src/HOL/Int.thy
 author wenzelm Fri Mar 28 19:43:54 2008 +0100 (2008-03-28) changeset 26462 dac4e2bce00d parent 26300 03def556e26e child 26507 6da615cef733 permissions -rw-r--r--
avoid rebinding of existing facts;
```     1 (*  Title:      Int.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4                 Tobias Nipkow, Florian Haftmann, TU Muenchen
```
```     5     Copyright   1994  University of Cambridge
```
```     6
```
```     7 *)
```
```     8
```
```     9 header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *}
```
```    10
```
```    11 theory Int
```
```    12 imports Equiv_Relations Nat Wellfounded_Relations
```
```    13 uses
```
```    14   ("Tools/numeral.ML")
```
```    15   ("Tools/numeral_syntax.ML")
```
```    16   ("~~/src/Provers/Arith/assoc_fold.ML")
```
```    17   "~~/src/Provers/Arith/cancel_numerals.ML"
```
```    18   "~~/src/Provers/Arith/combine_numerals.ML"
```
```    19   ("int_arith1.ML")
```
```    20 begin
```
```    21
```
```    22 subsection {* The equivalence relation underlying the integers *}
```
```    23
```
```    24 definition
```
```    25   intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
```
```    26 where
```
```    27   "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
```
```    28
```
```    29 typedef (Integ)
```
```    30   int = "UNIV//intrel"
```
```    31   by (auto simp add: quotient_def)
```
```    32
```
```    33 instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
```
```    34 begin
```
```    35
```
```    36 definition
```
```    37   Zero_int_def [code func del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
```
```    38
```
```    39 definition
```
```    40   One_int_def [code func del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
```
```    41
```
```    42 definition
```
```    43   add_int_def [code func del]: "z + w = Abs_Integ
```
```    44     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
```
```    45       intrel `` {(x + u, y + v)})"
```
```    46
```
```    47 definition
```
```    48   minus_int_def [code func del]:
```
```    49     "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
```
```    50
```
```    51 definition
```
```    52   diff_int_def [code func del]:  "z - w = z + (-w \<Colon> int)"
```
```    53
```
```    54 definition
```
```    55   mult_int_def [code func del]: "z * w = Abs_Integ
```
```    56     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
```
```    57       intrel `` {(x*u + y*v, x*v + y*u)})"
```
```    58
```
```    59 definition
```
```    60   le_int_def [code func del]:
```
```    61    "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)"
```
```    62
```
```    63 definition
```
```    64   less_int_def [code func del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```    65
```
```    66 definition
```
```    67   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
```
```    68
```
```    69 definition
```
```    70   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```    71
```
```    72 instance ..
```
```    73
```
```    74 end
```
```    75
```
```    76
```
```    77 subsection{*Construction of the Integers*}
```
```    78
```
```    79 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
```
```    80 by (simp add: intrel_def)
```
```    81
```
```    82 lemma equiv_intrel: "equiv UNIV intrel"
```
```    83 by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
```
```    84
```
```    85 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
```
```    86   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
```
```    87 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
```
```    88
```
```    89 text{*All equivalence classes belong to set of representatives*}
```
```    90 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
```
```    91 by (auto simp add: Integ_def intrel_def quotient_def)
```
```    92
```
```    93 text{*Reduces equality on abstractions to equality on representatives:
```
```    94   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
```
```    95 declare Abs_Integ_inject [simp,noatp]  Abs_Integ_inverse [simp,noatp]
```
```    96
```
```    97 text{*Case analysis on the representation of an integer as an equivalence
```
```    98       class of pairs of naturals.*}
```
```    99 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
```
```   100      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
```
```   101 apply (rule Abs_Integ_cases [of z])
```
```   102 apply (auto simp add: Integ_def quotient_def)
```
```   103 done
```
```   104
```
```   105
```
```   106 subsection {* Arithmetic Operations *}
```
```   107
```
```   108 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
```
```   109 proof -
```
```   110   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
```
```   111     by (simp add: congruent_def)
```
```   112   thus ?thesis
```
```   113     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
```
```   114 qed
```
```   115
```
```   116 lemma add:
```
```   117      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
```
```   118       Abs_Integ (intrel``{(x+u, y+v)})"
```
```   119 proof -
```
```   120   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
```
```   121         respects2 intrel"
```
```   122     by (simp add: congruent2_def)
```
```   123   thus ?thesis
```
```   124     by (simp add: add_int_def UN_UN_split_split_eq
```
```   125                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   126 qed
```
```   127
```
```   128 text{*Congruence property for multiplication*}
```
```   129 lemma mult_congruent2:
```
```   130      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
```
```   131       respects2 intrel"
```
```   132 apply (rule equiv_intrel [THEN congruent2_commuteI])
```
```   133  apply (force simp add: mult_ac, clarify)
```
```   134 apply (simp add: congruent_def mult_ac)
```
```   135 apply (rename_tac u v w x y z)
```
```   136 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
```
```   137 apply (simp add: mult_ac)
```
```   138 apply (simp add: add_mult_distrib [symmetric])
```
```   139 done
```
```   140
```
```   141 lemma mult:
```
```   142      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
```
```   143       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
```
```   144 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
```
```   145               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
```
```   146
```
```   147 text{*The integers form a @{text comm_ring_1}*}
```
```   148 instance int :: comm_ring_1
```
```   149 proof
```
```   150   fix i j k :: int
```
```   151   show "(i + j) + k = i + (j + k)"
```
```   152     by (cases i, cases j, cases k) (simp add: add add_assoc)
```
```   153   show "i + j = j + i"
```
```   154     by (cases i, cases j) (simp add: add_ac add)
```
```   155   show "0 + i = i"
```
```   156     by (cases i) (simp add: Zero_int_def add)
```
```   157   show "- i + i = 0"
```
```   158     by (cases i) (simp add: Zero_int_def minus add)
```
```   159   show "i - j = i + - j"
```
```   160     by (simp add: diff_int_def)
```
```   161   show "(i * j) * k = i * (j * k)"
```
```   162     by (cases i, cases j, cases k) (simp add: mult ring_simps)
```
```   163   show "i * j = j * i"
```
```   164     by (cases i, cases j) (simp add: mult ring_simps)
```
```   165   show "1 * i = i"
```
```   166     by (cases i) (simp add: One_int_def mult)
```
```   167   show "(i + j) * k = i * k + j * k"
```
```   168     by (cases i, cases j, cases k) (simp add: add mult ring_simps)
```
```   169   show "0 \<noteq> (1::int)"
```
```   170     by (simp add: Zero_int_def One_int_def)
```
```   171 qed
```
```   172
```
```   173 lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
```
```   174 by (induct m, simp_all add: Zero_int_def One_int_def add)
```
```   175
```
```   176
```
```   177 subsection {* The @{text "\<le>"} Ordering *}
```
```   178
```
```   179 lemma le:
```
```   180   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
```
```   181 by (force simp add: le_int_def)
```
```   182
```
```   183 lemma less:
```
```   184   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
```
```   185 by (simp add: less_int_def le order_less_le)
```
```   186
```
```   187 instance int :: linorder
```
```   188 proof
```
```   189   fix i j k :: int
```
```   190   show "(i < j) = (i \<le> j \<and> i \<noteq> j)"
```
```   191     by (simp add: less_int_def)
```
```   192   show "i \<le> i"
```
```   193     by (cases i) (simp add: le)
```
```   194   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
```
```   195     by (cases i, cases j, cases k) (simp add: le)
```
```   196   show "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
```
```   197     by (cases i, cases j) (simp add: le)
```
```   198   show "i \<le> j \<or> j \<le> i"
```
```   199     by (cases i, cases j) (simp add: le linorder_linear)
```
```   200 qed
```
```   201
```
```   202 instantiation int :: distrib_lattice
```
```   203 begin
```
```   204
```
```   205 definition
```
```   206   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
```
```   207
```
```   208 definition
```
```   209   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
```
```   210
```
```   211 instance
```
```   212   by intro_classes
```
```   213     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
```
```   214
```
```   215 end
```
```   216
```
```   217 instance int :: pordered_cancel_ab_semigroup_add
```
```   218 proof
```
```   219   fix i j k :: int
```
```   220   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
```
```   221     by (cases i, cases j, cases k) (simp add: le add)
```
```   222 qed
```
```   223
```
```   224
```
```   225 text{*Strict Monotonicity of Multiplication*}
```
```   226
```
```   227 text{*strict, in 1st argument; proof is by induction on k>0*}
```
```   228 lemma zmult_zless_mono2_lemma:
```
```   229      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
```
```   230 apply (induct "k", simp)
```
```   231 apply (simp add: left_distrib)
```
```   232 apply (case_tac "k=0")
```
```   233 apply (simp_all add: add_strict_mono)
```
```   234 done
```
```   235
```
```   236 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
```
```   237 apply (cases k)
```
```   238 apply (auto simp add: le add int_def Zero_int_def)
```
```   239 apply (rule_tac x="x-y" in exI, simp)
```
```   240 done
```
```   241
```
```   242 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
```
```   243 apply (cases k)
```
```   244 apply (simp add: less int_def Zero_int_def)
```
```   245 apply (rule_tac x="x-y" in exI, simp)
```
```   246 done
```
```   247
```
```   248 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
```
```   249 apply (drule zero_less_imp_eq_int)
```
```   250 apply (auto simp add: zmult_zless_mono2_lemma)
```
```   251 done
```
```   252
```
```   253 text{*The integers form an ordered integral domain*}
```
```   254 instance int :: ordered_idom
```
```   255 proof
```
```   256   fix i j k :: int
```
```   257   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
```
```   258     by (rule zmult_zless_mono2)
```
```   259   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
```
```   260     by (simp only: zabs_def)
```
```   261   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
```
```   262     by (simp only: zsgn_def)
```
```   263 qed
