
```  1135     with odd_nonzero show False by blast
```
```  1136   qed
```
```  1137   thus ?thesis
```
```  1138     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```  1139 qed
```
```  1140
```
```  1141 lemmas iszero_simps =
```
```  1142   iszero_0 not_iszero_1
```
```  1143   iszero_number_of_Pls nonzero_number_of_Min
```
```  1144   iszero_number_of_Bit0 iszero_number_of_Bit1
```
```  1145 (* iszero_number_of_Pls would never normally be used
```
```  1146    because its lhs simplifies to "iszero 0" *)
```
```  1147
```
```  1148 subsubsection {* The Less-Than Relation *}
```
```  1149
```
```  1150 lemma double_less_0_iff:
```
```  1151   "(a + a < 0) = (a < (0::'a::ordered_idom))"
```
```  1152 proof -
```
```  1153   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
```
```  1154   also have "... = (a < 0)"
```
```  1155     by (simp add: mult_less_0_iff zero_less_two
```
```  1156                   order_less_not_sym [OF zero_less_two])
```
```  1157   finally show ?thesis .
```
```  1158 qed
```
```  1159
```
```  1160 lemma odd_less_0:
```
```  1161   "(1 + z + z < 0) = (z < (0::int))";
```
```  1162 proof (cases z rule: int_cases)
```
```  1163   case (nonneg n)
```
```  1164   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```  1165                              le_imp_0_less [THEN order_less_imp_le])
```
```  1166 next
```
```  1167   case (neg n)
```
```  1168   thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
```
```  1169     add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
```
```  1170 qed
```
```  1171
```
```  1172 text {* Less-Than or Equals *}
```
```  1173
```
```  1174 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
```
```  1175
```
```  1176 lemmas le_number_of_eq_not_less =
```
```  1177   linorder_not_less [of "number_of w" "number_of v", symmetric,
```
```  1178   standard]
```
```  1179
```
```  1180
```
```  1181 text {* Absolute value (@{term abs}) *}
```
```  1182
```
```  1183 lemma abs_number_of:
```
```  1184   "abs(number_of x::'a::{ordered_idom,number_ring}) =
```
```  1185    (if number_of x < (0::'a) then -number_of x else number_of x)"
```
```  1186   by (simp add: abs_if)
```
```  1187
```
```  1188
```
```  1189 text {* Re-orientation of the equation nnn=x *}
```
```  1190
```
```  1191 lemma number_of_reorient:
```
```  1192   "(number_of w = x) = (x = number_of w)"
```
```  1193   by auto
```
```  1194
```
```  1195
```
```  1196 subsubsection {* Simplification of arithmetic operations on integer constants. *}
```
```  1197
```
```  1198 lemmas arith_extra_simps [standard, simp] =
```
```  1199   number_of_add [symmetric]
```
```  1200   number_of_minus [symmetric]
```
```  1201   numeral_m1_eq_minus_1 [symmetric]
```
```  1202   number_of_mult [symmetric]
```
```  1203   diff_number_of_eq abs_number_of
```
```  1204
```
```  1205 text {*
```
```  1206   For making a minimal simpset, one must include these default simprules.
```
```  1207   Also include @{text simp_thms}.
```
```  1208 *}
```
```  1209
```
```  1210 lemmas arith_simps =
```
```  1211   normalize_bin_simps pred_bin_simps succ_bin_simps
```
```  1212   add_bin_simps minus_bin_simps mult_bin_simps
```
```  1213   abs_zero abs_one arith_extra_simps
```
```  1214
```
```  1215 text {* Simplification of relational operations *}
```
```  1216
```
```  1217 lemma less_number_of [simp]:
```
```  1218   "(number_of x::'a::{ordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
```
```  1219   unfolding number_of_eq by (rule of_int_less_iff)
```
```  1220
```
```  1221 lemma le_number_of [simp]:
```
```  1222   "(number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
```
```  1223   unfolding number_of_eq by (rule of_int_le_iff)
```
```  1224
```
```  1225 lemma eq_number_of [simp]:
```
```  1226   "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
```
```  1227   unfolding number_of_eq by (rule of_int_eq_iff)
```
```  1228
```
```  1229 lemmas rel_simps [simp] =
```
```  1230   less_number_of less_bin_simps
```
```  1231   le_number_of le_bin_simps
```
```  1232   eq_number_of_eq eq_bin_simps
```
```  1233   iszero_simps
```
```  1234
```
```  1235
```
```  1236 subsubsection {* Simplification of arithmetic when nested to the right. *}
```
```  1237
```
```  1238 lemma add_number_of_left [simp]:
```
```  1239   "number_of v + (number_of w + z) =
```
```  1240    (number_of(v + w) + z::'a::number_ring)"
```
```  1241   by (simp add: add_assoc [symmetric])
```
```  1242
```
```  1243 lemma mult_number_of_left [simp]:
```
```  1244   "number_of v * (number_of w * z) =
```
```  1245    (number_of(v * w) * z::'a::number_ring)"
```
```  1246   by (simp add: mult_assoc [symmetric])
```
```  1247
```
```  1248 lemma add_number_of_diff1:
```
```  1249   "number_of v + (number_of w - c) =
```
```  1250   number_of(v + w) - (c::'a::number_ring)"
```
```  1251   by (simp add: diff_minus add_number_of_left)
```
```  1252
```
```  1253 lemma add_number_of_diff2 [simp]:
```
```  1254   "number_of v + (c - number_of w) =
```
```  1255    number_of (v + uminus w) + (c::'a::number_ring)"
```
```  1256 by (simp add: algebra_simps diff_number_of_eq [symmetric])
```
```  1257
```
```  1258
```
```  1259
```
```  1260
```
```  1261 subsection {* The Set of Integers *}
```
```  1262
```
```  1263 context ring_1
```
```  1264 begin
```
```  1265
```
```  1266 definition Ints  :: "'a set" where
```
```  1267   [code del]: "Ints = range of_int"
```
```  1268
```
```  1269 end
```
```  1270
```
```  1271 notation (xsymbols)
```
```  1272   Ints  ("\<int>")
```
```  1273
```
```  1274 context ring_1
```
```  1275 begin
```
```  1276
```
```  1277 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```  1278 apply (simp add: Ints_def)
```
```  1279 apply (rule range_eqI)
```
```  1280 apply (rule of_int_0 [symmetric])
```
```  1281 done
```
```  1282
```
```  1283 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```  1284 apply (simp add: Ints_def)
```
```  1285 apply (rule range_eqI)
```
```  1286 apply (rule of_int_1 [symmetric])
```
```  1287 done
```
```  1288
```
```  1289 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```  1290 apply (auto simp add: Ints_def)
```
```  1291 apply (rule range_eqI)
```
```  1292 apply (rule of_int_add [symmetric])
```
```  1293 done
```
```  1294
```
```  1295 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```  1296 apply (auto simp add: Ints_def)
```
```  1297 apply (rule range_eqI)
```
```  1298 apply (rule of_int_minus [symmetric])
```
