src/HOL/Divides.thy
 author wenzelm Tue Apr 17 00:30:44 2007 +0200 (2007-04-17) changeset 22718 936f7580937d parent 22473 753123c89d72 child 22744 5cbe966d67a2 permissions -rw-r--r--
tuned proofs;
1 (*  Title:      HOL/Divides.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1999  University of Cambridge
5 *)
7 header {* The division operators div, mod and the divides relation "dvd" *}
9 theory Divides
10 imports Datatype Power
11 begin
13 (*We use the same class for div and mod;
14   moreover, dvd is defined whenever multiplication is*)
15 class div = type +
16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
17   fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
18 begin
20 notation
21   div (infixl "\<^loc>div" 70)
23 notation
24   mod (infixl "\<^loc>mod" 70)
26 end
28 notation
29   div (infixl "div" 70)
31 notation
32   mod (infixl "mod" 70)
34 instance nat :: "Divides.div"
35   mod_def: "m mod n == wfrec (pred_nat^+)
36                           (%f j. if j<n | n=0 then j else f (j-n)) m"
37   div_def:   "m div n == wfrec (pred_nat^+)
38                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m" ..
40 definition
41   (*The definition of dvd is polymorphic!*)
42   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
43   dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)"
45 definition
46   quorem :: "(nat*nat) * (nat*nat) => bool" where
47   (*This definition helps prove the harder properties of div and mod.
48     It is copied from IntDiv.thy; should it be overloaded?*)
49   "quorem = (%((a,b), (q,r)).
50                     a = b*q + r &
51                     (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
55 subsection{*Initial Lemmas*}
57 lemmas wf_less_trans =
58        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
59                   standard]
61 lemma mod_eq: "(%m. m mod n) =
62               wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
65 lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
66                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
70 (** Aribtrary definitions for division by zero.  Useful to simplify
71     certain equations **)
73 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
74   by (rule div_eq [THEN wf_less_trans], simp)
76 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
77   by (rule mod_eq [THEN wf_less_trans], simp)
80 subsection{*Remainder*}
82 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
83   by (rule mod_eq [THEN wf_less_trans]) simp
85 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
86   apply (cases "n=0")
87    apply simp
88   apply (rule mod_eq [THEN wf_less_trans])
89   apply (simp add: cut_apply less_eq)
90   done
92 (*Avoids the ugly ~m<n above*)
93 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
94   by (simp add: mod_geq linorder_not_less)
96 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
99 lemma mod_1 [simp]: "m mod Suc 0 = 0"
100   by (induct m) (simp_all add: mod_geq)
102 lemma mod_self [simp]: "n mod n = (0::nat)"
103   by (cases "n = 0") (simp_all add: mod_geq)
105 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
106   apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
108   apply (subst mod_geq [symmetric], simp_all)
109   done
111 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
114 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
117 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
118   by (simp add: mult_commute mod_mult_self1)
120 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
121   apply (cases "n = 0", simp)
122   apply (cases "k = 0", simp)
123   apply (induct m rule: nat_less_induct)
124   apply (subst mod_if, simp)
125   apply (simp add: mod_geq diff_mult_distrib)
126   done
128 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
129   by (simp add: mult_commute [of k] mod_mult_distrib)
131 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
132   apply (cases "n = 0", simp)
133   apply (induct m, simp)
134   apply (rename_tac k)
135   apply (cut_tac m = "k * n" and n = n in mod_add_self2)
137   done
139 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
140   by (simp add: mult_commute mod_mult_self_is_0)
143 subsection{*Quotient*}
145 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
146   by (rule div_eq [THEN wf_less_trans], simp)
148 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
149   apply (rule div_eq [THEN wf_less_trans])
150   apply (simp add: cut_apply less_eq)
151   done
153 (*Avoids the ugly ~m<n above*)
154 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
155   by (simp add: div_geq linorder_not_less)
157 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
161 (*Main Result about quotient and remainder.*)
162 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
163   apply (cases "n = 0", simp)
164   apply (induct m rule: nat_less_induct)
165   apply (subst mod_if)
167   done
169 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
170   apply (cut_tac m = m and n = n in mod_div_equality)
172   done
174 subsection{*Simproc for Cancelling Div and Mod*}
176 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
179 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
182 ML
183 {*
184 structure CancelDivModData =
185 struct
187 val div_name = @{const_name Divides.div};
188 val mod_name = @{const_name Divides.mod};
189 val mk_binop = HOLogic.mk_binop;
190 val mk_sum = NatArithUtils.mk_sum;