```
```   264
```
```   265 instance int :: lordered_ring
```
```   266 proof
```
```   267   fix k :: int
```
```   268   show "abs k = sup k (- k)"
```
```   269     by (auto simp add: sup_int_def zabs_def max_def less_minus_self_iff [symmetric])
```
```   270 qed
```
```   271
```
```   272 lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
```
```   273 apply (cases w, cases z)
```
```   274 apply (simp add: less le add One_int_def)
```
```   275 done
```
```   276
```
```   277 lemma zless_iff_Suc_zadd:
```
```   278   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
```
```   279 apply (cases z, cases w)
```
```   280 apply (auto simp add: less add int_def)
```
```   281 apply (rename_tac a b c d)
```
```   282 apply (rule_tac x="a+d - Suc(c+b)" in exI)
```
```   283 apply arith
```
```   284 done
```
```   285
```
```   286 lemmas int_distrib =
```
```   287   left_distrib [of "z1::int" "z2" "w", standard]
```
```   288   right_distrib [of "w::int" "z1" "z2", standard]
```
```   289   left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```   290   right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```   291
```
```   292
```
```   293 subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
```
```   294
```
```   295 context ring_1
```
```   296 begin
```
```   297
```
```   298 definition
```
```   299   of_int :: "int \<Rightarrow> 'a"
```
```   300 where
```
```   301   [code func del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
```
```   302
```
```   303 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
```
```   304 proof -
```
```   305   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
```
```   306     by (simp add: congruent_def compare_rls of_nat_add [symmetric]
```
```   307             del: of_nat_add)
```
```   308   thus ?thesis
```
```   309     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
```
```   310 qed
```
```   311
```
```   312 lemma of_int_0 [simp]: "of_int 0 = 0"
```
```   313   by (simp add: of_int Zero_int_def)
```
```   314
```
```   315 lemma of_int_1 [simp]: "of_int 1 = 1"
```
```   316   by (simp add: of_int One_int_def)
```
```   317
```
```   318 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
```
```   319   by (cases w, cases z, simp add: compare_rls of_int OrderedGroup.compare_rls add)
```
```   320
```
```   321 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
```
```   322   by (cases z, simp add: compare_rls of_int minus)
```
```   323
```
```   324 lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
```
```   325   by (simp add: OrderedGroup.diff_minus diff_minus)
```
```   326
```
```   327 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
```
```   328 apply (cases w, cases z)
```
```   329 apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
```
```   330                  mult add_ac of_nat_mult)
```
```   331 done
```
```   332
```
```   333 text{*Collapse nested embeddings*}
```
```   334 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
```
```   335   by (induct n) auto
```
```   336
```
```   337 end
```
```   338
```
```   339 context ordered_idom
```
```   340 begin
```
```   341
```
```   342 lemma of_int_le_iff [simp]:
```
```   343   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
```
```   344   by (cases w, cases z, simp add: of_int le minus compare_rls of_nat_add [symmetric] del: of_nat_add)
```
```   345
```
```   346 text{*Special cases where either operand is zero*}
```
```   347 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
```
```   348 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
```
```   349
```
```   350 lemma of_int_less_iff [simp]:
```
```   351   "of_int w < of_int z \<longleftrightarrow> w < z"
```
```   352   by (simp add: not_le [symmetric] linorder_not_le [symmetric])
```
```   353
```
```   354 text{*Special cases where either operand is zero*}
```
```   355 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
```
```   356 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
```
```   357
```
```   358 end
```
```   359
```
```   360 text{*Class for unital rings with characteristic zero.
```
```   361  Includes non-ordered rings like the complex numbers.*}
```
```   362 class ring_char_0 = ring_1 + semiring_char_0
```
```   363 begin
```
```   364
```
```   365 lemma of_int_eq_iff [simp]:
```
```   366    "of_int w = of_int z \<longleftrightarrow> w = z"
```
```   367 apply (cases w, cases z, simp add: of_int)
```
```   368 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
```
```   369 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
```
```   370 done
```
```   371
```
```   372 text{*Special cases where either operand is zero*}
```
```   373 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
```
```   374 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
```
```   375
```
```   376 end
```
```   377
```
```   378 text{*Every @{text ordered_idom} has characteristic zero.*}
```
```   379 subclass (in ordered_idom) ring_char_0 by intro_locales
```
```   380
```
```   381 lemma of_int_eq_id [simp]: "of_int = id"
```
```   382 proof
```
```   383   fix z show "of_int z = id z"
```
```   384     by (cases z) (simp add: of_int add minus int_def diff_minus)
```
```   385 qed
```
```   386
```
```   387
```
```   388 subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
```
```   389
```
```   390 definition
```
```   391   nat :: "int \<Rightarrow> nat"
```
```   392 where
```
```   393   [code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
```
```   394
```
```   395 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
```
```   396 proof -
```
```   397   have "(\<lambda>(x,y). {x-y}) respects intrel"
```
```   398     by (simp add: congruent_def) arith
```
```   399   thus ?thesis
```
```   400     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
```
```   401 qed
```
```   402
```
```   403 lemma nat_int [simp]: "nat (of_nat n) = n"
```
```   404 by (simp add: nat int_def)
```
```   405
```
```   406 lemma nat_zero [simp]: "nat 0 = 0"
```
```   407 by (simp add: Zero_int_def nat)
```
```   408
```
```   409 lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
```
```   410 by (cases z, simp add: nat le int_def Zero_int_def)
```
```   411
```
```   412 corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
```
```   413 by simp
```
```   414
```
```   415 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
```
```   416 by (cases z, simp add: nat le Zero_int_def)
```
```   417
```
```   418 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
```
```   419 apply (cases w, cases z)
```
```   420 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
```
```   421 done
```
```   422
```
```   423 text{*An alternative condition is @{term "0 \<le> w"} *}
```
```   424 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
```
```   425 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   426
```
```   427 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
```
```   428 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
```
```   429
```
```   430 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
```
```   431 apply (cases w, cases z)
```
```   432 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
```
```   433 done
```
```   434
```
```   435 lemma nonneg_eq_int:
```
```   436   fixes z :: int
```
```   437   assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
```
```   438   shows P
```
```   439   using assms by (blast dest: nat_0_le sym)
```
```   440
```
```   441 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   442 by (cases w, simp add: nat le int_def Zero_int_def, arith)
```
```   443
```
```   444 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
```
```   445 by (simp only: eq_commute [of m] nat_eq_iff)
```
```   446
```
```   447 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
```
```   448 apply (cases w)
```
```   449 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
```
```   450 done
```
```   451
```
```   452 lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
```
```   453 by (auto simp add: nat_eq_iff2)
```
```   454
```
```   455 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
```
```   456 by (insert zless_nat_conj [of 0], auto)
```
```   457
```
```   458 lemma nat_add_distrib:
```
```   459      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
```
```   460 by (cases z, cases z', simp add: nat add le Zero_int_def)
```
```   461
```
```   462 lemma nat_diff_distrib:
```
```   463      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
```
```   464 by (cases z, cases z',
```
```   465     simp add: nat add minus diff_minus le Zero_int_def)
```
```   466
```
```   467 lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
```
```   468 by (simp add: int_def minus nat Zero_int_def)
```
```   469
```
```   470 lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
```
```   471 by (cases z, simp add: nat less int_def, arith)
```
```   472
```
```   473 context ring_1
```
```   474 begin
```
```   475
```
```   476 lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
```
```   477   by (cases z rule: eq_Abs_Integ)
```
```   478    (simp add: nat le of_int Zero_int_def of_nat_diff)
```
```   479
```
```   480 end
```
```   481
```
```   482
```
```   483 subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
```
```   484
```
```   485 lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
```
```   486 by (simp add: order_less_le del: of_nat_Suc)
```
```   487
```
```   488 lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
```
```   489 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
```
```   490
```
```   491 lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
```
```   492 by (simp add: minus_le_iff)
```
```   493
```
```   494 lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
```
```   495 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
```
```   496
```
```   497 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
```
```   498 by (subst le_minus_iff, simp del: of_nat_Suc)
```
```   499
```
```   500 lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
```
```   501 by (simp add: int_def le minus Zero_int_def)
```
```   502
```
```   503 lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
```
```   504 by (simp add: linorder_not_less)
```
```   505
```
```   506 lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
```
```   507 by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
```
```   508
```
```   509 lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
```
```   510 proof -
```
```   511   have "(w \<le> z) = (0 \<le> z - w)"
```
```   512     by (simp only: le_diff_eq add_0_left)
```
```   513   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
```
```   514     by (auto elim: zero_le_imp_eq_int)
```
```   515   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
```
```   516     by (simp only: group_simps)
```
```   517   finally show ?thesis .
```
```   518 qed
```
```   519
```
```   520 lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
```
```   521 by simp
```
```   522
```
```   523 lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
```
```   524 by simp
```
```   525
```
```   526 text{*This version is proved for all ordered rings, not just integers!
```
```   527       It is proved here because attribute @{text arith_split} is not available
```
```   528       in theory @{text Ring_and_Field}.
