src/HOL/Divides.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 47217 501b9bbd0d6e child 47268 262d96552e50 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Divides.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1999  University of Cambridge
4 *)
6 header {* The division operators div and mod *}
8 theory Divides
9 imports Nat_Transfer
10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"
11 begin
13 subsection {* Syntactic division operations *}
15 class div = dvd +
16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
17     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
20 subsection {* Abstract division in commutative semirings. *}
22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
23   assumes mod_div_equality: "a div b * b + a mod b = a"
24     and div_by_0 [simp]: "a div 0 = 0"
25     and div_0 [simp]: "0 div a = 0"
26     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
27     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
28 begin
30 text {* @{const div} and @{const mod} *}
32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
33   unfolding mult_commute [of b]
34   by (rule mod_div_equality)
36 lemma mod_div_equality': "a mod b + a div b * b = a"
37   using mod_div_equality [of a b]
40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
46 lemma mod_by_0 [simp]: "a mod 0 = a"
47   using mod_div_equality [of a zero] by simp
49 lemma mod_0 [simp]: "0 mod a = 0"
50   using mod_div_equality [of zero a] div_0 by simp
52 lemma div_mult_self2 [simp]:
53   assumes "b \<noteq> 0"
54   shows "(a + b * c) div b = c + a div b"
55   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
58 proof (cases "b = 0")
59   case True then show ?thesis by simp
60 next
61   case False
62   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
64   also from False div_mult_self1 [of b a c] have
65     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
67   finally have "a = a div b * b + (a + c * b) mod b"
69   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
71   then show ?thesis by simp
72 qed
74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
75   by (simp add: mult_commute [of b])
77 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
78   using div_mult_self2 [of b 0 a] by simp
80 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
81   using div_mult_self1 [of b 0 a] by simp
83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
84   using mod_mult_self2 [of 0 b a] by simp
86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
87   using mod_mult_self1 [of 0 a b] by simp
89 lemma div_by_1 [simp]: "a div 1 = a"
90   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
92 lemma mod_by_1 [simp]: "a mod 1 = 0"
93 proof -
94   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
95   then have "a + a mod 1 = a + 0" by simp
96   then show ?thesis by (rule add_left_imp_eq)
97 qed
99 lemma mod_self [simp]: "a mod a = 0"
100   using mod_mult_self2_is_0 [of 1] by simp
102 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
103   using div_mult_self2_is_id [of _ 1] by simp
106   assumes "b \<noteq> 0"
107   shows "(b + a) div b = a div b + 1"
108   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
111   assumes "b \<noteq> 0"
112   shows "(a + b) div b = a div b + 1"
116   "(b + a) mod b = a mod b"
120   "(a + b) mod b = a mod b"
121   using mod_mult_self1 [of a 1 b] by simp
123 lemma mod_div_decomp:
124   fixes a b
125   obtains q r where "q = a div b" and "r = a mod b"
126     and "a = q * b + r"
127 proof -
128   from mod_div_equality have "a = a div b * b + a mod b" by simp
129   moreover have "a div b = a div b" ..
130   moreover have "a mod b = a mod b" ..
131   note that ultimately show thesis by blast
132 qed
134 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
135 proof
136   assume "b mod a = 0"
137   with mod_div_equality [of b a] have "b div a * a = b" by simp
138   then have "b = a * (b div a)" unfolding mult_commute ..
139   then have "\<exists>c. b = a * c" ..
140   then show "a dvd b" unfolding dvd_def .
141 next
142   assume "a dvd b"
143   then have "\<exists>c. b = a * c" unfolding dvd_def .
144   then obtain c where "b = a * c" ..
145   then have "b mod a = a * c mod a" by simp
146   then have "b mod a = c * a mod a" by (simp add: mult_commute)
147   then show "b mod a = 0" by simp
148 qed
150 lemma mod_div_trivial [simp]: "a mod b div b = 0"
151 proof (cases "b = 0")
152   assume "b = 0"
153   thus ?thesis by simp
154 next
155   assume "b \<noteq> 0"
156   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
157     by (rule div_mult_self1 [symmetric])
158   also have "\<dots> = a div b"
159     by (simp only: mod_div_equality')
160   also have "\<dots> = a div b + 0"
161     by simp
162   finally show ?thesis
164 qed
166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
167 proof -
168   have "a mod b mod b = (a mod b + a div b * b) mod b"
169     by (simp only: mod_mult_self1)
170   also have "\<dots> = a mod b"
171     by (simp only: mod_div_equality')
172   finally show ?thesis .
173 qed
175 lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
176 by (rule dvd_eq_mod_eq_0[THEN iffD1])
178 lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
181 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
182 by (drule dvd_div_mult_self) (simp add: mult_commute)
184 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
185 apply (cases "a = 0")
186  apply simp
187 apply (auto simp: dvd_def mult_assoc)
188 done
190 lemma div_dvd_div[simp]:
191   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
192 apply (cases "a = 0")
193  apply simp
194 apply (unfold dvd_def)
195 apply auto
196  apply(blast intro:mult_assoc[symmetric])
198 done
200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
201   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
203   apply (simp only: dvd_add dvd_mult)
204   done
206 text {* Addition respects modular equivalence. *}
208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
209 proof -
210   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
211     by (simp only: mod_div_equality)
212   also have "\<dots> = (a mod c + b + a div c * c) mod c"
214   also have "\<dots> = (a mod c + b) mod c"
215     by (rule mod_mult_self1)
216   finally show ?thesis .
217 qed
219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
220 proof -
221   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
222     by (simp only: mod_div_equality)
223   also have "\<dots> = (a + b mod c + b div c * c) mod c"
225   also have "\<dots> = (a + b mod c) mod c"
226     by (rule mod_mult_self1)
227   finally show ?thesis .
228 qed
230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
234   assumes "a mod c = a' mod c"
235   assumes "b mod c = b' mod c"
236   shows "(a + b) mod c = (a' + b') mod c"
237 proof -
238   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
239     unfolding assms ..
240   thus ?thesis
241     by (simp only: mod_add_eq [symmetric])
242 qed
244 lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
245   \<Longrightarrow> (x + y) div z = x div z + y div z"
246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
248 text {* Multiplication respects modular equivalence. *}
250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
251 proof -
252   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
253     by (simp only: mod_div_equality)
254   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
255     by (simp only: algebra_simps)
256   also have "\<dots> = (a mod c * b) mod c"
257     by (rule mod_mult_self1)
258   finally show ?thesis .
259 qed
261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
262 proof -
263   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
264     by (simp only: mod_div_equality)
265   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
266     by (simp only: algebra_simps)
267   also have "\<dots> = (a * (b mod c)) mod c"
268     by (rule mod_mult_self1)
269   finally show ?thesis .
270 qed
272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
275 lemma mod_mult_cong:
276   assumes "a mod c = a' mod c"
277   assumes "b mod c = b' mod c"
278   shows "(a * b) mod c = (a' * b') mod c"
279 proof -
280   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
281     unfolding assms ..
282   thus ?thesis
283     by (simp only: mod_mult_eq [symmetric])
284 qed
286 text {* Exponentiation respects modular equivalence. *}
288 lemma power_mod: "(a mod b)^n mod b = a^n mod b"
289 apply (induct n, simp_all)
290 apply (rule mod_mult_right_eq [THEN trans])
291 apply (simp (no_asm_simp))
292 apply (rule mod_mult_eq [symmetric])
293 done
295 lemma mod_mod_cancel:
296   assumes "c dvd b"
297   shows "a mod b mod c = a mod c"
298 proof -
299   from c dvd b obtain k where "b = c * k"
300     by (rule dvdE)
301   have "a mod b mod c = a mod (c * k) mod c"
302     by (simp only: b = c * k)
303   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
304     by (simp only: mod_mult_self1)
305   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
306     by (simp only: add_ac mult_ac)
307   also have "\<dots> = a mod c"
308     by (simp only: mod_div_equality)
309   finally show ?thesis .