191 val dest_sum = NatArithUtils.dest_sum;
193 (*logic*)
195 val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
197 val trans = trans
199 val prove_eq_sums =
201   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
203 end;
205 structure CancelDivMod = CancelDivModFun(CancelDivModData);
207 val cancel_div_mod_proc = NatArithUtils.prep_simproc
208       ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
211 *}
214 (* a simple rearrangement of mod_div_equality: *)
215 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
216   by (cut_tac m = m and n = n in mod_div_equality2, arith)
218 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
219   apply (induct m rule: nat_less_induct)
220   apply (rename_tac m)
221   apply (case_tac "m<n", simp)
222   txt{*case @{term "n \<le> m"}*}
224   done
226 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
227   apply (drule mod_less_divisor [where m = m])
228   apply simp
229   done
231 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
232   by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
234 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
235   by (simp add: mult_commute div_mult_self_is_m)
237 (*mod_mult_distrib2 above is the counterpart for remainder*)
240 subsection{*Proving facts about Quotient and Remainder*}
242 lemma unique_quotient_lemma:
243      "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
244       ==> q' \<le> (q::nat)"
245   apply (rule leI)
248   done
250 lemma unique_quotient:
251      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
252       ==> q = q'"
253   apply (simp add: split_ifs quorem_def)
254   apply (blast intro: order_antisym
255     dest: order_eq_refl [THEN unique_quotient_lemma] sym)
256   done
258 lemma unique_remainder:
259      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
260       ==> r = r'"
261   apply (subgoal_tac "q = q'")
262    prefer 2 apply (blast intro: unique_quotient)
264   done
266 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
267   unfolding quorem_def by simp
269 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
270   by (simp add: quorem_div_mod [THEN unique_quotient])
272 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
273   by (simp add: quorem_div_mod [THEN unique_remainder])
275 (** A dividend of zero **)
277 lemma div_0 [simp]: "0 div m = (0::nat)"
278   by (cases "m = 0") simp_all
280 lemma mod_0 [simp]: "0 mod m = (0::nat)"
281   by (cases "m = 0") simp_all
283 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
285 lemma quorem_mult1_eq:
286      "[| quorem((b,c),(q,r));  0 < c |]
287       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
290 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
291   apply (cases "c = 0", simp)
292   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
293   done
295 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
296   apply (cases "c = 0", simp)
297   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
298   done
300 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
301   apply (rule trans)
302    apply (rule_tac s = "b*a mod c" in trans)
303     apply (rule_tac [2] mod_mult1_eq)
305   done
307 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
308   apply (rule mod_mult1_eq' [THEN trans])
309   apply (rule mod_mult1_eq)
310   done
312 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
315      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
316       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
319 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
321      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
322   apply (cases "c = 0", simp)
323   apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
324   done
326 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
327   apply (cases "c = 0", simp)
328   apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])
329   done
332 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
334 (** first, a lemma to bound the remainder **)
336 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
337   apply (cut_tac m = q and n = c in mod_less_divisor)
338   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
339   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
341   done
343 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
344       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
347 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
348   apply (cases "b = 0", simp)
349   apply (cases "c = 0", simp)
350   apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
351   done
353 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
354   apply (cases "b = 0", simp)
355   apply (cases "c = 0", simp)
356   apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
357   done
360 subsection{*Cancellation of Common Factors in Division*}
362 lemma div_mult_mult_lemma:
363     "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
364   by (auto simp add: div_mult2_eq)
366 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
367   apply (cases "b = 0")
368   apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
369   done
371 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
372   apply (drule div_mult_mult1)
373   apply (auto simp add: mult_commute)
374   done
377 (*Distribution of Factors over Remainders:
379 Could prove these as in Integ/IntDiv.ML, but we already have
380 mod_mult_distrib and mod_mult_distrib2 above!
