src/HOL/IntDef.thy
 author haftmann Fri Jul 20 14:28:01 2007 +0200 (2007-07-20) changeset 23879 4776af8be741 parent 23852 3736cdf9398b child 23950 f54c0e339061 permissions -rw-r--r--
split class abs from class minus
1 (*  Title:      IntDef.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1996  University of Cambridge
6 *)
8 header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
10 theory IntDef
11 imports Equiv_Relations Nat
12 begin
15 text {* the equivalence relation underlying the integers *}
17 definition
18   intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
19 where
20   "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
22 typedef (Integ)
23   int = "UNIV//intrel"
24   by (auto simp add: quotient_def)
26 instance int :: zero
27   Zero_int_def: "0 \<equiv> Abs_Integ (intrel `` {(0, 0)})" ..
29 instance int :: one
30   One_int_def: "1 \<equiv> Abs_Integ (intrel `` {(1, 0)})" ..
32 instance int :: plus
33   add_int_def: "z + w \<equiv> Abs_Integ
34     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
35       intrel `` {(x + u, y + v)})" ..
37 instance int :: minus
38   minus_int_def:
39     "- z \<equiv> Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
40   diff_int_def:  "z - w \<equiv> z + (-w)" ..
42 instance int :: times
43   mult_int_def: "z * w \<equiv>  Abs_Integ
44     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
45       intrel `` {(x*u + y*v, x*v + y*u)})" ..
47 instance int :: ord
48   le_int_def:
49    "z \<le> w \<equiv> \<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w"
50   less_int_def: "z < w \<equiv> z \<le> w \<and> z \<noteq> w" ..
52 lemmas [code func del] = Zero_int_def One_int_def add_int_def
53   minus_int_def mult_int_def le_int_def less_int_def
56 subsection{*Construction of the Integers*}
58 lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
61 lemma equiv_intrel: "equiv UNIV intrel"
62 by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
64 text{*Reduces equality of equivalence classes to the @{term intrel} relation:
65   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
66 lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
68 text{*All equivalence classes belong to set of representatives*}
69 lemma [simp]: "intrel``{(x,y)} \<in> Integ"
70 by (auto simp add: Integ_def intrel_def quotient_def)
72 text{*Reduces equality on abstractions to equality on representatives:
73   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
74 declare Abs_Integ_inject [simp]  Abs_Integ_inverse [simp]
76 text{*Case analysis on the representation of an integer as an equivalence
77       class of pairs of naturals.*}
78 lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
79      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
80 apply (rule Abs_Integ_cases [of z])
81 apply (auto simp add: Integ_def quotient_def)
82 done
85 subsection{*Arithmetic Operations*}
87 lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
88 proof -
89   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
91   thus ?thesis
92     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
93 qed
96      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
97       Abs_Integ (intrel``{(x+u, y+v)})"
98 proof -
99   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)
100         respects2 intrel"
102   thus ?thesis
104                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
105 qed
107 text{*Congruence property for multiplication*}
108 lemma mult_congruent2:
109      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
110       respects2 intrel"
111 apply (rule equiv_intrel [THEN congruent2_commuteI])
112  apply (force simp add: mult_ac, clarify)
113 apply (simp add: congruent_def mult_ac)
114 apply (rename_tac u v w x y z)
115 apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
118 done
120 lemma mult:
121      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
122       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
123 by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
124               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
126 text{*The integers form a @{text comm_ring_1}*}
127 instance int :: comm_ring_1
128 proof
129   fix i j k :: int
130   show "(i + j) + k = i + (j + k)"
132   show "i + j = j + i"
134   show "0 + i = i"
136   show "- i + i = 0"
138   show "i - j = i + - j"
140   show "(i * j) * k = i * (j * k)"
141     by (cases i, cases j, cases k) (simp add: mult ring_simps)
142   show "i * j = j * i"
143     by (cases i, cases j) (simp add: mult ring_simps)
144   show "1 * i = i"
145     by (cases i) (simp add: One_int_def mult)
146   show "(i + j) * k = i * k + j * k"
147     by (cases i, cases j, cases k) (simp add: add mult ring_simps)
148   show "0 \<noteq> (1::int)"
149     by (simp add: Zero_int_def One_int_def)
150 qed
152 abbreviation
153   int :: "nat \<Rightarrow> int"
154 where
155   "int \<equiv> of_nat"
157 lemma int_def: "int m = Abs_Integ (intrel `` {(m, 0)})"
161 subsection{*The @{text "\<le>"} Ordering*}
163 lemma le:
164   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
165 by (force simp add: le_int_def)
167 lemma less:
168   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
169 by (simp add: less_int_def le order_less_le)
171 instance int :: linorder
172 proof
173   fix i j k :: int
174   show "(i < j) = (i \<le> j \<and> i \<noteq> j)"
176   show "i \<le> i"
177     by (cases i) (simp add: le)
178   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
179     by (cases i, cases j, cases k) (simp add: le)
180   show "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