310 qed
312 lemma div_mult_div_if_dvd:
313   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
314   apply (cases "y = 0", simp)
315   apply (cases "z = 0", simp)
316   apply (auto elim!: dvdE simp add: algebra_simps)
317   apply (subst mult_assoc [symmetric])
319   done
321 lemma div_mult_swap:
322   assumes "c dvd b"
323   shows "a * (b div c) = (a * b) div c"
324 proof -
325   from assms have "b div c * (a div 1) = b * a div (c * 1)"
326     by (simp only: div_mult_div_if_dvd one_dvd)
327   then show ?thesis by (simp add: mult_commute)
328 qed
330 lemma div_mult_mult2 [simp]:
331   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
332   by (drule div_mult_mult1) (simp add: mult_commute)
334 lemma div_mult_mult1_if [simp]:
335   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
336   by simp_all
338 lemma mod_mult_mult1:
339   "(c * a) mod (c * b) = c * (a mod b)"
340 proof (cases "c = 0")
341   case True then show ?thesis by simp
342 next
343   case False
344   from mod_div_equality
345   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
346   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
347     = c * a + c * (a mod b)" by (simp add: algebra_simps)
348   with mod_div_equality show ?thesis by simp
349 qed
351 lemma mod_mult_mult2:
352   "(a * c) mod (b * c) = (a mod b) * c"
353   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
355 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
356   by (fact mod_mult_mult2 [symmetric])
358 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
359   by (fact mod_mult_mult1 [symmetric])
361 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
362   unfolding dvd_def by (auto simp add: mod_mult_mult1)
364 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
365 by (blast intro: dvd_mod_imp_dvd dvd_mod)
367 lemma div_power:
368   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
369 apply (induct n)
370  apply simp
372 done
374 lemma dvd_div_eq_mult:
375   assumes "a \<noteq> 0" and "a dvd b"
376   shows "b div a = c \<longleftrightarrow> b = c * a"
377 proof
378   assume "b = c * a"
379   then show "b div a = c" by (simp add: assms)
380 next
381   assume "b div a = c"
382   then have "b div a * a = c * a" by simp
383   moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)
384   ultimately show "b = c * a" by simp
385 qed
387 lemma dvd_div_div_eq_mult:
388   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
389   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
390   using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
392 end
394 class ring_div = semiring_div + comm_ring_1
395 begin
397 subclass ring_1_no_zero_divisors ..
399 text {* Negation respects modular equivalence. *}
401 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
402 proof -
403   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
404     by (simp only: mod_div_equality)
405   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
407   also have "\<dots> = (- (a mod b)) mod b"
408     by (rule mod_mult_self1)
409   finally show ?thesis .
410 qed
412 lemma mod_minus_cong:
413   assumes "a mod b = a' mod b"
414   shows "(- a) mod b = (- a') mod b"
415 proof -
416   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
417     unfolding assms ..
418   thus ?thesis
419     by (simp only: mod_minus_eq [symmetric])
420 qed
422 text {* Subtraction respects modular equivalence. *}
424 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
425   unfolding diff_minus
426   by (intro mod_add_cong mod_minus_cong) simp_all
428 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
429   unfolding diff_minus
430   by (intro mod_add_cong mod_minus_cong) simp_all
432 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
433   unfolding diff_minus
434   by (intro mod_add_cong mod_minus_cong) simp_all
436 lemma mod_diff_cong:
437   assumes "a mod c = a' mod c"
438   assumes "b mod c = b' mod c"
439   shows "(a - b) mod c = (a' - b') mod c"
440   unfolding diff_minus using assms
443 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
444 apply (case_tac "y = 0") apply simp
445 apply (auto simp add: dvd_def)
446 apply (subgoal_tac "-(y * k) = y * - k")
447  apply (erule ssubst)
448  apply (erule div_mult_self1_is_id)
449 apply simp
450 done
452 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
453 apply (case_tac "y = 0") apply simp
454 apply (auto simp add: dvd_def)
455 apply (subgoal_tac "y * k = -y * -k")
456  apply (erule ssubst)
457  apply (rule div_mult_self1_is_id)
458  apply simp
459 apply simp
460 done
462 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
463   using div_mult_mult1 [of "- 1" a b]
464   unfolding neg_equal_0_iff_equal by simp
466 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
467   using mod_mult_mult1 [of "- 1" a b] by simp
469 lemma div_minus_right: "a div (-b) = (-a) div b"
470   using div_minus_minus [of "-a" b] by simp
472 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
473   using mod_minus_minus [of "-a" b] by simp
475 lemma div_minus1_right [simp]: "a div (-1) = -a"
476   using div_minus_right [of a 1] by simp
478 lemma mod_minus1_right [simp]: "a mod (-1) = 0"
479   using mod_minus_right [of a 1] by simp
481 end
484 subsection {* Division on @{typ nat} *}
486 text {*
487   We define @{const div} and @{const mod} on @{typ nat} by means
488   of a characteristic relation with two input arguments
489   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
490   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
491 *}
493 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
494   "divmod_nat_rel m n qr \<longleftrightarrow>
495     m = fst qr * n + snd qr \<and>
496       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
498 text {* @{const divmod_nat_rel} is total: *}
500 lemma divmod_nat_rel_ex:
501   obtains q r where "divmod_nat_rel m n (q, r)"
502 proof (cases "n = 0")
503   case True  with that show thesis
504     by (auto simp add: divmod_nat_rel_def)
505 next
506   case False
507   have "\<exists>q r. m = q * n + r \<and> r < n"
508   proof (induct m)
509     case 0 with n \<noteq> 0
510     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
511     then show ?case by blast
512   next
513     case (Suc m) then obtain q' r'
514       where m: "m = q' * n + r'" and n: "r' < n" by auto
515     then show ?case proof (cases "Suc r' < n")
516       case True
517       from m n have "Suc m = q' * n + Suc r'" by simp
518       with True show ?thesis by blast
519     next
520       case False then have "n \<le> Suc r'" by auto
521       moreover from n have "Suc r' \<le> n" by auto
522       ultimately have "n = Suc r'" by auto
523       with m have "Suc m = Suc q' * n + 0" by simp
524       with n \<noteq> 0 show ?thesis by blast
525     qed
526   qed
527   with that show thesis
528     using n \<noteq> 0 by (auto simp add: divmod_nat_rel_def)
529 qed
531 text {* @{const divmod_nat_rel} is injective: *}
533 lemma divmod_nat_rel_unique:
534   assumes "divmod_nat_rel m n qr"
535     and "divmod_nat_rel m n qr'"
536   shows "qr = qr'"
537 proof (cases "n = 0")
538   case True with assms show ?thesis
539     by (cases qr, cases qr')
541 next
542   case False
543   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
544   apply (rule leI)
547   done
548   from n \<noteq> 0 assms have "fst qr = fst qr'"
549     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
550   moreover from this assms have "snd qr = snd qr'"
552   ultimately show ?thesis by (cases qr, cases qr') simp
553 qed
555 text {*
556   We instantiate divisibility on the natural numbers by
557   means of @{const divmod_nat_rel}:
558 *}
560 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
561   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
563 lemma divmod_nat_rel_divmod_nat:
564   "divmod_nat_rel m n (divmod_nat m n)"
565 proof -
566   from divmod_nat_rel_ex
567     obtain qr where rel: "divmod_nat_rel m n qr" .