```
```   529       But is it really better than just rewriting with @{text abs_if}?*}
```
```   530 lemma abs_split [arith_split,noatp]:
```
```   531      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
```
```   532 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
```
```   533
```
```   534 lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
```
```   535 apply (cases x)
```
```   536 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
```
```   537 apply (rule_tac x="y - Suc x" in exI, arith)
```
```   538 done
```
```   539
```
```   540
```
```   541 subsection {* Cases and induction *}
```
```   542
```
```   543 text{*Now we replace the case analysis rule by a more conventional one:
```
```   544 whether an integer is negative or not.*}
```
```   545
```
```   546 theorem int_cases [cases type: int, case_names nonneg neg]:
```
```   547   "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
```
```   548 apply (cases "z < 0", blast dest!: negD)
```
```   549 apply (simp add: linorder_not_less del: of_nat_Suc)
```
```   550 apply auto
```
```   551 apply (blast dest: nat_0_le [THEN sym])
```
```   552 done
```
```   553
```
```   554 theorem int_induct [induct type: int, case_names nonneg neg]:
```
```   555      "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
```
```   556   by (cases z rule: int_cases) auto
```
```   557
```
```   558 text{*Contributed by Brian Huffman*}
```
```   559 theorem int_diff_cases:
```
```   560   obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
```
```   561 apply (cases z rule: eq_Abs_Integ)
```
```   562 apply (rule_tac m=x and n=y in diff)
```
```   563 apply (simp add: int_def diff_def minus add)
```
```   564 done
```
```   565
```
```   566
```
```   567 subsection {* Binary representation *}
```
```   568
```
```   569 text {*
```
```   570   This formalization defines binary arithmetic in terms of the integers
```
```   571   rather than using a datatype. This avoids multiple representations (leading
```
```   572   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
```
```   573   int_of_binary}, for the numerical interpretation.
```
```   574
```
```   575   The representation expects that @{text "(m mod 2)"} is 0 or 1,
```
```   576   even if m is negative;
```
```   577   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
```
```   578   @{text "-5 = (-3)*2 + 1"}.
```
```   579
```
```   580   This two's complement binary representation derives from the paper
```
```   581   "An Efficient Representation of Arithmetic for Term Rewriting" by
```
```   582   Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
```
```   583   Springer LNCS 488 (240-251), 1991.
```
```   584 *}
```
```   585
```
```   586 definition
```
```   587   Pls :: int where
```
```   588   [code func del]: "Pls = 0"
```
```   589
```
```   590 definition
```
```   591   Min :: int where
```
```   592   [code func del]: "Min = - 1"
```
```   593
```
```   594 definition
```
```   595   Bit0 :: "int \<Rightarrow> int" where
```
```   596   [code func del]: "Bit0 k = k + k"
```
```   597
```
```   598 definition
```
```   599   Bit1 :: "int \<Rightarrow> int" where
```
```   600   [code func del]: "Bit1 k = 1 + k + k"
```
```   601
```
```   602 class number = type + -- {* for numeric types: nat, int, real, \dots *}
```
```   603   fixes number_of :: "int \<Rightarrow> 'a"
```
```   604
```
```   605 use "Tools/numeral.ML"
```
```   606
```
```   607 syntax
```
```   608   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```   609
```
```   610 use "Tools/numeral_syntax.ML"
```
```   611 setup NumeralSyntax.setup
```
```   612
```
```   613 abbreviation
```
```   614   "Numeral0 \<equiv> number_of Pls"
```
```   615
```
```   616 abbreviation
```
```   617   "Numeral1 \<equiv> number_of (Bit1 Pls)"
```
```   618
```
```   619 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
```
```   620   -- {* Unfold all @{text let}s involving constants *}
```
```   621   unfolding Let_def ..
```
```   622
```
```   623 definition
```
```   624   succ :: "int \<Rightarrow> int" where
```
```   625   [code func del]: "succ k = k + 1"
```
```   626
```
```   627 definition
```
```   628   pred :: "int \<Rightarrow> int" where
```
```   629   [code func del]: "pred k = k - 1"
```
```   630
```
```   631 lemmas
```
```   632   max_number_of [simp] = max_def
```
```   633     [of "number_of u" "number_of v", standard, simp]
```
```   634 and
```
```   635   min_number_of [simp] = min_def
```
```   636     [of "number_of u" "number_of v", standard, simp]
```
```   637   -- {* unfolding @{text minx} and @{text max} on numerals *}
```
```   638
```
```   639 lemmas numeral_simps =
```
```   640   succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
```
```   641
```
```   642 text {* Removal of leading zeroes *}
```
```   643
```
```   644 lemma Bit0_Pls [simp, code post]:
```
```   645   "Bit0 Pls = Pls"
```
```   646   unfolding numeral_simps by simp
```
```   647
```
```   648 lemma Bit1_Min [simp, code post]:
```
```   649   "Bit1 Min = Min"
```
```   650   unfolding numeral_simps by simp
```
```   651
```
```   652 lemmas normalize_bin_simps =
```
```   653   Bit0_Pls Bit1_Min
```
```   654
```
```   655
```
```   656 subsection {* The Functions @{term succ}, @{term pred} and @{term uminus} *}
```
```   657
```
```   658 lemma succ_Pls [simp]:
```
```   659   "succ Pls = Bit1 Pls"
```
```   660   unfolding numeral_simps by simp
```
```   661
```
```   662 lemma succ_Min [simp]:
```
```   663   "succ Min = Pls"
```
```   664   unfolding numeral_simps by simp
```
```   665
```
```   666 lemma succ_Bit0 [simp]:
```
```   667   "succ (Bit0 k) = Bit1 k"
```
```   668   unfolding numeral_simps by simp
```
```   669
```
```   670 lemma succ_Bit1 [simp]:
```
```   671   "succ (Bit1 k) = Bit0 (succ k)"
```
```   672   unfolding numeral_simps by simp
```
```   673
```
```   674 lemmas succ_bin_simps =
```
```   675   succ_Pls succ_Min succ_Bit0 succ_Bit1
```
```   676
```
```   677 lemma pred_Pls [simp]:
```
```   678   "pred Pls = Min"
```
```   679   unfolding numeral_simps by simp
```
```   680
```
```   681 lemma pred_Min [simp]:
```
```   682   "pred Min = Bit0 Min"
```
```   683   unfolding numeral_simps by simp
```
```   684
```
```   685 lemma pred_Bit0 [simp]:
```
```   686   "pred (Bit0 k) = Bit1 (pred k)"
```
```   687   unfolding numeral_simps by simp
```
```   688
```
```   689 lemma pred_Bit1 [simp]:
```
```   690   "pred (Bit1 k) = Bit0 k"
```
```   691   unfolding numeral_simps by simp
```
```   692
```
```   693 lemmas pred_bin_simps =
```
```   694   pred_Pls pred_Min pred_Bit0 pred_Bit1
```
```   695
```
```   696 lemma minus_Pls [simp]:
```
```   697   "- Pls = Pls"
```
```   698   unfolding numeral_simps by simp
```
```   699
```
```   700 lemma minus_Min [simp]:
```
```   701   "- Min = Bit1 Pls"
```
```   702   unfolding numeral_simps by simp
```
```   703
```
```   704 lemma minus_Bit0 [simp]:
```
```   705   "- (Bit0 k) = Bit0 (- k)"
```
```   706   unfolding numeral_simps by simp
```
```   707
```
```   708 lemma minus_Bit1 [simp]:
```
```   709   "- (Bit1 k) = Bit1 (pred (- k))"
```
```   710   unfolding numeral_simps by simp
```
```   711
```
```   712 lemmas minus_bin_simps =
```
```   713   minus_Pls minus_Min minus_Bit0 minus_Bit1
```
```   714
```
```   715
```
```   716 subsection {*
```
```   717   Binary Addition and Multiplication: @{term "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   718     and @{term "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   719 *}
```
```   720
```
```   721 lemma add_Pls [simp]:
```
```   722   "Pls + k = k"
```
```   723   unfolding numeral_simps by simp
```
```   724
```
```   725 lemma add_Min [simp]:
```
```   726   "Min + k = pred k"
```
```   727   unfolding numeral_simps by simp
```
```   728
```
```   729 lemma add_Bit0_Bit0 [simp]:
```
```   730   "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
```
```   731   unfolding numeral_simps by simp_all
```
```   732
```
```   733 lemma add_Bit0_Bit1 [simp]:
```
```   734   "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
```
```   735   unfolding numeral_simps by simp_all
```
```   736
```
```   737 lemma add_Bit1_Bit0 [simp]:
```
```   738   "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
```
```   739   unfolding numeral_simps by simp
```
```   740
```
```   741 lemma add_Bit1_Bit1 [simp]:
```
```   742   "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
```
```   743   unfolding numeral_simps by simp
```
```   744
```
```   745 lemma add_Pls_right [simp]:
```
```   746   "k + Pls = k"
```
```   747   unfolding numeral_simps by simp
```
```   748
```
```   749 lemma add_Min_right [simp]:
```
```   750   "k + Min = pred k"
```
```   751   unfolding numeral_simps by simp
```
```   752
```
```   753 lemmas add_bin_simps =
```
```   754   add_Pls add_Min add_Pls_right add_Min_right
```
```   755   add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
```
```   756
```
```   757 lemma mult_Pls [simp]:
```
```   758   "Pls * w = Pls"
```
```   759   unfolding numeral_simps by simp
```
```   760
```
```   761 lemma mult_Min [simp]:
```
```   762   "Min * k = - k"
```
```   763   unfolding numeral_simps by simp
```
```   764
```
```   765 lemma mult_Bit0 [simp]:
```
```   766   "(Bit0 k) * l = Bit0 (k * l)"
```
```   767   unfolding numeral_simps int_distrib by simp
```
```   768
```
```   769 lemma mult_Bit1 [simp]:
```
```   770   "(Bit1 k) * l = (Bit0 (k * l)) + l"
```
```   771   unfolding numeral_simps int_distrib by simp
```
```   772
```
```   773 lemmas mult_bin_simps =
```
```   774   mult_Pls mult_Min mult_Bit0 mult_Bit1
```
```   775
```
```   776
```
```   777 subsection {* Converting Numerals to Rings: @{term number_of} *}
```
```   778
```
```   779 class number_ring = number + comm_ring_1 +
```
```   780   assumes number_of_eq: "number_of k = of_int k"
```
```   781
```
```   782 text {* self-embedding of the integers *}
```
```   783
```
```   784 instantiation int :: number_ring
```
```   785 begin
```
```   786
```
```   787 definition
```
```   788   int_number_of_def [code func del]: "number_of w = (of_int w \<Colon> int)"
```
```   789
```
```   790 instance
```
```   791   by intro_classes (simp only: int_number_of_def)
```
```   792
```
```   793 end
```
```   794
```
```   795 lemma number_of_is_id:
```
```   796   "number_of (k::int) = k"
```
```   797   unfolding int_number_of_def by simp
```
```   798
```
```   799 lemma number_of_succ:
```
```   800   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
```
```   801   unfolding number_of_eq numeral_simps by simp
```
```   802
```
```   803 lemma number_of_pred:
```
```   804   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
```
```   805   unfolding number_of_eq numeral_simps by simp
```
```   806
```
```   807 lemma number_of_minus:
```
```   808   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
```
```   809   unfolding number_of_eq numeral_simps by simp
```
```   810
```
```   811 lemma number_of_add:
```
```   812   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
```
```   813   unfolding number_of_eq numeral_simps by simp
```
```   814
```
```   815 lemma number_of_mult:
```
```   816   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
```
```   817   unfolding number_of_eq numeral_simps by simp
```
```   818
```
```   819 text {*
```
```   820   The correctness of shifting.