181     by (cases i, cases j) (simp add: le)
182   show "i \<le> j \<or> j \<le> i"
183     by (cases i, cases j) (simp add: le linorder_linear)
184 qed
187 proof
188   fix i j k :: int
189   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
190     by (cases i, cases j, cases k) (simp add: le add)
191 qed
193 text{*Strict Monotonicity of Multiplication*}
195 text{*strict, in 1st argument; proof is by induction on k>0*}
196 lemma zmult_zless_mono2_lemma:
197      "(i::int)<j ==> 0<k ==> int k * i < int k * j"
198 apply (induct "k", simp)
200 apply (case_tac "k=0")
202 done
204 lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
205 apply (cases k)
207 apply (rule_tac x="x-y" in exI, simp)
208 done
210 lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
211 apply (cases k)
212 apply (simp add: less int_def Zero_int_def)
213 apply (rule_tac x="x-y" in exI, simp)
214 done
216 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
217 apply (drule zero_less_imp_eq_int)
218 apply (auto simp add: zmult_zless_mono2_lemma)
219 done
221 instance int :: abs
222   zabs_def: "\<bar>i\<Colon>int\<bar> \<equiv> if i < 0 then - i else i" ..
224 instance int :: distrib_lattice
225   "inf \<equiv> min"
226   "sup \<equiv> max"
227   by intro_classes
228     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
230 text{*The integers form an ordered integral domain*}
231 instance int :: ordered_idom
232 proof
233   fix i j k :: int
234   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
235     by (rule zmult_zless_mono2)
236   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
237     by (simp only: zabs_def)
238 qed
240 lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
241 apply (cases w, cases z)
243 done
246 subsection{*Magnitude of an Integer, as a Natural Number: @{term nat}*}
248 definition
249   nat :: "int \<Rightarrow> nat"
250 where
251   [code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
253 lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
254 proof -
255   have "(\<lambda>(x,y). {x-y}) respects intrel"
256     by (simp add: congruent_def) arith
257   thus ?thesis
258     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
259 qed
261 lemma nat_int [simp]: "nat (int n) = n"
262 by (simp add: nat int_def)
264 lemma nat_zero [simp]: "nat 0 = 0"
265 by (simp add: Zero_int_def nat)
267 lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
268 by (cases z, simp add: nat le int_def Zero_int_def)
270 corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
271 by simp
273 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
274 by (cases z, simp add: nat le Zero_int_def)
276 lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
277 apply (cases w, cases z)
278 apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
279 done
281 text{*An alternative condition is @{term "0 \<le> w"} *}
282 corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
283 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
285 corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
286 by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
288 lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
289 apply (cases w, cases z)
290 apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
291 done
293 lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
294 by (blast dest: nat_0_le sym)
296 lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
297 by (cases w, simp add: nat le int_def Zero_int_def, arith)
299 corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
300 by (simp only: eq_commute [of m] nat_eq_iff)
302 lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
303 apply (cases w)
304 apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
305 done
307 lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
308 by (auto simp add: nat_eq_iff2)
310 lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
311 by (insert zless_nat_conj [of 0], auto)
314      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
315 by (cases z, cases z', simp add: nat add le Zero_int_def)
317 lemma nat_diff_distrib:
318      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
319 by (cases z, cases z',
322 lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
323 by (simp add: int_def minus nat Zero_int_def)
325 lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
326 by (cases z, simp add: nat less int_def, arith)
329 subsection{*Lemmas about the Function @{term int} and Orderings*}
331 lemma negative_zless_0: "- (int (Suc n)) < 0"
332 by (simp add: order_less_le del: of_nat_Suc)
334 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
335 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
337 lemma negative_zle_0: "- int n \<le> 0"
340 lemma negative_zle [iff]: "- int n \<le> int m"
341 by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
343 lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
344 by (subst le_minus_iff, simp del: of_nat_Suc)
346 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
347 by (simp add: int_def le minus Zero_int_def)
349 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
352 lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
353 by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
355 lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
356 proof -
357   have "(w \<le> z) = (0 \<le> z - w)"
358     by (simp only: le_diff_eq add_0_left)
359   also have "\<dots> = (\<exists>n. z - w = int n)"
360     by (auto elim: zero_le_imp_eq_int)
361   also have "\<dots> = (\<exists>n. z = w + int n)"
362     by (simp only: group_simps)
363   finally show ?thesis .