568   then show ?thesis
569   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
570 qed
572 lemma divmod_nat_unique:
573   assumes "divmod_nat_rel m n qr"
574   shows "divmod_nat m n = qr"
575   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
577 instantiation nat :: semiring_div
578 begin
580 definition div_nat where
581   "m div n = fst (divmod_nat m n)"
583 lemma fst_divmod_nat [simp]:
584   "fst (divmod_nat m n) = m div n"
587 definition mod_nat where
588   "m mod n = snd (divmod_nat m n)"
590 lemma snd_divmod_nat [simp]:
591   "snd (divmod_nat m n) = m mod n"
594 lemma divmod_nat_div_mod:
595   "divmod_nat m n = (m div n, m mod n)"
598 lemma div_nat_unique:
599   assumes "divmod_nat_rel m n (q, r)"
600   shows "m div n = q"
601   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
603 lemma mod_nat_unique:
604   assumes "divmod_nat_rel m n (q, r)"
605   shows "m mod n = r"
606   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
608 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
609   using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
611 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
612   by (simp add: divmod_nat_unique divmod_nat_rel_def)
614 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
615   by (simp add: divmod_nat_unique divmod_nat_rel_def)
617 lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
618   by (simp add: divmod_nat_unique divmod_nat_rel_def)
620 lemma divmod_nat_step:
621   assumes "0 < n" and "n \<le> m"
622   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
623 proof (rule divmod_nat_unique)
624   have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
625     by (rule divmod_nat_rel)
626   thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
627     unfolding divmod_nat_rel_def using assms by auto
628 qed
630 text {* The ''recursion'' equations for @{const div} and @{const mod} *}
632 lemma div_less [simp]:
633   fixes m n :: nat
634   assumes "m < n"
635   shows "m div n = 0"
636   using assms divmod_nat_base by (simp add: prod_eq_iff)
638 lemma le_div_geq:
639   fixes m n :: nat
640   assumes "0 < n" and "n \<le> m"
641   shows "m div n = Suc ((m - n) div n)"
642   using assms divmod_nat_step by (simp add: prod_eq_iff)
644 lemma mod_less [simp]:
645   fixes m n :: nat
646   assumes "m < n"
647   shows "m mod n = m"
648   using assms divmod_nat_base by (simp add: prod_eq_iff)
650 lemma le_mod_geq:
651   fixes m n :: nat
652   assumes "n \<le> m"
653   shows "m mod n = (m - n) mod n"
654   using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
656 instance proof
657   fix m n :: nat
658   show "m div n * n + m mod n = m"
659     using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
660 next
661   fix m n q :: nat
662   assume "n \<noteq> 0"
663   then show "(q + m * n) div n = m + q div n"
664     by (induct m) (simp_all add: le_div_geq)
665 next
666   fix m n q :: nat
667   assume "m \<noteq> 0"
668   hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
669     unfolding divmod_nat_rel_def
670     by (auto split: split_if_asm, simp_all add: algebra_simps)
671   moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
672   ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
673   thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
674 next
675   fix n :: nat show "n div 0 = 0"
676     by (simp add: div_nat_def divmod_nat_zero)
677 next
678   fix n :: nat show "0 div n = 0"
679     by (simp add: div_nat_def divmod_nat_zero_left)
680 qed
682 end
684 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
685   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
686   by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
688 text {* Simproc for cancelling @{const div} and @{const mod} *}
690 ML {*
691 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
692 (
693   val div_name = @{const_name div};
694   val mod_name = @{const_name mod};
695   val mk_binop = HOLogic.mk_binop;
696   val mk_sum = Nat_Arith.mk_sum;
697   val dest_sum = Nat_Arith.dest_sum;
699   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
701   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
703 )
704 *}
706 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
709 subsubsection {* Quotient *}
711 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
712 by (simp add: le_div_geq linorder_not_less)
714 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
717 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
718 by simp
720 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
721 by simp
724 subsubsection {* Remainder *}
726 lemma mod_less_divisor [simp]:
727   fixes m n :: nat
728   assumes "n > 0"
729   shows "m mod n < (n::nat)"
730   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
732 lemma mod_less_eq_dividend [simp]:
733   fixes m n :: nat
734   shows "m mod n \<le> m"
736   from mod_div_equality have "m div n * n + m mod n = m" .
737   then show "m div n * n + m mod n \<le> m" by auto
738 qed
740 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
741 by (simp add: le_mod_geq linorder_not_less)
743 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
746 lemma mod_1 [simp]: "m mod Suc 0 = 0"
747 by (induct m) (simp_all add: mod_geq)
749 (* a simple rearrangement of mod_div_equality: *)
750 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
751   using mod_div_equality2 [of n m] by arith
753 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
754   apply (drule mod_less_divisor [where m = m])
755   apply simp
756   done
758 subsubsection {* Quotient and Remainder *}
760 lemma divmod_nat_rel_mult1_eq:
761   "divmod_nat_rel b c (q, r)
762    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
763 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
765 lemma div_mult1_eq:
766   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
767 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
770   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
771    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
772 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
774 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
776   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
777 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
779 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
780   apply (cut_tac m = q and n = c in mod_less_divisor)
781   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
782   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
784   done
786 lemma divmod_nat_rel_mult2_eq:
787   "divmod_nat_rel a b (q, r)
788    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
791 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
792 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
794 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
795 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
798 subsubsection {* Further Facts about Quotient and Remainder *}
800 lemma div_1 [simp]: "m div Suc 0 = m"
801 by (induct m) (simp_all add: div_geq)
803 (* Monotonicity of div in first argument *)
804 lemma div_le_mono [rule_format (no_asm)]:
805     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
806 apply (case_tac "k=0", simp)
807 apply (induct "n" rule: nat_less_induct, clarify)
808 apply (case_tac "n<k")
809 (* 1  case n<k *)
810 apply simp
811 (* 2  case n >= k *)
812 apply (case_tac "m<k")
813 (* 2.1  case m<k *)
814 apply simp
815 (* 2.2  case m>=k *)
816 apply (simp add: div_geq diff_le_mono)
817 done
819 (* Antimonotonicity of div in second argument *)
820 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
821 apply (subgoal_tac "0<n")
822  prefer 2 apply simp
823 apply (induct_tac k rule: nat_less_induct)
824 apply (rename_tac "k")
825 apply (case_tac "k<n", simp)
826 apply (subgoal_tac "~ (k<m) ")
827  prefer 2 apply simp
829 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
830  prefer 2
831  apply (blast intro: div_le_mono diff_le_mono2)
832 apply (rule le_trans, simp)
833 apply (simp)
834 done
836 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
837 apply (case_tac "n=0", simp)
838 apply (subgoal_tac "m div n \<le> m div 1", simp)
839 apply (rule div_le_mono2)
840 apply (simp_all (no_asm_simp))
841 done
843 (* Similar for "less than" *)
844 lemma div_less_dividend [simp]:
845   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
846 apply (induct m rule: nat_less_induct)
847 apply (rename_tac "m")
848 apply (case_tac "m<n", simp)
849 apply (subgoal_tac "0<n")
850  prefer 2 apply simp
852 apply (case_tac "n<m")
853  apply (subgoal_tac "(m-n) div n < (m-n) ")
854   apply (rule impI less_trans_Suc)+
855 apply assumption
856   apply (simp_all)
857 done
859 text{*A fact for the mutilated chess board*}
860 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
861 apply (case_tac "n=0", simp)
862 apply (induct "m" rule: nat_less_induct)
863 apply (case_tac "Suc (na) <n")
864 (* case Suc(na) < n *)
865 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
866 (* case n \<le> Suc(na) *)
867 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
868 apply (auto simp add: Suc_diff_le le_mod_geq)
869 done
871 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
872 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
874 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
876 (*Loses information, namely we also have r<d provided d is nonzero*)
877 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
878   apply (cut_tac a = m in mod_div_equality)
880   apply (blast intro: sym)
881   done
883 lemma split_div:
884  "P(n div k :: nat) =
885  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
886  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
887 proof
888   assume P: ?P
889   show ?Q
890   proof (cases)
891     assume "k = 0"
892     with P show ?Q by simp
893   next
894     assume not0: "k \<noteq> 0"
895     thus ?Q
896     proof (simp, intro allI impI)
897       fix i j
898       assume n: "n = k*i + j" and j: "j < k"
899       show "P i"
900       proof (cases)
901         assume "i = 0"
902         with n j P show "P i" by simp
903       next
904         assume "i \<noteq> 0"
906       qed
907     qed
908   qed
909 next
910   assume Q: ?Q
911   show ?P
912   proof (cases)
913     assume "k = 0"
914     with Q show ?P by simp
915   next
916     assume not0: "k \<noteq> 0"
917     with Q have R: ?R by simp
918     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
919     show ?P by simp
920   qed
921 qed
923 lemma split_div_lemma:
924   assumes "0 < n"
925   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
926 proof
927   assume ?rhs
928   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
929   then have A: "n * q \<le> m" by simp
930   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
931   then have "m < m + (n - (m mod n))" by simp
932   then have "m < n + (m - (m mod n))" by simp
933   with nq have "m < n + n * q" by simp
934   then have B: "m < n * Suc q" by simp
935   from A B show ?lhs ..