382 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
383 qed "mod_mult_mult1";
385 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
386 qed "mod_mult_mult2";
387  ***)
389 subsection{*Further Facts about Quotient and Remainder*}
391 lemma div_1 [simp]: "m div Suc 0 = m"
392   by (induct m) (simp_all add: div_geq)
394 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
397 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
398   apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
400   apply (subst div_geq [symmetric], simp_all)
401   done
403 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
406 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
408   apply (subst div_mult1_eq, simp)
409   done
411 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
412   by (simp add: mult_commute div_mult_self1)
415 (* Monotonicity of div in first argument *)
416 lemma div_le_mono [rule_format (no_asm)]:
417     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
418 apply (case_tac "k=0", simp)
419 apply (induct "n" rule: nat_less_induct, clarify)
420 apply (case_tac "n<k")
421 (* 1  case n<k *)
422 apply simp
423 (* 2  case n >= k *)
424 apply (case_tac "m<k")
425 (* 2.1  case m<k *)
426 apply simp
427 (* 2.2  case m>=k *)
428 apply (simp add: div_geq diff_le_mono)
429 done
431 (* Antimonotonicity of div in second argument *)
432 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
433 apply (subgoal_tac "0<n")
434  prefer 2 apply simp
435 apply (induct_tac k rule: nat_less_induct)
436 apply (rename_tac "k")
437 apply (case_tac "k<n", simp)
438 apply (subgoal_tac "~ (k<m) ")
439  prefer 2 apply simp
441 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
442  prefer 2
443  apply (blast intro: div_le_mono diff_le_mono2)
444 apply (rule le_trans, simp)
445 apply (simp)
446 done
448 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
449 apply (case_tac "n=0", simp)
450 apply (subgoal_tac "m div n \<le> m div 1", simp)
451 apply (rule div_le_mono2)
452 apply (simp_all (no_asm_simp))
453 done
455 (* Similar for "less than" *)
456 lemma div_less_dividend [rule_format]:
457      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
458 apply (induct_tac m rule: nat_less_induct)
459 apply (rename_tac "m")
460 apply (case_tac "m<n", simp)
461 apply (subgoal_tac "0<n")
462  prefer 2 apply simp
464 apply (case_tac "n<m")
465  apply (subgoal_tac "(m-n) div n < (m-n) ")
466   apply (rule impI less_trans_Suc)+
467 apply assumption
468   apply (simp_all)
469 done
471 declare div_less_dividend [simp]
473 text{*A fact for the mutilated chess board*}
474 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
475 apply (case_tac "n=0", simp)
476 apply (induct "m" rule: nat_less_induct)
477 apply (case_tac "Suc (na) <n")
478 (* case Suc(na) < n *)
479 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
480 (* case n \<le> Suc(na) *)
481 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
482 apply (auto simp add: Suc_diff_le le_mod_geq)
483 done
485 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
486   by (cases "n = 0") auto
488 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
489   by (cases "n = 0") auto
492 subsection{*The Divides Relation*}
494 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
495   unfolding dvd_def by blast
497 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
498   unfolding dvd_def by blast
500 lemma dvd_0_right [iff]: "m dvd (0::nat)"
501   unfolding dvd_def by (blast intro: mult_0_right [symmetric])
503 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
504   by (force simp add: dvd_def)
506 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
507   by (blast intro: dvd_0_left)
509 lemma dvd_1_left [iff]: "Suc 0 dvd k"
510   unfolding dvd_def by simp
512 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
515 lemma dvd_refl [simp]: "m dvd (m::nat)"
516   unfolding dvd_def by (blast intro: mult_1_right [symmetric])
518 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
519   unfolding dvd_def by (blast intro: mult_assoc)
521 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
522   unfolding dvd_def
523   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
525 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
526   unfolding dvd_def
527   by (blast intro: add_mult_distrib2 [symmetric])
529 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
530   unfolding dvd_def
531   by (blast intro: diff_mult_distrib2 [symmetric])
533 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
534   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
536   done
538 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
539   by (drule_tac m = m in dvd_diff, auto)
541 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
542   unfolding dvd_def by (blast intro: mult_left_commute)
544 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
545   apply (subst mult_commute)
546   apply (erule dvd_mult)
547   done
549 lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
550   by (rule dvd_refl [THEN dvd_mult])
552 lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
553   by (rule dvd_refl [THEN dvd_mult2])
555 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
556   apply (rule iffI)
558    apply (rule_tac [2] dvd_refl)
559   apply (subgoal_tac "n = (n+k) -k")
560    prefer 2 apply simp
561   apply (erule ssubst)
562   apply (erule dvd_diff)
563   apply (rule dvd_refl)
564   done
566 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
567   unfolding dvd_def
568   apply (case_tac "n = 0", auto)
569   apply (blast intro: mod_mult_distrib2 [symmetric])
570   done
572 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