```  1299 done
```
```  1300
```
```  1301 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```  1302 apply (auto simp add: Ints_def)
```
```  1303 apply (rule range_eqI)
```
```  1304 apply (rule of_int_mult [symmetric])
```
```  1305 done
```
```  1306
```
```  1307 lemma Ints_cases [cases set: Ints]:
```
```  1308   assumes "q \<in> \<int>"
```
```  1309   obtains (of_int) z where "q = of_int z"
```
```  1310   unfolding Ints_def
```
```  1311 proof -
```
```  1312   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```  1313   then obtain z where "q = of_int z" ..
```
```  1314   then show thesis ..
```
```  1315 qed
```
```  1316
```
```  1317 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```  1318   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```  1319   by (rule Ints_cases) auto
```
```  1320
```
```  1321 end
```
```  1322
```
```  1323 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a-b \<in> \<int>"
```
```  1324 apply (auto simp add: Ints_def)
```
```  1325 apply (rule range_eqI)
```
```  1326 apply (rule of_int_diff [symmetric])
```
```  1327 done
```
```  1328
```
```  1329 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```  1330
```
```  1331 lemma Ints_double_eq_0_iff:
```
```  1332   assumes in_Ints: "a \<in> Ints"
```
```  1333   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```  1334 proof -
```
```  1335   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1336   then obtain z where a: "a = of_int z" ..
```
```  1337   show ?thesis
```
```  1338   proof
```
```  1339     assume "a = 0"
```
```  1340     thus "a + a = 0" by simp
```
```  1341   next
```
```  1342     assume eq: "a + a = 0"
```
```  1343     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1344     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```  1345     hence "z = 0" by (simp only: double_eq_0_iff)
```
```  1346     thus "a = 0" by (simp add: a)
```
```  1347   qed
```
```  1348 qed
```
```  1349
```
```  1350 lemma Ints_odd_nonzero:
```
```  1351   assumes in_Ints: "a \<in> Ints"
```
```  1352   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```  1353 proof -
```
```  1354   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1355   then obtain z where a: "a = of_int z" ..
```
```  1356   show ?thesis
```
```  1357   proof
```
```  1358     assume eq: "1 + a + a = 0"
```
```  1359     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1360     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```  1361     with odd_nonzero show False by blast
```
```  1362   qed
```
```  1363 qed
```
```  1364
```
```  1365 lemma Ints_number_of:
```
```  1366   "(number_of w :: 'a::number_ring) \<in> Ints"
```
```  1367   unfolding number_of_eq Ints_def by simp
```
```  1368
```
```  1369 lemma Ints_odd_less_0:
```
```  1370   assumes in_Ints: "a \<in> Ints"
```
```  1371   shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
```
```  1372 proof -
```
```  1373   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1374   then obtain z where a: "a = of_int z" ..
```
```  1375   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```  1376     by (simp add: a)
```
```  1377   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```  1378   also have "... = (a < 0)" by (simp add: a)
```
```  1379   finally show ?thesis .
```
```  1380 qed
```
```  1381
```
```  1382
```
```  1383 subsection {* @{term setsum} and @{term setprod} *}
```
```  1384
```
```  1385 text {*By Jeremy Avigad*}
```
```  1386
```
```  1387 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```  1388   apply (cases "finite A")
```
```  1389   apply (erule finite_induct, auto)
```
```  1390   done
```
```  1391
```
```  1392 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```  1393   apply (cases "finite A")
```
```  1394   apply (erule finite_induct, auto)
```
```  1395   done
```
```  1396
```
```  1397 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```  1398   apply (cases "finite A")
```
```  1399   apply (erule finite_induct, auto simp add: of_nat_mult)
```
```  1400   done
```
```  1401
```
```  1402 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```  1403   apply (cases "finite A")
```
```  1404   apply (erule finite_induct, auto)
```
```  1405   done
```
```  1406
```
```  1407 lemma setprod_nonzero_nat:
```
```  1408     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
```
```  1409   by (rule setprod_nonzero, auto)
```
```  1410
```
```  1411 lemma setprod_zero_eq_nat:
```
```  1412     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
```
```  1413   by (rule setprod_zero_eq, auto)
```
```  1414
```
```  1415 lemma setprod_nonzero_int:
```
```  1416     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
```
```  1417   by (rule setprod_nonzero, auto)
```
```  1418
```
```  1419 lemma setprod_zero_eq_int:
```
```  1420     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
```
```  1421   by (rule setprod_zero_eq, auto)
```
```  1422
```
```  1423 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```  1424 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```  1425
```
```  1426
```
```  1427 subsection{*Inequality Reasoning for the Arithmetic Simproc*}
```
```  1428
```
```  1429 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
```
```  1430 by simp
```
```  1431
```
```  1432 lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
```
```  1433 by simp
```
```  1434
```
```  1435 lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
```
```  1436 by simp
```
```  1437
```
```  1438 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
```
```  1439 by simp
```
```  1440
```
```  1441 lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
```
```  1442 by simp
```
```  1443
```
```  1444 lemma inverse_numeral_1:
```
```  1445   "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
```
```  1446 by simp
```
```  1447
```
```  1448 text{*Theorem lists for the cancellation simprocs. The use of binary numerals
```
```  1449 for 0 and 1 reduces the number of special cases.*}
```
```  1450
```
```  1451 lemmas add_0s = add_numeral_0 add_numeral_0_right
```
```  1452 lemmas mult_1s = mult_numeral_1 mult_numeral_1_right
```
```  1453                  mult_minus1 mult_minus1_right
```
```  1454
```
```  1455
```
```  1456 subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
```
```  1457
```
```  1458 text{*Arithmetic computations are defined for binary literals, which leaves 0
```
```  1459 and 1 as special cases. Addition already has rules for 0, but not 1.