936 next
937   assume P: ?lhs
938   then have "divmod_nat_rel m n (q, m - n * q)"
939     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
940   with divmod_nat_rel_unique divmod_nat_rel [of m n]
941   have "(q, m - n * q) = (m div n, m mod n)" by auto
942   then show ?rhs by simp
943 qed
945 theorem split_div':
946   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
947    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
948   apply (case_tac "0 < n")
949   apply (simp only: add: split_div_lemma)
950   apply simp_all
951   done
953 lemma split_mod:
954  "P(n mod k :: nat) =
955  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
956  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
957 proof
958   assume P: ?P
959   show ?Q
960   proof (cases)
961     assume "k = 0"
962     with P show ?Q by simp
963   next
964     assume not0: "k \<noteq> 0"
965     thus ?Q
966     proof (simp, intro allI impI)
967       fix i j
968       assume "n = k*i + j" "j < k"
970     qed
971   qed
972 next
973   assume Q: ?Q
974   show ?P
975   proof (cases)
976     assume "k = 0"
977     with Q show ?P by simp
978   next
979     assume not0: "k \<noteq> 0"
980     with Q have R: ?R by simp
981     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
982     show ?P by simp
983   qed
984 qed
986 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
987   using mod_div_equality [of m n] by arith
989 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
990   using mod_div_equality [of m n] by arith
991 (* FIXME: very similar to mult_div_cancel *)
994 subsubsection {* An induction'' law for modulus arithmetic. *}
996 lemma mod_induct_0:
997   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
998   and base: "P i" and i: "i<p"
999   shows "P 0"
1000 proof (rule ccontr)
1001   assume contra: "\<not>(P 0)"
1002   from i have p: "0<p" by simp
1003   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
1004   proof
1005     fix k
1006     show "?A k"
1007     proof (induct k)
1008       show "?A 0" by simp  -- "by contradiction"
1009     next
1010       fix n
1011       assume ih: "?A n"
1012       show "?A (Suc n)"
1013       proof (clarsimp)
1014         assume y: "P (p - Suc n)"
1015         have n: "Suc n < p"
1016         proof (rule ccontr)
1017           assume "\<not>(Suc n < p)"
1018           hence "p - Suc n = 0"
1019             by simp
1020           with y contra show "False"
1021             by simp
1022         qed
1023         hence n2: "Suc (p - Suc n) = p-n" by arith
1024         from p have "p - Suc n < p" by arith
1025         with y step have z: "P ((Suc (p - Suc n)) mod p)"
1026           by blast
1027         show "False"
1028         proof (cases "n=0")
1029           case True
1030           with z n2 contra show ?thesis by simp
1031         next
1032           case False
1033           with p have "p-n < p" by arith
1034           with z n2 False ih show ?thesis by simp
1035         qed
1036       qed
1037     qed
1038   qed
1039   moreover
1040   from i obtain k where "0<k \<and> i+k=p"
1042   hence "0<k \<and> i=p-k" by auto
1043   moreover
1044   note base
1045   ultimately
1046   show "False" by blast
1047 qed
1049 lemma mod_induct:
1050   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
1051   and base: "P i" and i: "i<p" and j: "j<p"
1052   shows "P j"
1053 proof -
1054   have "\<forall>j<p. P j"
1055   proof
1056     fix j
1057     show "j<p \<longrightarrow> P j" (is "?A j")
1058     proof (induct j)
1059       from step base i show "?A 0"
1060         by (auto elim: mod_induct_0)
1061     next
1062       fix k
1063       assume ih: "?A k"
1064       show "?A (Suc k)"
1065       proof
1066         assume suc: "Suc k < p"
1067         hence k: "k<p" by simp
1068         with ih have "P k" ..
1069         with step k have "P (Suc k mod p)"
1070           by blast
1071         moreover
1072         from suc have "Suc k mod p = Suc k"
1073           by simp
1074         ultimately
1075         show "P (Suc k)" by simp
1076       qed
1077     qed
1078   qed
1079   with j show ?thesis by blast
1080 qed
1082 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
1083   by (simp add: numeral_2_eq_2 le_div_geq)
1085 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
1086   by (simp add: numeral_2_eq_2 le_mod_geq)
1088 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
1089 by (simp add: mult_2 [symmetric])
1091 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
1092 proof -
1093   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
1094   moreover have "m mod 2 < 2" by simp
1095   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
1096   then show ?thesis by auto
1097 qed
1099 text{*These lemmas collapse some needless occurrences of Suc:
1100     at least three Sucs, since two and fewer are rewritten back to Suc again!
1101     We already have some rules to simplify operands smaller than 3.*}
1103 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
1106 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
1109 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
1112 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
1115 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
1116 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
1119 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
1120 apply (induct "m")
1122 done
1124 declare Suc_times_mod_eq [of "numeral w", simp] for w
1126 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
1129 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
1130 by (cases n) simp_all
1132 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
1133 proof -
1134   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
1135   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
1136 qed
1138   (* Potential use of algebra : Equality modulo n*)
1139 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
1142 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
1143 proof -
1144   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
1145   also have "... = Suc m mod n" by (rule mod_mult_self3)
1146   finally show ?thesis .
1147 qed
1149 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
1150 apply (subst mod_Suc [of m])
1151 apply (subst mod_Suc [of "m mod n"], simp)
1152 done
1154 lemma mod_2_not_eq_zero_eq_one_nat:
1155   fixes n :: nat
1156   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
1157   by simp
1160 subsection {* Division on @{typ int} *}
1162 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
1163     --{*definition of quotient and remainder*}
1164   "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
1165     (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
1167 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
1168     --{*for the division algorithm*}
1169     "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
1170                          else (2 * q, r))"
1172 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
1173 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1174   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
1175      else adjust b (posDivAlg a (2 * b)))"
1176 by auto
1177 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
1180 text{*algorithm for the case @{text "a<0, b>0"}*}
1181 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1182   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
1183      else adjust b (negDivAlg a (2 * b)))"
1184 by auto
1185 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
1188 text{*algorithm for the general case @{term "b\<noteq>0"}*}
1190 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1191     --{*The full division algorithm considers all possible signs for a, b
1192        including the special case @{text "a=0, b<0"} because
1193        @{term negDivAlg} requires @{term "a<0"}.*}
1194   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
1195                   else if a = 0 then (0, 0)
1196                        else apsnd uminus (negDivAlg (-a) (-b))
1197                else
1198                   if 0 < b then negDivAlg a b
1199                   else apsnd uminus (posDivAlg (-a) (-b)))"
1201 instantiation int :: Divides.div
1202 begin
1204 definition div_int where
1205   "a div b = fst (divmod_int a b)"
1207 lemma fst_divmod_int [simp]:
1208   "fst (divmod_int a b) = a div b"
1211 definition mod_int where
1212   "a mod b = snd (divmod_int a b)"
1214 lemma snd_divmod_int [simp]:
1215   "snd (divmod_int a b) = a mod b"
1218 instance ..