```
```   821   But it doesn't seem to give a measurable speed-up.
```
```   822 *}
```
```   823
```
```   824 lemma double_number_of_Bit0:
```
```   825   "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
```
```   826   unfolding number_of_eq numeral_simps left_distrib by simp
```
```   827
```
```   828 text {*
```
```   829   Converting numerals 0 and 1 to their abstract versions.
```
```   830 *}
```
```   831
```
```   832 lemma numeral_0_eq_0 [simp]:
```
```   833   "Numeral0 = (0::'a::number_ring)"
```
```   834   unfolding number_of_eq numeral_simps by simp
```
```   835
```
```   836 lemma numeral_1_eq_1 [simp]:
```
```   837   "Numeral1 = (1::'a::number_ring)"
```
```   838   unfolding number_of_eq numeral_simps by simp
```
```   839
```
```   840 text {*
```
```   841   Special-case simplification for small constants.
```
```   842 *}
```
```   843
```
```   844 text{*
```
```   845   Unary minus for the abstract constant 1. Cannot be inserted
```
```   846   as a simprule until later: it is @{text number_of_Min} re-oriented!
```
```   847 *}
```
```   848
```
```   849 lemma numeral_m1_eq_minus_1:
```
```   850   "(-1::'a::number_ring) = - 1"
```
```   851   unfolding number_of_eq numeral_simps by simp
```
```   852
```
```   853 lemma mult_minus1 [simp]:
```
```   854   "-1 * z = -(z::'a::number_ring)"
```
```   855   unfolding number_of_eq numeral_simps by simp
```
```   856
```
```   857 lemma mult_minus1_right [simp]:
```
```   858   "z * -1 = -(z::'a::number_ring)"
```
```   859   unfolding number_of_eq numeral_simps by simp
```
```   860
```
```   861 (*Negation of a coefficient*)
```
```   862 lemma minus_number_of_mult [simp]:
```
```   863    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
```
```   864    unfolding number_of_eq by simp
```
```   865
```
```   866 text {* Subtraction *}
```
```   867
```
```   868 lemma diff_number_of_eq:
```
```   869   "number_of v - number_of w =
```
```   870     (number_of (v + uminus w)::'a::number_ring)"
```
```   871   unfolding number_of_eq by simp
```
```   872
```
```   873 lemma number_of_Pls:
```
```   874   "number_of Pls = (0::'a::number_ring)"
```
```   875   unfolding number_of_eq numeral_simps by simp
```
```   876
```
```   877 lemma number_of_Min:
```
```   878   "number_of Min = (- 1::'a::number_ring)"
```
```   879   unfolding number_of_eq numeral_simps by simp
```
```   880
```
```   881 lemma number_of_Bit0:
```
```   882   "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
```
```   883   unfolding number_of_eq numeral_simps by simp
```
```   884
```
```   885 lemma number_of_Bit1:
```
```   886   "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
```
```   887   unfolding number_of_eq numeral_simps by simp
```
```   888
```
```   889
```
```   890 subsection {* Equality of Binary Numbers *}
```
```   891
```
```   892 text {* First version by Norbert Voelker *}
```
```   893
```
```   894 definition
```
```   895   neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
```
```   896 where
```
```   897   "neg Z \<longleftrightarrow> Z < 0"
```
```   898
```
```   899 definition (*for simplifying equalities*)
```
```   900   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
```
```   901 where
```
```   902   "iszero z \<longleftrightarrow> z = 0"
```
```   903
```
```   904 lemma not_neg_int [simp]: "~ neg (of_nat n)"
```
```   905 by (simp add: neg_def)
```
```   906
```
```   907 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
```
```   908 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
```
```   909
```
```   910 lemmas neg_eq_less_0 = neg_def
```
```   911
```
```   912 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
```
```   913 by (simp add: neg_def linorder_not_less)
```
```   914
```
```   915 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
```
```   916
```
```   917 lemma not_neg_0: "~ neg 0"
```
```   918 by (simp add: One_int_def neg_def)
```
```   919
```
```   920 lemma not_neg_1: "~ neg 1"
```
```   921 by (simp add: neg_def linorder_not_less zero_le_one)
```
```   922
```
```   923 lemma iszero_0: "iszero 0"
```
```   924 by (simp add: iszero_def)
```
```   925
```
```   926 lemma not_iszero_1: "~ iszero 1"
```
```   927 by (simp add: iszero_def eq_commute)
```
```   928
```
```   929 lemma neg_nat: "neg z ==> nat z = 0"
```
```   930 by (simp add: neg_def order_less_imp_le)
```
```   931
```
```   932 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
```
```   933 by (simp add: linorder_not_less neg_def)
```
```   934
```
```   935 lemma eq_number_of_eq:
```
```   936   "((number_of x::'a::number_ring) = number_of y) =
```
```   937    iszero (number_of (x + uminus y) :: 'a)"
```
```   938   unfolding iszero_def number_of_add number_of_minus
```
```   939   by (simp add: compare_rls)
```
```   940
```
```   941 lemma iszero_number_of_Pls:
```
```   942   "iszero ((number_of Pls)::'a::number_ring)"
```
```   943   unfolding iszero_def numeral_0_eq_0 ..
```
```   944
```
```   945 lemma nonzero_number_of_Min:
```
```   946   "~ iszero ((number_of Min)::'a::number_ring)"
```
```   947   unfolding iszero_def numeral_m1_eq_minus_1 by simp
```
```   948
```
```   949
```
```   950 subsection {* Comparisons, for Ordered Rings *}
```
```   951
```
```   952 lemmas double_eq_0_iff = double_zero
```
```   953
```
```   954 lemma le_imp_0_less:
```
```   955   assumes le: "0 \<le> z"
```
```   956   shows "(0::int) < 1 + z"
```
```   957 proof -
```
```   958   have "0 \<le> z" by fact
```
```   959   also have "... < z + 1" by (rule less_add_one)
```
```   960   also have "... = 1 + z" by (simp add: add_ac)
```
```   961   finally show "0 < 1 + z" .
```
```   962 qed
```
```   963
```
```   964 lemma odd_nonzero:
```
```   965   "1 + z + z \<noteq> (0::int)";
```
```   966 proof (cases z rule: int_cases)
```
```   967   case (nonneg n)
```
```   968   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```   969   thus ?thesis using  le_imp_0_less [OF le]
```
```   970     by (auto simp add: add_assoc)
```
```   971 next
```
```   972   case (neg n)
```
```   973   show ?thesis
```
```   974   proof
```
```   975     assume eq: "1 + z + z = 0"
```
```   976     have "(0::int) < 1 + (of_nat n + of_nat n)"
```
```   977       by (simp add: le_imp_0_less add_increasing)
```
```   978     also have "... = - (1 + z + z)"
```
```   979       by (simp add: neg add_assoc [symmetric])
```
```   980     also have "... = 0" by (simp add: eq)
```
```   981     finally have "0<0" ..