364 qed
366 lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
367 by simp
369 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
370 by simp
372 lemma abs_of_nat [simp]: "\<bar>of_nat n::'a::ordered_idom\<bar> = of_nat n"
373 by (rule  of_nat_0_le_iff [THEN abs_of_nonneg]) (* belongs in Nat.thy *)
375 text{*This version is proved for all ordered rings, not just integers!
376       It is proved here because attribute @{text arith_split} is not available
377       in theory @{text Ring_and_Field}.
378       But is it really better than just rewriting with @{text abs_if}?*}
379 lemma abs_split [arith_split]:
380      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
381 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
384 subsection {* Constants @{term neg} and @{term iszero} *}
386 definition
387   neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
388 where
389   [code inline]: "neg Z \<longleftrightarrow> Z < 0"
391 definition (*for simplifying equalities*)
392   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
393 where
394   "iszero z \<longleftrightarrow> z = 0"
396 lemma not_neg_int [simp]: "~ neg (int n)"
399 lemma neg_zminus_int [simp]: "neg (- (int (Suc n)))"
400 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
402 lemmas neg_eq_less_0 = neg_def
404 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
405 by (simp add: neg_def linorder_not_less)
408 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
410 lemma not_neg_0: "~ neg 0"
411 by (simp add: One_int_def neg_def)
413 lemma not_neg_1: "~ neg 1"
414 by (simp add: neg_def linorder_not_less zero_le_one)
416 lemma iszero_0: "iszero 0"
419 lemma not_iszero_1: "~ iszero 1"
420 by (simp add: iszero_def eq_commute)
422 lemma neg_nat: "neg z ==> nat z = 0"
423 by (simp add: neg_def order_less_imp_le)
425 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
426 by (simp add: linorder_not_less neg_def)
429 subsection{*Embedding of the Integers into any @{text ring_1}: @{term of_int}*}
431 constdefs
432   of_int :: "int => 'a::ring_1"
433   "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
434 lemmas [code func del] = of_int_def
436 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
437 proof -
438   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
441   thus ?thesis
442     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
443 qed
445 lemma of_int_0 [simp]: "of_int 0 = 0"
446 by (simp add: of_int Zero_int_def)
448 lemma of_int_1 [simp]: "of_int 1 = 1"
449 by (simp add: of_int One_int_def)
451 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
454 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
455 by (cases z, simp add: compare_rls of_int minus)
457 lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
460 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
461 apply (cases w, cases z)
462 apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
464 done
466 lemma of_int_le_iff [simp]:
467      "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
468 apply (cases w)
469 apply (cases z)
472 done
474 text{*Special cases where either operand is zero*}
475 lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
476 lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
479 lemma of_int_less_iff [simp]:
480      "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
481 by (simp add: linorder_not_le [symmetric])
483 text{*Special cases where either operand is zero*}
484 lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
485 lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
487 text{*Class for unital rings with characteristic zero.
488  Includes non-ordered rings like the complex numbers.*}
489 axclass ring_char_0 < ring_1, semiring_char_0
491 lemma of_int_eq_iff [simp]:
492      "(of_int w = (of_int z::'a::ring_char_0)) = (w = z)"
493 apply (cases w, cases z, simp add: of_int)
494 apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
495 apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
496 done
498 text{*Every @{text ordered_idom} has characteristic zero.*}
499 instance ordered_idom < ring_char_0 ..
501 text{*Special cases where either operand is zero*}
502 lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
503 lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
505 lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
506 proof
507   fix z
508   show "of_int z = id z"
509     by (cases z)
511 qed
513 lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
514 by (cases z rule: eq_Abs_Integ)
515    (simp add: nat le of_int Zero_int_def of_nat_diff)
518 subsection{*The Set of Integers*}
520 constdefs
521   Ints  :: "'a::ring_1 set"
522   "Ints == range of_int"
524 notation (xsymbols)
525   Ints  ("\<int>")
527 lemma Ints_0 [simp]: "0 \<in> Ints"
529 apply (rule range_eqI)
530 apply (rule of_int_0 [symmetric])
531 done
533 lemma Ints_1 [simp]: "1 \<in> Ints"
535 apply (rule range_eqI)
536 apply (rule of_int_1 [symmetric])
537 done
539 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
540 apply (auto simp add: Ints_def)
541 apply (rule range_eqI)
543 done
545 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
546 apply (auto simp add: Ints_def)
547 apply (rule range_eqI)
548 apply (rule of_int_minus [symmetric])
549 done
551 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
552 apply (auto simp add: Ints_def)
553 apply (rule range_eqI)
554 apply (rule of_int_diff [symmetric])
555 done
557 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
558 apply (auto simp add: Ints_def)
559 apply (rule range_eqI)
560 apply (rule of_int_mult [symmetric])
561 done
563 text{*Collapse nested embeddings*}
564 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
565 by (induct n, auto)
567 lemma Ints_cases [cases set: Ints]:
568   assumes "q \<in> \<int>"
569   obtains (of_int) z where "q = of_int z"
570   unfolding Ints_def
571 proof -
572   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
573   then obtain z where "q = of_int z" ..