1220 end
1222 lemma divmod_int_mod_div:
1223   "divmod_int p q = (p div q, p mod q)"
1226 text{*
1227 Here is the division algorithm in ML:
1229 \begin{verbatim}
1230     fun posDivAlg (a,b) =
1231       if a<b then (0,a)
1232       else let val (q,r) = posDivAlg(a, 2*b)
1233                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1234            end
1236     fun negDivAlg (a,b) =
1237       if 0\<le>a+b then (~1,a+b)
1238       else let val (q,r) = negDivAlg(a, 2*b)
1239                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1240            end;
1242     fun negateSnd (q,r:int) = (q,~r);
1244     fun divmod (a,b) = if 0\<le>a then
1245                           if b>0 then posDivAlg (a,b)
1246                            else if a=0 then (0,0)
1247                                 else negateSnd (negDivAlg (~a,~b))
1248                        else
1249                           if 0<b then negDivAlg (a,b)
1250                           else        negateSnd (posDivAlg (~a,~b));
1251 \end{verbatim}
1252 *}
1255 subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
1257 lemma unique_quotient_lemma:
1258      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]
1259       ==> q' \<le> (q::int)"
1260 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
1261  prefer 2 apply (simp add: right_diff_distrib)
1262 apply (subgoal_tac "0 < b * (1 + q - q') ")
1263 apply (erule_tac [2] order_le_less_trans)
1264  prefer 2 apply (simp add: right_diff_distrib right_distrib)
1265 apply (subgoal_tac "b * q' < b * (1 + q) ")
1266  prefer 2 apply (simp add: right_diff_distrib right_distrib)
1268 done
1270 lemma unique_quotient_lemma_neg:
1271      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]
1272       ==> q \<le> (q'::int)"
1273 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
1274     auto)
1276 lemma unique_quotient:
1277      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]
1278       ==> q = q'"
1279 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
1280 apply (blast intro: order_antisym
1281              dest: order_eq_refl [THEN unique_quotient_lemma]
1282              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
1283 done
1286 lemma unique_remainder:
1287      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]
1288       ==> r = r'"
1289 apply (subgoal_tac "q = q'")
1291 apply (blast intro: unique_quotient)
1292 done
1295 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
1297 text{*And positive divisors*}
1300      "adjust b (q, r) =
1301       (let diff = r - b in
1302         if 0 \<le> diff then (2 * q + 1, diff)
1303                      else (2*q, r))"
1306 declare posDivAlg.simps [simp del]
1308 text{*use with a simproc to avoid repeatedly proving the premise*}
1309 lemma posDivAlg_eqn:
1310      "0 < b ==>
1311       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
1312 by (rule posDivAlg.simps [THEN trans], simp)
1314 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
1315 theorem posDivAlg_correct:
1316   assumes "0 \<le> a" and "0 < b"
1317   shows "divmod_int_rel a b (posDivAlg a b)"
1318   using assms
1319   apply (induct a b rule: posDivAlg.induct)
1320   apply auto
1322   apply (subst posDivAlg_eqn, simp add: right_distrib)
1323   apply (case_tac "a < b")
1324   apply simp_all
1325   apply (erule splitE)
1326   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
1327   done
1330 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
1332 text{*And positive divisors*}
1334 declare negDivAlg.simps [simp del]
1336 text{*use with a simproc to avoid repeatedly proving the premise*}
1337 lemma negDivAlg_eqn:
1338      "0 < b ==>
1339       negDivAlg a b =
1340        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
1341 by (rule negDivAlg.simps [THEN trans], simp)
1343 (*Correctness of negDivAlg: it computes quotients correctly
1344   It doesn't work if a=0 because the 0/b equals 0, not -1*)
1345 lemma negDivAlg_correct:
1346   assumes "a < 0" and "b > 0"
1347   shows "divmod_int_rel a b (negDivAlg a b)"
1348   using assms
1349   apply (induct a b rule: negDivAlg.induct)
1350   apply (auto simp add: linorder_not_le)
1352   apply (subst negDivAlg_eqn, assumption)
1353   apply (case_tac "a + b < (0\<Colon>int)")
1354   apply simp_all
1355   apply (erule splitE)
1356   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
1357   done
1360 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
1362 (*the case a=0*)
1363 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
1364 by (auto simp add: divmod_int_rel_def linorder_neq_iff)
1366 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
1367 by (subst posDivAlg.simps, auto)
1369 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
1370 by (subst posDivAlg.simps, auto)
1372 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
1373 by (subst negDivAlg.simps, auto)
1375 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
1376 by (auto simp add: divmod_int_rel_def)
1378 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
1379 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
1380 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
1381                     posDivAlg_correct negDivAlg_correct)
1383 lemma divmod_int_unique:
1384   assumes "divmod_int_rel a b qr"
1385   shows "divmod_int a b = qr"
1386   using assms divmod_int_correct [of a b]
1387   using unique_quotient [of a b] unique_remainder [of a b]
1388   by (metis pair_collapse)
1390 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
1391   using divmod_int_correct by (simp add: divmod_int_mod_div)
1393 lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
1394   by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
1396 lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
1397   by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
1399 instance int :: ring_div
1400 proof
1401   fix a b :: int
1402   show "a div b * b + a mod b = a"
1403     using divmod_int_rel_div_mod [of a b]
1404     unfolding divmod_int_rel_def by (simp add: mult_commute)
1405 next
1406   fix a b c :: int
1407   assume "b \<noteq> 0"
1408   hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
1409     using divmod_int_rel_div_mod [of a b]
1410     unfolding divmod_int_rel_def by (auto simp: algebra_simps)
1411   thus "(a + c * b) div b = c + a div b"
1412     by (rule div_int_unique)
1413 next
1414   fix a b c :: int
1415   assume "c \<noteq> 0"
1416   hence "\<And>q r. divmod_int_rel a b (q, r)
1417     \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
1418     unfolding divmod_int_rel_def
1419     by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
1420       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
1421       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
1422   hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
1423     using divmod_int_rel_div_mod [of a b] .