```
```   982     thus False by blast
```
```   983   qed
```
```   984 qed
```
```   985
```
```   986 lemma iszero_number_of_Bit0:
```
```   987   "iszero (number_of (Bit0 w)::'a) =
```
```   988    iszero (number_of w::'a::{ring_char_0,number_ring})"
```
```   989 proof -
```
```   990   have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
```
```   991   proof -
```
```   992     assume eq: "of_int w + of_int w = (0::'a)"
```
```   993     then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
```
```   994     then have "w + w = 0" by (simp only: of_int_eq_iff)
```
```   995     then show "w = 0" by (simp only: double_eq_0_iff)
```
```   996   qed
```
```   997   thus ?thesis
```
```   998     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```   999 qed
```
```  1000
```
```  1001 lemma iszero_number_of_Bit1:
```
```  1002   "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
```
```  1003 proof -
```
```  1004   have "1 + of_int w + of_int w \<noteq> (0::'a)"
```
```  1005   proof
```
```  1006     assume eq: "1 + of_int w + of_int w = (0::'a)"
```
```  1007     hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp
```
```  1008     hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
```
```  1009     with odd_nonzero show False by blast
```
```  1010   qed
```
```  1011   thus ?thesis
```
```  1012     by (auto simp add: iszero_def number_of_eq numeral_simps)
```
```  1013 qed
```
```  1014
```
```  1015
```
```  1016 subsection {* The Less-Than Relation *}
```
```  1017
```
```  1018 lemma less_number_of_eq_neg:
```
```  1019   "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
```
```  1020   = neg (number_of (x + uminus y) :: 'a)"
```
```  1021 apply (subst less_iff_diff_less_0)
```
```  1022 apply (simp add: neg_def diff_minus number_of_add number_of_minus)
```
```  1023 done
```
```  1024
```
```  1025 text {*
```
```  1026   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```  1027   @{term Numeral0} IS @{term "number_of Pls"}
```
```  1028 *}
```
```  1029
```
```  1030 lemma not_neg_number_of_Pls:
```
```  1031   "~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
```
```  1032   by (simp add: neg_def numeral_0_eq_0)
```
```  1033
```
```  1034 lemma neg_number_of_Min:
```
```  1035   "neg (number_of Min ::'a::{ordered_idom,number_ring})"
```
```  1036   by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
```
```  1037
```
```  1038 lemma double_less_0_iff:
```
```  1039   "(a + a < 0) = (a < (0::'a::ordered_idom))"
```
```  1040 proof -
```
```  1041   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
```
```  1042   also have "... = (a < 0)"
```
```  1043     by (simp add: mult_less_0_iff zero_less_two
```
```  1044                   order_less_not_sym [OF zero_less_two])
```
```  1045   finally show ?thesis .
```
```  1046 qed
```
```  1047
```
```  1048 lemma odd_less_0:
```
```  1049   "(1 + z + z < 0) = (z < (0::int))";
```
```  1050 proof (cases z rule: int_cases)
```
```  1051   case (nonneg n)
```
```  1052   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```  1053                              le_imp_0_less [THEN order_less_imp_le])
```
```  1054 next
```
```  1055   case (neg n)
```
```  1056   thus ?thesis by (simp del: of_nat_Suc of_nat_add
```
```  1057     add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
```
```  1058 qed
```
```  1059
```
```  1060 lemma neg_number_of_Bit0:
```
```  1061   "neg (number_of (Bit0 w)::'a) =
```
```  1062   neg (number_of w :: 'a::{ordered_idom,number_ring})"
```
```  1063 by (simp add: neg_def number_of_eq numeral_simps double_less_0_iff)
```
```  1064
```
```  1065 lemma neg_number_of_Bit1:
```
```  1066   "neg (number_of (Bit1 w)::'a) =
```
```  1067   neg (number_of w :: 'a::{ordered_idom,number_ring})"
```
```  1068 proof -
```
```  1069   have "((1::'a) + of_int w + of_int w < 0) = (of_int (1 + w + w) < (of_int 0 :: 'a))"
```
```  1070     by simp
```
```  1071   also have "... = (w < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```  1072   finally show ?thesis
```
```  1073   by (simp add: neg_def number_of_eq numeral_simps)
```
```  1074 qed
```
```  1075
```
```  1076
```
```  1077 text {* Less-Than or Equals *}
```
```  1078
```
```  1079 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
```
```  1080
```
```  1081 lemmas le_number_of_eq_not_less =
```
```  1082   linorder_not_less [of "number_of w" "number_of v", symmetric,
```
```  1083   standard]
```
```  1084
```
```  1085 lemma le_number_of_eq:
```
```  1086     "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
```
```  1087      = (~ (neg (number_of (y + uminus x) :: 'a)))"
```
```  1088 by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
```
```  1089
```
```  1090
```
```  1091 text {* Absolute value (@{term abs}) *}
```
```  1092
```
```  1093 lemma abs_number_of:
```
```  1094   "abs(number_of x::'a::{ordered_idom,number_ring}) =
```
```  1095    (if number_of x < (0::'a) then -number_of x else number_of x)"
```
```  1096   by (simp add: abs_if)
```
```  1097
```
```  1098
```
```  1099 text {* Re-orientation of the equation nnn=x *}
```
```  1100
```
```  1101 lemma number_of_reorient:
```
```  1102   "(number_of w = x) = (x = number_of w)"
```
```  1103   by auto
```
```  1104
```
```  1105
```
```  1106 subsection {* Simplification of arithmetic operations on integer constants. *}
```
```  1107
```
```  1108 lemmas arith_extra_simps [standard, simp] =
```
```  1109   number_of_add [symmetric]
```
```  1110   number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
```
```  1111   number_of_mult [symmetric]
```
```  1112   diff_number_of_eq abs_number_of
```
```  1113
```
```  1114 text {*
```
```  1115   For making a minimal simpset, one must include these default simprules.
```
```  1116   Also include @{text simp_thms}.
```
```  1117 *}
```
```  1118
```
```  1119 lemmas arith_simps =
```
```  1120   normalize_bin_simps pred_bin_simps succ_bin_simps
```
```  1121   add_bin_simps minus_bin_simps mult_bin_simps
```
```  1122   abs_zero abs_one arith_extra_simps
```
```  1123
```
```  1124 text {* Simplification of relational operations *}
```
```  1125
```
```  1126 lemmas rel_simps [simp] =
```
```  1127   eq_number_of_eq iszero_0 nonzero_number_of_Min
```
```  1128   iszero_number_of_Bit0 iszero_number_of_Bit1
```
```  1129   less_number_of_eq_neg
```
```  1130   not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
```
```  1131   neg_number_of_Min neg_number_of_Bit0 neg_number_of_Bit1
```
```  1132   le_number_of_eq
```
```  1133 (* iszero_number_of_Pls would never be used
```
```  1134    because its lhs simplifies to "iszero 0" *)
```
```  1135
```
```  1136
```
```  1137 subsection {* Simplification of arithmetic when nested to the right. *}
```
```  1138
```
```  1139 lemma add_number_of_left [simp]:
```
```  1140   "number_of v + (number_of w + z) =
```
```  1141    (number_of(v + w) + z::'a::number_ring)"
```
```  1142   by (simp add: add_assoc [symmetric])
```
```  1143
```
```  1144 lemma mult_number_of_left [simp]:
```
```  1145   "number_of v * (number_of w * z) =
```
```  1146    (number_of(v * w) * z::'a::number_ring)"
```
```  1147   by (simp add: mult_assoc [symmetric])
```
```  1148
```
```  1149 lemma add_number_of_diff1:
```
```  1150   "number_of v + (number_of w - c) =
```
```  1151   number_of(v + w) - (c::'a::number_ring)"
```
```  1152   by (simp add: diff_minus add_number_of_left)
```
```  1153
```
```  1154 lemma add_number_of_diff2 [simp]:
```
```  1155   "number_of v + (c - number_of w) =
```
```  1156    number_of (v + uminus w) + (c::'a::number_ring)"
```
```  1157 apply (subst diff_number_of_eq [symmetric])
```
```  1158 apply (simp only: compare_rls)
```
```  1159 done
```
```  1160
```
```  1161
```
```  1162 subsection {* The Set of Integers *}
```
```  1163
```
```  1164 context ring_1
```
```  1165 begin
```
```  1166
```
```  1167 definition
```
```  1168   Ints  :: "'a set"
```
```  1169 where
```
```  1170   "Ints = range of_int"
```
```  1171
```
```  1172 end
```
```  1173
```
```  1174 notation (xsymbols)
```
```  1175   Ints  ("\<int>")
```
```  1176
```
```  1177 context ring_1
```
```  1178 begin
```
```  1179
```
```  1180 lemma Ints_0 [simp]: "0 \<in> \<int>"
```
```  1181 apply (simp add: Ints_def)
```
```  1182 apply (rule range_eqI)
```
```  1183 apply (rule of_int_0 [symmetric])
```
```  1184 done
```
```  1185
```
```  1186 lemma Ints_1 [simp]: "1 \<in> \<int>"
```
```  1187 apply (simp add: Ints_def)
```
```  1188 apply (rule range_eqI)
```
```  1189 apply (rule of_int_1 [symmetric])
```
```  1190 done
```
```  1191
```
```  1192 lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
```
```  1193 apply (auto simp add: Ints_def)
```
```  1194 apply (rule range_eqI)
```
```  1195 apply (rule of_int_add [symmetric])
```
```  1196 done
```
```  1197
```
```  1198 lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
```
```  1199 apply (auto simp add: Ints_def)
```
```  1200 apply (rule range_eqI)
```
```  1201 apply (rule of_int_minus [symmetric])
```
```  1202 done
```
```  1203
```
```  1204 lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
```
```  1205 apply (auto simp add: Ints_def)
```
```  1206 apply (rule range_eqI)
```
```  1207 apply (rule of_int_mult [symmetric])
```
```  1208 done
```
```  1209
```
```  1210 lemma Ints_cases [cases set: Ints]:
```
```  1211   assumes "q \<in> \<int>"
```
```  1212   obtains (of_int) z where "q = of_int z"
```
```  1213   unfolding Ints_def
```
```  1214 proof -
```
```  1215   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
```
```  1216   then obtain z where "q = of_int z" ..
```
```  1217   then show thesis ..