```
```  1460 Multiplication and unary minus already have rules for both 0 and 1.*}
```
```  1461
```
```  1462
```
```  1463 lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
```
```  1464 by simp
```
```  1465
```
```  1466
```
```  1467 lemmas add_number_of_eq = number_of_add [symmetric]
```
```  1468
```
```  1469 text{*Allow 1 on either or both sides*}
```
```  1470 lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
```
```  1471 by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
```
```  1472
```
```  1473 lemmas add_special =
```
```  1474     one_add_one_is_two
```
```  1475     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1476     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
```
```  1477
```
```  1478 text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
```
```  1479 lemmas diff_special =
```
```  1480     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1481     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
```
```  1482
```
```  1483 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1484 lemmas eq_special =
```
```  1485     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
```
```  1486     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1487     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
```
```  1488     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
```
```  1489
```
```  1490 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1491 lemmas less_special =
```
```  1492   binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]
```
```  1493   binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]
```
```  1494   binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]
```
```  1495   binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]
```
```  1496
```
```  1497 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1498 lemmas le_special =
```
```  1499     binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]
```
```  1500     binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]
```
```  1501     binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]
```
```  1502     binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]
```
```  1503
```
```  1504 lemmas arith_special[simp] =
```
```  1505        add_special diff_special eq_special less_special le_special
```
```  1506
```
```  1507
```
```  1508 lemma min_max_01: "min (0::int) 1 = 0 & min (1::int) 0 = 0 &
```
```  1509                    max (0::int) 1 = 1 & max (1::int) 0 = 1"
```
```  1510 by(simp add:min_def max_def)
```
```  1511
```
```  1512 lemmas min_max_special[simp] =
```
```  1513  min_max_01
```
```  1514  max_def[of "0::int" "number_of v", standard, simp]
```
```  1515  min_def[of "0::int" "number_of v", standard, simp]
```
```  1516  max_def[of "number_of u" "0::int", standard, simp]
```
```  1517  min_def[of "number_of u" "0::int", standard, simp]
```
```  1518  max_def[of "1::int" "number_of v", standard, simp]
```
```  1519  min_def[of "1::int" "number_of v", standard, simp]
```
```  1520  max_def[of "number_of u" "1::int", standard, simp]
```
```  1521  min_def[of "number_of u" "1::int", standard, simp]
```
```  1522
```
```  1523 text {* Legacy theorems *}
```
```  1524
```
```  1525 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```  1526 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```  1527
```
```  1528 subsection {* Setting up simplification procedures *}
```
```  1529
```
```  1530 lemmas int_arith_rules =
```
```  1531   neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1
```
```  1532   minus_zero diff_minus left_minus right_minus
```
```  1533   mult_zero_left mult_zero_right mult_Bit1 mult_1_right
```
```  1534   mult_minus_left mult_minus_right
```
```  1535   minus_add_distrib minus_minus mult_assoc
```
```  1536   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
```
```  1537   of_int_0 of_int_1 of_int_add of_int_mult
```
```  1538
```
```  1539 use "Tools/int_arith.ML"
```
```  1540 declaration {* K Int_Arith.setup *}
```
```  1541
```
```  1542
```
```  1543 subsection{*Lemmas About Small Numerals*}
```
```  1544
```
```  1545 lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
```
```  1546 proof -
```
```  1547   have "(of_int -1 :: 'a) = of_int (- 1)" by simp
```
```  1548   also have "... = - of_int 1" by (simp only: of_int_minus)
```
```  1549   also have "... = -1" by simp
```
```  1550   finally show ?thesis .
```
```  1551 qed
```
```  1552
```
```  1553 lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
```
```  1554 by (simp add: abs_if)
```
```  1555
```
```  1556 lemma abs_power_minus_one [simp]:
```
```  1557      "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
```
```  1558 by (simp add: power_abs)
```
```  1559
```
```  1560 lemma of_int_number_of_eq [simp]:
```
```  1561      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
```
```  1562 by (simp add: number_of_eq)
```
```  1563
```
```  1564 text{*Lemmas for specialist use, NOT as default simprules*}
```
```  1565 lemma mult_2: "2 * z = (z+z::'a::number_ring)"
```
```  1566 proof -
```
```  1567   have "2*z = (1 + 1)*z" by simp
```
```  1568   also have "... = z+z" by (simp add: left_distrib)
```
```  1569   finally show ?thesis .