1424   thus "(c * a) div (c * b) = a div b"
1425     by (rule div_int_unique)
1426 next
1427   fix a :: int show "a div 0 = 0"
1428     by (rule div_int_unique, simp add: divmod_int_rel_def)
1429 next
1430   fix a :: int show "0 div a = 0"
1431     by (rule div_int_unique, auto simp add: divmod_int_rel_def)
1432 qed
1434 text{*Basic laws about division and remainder*}
1436 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
1437   by (fact mod_div_equality2 [symmetric])
1439 text {* Tool setup *}
1441 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
1444 ML {*
1445 structure Cancel_Div_Mod_Int = Cancel_Div_Mod
1446 (
1447   val div_name = @{const_name div};
1448   val mod_name = @{const_name mod};
1449   val mk_binop = HOLogic.mk_binop;
1450   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
1451   val dest_sum = Arith_Data.dest_sum;
1453   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
1455   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
1457 )
1458 *}
1460 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
1462 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
1463   using divmod_int_correct [of a b]
1464   by (auto simp add: divmod_int_rel_def prod_eq_iff)
1466 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
1467    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
1469 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
1470   using divmod_int_correct [of a b]
1471   by (auto simp add: divmod_int_rel_def prod_eq_iff)
1473 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
1474    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
1477 subsubsection {* General Properties of div and mod *}
1479 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
1480 apply (rule div_int_unique)
1481 apply (auto simp add: divmod_int_rel_def)
1482 done
1484 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
1485 apply (rule div_int_unique)
1486 apply (auto simp add: divmod_int_rel_def)
1487 done
1489 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
1490 apply (rule div_int_unique)
1491 apply (auto simp add: divmod_int_rel_def)
1492 done
1494 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
1496 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
1497 apply (rule_tac q = 0 in mod_int_unique)
1498 apply (auto simp add: divmod_int_rel_def)
1499 done
1501 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
1502 apply (rule_tac q = 0 in mod_int_unique)
1503 apply (auto simp add: divmod_int_rel_def)
1504 done
1506 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
1507 apply (rule_tac q = "-1" in mod_int_unique)
1508 apply (auto simp add: divmod_int_rel_def)
1509 done
1511 text{*There is no @{text mod_neg_pos_trivial}.*}
1514 subsubsection {* Laws for div and mod with Unary Minus *}
1516 lemma zminus1_lemma:
1517      "divmod_int_rel a b (q, r) ==> b \<noteq> 0
1518       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,
1519                           if r=0 then 0 else b-r)"
1520 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
1523 lemma zdiv_zminus1_eq_if:
1524      "b \<noteq> (0::int)
1525       ==> (-a) div b =
1526           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
1527 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
1529 lemma zmod_zminus1_eq_if:
1530      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
1531 apply (case_tac "b = 0", simp)
1532 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
1533 done
1535 lemma zmod_zminus1_not_zero:
1536   fixes k l :: int
1537   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
1538   unfolding zmod_zminus1_eq_if by auto
1540 lemma zdiv_zminus2_eq_if:
1541      "b \<noteq> (0::int)
1542       ==> a div (-b) =
1543           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
1544 by (simp add: zdiv_zminus1_eq_if div_minus_right)
1546 lemma zmod_zminus2_eq_if:
1547      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
1548 by (simp add: zmod_zminus1_eq_if mod_minus_right)
1550 lemma zmod_zminus2_not_zero:
1551   fixes k l :: int
1552   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
1553   unfolding zmod_zminus2_eq_if by auto
1556 subsubsection {* Computation of Division and Remainder *}
1558 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
1559 by (simp add: div_int_def divmod_int_def)
1561 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
1562 by (simp add: mod_int_def divmod_int_def)
1564 text{*a positive, b positive *}
1566 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
1567 by (simp add: div_int_def divmod_int_def)
1569 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
1570 by (simp add: mod_int_def divmod_int_def)
1572 text{*a negative, b positive *}
1574 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
1575 by (simp add: div_int_def divmod_int_def)
1577 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
1578 by (simp add: mod_int_def divmod_int_def)
1580 text{*a positive, b negative *}
1582 lemma div_pos_neg:
1583      "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
1584 by (simp add: div_int_def divmod_int_def)
1586 lemma mod_pos_neg:
1587      "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
1588 by (simp add: mod_int_def divmod_int_def)
1590 text{*a negative, b negative *}
1592 lemma div_neg_neg:
1593      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
1594 by (simp add: div_int_def divmod_int_def)
1596 lemma mod_neg_neg:
1597      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
1598 by (simp add: mod_int_def divmod_int_def)
1600 text {*Simplify expresions in which div and mod combine numerical constants*}
1602 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
1603   by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
1605 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
1606   by (rule div_int_unique [of a b q r],
1609 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
1610   by (rule mod_int_unique [of a b q r],
1613 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
1614   by (rule mod_int_unique [of a b q r],
1617 (* simprocs adapted from HOL/ex/Binary.thy *)
1618 ML {*
1619 local
1620   val mk_number = HOLogic.mk_number HOLogic.intT
1621   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
1622   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
1623   val zero = @{term "0 :: int"}
1624   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
1625   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
1626   val simps = @{thms arith_simps} @ @{thms rel_simps} @
1627     map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
1628   fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
1629     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));
1630   fun binary_proc proc ss ct =
1631     (case Thm.term_of ct of
1632       _ $t$ u =>
1633       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of
1634         SOME args => proc (Simplifier.the_context ss) args
1635       | NONE => NONE)
1636     | _ => NONE);
1637 in
1638   fun divmod_proc posrule negrule =
1639     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
1640       if b = 0 then NONE else let
1641         val (q, r) = pairself mk_number (Integer.div_mod a b)
1642         val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)
1643         val (goal2, goal3, rule) = if b > 0
1644           then (le $zero$ r, less $r$ u, posrule RS eq_reflection)
1645           else (le $r$ zero, less $u$ r, negrule RS eq_reflection)
1646       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
1647 end
1648 *}
1650 simproc_setup binary_int_div
1651   ("numeral m div numeral n :: int" |
1652    "numeral m div neg_numeral n :: int" |
1653    "neg_numeral m div numeral n :: int" |
1654    "neg_numeral m div neg_numeral n :: int") =
1655   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
1657 simproc_setup binary_int_mod
1658   ("numeral m mod numeral n :: int" |
1659    "numeral m mod neg_numeral n :: int" |
1660    "neg_numeral m mod numeral n :: int" |
1661    "neg_numeral m mod neg_numeral n :: int") =
1662   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
1664 lemmas posDivAlg_eqn_numeral [simp] =
1665     posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
1667 lemmas negDivAlg_eqn_numeral [simp] =
1668     negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
1671 text{*Special-case simplification *}
1673 (** The last remaining special cases for constant arithmetic:
1674     1 div z and 1 mod z **)
1676 lemmas div_pos_pos_1_numeral [simp] =
1677   div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
1679 lemmas div_pos_neg_1_numeral [simp] =
1680   div_pos_neg [OF zero_less_one, of "neg_numeral w",
1681   OF neg_numeral_less_zero] for w
1683 lemmas mod_pos_pos_1_numeral [simp] =
1684   mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
1686 lemmas mod_pos_neg_1_numeral [simp] =
1687   mod_pos_neg [OF zero_less_one, of "neg_numeral w",
1688   OF neg_numeral_less_zero] for w
1690 lemmas posDivAlg_eqn_1_numeral [simp] =
1691     posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
1693 lemmas negDivAlg_eqn_1_numeral [simp] =
1694     negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
1697 subsubsection {* Monotonicity in the First Argument (Dividend) *}
1699 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
1700 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1701 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1702 apply (rule unique_quotient_lemma)
1703 apply (erule subst)
1704 apply (erule subst, simp_all)
1705 done
1707 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
1708 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1709 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1710 apply (rule unique_quotient_lemma_neg)
1711 apply (erule subst)
1712 apply (erule subst, simp_all)
1713 done
1716 subsubsection {* Monotonicity in the Second Argument (Divisor) *}
1718 lemma q_pos_lemma:
1719      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
1720 apply (subgoal_tac "0 < b'* (q' + 1) ")
1723 done
1725 lemma zdiv_mono2_lemma:
1726      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
1727          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
1728       ==> q \<le> (q'::int)"
1729 apply (frule q_pos_lemma, assumption+)
1730 apply (subgoal_tac "b*q < b* (q' + 1) ")
1732 apply (subgoal_tac "b*q = r' - r + b'*q'")
1733  prefer 2 apply simp
1734 apply (simp (no_asm_simp) add: right_distrib)
1736 apply (rule mult_right_mono, auto)
1737 done
1739 lemma zdiv_mono2:
1740      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
1741 apply (subgoal_tac "b \<noteq> 0")
1742  prefer 2 apply arith
1743 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1744 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1745 apply (rule zdiv_mono2_lemma)
1746 apply (erule subst)
1747 apply (erule subst, simp_all)
1748 done
1750 lemma q_neg_lemma:
1751      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