```
```  1218 qed
```
```  1219
```
```  1220 lemma Ints_induct [case_names of_int, induct set: Ints]:
```
```  1221   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
```
```  1222   by (rule Ints_cases) auto
```
```  1223
```
```  1224 end
```
```  1225
```
```  1226 lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a-b \<in> \<int>"
```
```  1227 apply (auto simp add: Ints_def)
```
```  1228 apply (rule range_eqI)
```
```  1229 apply (rule of_int_diff [symmetric])
```
```  1230 done
```
```  1231
```
```  1232 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```  1233
```
```  1234 lemma Ints_double_eq_0_iff:
```
```  1235   assumes in_Ints: "a \<in> Ints"
```
```  1236   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```  1237 proof -
```
```  1238   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1239   then obtain z where a: "a = of_int z" ..
```
```  1240   show ?thesis
```
```  1241   proof
```
```  1242     assume "a = 0"
```
```  1243     thus "a + a = 0" by simp
```
```  1244   next
```
```  1245     assume eq: "a + a = 0"
```
```  1246     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1247     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```  1248     hence "z = 0" by (simp only: double_eq_0_iff)
```
```  1249     thus "a = 0" by (simp add: a)
```
```  1250   qed
```
```  1251 qed
```
```  1252
```
```  1253 lemma Ints_odd_nonzero:
```
```  1254   assumes in_Ints: "a \<in> Ints"
```
```  1255   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```  1256 proof -
```
```  1257   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1258   then obtain z where a: "a = of_int z" ..
```
```  1259   show ?thesis
```
```  1260   proof
```
```  1261     assume eq: "1 + a + a = 0"
```
```  1262     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```  1263     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```  1264     with odd_nonzero show False by blast
```
```  1265   qed
```
```  1266 qed
```
```  1267
```
```  1268 lemma Ints_number_of:
```
```  1269   "(number_of w :: 'a::number_ring) \<in> Ints"
```
```  1270   unfolding number_of_eq Ints_def by simp
```
```  1271
```
```  1272 lemma Ints_odd_less_0:
```
```  1273   assumes in_Ints: "a \<in> Ints"
```
```  1274   shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
```
```  1275 proof -
```
```  1276   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```  1277   then obtain z where a: "a = of_int z" ..
```
```  1278   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```  1279     by (simp add: a)
```
```  1280   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```  1281   also have "... = (a < 0)" by (simp add: a)
```
```  1282   finally show ?thesis .
```
```  1283 qed
```
```  1284
```
```  1285
```
```  1286 subsection {* @{term setsum} and @{term setprod} *}
```
```  1287
```
```  1288 text {*By Jeremy Avigad*}
```
```  1289
```
```  1290 lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
```
```  1291   apply (cases "finite A")
```
```  1292   apply (erule finite_induct, auto)
```
```  1293   done
```
```  1294
```
```  1295 lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
```
```  1296   apply (cases "finite A")
```
```  1297   apply (erule finite_induct, auto)
```
```  1298   done
```
```  1299
```
```  1300 lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
```
```  1301   apply (cases "finite A")
```
```  1302   apply (erule finite_induct, auto simp add: of_nat_mult)
```
```  1303   done
```
```  1304
```
```  1305 lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
```
```  1306   apply (cases "finite A")
```
```  1307   apply (erule finite_induct, auto)
```
```  1308   done
```
```  1309
```
```  1310 lemma setprod_nonzero_nat:
```
```  1311     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
```
```  1312   by (rule setprod_nonzero, auto)
```
```  1313
```
```  1314 lemma setprod_zero_eq_nat:
```
```  1315     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
```
```  1316   by (rule setprod_zero_eq, auto)
```
```  1317
```
```  1318 lemma setprod_nonzero_int:
```
```  1319     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
```
```  1320   by (rule setprod_nonzero, auto)
```
```  1321
```
```  1322 lemma setprod_zero_eq_int:
```
```  1323     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
```
```  1324   by (rule setprod_zero_eq, auto)
```
```  1325
```
```  1326 lemmas int_setsum = of_nat_setsum [where 'a=int]
```
```  1327 lemmas int_setprod = of_nat_setprod [where 'a=int]
```
```  1328
```
```  1329
```
```  1330 subsection{*Inequality Reasoning for the Arithmetic Simproc*}
```
```  1331
```
```  1332 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
```
```  1333 by simp
```
```  1334
```
```  1335 lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
```
```  1336 by simp
```
```  1337
```
```  1338 lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
```
```  1339 by simp
```
```  1340
```
```  1341 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
```
```  1342 by simp
```
```  1343
```
```  1344 lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
```
```  1345 by simp
```
```  1346
```
```  1347 lemma inverse_numeral_1:
```
```  1348   "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
```
```  1349 by simp
```
```  1350
```
```  1351 text{*Theorem lists for the cancellation simprocs. The use of binary numerals
```
```  1352 for 0 and 1 reduces the number of special cases.*}
```
```  1353
```
```  1354 lemmas add_0s = add_numeral_0 add_numeral_0_right
```
```  1355 lemmas mult_1s = mult_numeral_1 mult_numeral_1_right
```
```  1356                  mult_minus1 mult_minus1_right
```
```  1357
```
```  1358
```
```  1359 subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
```
```  1360
```
```  1361 text{*Arithmetic computations are defined for binary literals, which leaves 0
```
```  1362 and 1 as special cases. Addition already has rules for 0, but not 1.
```
```  1363 Multiplication and unary minus already have rules for both 0 and 1.*}
```
```  1364
```
```  1365
```
```  1366 lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
```
```  1367 by simp
```
```  1368
```
```  1369
```
```  1370 lemmas add_number_of_eq = number_of_add [symmetric]
```
```  1371
```
```  1372 text{*Allow 1 on either or both sides*}
```
```  1373 lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
```
```  1374 by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
```
```  1375
```
```  1376 lemmas add_special =
```
```  1377     one_add_one_is_two
```
```  1378     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1379     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
```
```  1380
```
```  1381 text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
```
```  1382 lemmas diff_special =
```
```  1383     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1384     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
```
```  1385
```
```  1386 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1387 lemmas eq_special =
```
```  1388     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
```
```  1389     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1390     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
```
```  1391     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
```
```  1392
```
```  1393 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1394 lemmas less_special =
```
```  1395   binop_eq [of "op <", OF less_number_of_eq_neg numeral_0_eq_0 refl, standard]
```
```  1396   binop_eq [of "op <", OF less_number_of_eq_neg numeral_1_eq_1 refl, standard]
```
```  1397   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_0_eq_0, standard]
```
```  1398   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_1_eq_1, standard]
```
```  1399
```
```  1400 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```  1401 lemmas le_special =
```
```  1402     binop_eq [of "op \<le>", OF le_number_of_eq numeral_0_eq_0 refl, standard]
```
```  1403     binop_eq [of "op \<le>", OF le_number_of_eq numeral_1_eq_1 refl, standard]
```
```  1404     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_0_eq_0, standard]
```
```  1405     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_1_eq_1, standard]
```
```  1406
```
```  1407 lemmas arith_special[simp] =
```
```  1408        add_special diff_special eq_special less_special le_special
```
```  1409
```
```  1410
```
```  1411 lemma min_max_01: "min (0::int) 1 = 0 & min (1::int) 0 = 0 &
```
```  1412                    max (0::int) 1 = 1 & max (1::int) 0 = 1"
```
```  1413 by(simp add:min_def max_def)
```
```  1414
```
```  1415 lemmas min_max_special[simp] =
```
```  1416  min_max_01
```
```  1417  max_def[of "0::int" "number_of v", standard, simp]
```
```  1418  min_def[of "0::int" "number_of v", standard, simp]
```
```  1419  max_def[of "number_of u" "0::int", standard, simp]
```
```  1420  min_def[of "number_of u" "0::int", standard, simp]
```
```  1421  max_def[of "1::int" "number_of v", standard, simp]
```
```  1422  min_def[of "1::int" "number_of v", standard, simp]
```
```  1423  max_def[of "number_of u" "1::int", standard, simp]
```
```  1424  min_def[of "number_of u" "1::int", standard, simp]
```
```  1425
```
```  1426 text {* Legacy theorems *}
```
```  1427
```
```  1428 lemmas zle_int = of_nat_le_iff [where 'a=int]
```
```  1429 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
```
```  1430
```
```  1431 use "~~/src/Provers/Arith/assoc_fold.ML"
```
```  1432 use "int_arith1.ML"
```
```  1433 declaration {* K int_arith_setup *}
```
```  1434
```
```  1435
```
```  1436 subsection{*Lemmas About Small Numerals*}
```
```  1437
```
```  1438 lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
```
```  1439 proof -
```
```  1440   have "(of_int -1 :: 'a) = of_int (- 1)" by simp
```
```  1441   also have "... = - of_int 1" by (simp only: of_int_minus)
```
```  1442   also have "... = -1" by simp
```
```  1443   finally show ?thesis .
```
```  1444 qed
```
```  1445
```
```  1446 lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
```
```  1447 by (simp add: abs_if)
```
```  1448
```
```  1449 lemma abs_power_minus_one [simp]:
```
```  1450      "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
```
```  1451 by (simp add: power_abs)
```
```  1452
```
```  1453 lemma of_int_number_of_eq:
```
```  1454      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
```
```  1455 by (simp add: number_of_eq)
```
```  1456
```
```  1457 text{*Lemmas for specialist use, NOT as default simprules*}
```
```  1458 lemma mult_2: "2 * z = (z+z::'a::number_ring)"
```
```  1459 proof -
```
```  1460   have "2*z = (1 + 1)*z" by simp
```
```  1461   also have "... = z+z" by (simp add: left_distrib)
```
```  1462   finally show ?thesis .