```
```  1570 qed
```
```  1571
```
```  1572 lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
```
```  1573 by (subst mult_commute, rule mult_2)
```
```  1574
```
```  1575
```
```  1576 subsection{*More Inequality Reasoning*}
```
```  1577
```
```  1578 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```  1579 by arith
```
```  1580
```
```  1581 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```  1582 by arith
```
```  1583
```
```  1584 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```  1585 by arith
```
```  1586
```
```  1587 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```  1588 by arith
```
```  1589
```
```  1590 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```  1591 by arith
```
```  1592
```
```  1593
```
```  1594 subsection{*The functions @{term nat} and @{term int}*}
```
```  1595
```
```  1596 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
```
```  1597   @{term "w + - z"}*}
```
```  1598 declare Zero_int_def [symmetric, simp]
```
```  1599 declare One_int_def [symmetric, simp]
```
```  1600
```
```  1601 lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
```
```  1602
```
```  1603 lemma nat_0: "nat 0 = 0"
```
```  1604 by (simp add: nat_eq_iff)
```
```  1605
```
```  1606 lemma nat_1: "nat 1 = Suc 0"
```
```  1607 by (subst nat_eq_iff, simp)
```
```  1608
```
```  1609 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```  1610 by (subst nat_eq_iff, simp)
```
```  1611
```
```  1612 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```  1613 apply (insert zless_nat_conj [of 1 z])
```
```  1614 apply (auto simp add: nat_1)
```
```  1615 done
```
```  1616
```
```  1617 text{*This simplifies expressions of the form @{term "int n = z"} where
```
```  1618       z is an integer literal.*}
```
```  1619 lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
```
```  1620
```
```  1621 lemma split_nat [arith_split]:
```
```  1622   "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```  1623   (is "?P = (?L & ?R)")
```
```  1624 proof (cases "i < 0")
```
```  1625   case True thus ?thesis by auto
```
```  1626 next
```
```  1627   case False
```
```  1628   have "?P = ?L"
```
```  1629   proof
```
```  1630     assume ?P thus ?L using False by clarsimp
```
```  1631   next
```
```  1632     assume ?L thus ?P using False by simp
```
```  1633   qed
```
```  1634   with False show ?thesis by simp
```
```  1635 qed
```
```  1636
```
```  1637 context ring_1
```
```  1638 begin
```
```  1639
```
```  1640 lemma of_int_of_nat [nitpick_const_simp]:
```
```  1641   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```  1642 proof (cases "k < 0")
```
```  1643   case True then have "0 \<le> - k" by simp
```
```  1644   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```  1645   with True show ?thesis by simp
```
```  1646 next
```
```  1647   case False then show ?thesis by (simp add: not_less of_nat_nat)
```
```  1648 qed
```
```  1649
```
```  1650 end
```
```  1651
```
```  1652 lemma nat_mult_distrib:
```
```  1653   fixes z z' :: int
```
```  1654   assumes "0 \<le> z"
```
```  1655   shows "nat (z * z') = nat z * nat z'"
```
```  1656 proof (cases "0 \<le> z'")
```
```  1657   case False with assms have "z * z' \<le> 0"
```
```  1658     by (simp add: not_le mult_le_0_iff)
```
```  1659   then have "nat (z * z') = 0" by simp
```
```  1660   moreover from False have "nat z' = 0" by simp
```
```  1661   ultimately show ?thesis by simp
```
```  1662 next
```
```  1663   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1664   show ?thesis
```
```  1665     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1666       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1667          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1668 qed
```
```  1669
```
```  1670 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```  1671 apply (rule trans)
```
```  1672 apply (rule_tac [2] nat_mult_distrib, auto)
```
```  1673 done
```
```  1674
```
```  1675 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```  1676 apply (cases "z=0 | w=0")
```
```  1677 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1678                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1679 done
```
```  1680
```
```  1681
```
```  1682 subsection "Induction principles for int"
```
```  1683
```
```  1684 text{*Well-founded segments of the integers*}
```
```  1685
```
```  1686 definition
```
```  1687   int_ge_less_than  ::  "int => (int * int) set"
```
```  1688 where
```
```  1689   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
```
```  1690
```
```  1691 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1692 proof -
```
```  1693   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
```
```  1694     by (auto simp add: int_ge_less_than_def)
```
```  1695   thus ?thesis
```
```  1696     by (rule wf_subset [OF wf_measure])
```
```  1697 qed
```
```  1698
```
```  1699 text{*This variant looks odd, but is typical of the relations suggested
```
```  1700 by RankFinder.*}
```
```  1701
```
```  1702 definition
```
```  1703   int_ge_less_than2 ::  "int => (int * int) set"
```
```  1704 where
```
```  1705   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
```
```  1706
```
```  1707 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1708 proof -
```
```  1709   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
```
```  1710     by (auto simp add: int_ge_less_than2_def)
```
```  1711   thus ?thesis
```
```  1712     by (rule wf_subset [OF wf_measure])
```
```  1713 qed
```
```  1714
```
```  1715 abbreviation
```
```  1716   int :: "nat \<Rightarrow> int"
```
```  1717 where
```
```  1718   "int \<equiv> of_nat"
```
```  1719
```
```  1720 (* `set:int': dummy construction *)
```
```  1721 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1722   fixes i :: int
```
```  1723   assumes ge: "k \<le> i" and
```
```  1724     base: "P k" and
```
```  1725     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1726   shows "P i"
```
```  1727 proof -
```
```  1728   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```  1729     proof (induct n)