1752 apply (subgoal_tac "b'*q' < 0")
1753  apply (simp add: mult_less_0_iff, arith)
1754 done
1756 lemma zdiv_mono2_neg_lemma:
1757      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
1758          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
1759       ==> q' \<le> (q::int)"
1760 apply (frule q_neg_lemma, assumption+)
1761 apply (subgoal_tac "b*q' < b* (q + 1) ")
1764 apply (subgoal_tac "b*q' \<le> b'*q'")
1765  prefer 2 apply (simp add: mult_right_mono_neg, arith)
1766 done
1768 lemma zdiv_mono2_neg:
1769      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
1770 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1771 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1772 apply (rule zdiv_mono2_neg_lemma)
1773 apply (erule subst)
1774 apply (erule subst, simp_all)
1775 done
1778 subsubsection {* More Algebraic Laws for div and mod *}
1780 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
1782 lemma zmult1_lemma:
1783      "[| divmod_int_rel b c (q, r) |]
1784       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
1785 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
1787 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
1788 apply (case_tac "c = 0", simp)
1789 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
1790 done
1792 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
1795      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]
1796       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
1797 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
1799 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
1801      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
1802 apply (case_tac "c = 0", simp)
1803 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
1804 done
1806 lemma posDivAlg_div_mod:
1807   assumes "k \<ge> 0"
1808   and "l \<ge> 0"
1809   shows "posDivAlg k l = (k div l, k mod l)"
1810 proof (cases "l = 0")
1811   case True then show ?thesis by (simp add: posDivAlg.simps)
1812 next
1813   case False with assms posDivAlg_correct
1814     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
1815     by simp
1816   from div_int_unique [OF this] mod_int_unique [OF this]
1817   show ?thesis by simp
1818 qed
1820 lemma negDivAlg_div_mod:
1821   assumes "k < 0"
1822   and "l > 0"
1823   shows "negDivAlg k l = (k div l, k mod l)"
1824 proof -
1825   from assms have "l \<noteq> 0" by simp
1826   from assms negDivAlg_correct
1827     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
1828     by simp
1829   from div_int_unique [OF this] mod_int_unique [OF this]
1830   show ?thesis by simp
1831 qed
1833 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
1834 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
1836 (* REVISIT: should this be generalized to all semiring_div types? *)
1837 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
1839 lemma zmod_zdiv_equality':
1840   "(m\<Colon>int) mod n = m - (m div n) * n"
1841   using mod_div_equality [of m n] by arith
1844 subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
1846 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
1847   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
1848   to cause particular problems.*)
1850 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
1852 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
1853 apply (subgoal_tac "b * (c - q mod c) < r * 1")
1855 apply (rule order_le_less_trans)
1856  apply (erule_tac [2] mult_strict_right_mono)
1857  apply (rule mult_left_mono_neg)
1859  apply (simp)
1860 apply (simp)
1861 done
1863 lemma zmult2_lemma_aux2:
1864      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
1865 apply (subgoal_tac "b * (q mod c) \<le> 0")
1866  apply arith
1868 done
1870 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
1871 apply (subgoal_tac "0 \<le> b * (q mod c) ")
1872 apply arith
1874 done
1876 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
1877 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
1879 apply (rule order_less_le_trans)
1880  apply (erule mult_strict_right_mono)
1881  apply (rule_tac [2] mult_left_mono)
1882   apply simp
1884 apply simp
1885 done
1887 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]
1888       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
1889 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
1890                    zero_less_mult_iff right_distrib [symmetric]
1891                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
1893 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
1894 apply (case_tac "b = 0", simp)
1895 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
1896 done
1898 lemma zmod_zmult2_eq:
1899      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
1900 apply (case_tac "b = 0", simp)
1901 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
1902 done
1904 lemma div_pos_geq:
1905   fixes k l :: int
1906   assumes "0 < l" and "l \<le> k"
1907   shows "k div l = (k - l) div l + 1"
1908 proof -
1909   have "k = (k - l) + l" by simp
1910   then obtain j where k: "k = j + l" ..
1911   with assms show ?thesis by simp
1912 qed
1914 lemma mod_pos_geq:
1915   fixes k l :: int
1916   assumes "0 < l" and "l \<le> k"
1917   shows "k mod l = (k - l) mod l"
1918 proof -
1919   have "k = (k - l) + l" by simp
1920   then obtain j where k: "k = j + l" ..
1921   with assms show ?thesis by simp
1922 qed
1925 subsubsection {* Splitting Rules for div and mod *}
1927 text{*The proofs of the two lemmas below are essentially identical*}
1929 lemma split_pos_lemma:
1930  "0<k ==>
1931     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
1932 apply (rule iffI, clarify)
1933  apply (erule_tac P="P ?x ?y" in rev_mp)
1936  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
1937 txt{*converse direction*}
1938 apply (drule_tac x = "n div k" in spec)
1939 apply (drule_tac x = "n mod k" in spec, simp)
1940 done
1942 lemma split_neg_lemma:
1943  "k<0 ==>
1944     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
1945 apply (rule iffI, clarify)
1946  apply (erule_tac P="P ?x ?y" in rev_mp)
1949  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
1950 txt{*converse direction*}
1951 apply (drule_tac x = "n div k" in spec)
1952 apply (drule_tac x = "n mod k" in spec, simp)
1953 done
1955 lemma split_zdiv:
1956  "P(n div k :: int) =
1957   ((k = 0 --> P 0) &
1958    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
1959    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
1960 apply (case_tac "k=0", simp)
1961 apply (simp only: linorder_neq_iff)
1962 apply (erule disjE)
1963  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
1964                       split_neg_lemma [of concl: "%x y. P x"])
1965 done
1967 lemma split_zmod:
1968  "P(n mod k :: int) =
1969   ((k = 0 --> P n) &
1970    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
1971    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
1972 apply (case_tac "k=0", simp)
1973 apply (simp only: linorder_neq_iff)
1974 apply (erule disjE)
1975  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
1976                       split_neg_lemma [of concl: "%x y. P y"])
1977 done
1979 text {* Enable (lin)arith to deal with @{const div} and @{const mod}
1980   when these are applied to some constant that is of the form
1981   @{term "numeral k"}: *}
1982 declare split_zdiv [of _ _ "numeral k", arith_split] for k
1983 declare split_zmod [of _ _ "numeral k", arith_split] for k
1986 subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
1988 lemma pos_divmod_int_rel_mult_2:
1989   assumes "0 \<le> b"
1990   assumes "divmod_int_rel a b (q, r)"
1991   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
1992   using assms unfolding divmod_int_rel_def by auto
1994 lemma neg_divmod_int_rel_mult_2:
1995   assumes "b \<le> 0"
1996   assumes "divmod_int_rel (a + 1) b (q, r)"
1997   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
1998   using assms unfolding divmod_int_rel_def by auto
2000 text{*computing div by shifting *}
2002 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
2003   using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
2004   by (rule div_int_unique)
2006 lemma neg_zdiv_mult_2:
2007   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
2008   using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
2009   by (rule div_int_unique)
2011 (* FIXME: add rules for negative numerals *)
2012 lemma zdiv_numeral_Bit0 [simp]:
2013   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
2014     numeral v div (numeral w :: int)"
2015   unfolding numeral.simps unfolding mult_2 [symmetric]
2016   by (rule div_mult_mult1, simp)
2018 lemma zdiv_numeral_Bit1 [simp]:
2019   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
2020     (numeral v div (numeral w :: int))"
2021   unfolding numeral.simps
2022   unfolding mult_2 [symmetric] add_commute [of _ 1]
2023   by (rule pos_zdiv_mult_2, simp)
2025 lemma pos_zmod_mult_2:
2026   fixes a b :: int
2027   assumes "0 \<le> a"
2028   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
2029   using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
2030   by (rule mod_int_unique)
2032 lemma neg_zmod_mult_2:
2033   fixes a b :: int
2034   assumes "a \<le> 0"
2035   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
2036   using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
2037   by (rule mod_int_unique)
2039 (* FIXME: add rules for negative numerals *)
2040 lemma zmod_numeral_Bit0 [simp]:
2041   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
2042     (2::int) * (numeral v mod numeral w)"
2043   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
2044   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
2046 lemma zmod_numeral_Bit1 [simp]:
2047   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
2048     2 * (numeral v mod numeral w) + (1::int)"
2049   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
2050   unfolding mult_2 [symmetric] add_commute [of _ 1]
2051   by (rule pos_zmod_mult_2, simp)
2053 lemma zdiv_eq_0_iff:
2054  "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
2055 proof
2056   assume ?L
2057   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
2058   with ?L` show ?R by blast
2059 next
2060   assume ?R thus ?L
2061     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
2062 qed
2065 subsubsection {* Quotients of Signs *}
2067 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
2068 apply (subgoal_tac "a div b \<le> -1", force)
2069 apply (rule order_trans)
2070 apply (rule_tac a' = "-1" in zdiv_mono1)
2071 apply (auto simp add: div_eq_minus1)
2072 done
2074 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
2075 by (drule zdiv_mono1_neg, auto)
2077 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
2078 by (drule zdiv_mono1, auto)
2080 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
2081 conditional upon the sign of @{text a} or @{text b}. There are many more.