```
```  1463 qed
```
```  1464
```
```  1465 lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
```
```  1466 by (subst mult_commute, rule mult_2)
```
```  1467
```
```  1468
```
```  1469 subsection{*More Inequality Reasoning*}
```
```  1470
```
```  1471 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```  1472 by arith
```
```  1473
```
```  1474 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```  1475 by arith
```
```  1476
```
```  1477 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```  1478 by arith
```
```  1479
```
```  1480 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```  1481 by arith
```
```  1482
```
```  1483 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```  1484 by arith
```
```  1485
```
```  1486
```
```  1487 subsection{*The Functions @{term nat} and @{term int}*}
```
```  1488
```
```  1489 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
```
```  1490   @{term "w + - z"}*}
```
```  1491 declare Zero_int_def [symmetric, simp]
```
```  1492 declare One_int_def [symmetric, simp]
```
```  1493
```
```  1494 lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
```
```  1495
```
```  1496 lemma nat_0: "nat 0 = 0"
```
```  1497 by (simp add: nat_eq_iff)
```
```  1498
```
```  1499 lemma nat_1: "nat 1 = Suc 0"
```
```  1500 by (subst nat_eq_iff, simp)
```
```  1501
```
```  1502 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```  1503 by (subst nat_eq_iff, simp)
```
```  1504
```
```  1505 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```  1506 apply (insert zless_nat_conj [of 1 z])
```
```  1507 apply (auto simp add: nat_1)
```
```  1508 done
```
```  1509
```
```  1510 text{*This simplifies expressions of the form @{term "int n = z"} where
```
```  1511       z is an integer literal.*}
```
```  1512 lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
```
```  1513
```
```  1514 lemma split_nat [arith_split]:
```
```  1515   "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```  1516   (is "?P = (?L & ?R)")
```
```  1517 proof (cases "i < 0")
```
```  1518   case True thus ?thesis by auto
```
```  1519 next
```
```  1520   case False
```
```  1521   have "?P = ?L"
```
```  1522   proof
```
```  1523     assume ?P thus ?L using False by clarsimp
```
```  1524   next
```
```  1525     assume ?L thus ?P using False by simp
```
```  1526   qed
```
```  1527   with False show ?thesis by simp
```
```  1528 qed
```
```  1529
```
```  1530 context ring_1
```
```  1531 begin
```
```  1532
```
```  1533 lemma of_int_of_nat:
```
```  1534   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
```
```  1535 proof (cases "k < 0")
```
```  1536   case True then have "0 \<le> - k" by simp
```
```  1537   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
```
```  1538   with True show ?thesis by simp
```
```  1539 next
```
```  1540   case False then show ?thesis by (simp add: not_less of_nat_nat)
```
```  1541 qed
```
```  1542
```
```  1543 end
```
```  1544
```
```  1545 lemma nat_mult_distrib:
```
```  1546   fixes z z' :: int
```
```  1547   assumes "0 \<le> z"
```
```  1548   shows "nat (z * z') = nat z * nat z'"
```
```  1549 proof (cases "0 \<le> z'")
```
```  1550   case False with assms have "z * z' \<le> 0"
```
```  1551     by (simp add: not_le mult_le_0_iff)
```
```  1552   then have "nat (z * z') = 0" by simp
```
```  1553   moreover from False have "nat z' = 0" by simp
```
```  1554   ultimately show ?thesis by simp
```
```  1555 next
```
```  1556   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
```
```  1557   show ?thesis
```
```  1558     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
```
```  1559       (simp only: of_nat_mult of_nat_nat [OF True]
```
```  1560          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
```
```  1561 qed
```
```  1562
```
```  1563 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```  1564 apply (rule trans)
```
```  1565 apply (rule_tac [2] nat_mult_distrib, auto)
```
```  1566 done
```
```  1567
```
```  1568 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```  1569 apply (cases "z=0 | w=0")
```
```  1570 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```  1571                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```  1572 done
```
```  1573
```
```  1574
```
```  1575 subsection "Induction principles for int"
```
```  1576
```
```  1577 text{*Well-founded segments of the integers*}
```
```  1578
```
```  1579 definition
```
```  1580   int_ge_less_than  ::  "int => (int * int) set"
```
```  1581 where
```
```  1582   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
```
```  1583
```
```  1584 theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
```
```  1585 proof -
```
```  1586   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
```
```  1587     by (auto simp add: int_ge_less_than_def)
```
```  1588   thus ?thesis
```
```  1589     by (rule wf_subset [OF wf_measure])
```
```  1590 qed
```
```  1591
```
```  1592 text{*This variant looks odd, but is typical of the relations suggested
```
```  1593 by RankFinder.*}
```
```  1594
```
```  1595 definition
```
```  1596   int_ge_less_than2 ::  "int => (int * int) set"
```
```  1597 where
```
```  1598   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
```
```  1599
```
```  1600 theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
```
```  1601 proof -
```
```  1602   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
```
```  1603     by (auto simp add: int_ge_less_than2_def)
```
```  1604   thus ?thesis
```
```  1605     by (rule wf_subset [OF wf_measure])
```
```  1606 qed
```
```  1607
```
```  1608 abbreviation
```
```  1609   int :: "nat \<Rightarrow> int"
```
```  1610 where
```
```  1611   "int \<equiv> of_nat"
```
```  1612
```
```  1613 (* `set:int': dummy construction *)
```
```  1614 theorem int_ge_induct [case_names base step, induct set: int]:
```
```  1615   fixes i :: int
```
```  1616   assumes ge: "k \<le> i" and
```
```  1617     base: "P k" and
```
```  1618     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
```
```  1619   shows "P i"
```
```  1620 proof -
```
```  1621   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```  1622     proof (induct n)
```
```  1623       case 0
```
```  1624       hence "i = k" by arith
```
```  1625       thus "P i" using base by simp
```
```  1626     next
```
```  1627       case (Suc n)
```
```  1628       then have "n = nat((i - 1) - k)" by arith
```
```  1629       moreover
```
```  1630       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```  1631       ultimately
```
```  1632       have "P(i - 1)" by(rule Suc.hyps)
```
```  1633       from step[OF ki1 this] show ?case by simp
```
```  1634     qed
```
```  1635   }
```
```  1636   with ge show ?thesis by fast
```
```  1637 qed
```
```  1638
```
```  1639 (* `set:int': dummy construction *)
```
```  1640 theorem int_gr_induct [case_names base step, induct set: int]:
```
```  1641   assumes gr: "k < (i::int)" and
```
```  1642         base: "P(k+1)" and
```
```  1643         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```  1644   shows "P i"
```
```  1645 apply(rule int_ge_induct[of "k + 1"])
```
```  1646   using gr apply arith
```
```  1647  apply(rule base)
```
```  1648 apply (rule step, simp+)
```
```  1649 done
```
```  1650
```
```  1651 theorem int_le_induct[consumes 1,case_names base step]:
```
```  1652   assumes le: "i \<le> (k::int)" and
```
```  1653         base: "P(k)" and
```
```  1654         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1655   shows "P i"
```
```  1656 proof -
```
```  1657   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
```
```  1658     proof (induct n)
```
```  1659       case 0
```
```  1660       hence "i = k" by arith
```
```  1661       thus "P i" using base by simp
```
```  1662     next
```
```  1663       case (Suc n)
```
```  1664       hence "n = nat(k - (i+1))" by arith
```
```  1665       moreover
```
```  1666       have ki1: "i + 1 \<le> k" using Suc.prems by arith
```
```  1667       ultimately
```
```  1668       have "P(i+1)" by(rule Suc.hyps)
```
```  1669       from step[OF ki1 this] show ?case by simp
```
```  1670     qed
```
```  1671   }
```
```  1672   with le show ?thesis by fast
```
```  1673 qed
```
```  1674
```
```  1675 theorem int_less_induct [consumes 1,case_names base step]:
```
```  1676   assumes less: "(i::int) < k" and
```
```  1677         base: "P(k - 1)" and
```
```  1678         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```  1679   shows "P i"
```
```  1680 apply(rule int_le_induct[of _ "k - 1"])
```
```  1681   using less apply arith
```
```  1682  apply(rule base)
```
```  1683 apply (rule step, simp+)
```
```  1684 done
```
```  1685
```
```  1686 subsection{*Intermediate value theorems*}
```
```  1687
```
```  1688 lemma int_val_lemma:
```
```  1689      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```  1690       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```  1691 apply (induct_tac "n", simp)
```
```  1692 apply (intro strip)
```
```  1693 apply (erule impE, simp)
```
```  1694 apply (erule_tac x = n in allE, simp)
```
```  1695 apply (case_tac "k = f (n+1) ")
```
```  1696  apply force
```
```  1697 apply (erule impE)
```
```  1698  apply (simp add: abs_if split add: split_if_asm)
```
```  1699 apply (blast intro: le_SucI)
```
```  1700 done
```
```  1701
```
```  1702 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```  1703
```
```  1704 lemma nat_intermed_int_val:
```
```  1705      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```  1706          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```  1707 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```  1708        in int_val_lemma)
```
```  1709 apply simp
```
```  1710 apply (erule exE)
```
```  1711 apply (rule_tac x = "i+m" in exI, arith)
```
```  1712 done
```
```  1713
```
```  1714
```
```  1715 subsection{*Products and 1, by T. M. Rasmussen*}
```
```  1716
```
```  1717 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```  1718 by arith
```
```  1719
```