```
```  1730       case 0
```
```  1731       hence "i = k" by arith
```
```  1732       thus "P i" using base by simp
```
```  1733     next
```
```  1734       case (Suc n)
```
```  1735       then have "n = nat((i - 1) - k)" by arith
```
```  1736       moreover
```
```  1737       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```  1738       ultimately
```
```  1739       have "P(i - 1)" by(rule Suc.hyps)
```
```  1740       from step[OF ki1 this] show ?case by simp
```
```  1741     qed
```
```  1742   }
```
```  1743   with ge show ?thesis by fast
```
```  1744 qed
```
```  1745
```
```  1746 (* `set:int': dummy construction *)
```
```  1747 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1748   assumes gr: "k < (i::int)" and
```
```  1749         base: "P(k+1)" and
```
```  1750         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```  1751   shows "P i"
```
```  1752 apply(rule int_ge_induct[of "k + 1"])
```
```  1753   using gr apply arith
```
```  1754  apply(rule base)
```
```  1755 apply (rule step, simp+)
```
```  1756 done
```
```  1757
```
```  1758 theorem int_le_induct[consumes 1,case_names base step]:
```
```  1759   assumes le: "i \<le> (k::int)" and
```
```  1760         base: "P(k)" and
```
```  1761         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1762   shows "P i"
```
```  1763 proof -
```
```  1764   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
```
```  1765     proof (induct n)
```
```  1766       case 0
```
```  1767       hence "i = k" by arith
```
```  1768       thus "P i" using base by simp
```
```  1769     next
```
```  1770       case (Suc n)
```
```  1771       hence "n = nat(k - (i+1))" by arith
```
```  1772       moreover
```
```  1773       have ki1: "i + 1 \<le> k" using Suc.prems by arith
```
```  1774       ultimately
```
```  1775       have "P(i+1)" by(rule Suc.hyps)
```
```  1776       from step[OF ki1 this] show ?case by simp
```
```  1777     qed
```
```  1778   }
```
```  1779   with le show ?thesis by fast
```
```  1780 qed
```
```  1781
```
```  1782 theorem int_less_induct [consumes 1,case_names base step]:
```
```  1783   assumes less: "(i::int) < k" and
```
```  1784         base: "P(k - 1)" and
```
```  1785         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1786   shows "P i"
```
```  1787 apply(rule int_le_induct[of _ "k - 1"])
```
```  1788   using less apply arith
```
```  1789  apply(rule base)
```
```  1790 apply (rule step, simp+)
```
```  1791 done
```
```  1792
```
```  1793 subsection{*Intermediate value theorems*}
```
```  1794
```
```  1795 lemma int_val_lemma:
```
```  1796      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```  1797       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```  1798 unfolding One_nat_def
```
```  1799 apply (induct n, simp)
```
```  1800 apply (intro strip)
```
```  1801 apply (erule impE, simp)
```
```  1802 apply (erule_tac x = n in allE, simp)
```
```  1803 apply (case_tac "k = f (Suc n)")
```
```  1804 apply force
```
```  1805 apply (erule impE)
```
```  1806  apply (simp add: abs_if split add: split_if_asm)
```
```  1807 apply (blast intro: le_SucI)
```
```  1808 done
```
```  1809
```
```  1810 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1811
```
```  1812 lemma nat_intermed_int_val:
```
```  1813      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```  1814          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```  1815 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```  1816        in int_val_lemma)
```
```  1817 unfolding One_nat_def
```
```  1818 apply simp
```
```  1819 apply (erule exE)
```
```  1820 apply (rule_tac x = "i+m" in exI, arith)
```
```  1821 done
```
```  1822
```
```  1823
```
```  1824 subsection{*Products and 1, by T. M. Rasmussen*}
```
```  1825
```
```  1826 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```  1827 by arith
```
```  1828
```
```  1829 lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
```
```  1830 apply (cases "\<bar>n\<bar>=1")
```
```  1831 apply (simp add: abs_mult)
```
```  1832 apply (rule ccontr)
```
```  1833 apply (auto simp add: linorder_neq_iff abs_mult)
```
```  1834 apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
```
```  1835  prefer 2 apply arith
```
```  1836 apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp)
```
```  1837 apply (rule mult_mono, auto)
```
```  1838 done
```
```  1839
```
```  1840 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```  1841 by (insert abs_zmult_eq_1 [of m n], arith)
```
```  1842
```
```  1843 lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
```
```  1844 apply (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1845 apply (simp add: mult_commute [of m])
```
```  1846 apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1847 done
```
```  1848
```
```  1849 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```  1850 apply (rule iffI)
```
```  1851  apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1852  apply (simp add: mult_commute [of m])
```
```  1853  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1854 done
```
```  1855
```
```  1856 (* Could be simplified but Presburger only becomes available too late *)
```
```  1857 lemma infinite_UNIV_int: "~finite(UNIV::int set)"
```
```  1858 proof
```
```  1859   assume "finite(UNIV::int set)"
```
```  1860   moreover have "~(EX i::int. 2*i = 1)"
```
```  1861     by (auto simp: pos_zmult_eq_1_iff)
```
```  1862   ultimately show False using finite_UNIV_inj_surj[of "%n::int. n+n"]
```
```  1863     by (simp add:inj_on_def surj_def) (blast intro:sym)
```
```  1864 qed
```
```  1865
```
```  1866
```
```  1867 subsection {* Integer Powers *}
```
```  1868
```
```  1869 instantiation int :: recpower
```
```  1870 begin
```
```  1871
```
```  1872 primrec power_int where
```
```  1873   "p ^ 0 = (1\<Colon>int)"
```
```  1874   | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
```
```  1875
```
```  1876 instance proof
```
```  1877   fix z :: int
```
```  1878   fix n :: nat
```
```  1879   show "z ^ 0 = 1" by simp