574   then show thesis ..
575 qed
577 lemma Ints_induct [case_names of_int, induct set: Ints]:
578   "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
579   by (rule Ints_cases) auto
582 subsection {* Further properties *}
584 text{*Now we replace the case analysis rule by a more conventional one:
585 whether an integer is negative or not.*}
588     "(w < z) = (\<exists>n. z = w + int (Suc n))"
589 apply (cases z, cases w)
591 apply (rename_tac a b c d)
592 apply (rule_tac x="a+d - Suc(c+b)" in exI)
593 apply arith
594 done
596 lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
597 apply (cases x)
598 apply (auto simp add: le minus Zero_int_def int_def order_less_le)
599 apply (rule_tac x="y - Suc x" in exI, arith)
600 done
602 theorem int_cases [cases type: int, case_names nonneg neg]:
603      "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
604 apply (cases "z < 0", blast dest!: negD)
605 apply (simp add: linorder_not_less del: of_nat_Suc)
606 apply (blast dest: nat_0_le [THEN sym])
607 done
609 theorem int_induct [induct type: int, case_names nonneg neg]:
610      "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
611   by (cases z rule: int_cases) auto
613 text{*Contributed by Brian Huffman*}
614 theorem int_diff_cases [case_names diff]:
615 assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
616 apply (cases z rule: eq_Abs_Integ)
617 apply (rule_tac m=x and n=y in prem)
619 done
622 subsection {* Legacy theorems *}
624 lemmas zminus_zminus = minus_minus [of "?z::int"]
625 lemmas zminus_0 = minus_zero [where 'a=int]
631 lemmas zmult_ac = OrderedGroup.mult_ac
634 lemmas zadd_zminus_inverse2 = left_minus [of "?z::int"]
635 lemmas zmult_zminus = mult_minus_left [of "?z::int" "?w"]
636 lemmas zmult_commute = mult_commute [of "?z::int" "?w"]
637 lemmas zmult_assoc = mult_assoc [of "?z1.0::int" "?z2.0" "?z3.0"]
638 lemmas zadd_zmult_distrib = left_distrib [of "?z1.0::int" "?z2.0" "?w"]
639 lemmas zadd_zmult_distrib2 = right_distrib [of "?w::int" "?z1.0" "?z2.0"]
640 lemmas zdiff_zmult_distrib = left_diff_distrib [of "?z1.0::int" "?z2.0" "?w"]
641 lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "?w::int" "?z1.0" "?z2.0"]
643 lemmas int_distrib =
645   zdiff_zmult_distrib zdiff_zmult_distrib2
647 lemmas zmult_1 = mult_1_left [of "?z::int"]
648 lemmas zmult_1_right = mult_1_right [of "?z::int"]
650 lemmas zle_refl = order_refl [of "?w::int"]
651 lemmas zle_trans = order_trans [where 'a=int and x="?i" and y="?j" and z="?k"]
652 lemmas zle_anti_sym = order_antisym [of "?z::int" "?w"]
653 lemmas zle_linear = linorder_linear [of "?z::int" "?w"]
654 lemmas zless_linear = linorder_less_linear [where 'a = int]
660 lemmas int_0_less_1 = zero_less_one [where 'a=int]
661 lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
663 lemmas inj_int = inj_of_nat [where 'a=int]
664 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
666 lemmas int_mult = of_nat_mult [where 'a=int]
667 lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
668 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="?n"]
669 lemmas zless_int = of_nat_less_iff [where 'a=int]
670 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="?k"]
671 lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
672 lemmas zle_int = of_nat_le_iff [where 'a=int]
673 lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
674 lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="?n"]
675 lemmas int_0 = of_nat_0 [where ?'a_1.0=int]
676 lemmas int_1 = of_nat_1 [where 'a=int]
677 lemmas int_Suc = of_nat_Suc [where ?'a_1.0=int]
678 lemmas abs_int_eq = abs_of_nat [where 'a=int and n="?m"]
679 lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
680 lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
681 lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq]
682 lemmas int_eq_of_nat = TrueI
684 abbreviation
685   int_of_nat :: "nat \<Rightarrow> int"
686 where
687   "int_of_nat \<equiv> of_nat"
689 end