```  1720 lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
```
```  1721 apply (cases "\<bar>n\<bar>=1")
```
```  1722 apply (simp add: abs_mult)
```
```  1723 apply (rule ccontr)
```
```  1724 apply (auto simp add: linorder_neq_iff abs_mult)
```
```  1725 apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
```
```  1726  prefer 2 apply arith
```
```  1727 apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp)
```
```  1728 apply (rule mult_mono, auto)
```
```  1729 done
```
```  1730
```
```  1731 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```  1732 by (insert abs_zmult_eq_1 [of m n], arith)
```
```  1733
```
```  1734 lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
```
```  1735 apply (auto dest: pos_zmult_eq_1_iff_lemma)
```
```  1736 apply (simp add: mult_commute [of m])
```
```  1737 apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1738 done
```
```  1739
```
```  1740 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```  1741 apply (rule iffI)
```
```  1742  apply (frule pos_zmult_eq_1_iff_lemma)
```
```  1743  apply (simp add: mult_commute [of m])
```
```  1744  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```  1745 done
```
```  1746
```
```  1747 (* Could be simplified but Presburger only becomes available too late *)
```
```  1748 lemma infinite_UNIV_int: "~finite(UNIV::int set)"
```
```  1749 proof
```
```  1750   assume "finite(UNIV::int set)"
```
```  1751   moreover have "~(EX i::int. 2*i = 1)"
```
```  1752     by (auto simp: pos_zmult_eq_1_iff)
```
```  1753   ultimately show False using finite_UNIV_inj_surj[of "%n::int. n+n"]
```
```  1754     by (simp add:inj_on_def surj_def) (blast intro:sym)
```
```  1755 qed
```
```  1756
```
```  1757
```
```  1758 subsection{*Integer Powers*}
```
```  1759
```
```  1760 instantiation int :: recpower
```
```  1761 begin
```
```  1762
```
```  1763 primrec power_int where
```
```  1764   "p ^ 0 = (1\<Colon>int)"
```
```  1765   | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
```
```  1766
```
```  1767 instance proof
```
```  1768   fix z :: int
```
```  1769   fix n :: nat
```
```  1770   show "z ^ 0 = 1" by simp
```
```  1771   show "z ^ Suc n = z * (z ^ n)" by simp
```
```  1772 qed
```
```  1773
```
```  1774 end
```
```  1775
```
```  1776 lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
```
```  1777   by (rule Power.power_add)
```
```  1778
```
```  1779 lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
```
```  1780   by (rule Power.power_mult [symmetric])
```
```  1781
```
```  1782 lemma zero_less_zpower_abs_iff [simp]:
```
```  1783   "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
```
```  1784   by (induct n) (auto simp add: zero_less_mult_iff)
```
```  1785
```
```  1786 lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
```
```  1787   by (induct n) (auto simp add: zero_le_mult_iff)
```
```  1788
```
```  1789 lemma of_int_power:
```
```  1790   "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
```
```  1791   by (induct n) (simp_all add: power_Suc)
```
```  1792
```
```  1793 lemma int_power: "int (m^n) = (int m) ^ n"
```
```  1794   by (rule of_nat_power)
```
```  1795
```
```  1796 lemmas zpower_int = int_power [symmetric]
```
```  1797
```
```  1798 subsection {* Configuration of the code generator *}
```
```  1799
```
```  1800 instance int :: eq ..
```
```  1801
```
```  1802 code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
```
```  1803
```
```  1804 definition
```
```  1805   int_aux :: "nat \<Rightarrow> int \<Rightarrow> int" where
```
```  1806   [code func del]: "int_aux = of_nat_aux"
```
```  1807
```
```  1808 lemmas int_aux_code = of_nat_aux_code [where ?'a = int, simplified int_aux_def [symmetric], code]
```
```  1809
```
```  1810 lemma [code, code unfold, code inline del]:
```
```  1811   "of_nat n = int_aux n 0"
```
```  1812   by (simp add: int_aux_def of_nat_aux_def)
```
```  1813
```
```  1814 definition
```
```  1815   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
```
```  1816   "nat_aux i n = nat i + n"
```
```  1817
```
```  1818 lemma [code]:
```
```  1819   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
```
```  1820   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
```
```  1821     dest: zless_imp_add1_zle)
```
```  1822
```
```  1823 lemma [code]: "nat i = nat_aux i 0"
```
```  1824   by (simp add: nat_aux_def)
```
```  1825
```
```  1826 hide (open) const int_aux nat_aux
```
```  1827
```
```  1828 lemma zero_is_num_zero [code func, code inline, symmetric, code post]:
```
```  1829   "(0\<Colon>int) = Numeral0"
```
```  1830   by simp
```
```  1831
```
```  1832 lemma one_is_num_one [code func, code inline, symmetric, code post]:
```
```  1833   "(1\<Colon>int) = Numeral1"
```
```  1834   by simp
```
```  1835
```
```  1836 code_modulename SML
```
```  1837   Int Integer
```
```  1838
```
```  1839 code_modulename OCaml
```
```  1840   Int Integer
```
```  1841
```
```  1842 code_modulename Haskell
```
```  1843   Int Integer
```
```  1844
```
```  1845 types_code
```
```  1846   "int" ("int")
```
```  1847 attach (term_of) {*
```
```  1848 val term_of_int = HOLogic.mk_number HOLogic.intT;
```
```  1849 *}
```
```  1850 attach (test) {*
```
```  1851 fun gen_int i =
```
```  1852   let val j = one_of [~1, 1] * random_range 0 i
```
```  1853   in (j, fn () => term_of_int j) end;
```
```  1854 *}
```
```  1855
```
```  1856 setup {*
```
```  1857 let
```
```  1858
```
```  1859 fun strip_number_of (@{term "Int.number_of :: int => int"} \$ t) = t
```
```  1860   | strip_number_of t = t;
```
```  1861
```
```  1862 fun numeral_codegen thy defs gr dep module b t =
```
```  1863   let val i = HOLogic.dest_numeral (strip_number_of t)
```
```  1864   in
```
```  1865     SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, HOLogic.intT)),
```
```  1866       Pretty.str (string_of_int i))
```
```  1867   end handle TERM _ => NONE;
```
```  1868
```
```  1869 in
```
```  1870
```
```  1871 Codegen.add_codegen "numeral_codegen" numeral_codegen
```
```  1872
```
```  1873 end
```
```  1874 *}
```
```  1875
```
```  1876 consts_code
```
```  1877   "number_of :: int \<Rightarrow> int"    ("(_)")
```
```  1878   "0 :: int"                   ("0")
```
```  1879   "1 :: int"                   ("1")
```
```  1880   "uminus :: int => int"       ("~")
```
```  1881   "op + :: int => int => int"  ("(_ +/ _)")
```
```  1882   "op * :: int => int => int"  ("(_ */ _)")
```
```  1883   "op \<le> :: int => int => bool" ("(_ <=/ _)")
```
```  1884   "op < :: int => int => bool" ("(_ </ _)")
```
```  1885
```
```  1886 quickcheck_params [default_type = int]
```
```  1887
```
```  1888 (*setup continues in theory Presburger*)
```
```  1889
```
```  1890 hide (open) const Pls Min Bit0 Bit1 succ pred
```
```  1891
```
```  1892
```
```  1893 subsection {* Legacy theorems *}
```
```  1894
```
```  1895 lemmas zminus_zminus = minus_minus [of "z::int", standard]
```
```  1896 lemmas zminus_0 = minus_zero [where 'a=int]
```
```  1897 lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
```
```  1898 lemmas zadd_commute = add_commute [of "z::int" "w", standard]
```
```  1899 lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
```
```  1900 lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
```
```  1901 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
```
```  1902 lemmas zmult_ac = OrderedGroup.mult_ac
```
```  1903 lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard]
```
```  1904 lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard]
```
```  1905 lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
```
```  1906 lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
```
```  1907 lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
```
```  1908 lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
```
```  1909 lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
```
```  1910 lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
```
```  1911 lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
```
```  1912 lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
```
```  1913
```
```  1914 lemmas zmult_1 = mult_1_left [of "z::int", standard]
```
```  1915 lemmas zmult_1_right = mult_1_right [of "z::int", standard]
```
```  1916
```
```  1917 lemmas zle_refl = order_refl [of "w::int", standard]
```
```  1918 lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
```
```  1919 lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard]
```
```  1920 lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
```
```  1921 lemmas zless_linear = linorder_less_linear [where 'a = int]
```
```  1922
```
```  1923 lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
```
```  1924 lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
```
```  1925 lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
```
```  1926
```
```  1927 lemmas int_0_less_1 = zero_less_one [where 'a=int]
```
```  1928 lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
```
```  1929
```
```  1930 lemmas inj_int = inj_of_nat [where 'a=int]
```
```  1931 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
```
```  1932 lemmas int_mult = of_nat_mult [where 'a=int]
```
```  1933 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
```
```  1934 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
```
```  1935 lemmas zless_int = of_nat_less_iff [where 'a=int]
```
```  1936 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
```
```  1937 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
```
```  1938 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
```
```  1939 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
```
```  1940 lemmas int_0 = of_nat_0 [where 'a=int]
```
```  1941 lemmas int_1 = of_nat_1 [where 'a=int]
```
```  1942 lemmas int_Suc = of_nat_Suc [where 'a=int]
```
```  1943 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
```
```  1944 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
```
```  1945 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
```
```  1946 lemmas zless_le = less_int_def
```
```  1947 lemmas int_eq_of_nat = TrueI
```
```  1948
```
```  1949 end
```