```
```  1880   show "z ^ Suc n = z * (z ^ n)" by simp
```
```  1881 qed
```
```  1882
```
```  1883 declare power_int.simps [simp del]
```
```  1884
```
```  1885 end
```
```  1886
```
```  1887 lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
```
```  1888   by (rule Power.power_add)
```
```  1889
```
```  1890 lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
```
```  1891   by (rule Power.power_mult [symmetric])
```
```  1892
```
```  1893 lemma zero_less_zpower_abs_iff [simp]:
```
```  1894   "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
```
```  1895   by (induct n) (auto simp add: zero_less_mult_iff)
```
```  1896
```
```  1897 lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
```
```  1898   by (induct n) (auto simp add: zero_le_mult_iff)
```
```  1899
```
```  1900 lemma of_int_power:
```
```  1901   "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
```
```  1902   by (induct n) simp_all
```
```  1903
```
```  1904 lemma int_power: "int (m^n) = (int m) ^ n"
```
```  1905   by (rule of_nat_power)
```
```  1906
```
```  1907 lemmas zpower_int = int_power [symmetric]
```
```  1908
```
```  1909
```
```  1910 subsection {* Further theorems on numerals *}
```
```  1911
```
```  1912 subsubsection{*Special Simplification for Constants*}
```
```  1913
```
```  1914 text{*These distributive laws move literals inside sums and differences.*}
```
```  1915
```
```  1916 lemmas left_distrib_number_of [simp] =
```
```  1917   left_distrib [of _ _ "number_of v", standard]
```
```  1918
```
```  1919 lemmas right_distrib_number_of [simp] =
```
```  1920   right_distrib [of "number_of v", standard]
```
```  1921
```
```  1922 lemmas left_diff_distrib_number_of [simp] =
```
```  1923   left_diff_distrib [of _ _ "number_of v", standard]
```
```  1924
```
```  1925 lemmas right_diff_distrib_number_of [simp] =
```
```  1926   right_diff_distrib [of "number_of v", standard]
```
```  1927
```
```  1928 text{*These are actually for fields, like real: but where else to put them?*}
```
```  1929
```
```  1930 lemmas zero_less_divide_iff_number_of [simp, noatp] =
```
```  1931   zero_less_divide_iff [of "number_of w", standard]
```
```  1932
```
```  1933 lemmas divide_less_0_iff_number_of [simp, noatp] =
```
```  1934   divide_less_0_iff [of "number_of w", standard]
```
```  1935
```
```  1936 lemmas zero_le_divide_iff_number_of [simp, noatp] =
```
```  1937   zero_le_divide_iff [of "number_of w", standard]
```
```  1938
```
```  1939 lemmas divide_le_0_iff_number_of [simp, noatp] =
```
```  1940   divide_le_0_iff [of "number_of w", standard]
```
```  1941
```
```  1942
```
```  1943 text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
```
```  1944   strange, but then other simprocs simplify the quotient.*}
```
```  1945
```
```  1946 lemmas inverse_eq_divide_number_of [simp] =
```
```  1947   inverse_eq_divide [of "number_of w", standard]
```
```  1948
```
```  1949 text {*These laws simplify inequalities, moving unary minus from a term
```
```  1950 into the literal.*}
```
```  1951
```
```  1952 lemmas less_minus_iff_number_of [simp, noatp] =
```
```  1953   less_minus_iff [of "number_of v", standard]
```
```  1954
```
```  1955 lemmas le_minus_iff_number_of [simp, noatp] =
```
```  1956   le_minus_iff [of "number_of v", standard]
```
```  1957
```
```  1958 lemmas equation_minus_iff_number_of [simp, noatp] =
```
```  1959   equation_minus_iff [of "number_of v", standard]
```
```  1960
```
```  1961 lemmas minus_less_iff_number_of [simp, noatp] =
```
```  1962   minus_less_iff [of _ "number_of v", standard]
```
```  1963
```
```  1964 lemmas minus_le_iff_number_of [simp, noatp] =
```
```  1965   minus_le_iff [of _ "number_of v", standard]
```
```  1966
```
```  1967 lemmas minus_equation_iff_number_of [simp, noatp] =
```
```  1968   minus_equation_iff [of _ "number_of v", standard]
```
```  1969
```
```  1970
```
```  1971 text{*To Simplify Inequalities Where One Side is the Constant 1*}
```
```  1972
```
```  1973 lemma less_minus_iff_1 [simp,noatp]:
```
```  1974   fixes b::"'b::{ordered_idom,number_ring}"
```
```  1975   shows "(1 < - b) = (b < -1)"
```
```  1976 by auto
```
```  1977
```
```  1978 lemma le_minus_iff_1 [simp,noatp]:
```
```  1979   fixes b::"'b::{ordered_idom,number_ring}"
```
```  1980   shows "(1 \<le> - b) = (b \<le> -1)"
```
```  1981 by auto
```
```  1982
```
```  1983 lemma equation_minus_iff_1 [simp,noatp]:
```
```  1984   fixes b::"'b::number_ring"
```
```  1985   shows "(1 = - b) = (b = -1)"
```
```  1986 by (subst equation_minus_iff, auto)
```
```  1987
```
```  1988 lemma minus_less_iff_1 [simp,noatp]:
```
```  1989   fixes a::"'b::{ordered_idom,number_ring}"
```
```  1990   shows "(- a < 1) = (-1 < a)"
```
```  1991 by auto
```
```  1992
```
```  1993 lemma minus_le_iff_1 [simp,noatp]:
```
```  1994   fixes a::"'b::{ordered_idom,number_ring}"
```
```  1995   shows "(- a \<le> 1) = (-1 \<le> a)"
```
```  1996 by auto
```
```  1997
```
```  1998 lemma minus_equation_iff_1 [simp,noatp]:
```
```  1999   fixes a::"'b::number_ring"
```
```  2000   shows "(- a = 1) = (a = -1)"
```
```  2001 by (subst minus_equation_iff, auto)
```
```  2002
```
```  2003
```
```  2004 text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
```
```  2005
```
```  2006 lemmas mult_less_cancel_left_number_of [simp, noatp] =
```
```  2007   mult_less_cancel_left [of "number_of v", standard]
```
```  2008
```
```  2009 lemmas mult_less_cancel_right_number_of [simp, noatp] =
```
```  2010   mult_less_cancel_right [of _ "number_of v", standard]
```
```  2011
```
```  2012 lemmas mult_le_cancel_left_number_of [simp, noatp] =
```
```  2013   mult_le_cancel_left [of "number_of v", standard]
```
```  2014
```
```  2015 lemmas mult_le_cancel_right_number_of [simp, noatp] =
```
```  2016   mult_le_cancel_right [of _ "number_of v", standard]
```
```  2017
```
```  2018
```
```  2019 text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
```
```  2020
```
```  2021 lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
```
```  2022 lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