2082 They should all be simp rules unless that causes too much search. *}
2084 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
2085 apply auto
2086 apply (drule_tac [2] zdiv_mono1)
2087 apply (auto simp add: linorder_neq_iff)
2088 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
2089 apply (blast intro: div_neg_pos_less0)
2090 done
2092 lemma neg_imp_zdiv_nonneg_iff:
2093   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
2094 apply (subst div_minus_minus [symmetric])
2095 apply (subst pos_imp_zdiv_nonneg_iff, auto)
2096 done
2098 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
2099 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
2100 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
2102 lemma pos_imp_zdiv_pos_iff:
2103   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
2104 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
2105 by arith
2107 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
2108 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
2109 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
2111 lemma nonneg1_imp_zdiv_pos_iff:
2112   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
2113 apply rule
2114  apply rule
2115   using div_pos_pos_trivial[of a b]apply arith
2116  apply(cases "b=0")apply simp
2117  using div_nonneg_neg_le0[of a b]apply arith
2118 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
2119 done
2121 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
2122 apply (rule split_zmod[THEN iffD2])
2123 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
2124 done
2127 subsubsection {* The Divides Relation *}
2129 lemmas zdvd_iff_zmod_eq_0_numeral [simp] =
2130   dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]
2131   dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]
2132   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]
2133   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y
2135 lemmas dvd_eq_mod_eq_0_numeral [simp] =
2136   dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
2139 subsubsection {* Further properties *}
2141 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
2142   using zmod_zdiv_equality[where a="m" and b="n"]
2143   by (simp add: algebra_simps) (* FIXME: generalize *)
2145 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
2146 apply (subst split_div, auto)
2147 apply (subst split_zdiv, auto)
2148 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
2149 apply (auto simp add: divmod_int_rel_def of_nat_mult)
2150 done
2152 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
2153 apply (subst split_mod, auto)
2154 apply (subst split_zmod, auto)
2155 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia
2156        in unique_remainder)
2157 apply (auto simp add: divmod_int_rel_def of_nat_mult)
2158 done
2160 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
2161 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
2163 text{*Suggested by Matthias Daum*}
2164 lemma int_power_div_base:
2165      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
2166 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
2167  apply (erule ssubst)
2169  apply simp_all
2170 done
2172 text {* by Brian Huffman *}
2173 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
2174 by (rule mod_minus_eq [symmetric])
2176 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
2177 by (rule mod_diff_left_eq [symmetric])
2179 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
2180 by (rule mod_diff_right_eq [symmetric])
2182 lemmas zmod_simps =
2185   mod_mult_right_eq[symmetric]
2186   mod_mult_left_eq [symmetric]
2187   power_mod
2188   zminus_zmod zdiff_zmod_left zdiff_zmod_right
2190 text {* Distributive laws for function @{text nat}. *}
2192 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
2193 apply (rule linorder_cases [of y 0])
2195 apply simp
2196 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
2197 done
2199 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
2200 lemma nat_mod_distrib:
2201   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
2202 apply (case_tac "y = 0", simp)
2203 apply (simp add: nat_eq_iff zmod_int)
2204 done
2206 text  {* transfer setup *}
2208 lemma transfer_nat_int_functions:
2209     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
2210     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
2211   by (auto simp add: nat_div_distrib nat_mod_distrib)
2213 lemma transfer_nat_int_function_closures:
2214     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
2215     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
2216   apply (cases "y = 0")
2217   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
2218   apply (cases "y = 0")
2219   apply auto
2220 done
2222 declare transfer_morphism_nat_int [transfer add return:
2223   transfer_nat_int_functions
2224   transfer_nat_int_function_closures
2225 ]
2227 lemma transfer_int_nat_functions:
2228     "(int x) div (int y) = int (x div y)"
2229     "(int x) mod (int y) = int (x mod y)"
2230   by (auto simp add: zdiv_int zmod_int)
2232 lemma transfer_int_nat_function_closures:
2233     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
2234     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
2235   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
2237 declare transfer_morphism_int_nat [transfer add return:
2238   transfer_int_nat_functions
2239   transfer_int_nat_function_closures
2240 ]
2242 text{*Suggested by Matthias Daum*}
2243 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
2244 apply (subgoal_tac "nat x div nat k < nat x")
2245  apply (simp add: nat_div_distrib [symmetric])
2246 apply (rule Divides.div_less_dividend, simp_all)
2247 done
2249 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
2250 proof
2251   assume H: "x mod n = y mod n"
2252   hence "x mod n - y mod n = 0" by simp
2253   hence "(x mod n - y mod n) mod n = 0" by simp
2254   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
2255   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
2256 next
2257   assume H: "n dvd x - y"
2258   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
2259   hence "x = n*k + y" by simp
2260   hence "x mod n = (n*k + y) mod n" by simp
2261   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
2262 qed
2264 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
2265   shows "\<exists>q. x = y + n * q"
2266 proof-
2267   from xy have th: "int x - int y = int (x - y)" by simp
2268   from xyn have "int x mod int n = int y mod int n"
2269     by (simp add: zmod_int [symmetric])
2270   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
2271   hence "n dvd x - y" by (simp add: th zdvd_int)
2272   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
2273 qed
2275 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
2276   (is "?lhs = ?rhs")
2277 proof
2278   assume H: "x mod n = y mod n"
2279   {assume xy: "x \<le> y"
2280     from H have th: "y mod n = x mod n" by simp
2281     from nat_mod_eq_lemma[OF th xy] have ?rhs
2282       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
2283   moreover
2284   {assume xy: "y \<le> x"
2285     from nat_mod_eq_lemma[OF H xy] have ?rhs
2286       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
2287   ultimately  show ?rhs using linear[of x y] by blast
2288 next
2289   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
2290   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
2291   thus  ?lhs by simp
2292 qed
2294 lemma div_nat_numeral [simp]:
2295   "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
2298 lemma one_div_nat_numeral [simp]:
2299   "Suc 0 div numeral v' = nat (1 div numeral v')"
2300   by (subst nat_div_distrib, simp_all)
2302 lemma mod_nat_numeral [simp]:
2303   "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
2306 lemma one_mod_nat_numeral [simp]:
2307   "Suc 0 mod numeral v' = nat (1 mod numeral v')"
2308   by (subst nat_mod_distrib) simp_all
2310 lemma mod_2_not_eq_zero_eq_one_int:
2311   fixes k :: int
2312   shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"
2313   by auto
2316 subsubsection {* Tools setup *}
2318 text {* Nitpick *}
2320 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
2323 subsubsection {* Code generation *}
2325 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
2326   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
2328 lemma pdivmod_posDivAlg [code]:
2329   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
2330 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
2332 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
2333   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
2334     then pdivmod k l
2335     else (let (r, s) = pdivmod k l in
2336        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
2337 proof -
2338   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
2339   show ?thesis
2340     by (simp add: divmod_int_mod_div pdivmod_def)
2341       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
2342       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
2343 qed
2345 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
2346   apsnd ((op *) (sgn l)) (if sgn k = sgn l
2347     then pdivmod k l
2348     else (let (r, s) = pdivmod k l in
2349       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
2350 proof -
2351   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
2352     by (auto simp add: not_less sgn_if)
2353   then show ?thesis by (simp add: divmod_int_pdivmod)
2354 qed
2356 code_modulename SML
2357   Divides Arith
2359 code_modulename OCaml
2360   Divides Arith