```
```  2023 lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
```
```  2024 lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
```
```  2025 lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
```
```  2026 lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
```
```  2027
```
```  2028
```
```  2029 subsubsection{*Optional Simplification Rules Involving Constants*}
```
```  2030
```
```  2031 text{*Simplify quotients that are compared with a literal constant.*}
```
```  2032
```
```  2033 lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
```
```  2034 lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
```
```  2035 lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
```
```  2036 lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
```
```  2037 lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
```
```  2038 lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
```
```  2039
```
```  2040
```
```  2041 text{*Not good as automatic simprules because they cause case splits.*}
```
```  2042 lemmas divide_const_simps =
```
```  2043   le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
```
```  2044   divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
```
```  2045   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
```
```  2046
```
```  2047 text{*Division By @{text "-1"}*}
```
```  2048
```
```  2049 lemma divide_minus1 [simp]:
```
```  2050      "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
```
```  2051 by simp
```
```  2052
```
```  2053 lemma minus1_divide [simp]:
```
```  2054      "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
```
```  2055 by (simp add: divide_inverse inverse_minus_eq)
```
```  2056
```
```  2057 lemma half_gt_zero_iff:
```
```  2058      "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
```
```  2059 by auto
```
```  2060
```
```  2061 lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
```
```  2062
```
```  2063
```
```  2064 subsection {* Configuration of the code generator *}
```
```  2065
```
```  2066 code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
```
```  2067
```
```  2068 lemmas pred_succ_numeral_code [code] =
```
```  2069   pred_bin_simps succ_bin_simps
```
```  2070
```
```  2071 lemmas plus_numeral_code [code] =
```
```  2072   add_bin_simps
```
```  2073   arith_extra_simps(1) [where 'a = int]
```
```  2074
```
```  2075 lemmas minus_numeral_code [code] =
```
```  2076   minus_bin_simps
```
```  2077   arith_extra_simps(2) [where 'a = int]
```
```  2078   arith_extra_simps(5) [where 'a = int]
```
```  2079
```
```  2080 lemmas times_numeral_code [code] =
```
```  2081   mult_bin_simps
```
```  2082   arith_extra_simps(4) [where 'a = int]
```
```  2083
```
```  2084 instantiation int :: eq
```
```  2085 begin
```
```  2086
```
```  2087 definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)"
```
```  2088
```
```  2089 instance by default (simp add: eq_int_def)
```
```  2090
```
```  2091 end
```
```  2092
```
```  2093 lemma eq_number_of_int_code [code]:
```
```  2094   "eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l"
```
```  2095   unfolding eq_int_def number_of_is_id ..
```
```  2096
```
```  2097 lemma eq_int_code [code]:
```
```  2098   "eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True"
```
```  2099   "eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False"
```
```  2100   "eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2"
```
```  2101   "eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
```
```  2102   "eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False"
```
```  2103   "eq_class.eq Int.Min Int.Min \<longleftrightarrow> True"
```
```  2104   "eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
```
```  2105   "eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2"
```
```  2106   "eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls"
```
```  2107   "eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
```
```  2108   "eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
```
```  2109   "eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min"
```
```  2110   "eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2"
```
```  2111   "eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
```
```  2112   "eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
```
```  2113   "eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2"
```
```  2114   unfolding eq_equals by simp_all
```
```  2115
```
```  2116 lemma eq_int_refl [code nbe]:
```
```  2117   "eq_class.eq (k::int) k \<longleftrightarrow> True"
```
```  2118   by (rule HOL.eq_refl)
```
```  2119
```
```  2120 lemma less_eq_number_of_int_code [code]:
```
```  2121   "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
```
```  2122   unfolding number_of_is_id ..
```
```  2123
```
```  2124 lemma less_eq_int_code [code]:
```
```  2125   "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
```
```  2126   "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
```
```  2127   "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2128   "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2129   "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
```
```  2130   "Int.Min \<le> Int.Min \<longleftrightarrow> True"
```
```  2131   "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
```
```  2132   "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
```
```  2133   "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
```
```  2134   "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2135   "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2136   "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2137   "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2138   "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2139   "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2140   "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2141   by simp_all
```
```  2142
```
```  2143 lemma less_number_of_int_code [code]:
```
```  2144   "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
```
```  2145   unfolding number_of_is_id ..
```
```  2146
```
```  2147 lemma less_int_code [code]:
```
```  2148   "Int.Pls < Int.Pls \<longleftrightarrow> False"
```
```  2149   "Int.Pls < Int.Min \<longleftrightarrow> False"
```
```  2150   "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
```
```  2151   "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
```
```  2152   "Int.Min < Int.Pls \<longleftrightarrow> True"
```
```  2153   "Int.Min < Int.Min \<longleftrightarrow> False"
```
```  2154   "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
```
```  2155   "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
```
```  2156   "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2157   "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
```
```  2158   "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
```
```  2159   "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
```
```  2160   "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2161   "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
```
```  2162   "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
```
```  2163   "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
```
```  2164   by simp_all
```
```  2165
```
```  2166 definition
```
```  2167   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
```
```  2168   "nat_aux i n = nat i + n"
```
```  2169
```
```  2170 lemma [code]:
```
```  2171   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
```
```  2172   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
```
```  2173     dest: zless_imp_add1_zle)
```
```  2174
```
```  2175 lemma [code]: "nat i = nat_aux i 0"
```
```  2176   by (simp add: nat_aux_def)
```
```  2177
```
```  2178 hide (open) const nat_aux
```
```  2179
```
```  2180 lemma zero_is_num_zero [code, code inline, symmetric, code post]:
```
```  2181   "(0\<Colon>int) = Numeral0"
```
```  2182   by simp
```
```  2183
```
```  2184 lemma one_is_num_one [code, code inline, symmetric, code post]:
```
```  2185   "(1\<Colon>int) = Numeral1"
```
```  2186   by simp
```
```  2187
```
```  2188 code_modulename SML
```
```  2189   Int Integer
```
```  2190
```
```  2191 code_modulename OCaml
```
```  2192   Int Integer
```
```  2193
```
```  2194 code_modulename Haskell
```
```  2195   Int Integer
```
```  2196
```
```  2197 types_code
```
```  2198   "int" ("int")
```
```  2199 attach (term_of) {*
```
```  2200 val term_of_int = HOLogic.mk_number HOLogic.intT;
```
```  2201 *}
```
```  2202 attach (test) {*
```
```  2203 fun gen_int i =
```
```  2204   let val j = one_of [~1, 1] * random_range 0 i
```
```  2205   in (j, fn () => term_of_int j) end;
```
```  2206 *}
```
```  2207
```
```  2208 setup {*
```
```  2209 let
```
```  2210
```
```  2211 fun strip_number_of (@{term "Int.number_of :: int => int"} \$ t) = t
```
```  2212   | strip_number_of t = t;
```
```  2213
```
```  2214 fun numeral_codegen thy defs dep module b t gr =
```
```  2215   let val i = HOLogic.dest_numeral (strip_number_of t)
```
```  2216   in
```
```  2217     SOME (Codegen.str (string_of_int i),
```
```  2218       snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr))
```
```  2219   end handle TERM _ => NONE;
```
```  2220
```
```  2221 in
```
```  2222
```
```  2223 Codegen.add_codegen "numeral_codegen" numeral_codegen
```
```  2224
```
```  2225 end
```
```  2226 *}
```
```  2227
```
```  2228 consts_code
```
```  2229   "number_of :: int \<Rightarrow> int"    ("(_)")
```
```  2230   "0 :: int"                   ("0")
```
```  2231   "1 :: int"                   ("1")
```
```  2232   "uminus :: int => int"       ("~")
```
```  2233   "op + :: int => int => int"  ("(_ +/ _)")
```
```  2234   "op * :: int => int => int"  ("(_ */ _)")
```
```  2235   "op \<le> :: int => int => bool" ("(_ <=/ _)")
```
```  2236   "op < :: int => int => bool" ("(_ </ _)")
```
```  2237
```
```  2238 quickcheck_params [default_type = int]
```
```  2239
```
```  2240 hide (open) const Pls Min Bit0 Bit1 succ pred
```
```  2241
```
```  2242
```
```  2243 subsection {* Legacy theorems *}
```
```  2244
```
```  2245 lemmas zminus_zminus = minus_minus [of "z::int", standard]
```
```  2246 lemmas zminus_0 = minus_zero [where 'a=int]
```
```  2247 lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
```
```  2248 lemmas zadd_commute = add_commute [of "z::int" "w", standard]
```
```  2249 lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
```
```  2250 lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
```
```  2251 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
```
```  2252 lemmas zmult_ac = OrderedGroup.mult_ac
```
```  2253 lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard]
```
```  2254 lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard]
```
```  2255 lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
```
```  2256 lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
```
```  2257 lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
```
```  2258 lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
```
```  2259 lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
```
```  2260 lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
```
```  2261 lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```  2262 lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```  2263
```
```  2264 lemmas zmult_1 = mult_1_left [of "z::int", standard]
```
```  2265 lemmas zmult_1_right = mult_1_right [of "z::int", standard]
```
```  2266
```
```  2267 lemmas zle_refl = order_refl [of "w::int", standard]
```
```  2268 lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
```
```  2269 lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard]
```
```  2270 lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
```
```  2271 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```  2272
```
```  2273 lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
```
```  2274 lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
```
```  2275 lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
```
```  2276
```
```  2277 lemmas int_0_less_1 = zero_less_one [where 'a=int]
```
```  2278 lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
```
```  2279
```
```  2280 lemmas inj_int = inj_of_nat [where 'a=int]
```
```  2281 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```  2282 lemmas int_mult = of_nat_mult [where 'a=int]
```
```  2283 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```  2284 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
```
```  2285 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```  2286 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
```
```  2287 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```  2288 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```  2289 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
```
```  2290 lemmas int_0 = of_nat_0 [where 'a=int]
```
```  2291 lemmas int_1 = of_nat_1 [where 'a=int]
```
```  2292 lemmas int_Suc = of_nat_Suc [where 'a=int]
```
```  2293 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
```
```  2294 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```  2295 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```  2296 lemmas zless_le = less_int_def
```
```  2297 lemmas int_eq_of_nat = TrueI
```
```  2298
```
```  2299 end
```