573   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
575   apply (simp only: dvd_add dvd_mult)
576   done
578 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
579   by (blast intro: dvd_mod_imp_dvd dvd_mod)
581 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
582   unfolding dvd_def
583   apply (erule exE)
585   done
587 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
588   apply auto
589    apply (subgoal_tac "m*n dvd m*1")
590    apply (drule dvd_mult_cancel, auto)
591   done
593 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
594   apply (subst mult_commute)
595   apply (erule dvd_mult_cancel1)
596   done
598 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
599   apply (unfold dvd_def, clarify)
600   apply (rule_tac x = "k*ka" in exI)
602   done
604 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
605   by (simp add: dvd_def mult_assoc, blast)
607 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
608   apply (unfold dvd_def, clarify)
609   apply (rule_tac x = "i*k" in exI)
611   done
613 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
614   apply (unfold dvd_def, clarify)
615   apply (simp_all (no_asm_use) add: zero_less_mult_iff)
616   apply (erule conjE)
617   apply (rule le_trans)
618    apply (rule_tac [2] le_refl [THEN mult_le_mono])
619    apply (erule_tac [2] Suc_leI, simp)
620   done
622 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
623   apply (unfold dvd_def)
624   apply (case_tac "k=0", simp, safe)
626   apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
627   apply (subst mult_commute, simp)
628   done
630 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
631   apply (subgoal_tac "m mod n = 0")
633   apply (simp only: dvd_eq_mod_eq_0)
634   done
636 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
637   apply (unfold dvd_def)
638   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
640   done
642 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
643   by (induct n) auto
645 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
646   apply (induct j)
648   apply (blast dest!: dvd_mult_right)
649   done
651 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
652   apply (rule power_le_imp_le_exp, assumption)
653   apply (erule dvd_imp_le, simp)
654   done
656 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
657   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
659 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
661 (*Loses information, namely we also have r<d provided d is nonzero*)
662 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
663   apply (cut_tac m = m in mod_div_equality)
665   apply (blast intro: sym)
666   done
669 lemma split_div:
670  "P(n div k :: nat) =
671  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
672  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
673 proof
674   assume P: ?P
675   show ?Q
676   proof (cases)
677     assume "k = 0"
678     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
679   next
680     assume not0: "k \<noteq> 0"
681     thus ?Q
682     proof (simp, intro allI impI)
683       fix i j
684       assume n: "n = k*i + j" and j: "j < k"
685       show "P i"
686       proof (cases)
687         assume "i = 0"
688         with n j P show "P i" by simp
689       next
690         assume "i \<noteq> 0"
692       qed
693     qed
694   qed
695 next
696   assume Q: ?Q
697   show ?P
698   proof (cases)
699     assume "k = 0"
700     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
701   next
702     assume not0: "k \<noteq> 0"
703     with Q have R: ?R by simp
704     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
705     show ?P by simp
706   qed
707 qed
709 lemma split_div_lemma:
710   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
711   apply (rule iffI)
712   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
713 prefer 3; apply assumption
714   apply (simp_all add: quorem_def) apply arith
715   apply (rule conjI)
716   apply (rule_tac P="%x. n * (m div n) \<le> x" in
717     subst [OF mod_div_equality [of _ n]])
718   apply (simp only: add: mult_ac)
719   apply (rule_tac P="%x. x < n + n * (m div n)" in
720     subst [OF mod_div_equality [of _ n]])
723   done
725 theorem split_div':
726   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
727    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
728   apply (case_tac "0 < n")
729   apply (simp only: add: split_div_lemma)
731   done
733 lemma split_mod:
734  "P(n mod k :: nat) =
735  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
736  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
737 proof
738   assume P: ?P
739   show ?Q
740   proof (cases)
741     assume "k = 0"
742     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
743   next
744     assume not0: "k \<noteq> 0"
745     thus ?Q
746     proof (simp, intro allI impI)
747       fix i j
748       assume "n = k*i + j" "j < k"
750     qed
751   qed
752 next
753   assume Q: ?Q
754   show ?P
755   proof (cases)
756     assume "k = 0"
757     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
758   next
759     assume not0: "k \<noteq> 0"
760     with Q have R: ?R by simp
761     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
762     show ?P by simp
763   qed
764 qed
766 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
767   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
768     subst [OF mod_div_equality [of _ n]])
769   apply arith
770   done
772 subsection {*An ``induction'' law for modulus arithmetic.*}
774 lemma mod_induct_0:
775   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
776   and base: "P i" and i: "i<p"
777   shows "P 0"
778 proof (rule ccontr)
779   assume contra: "\<not>(P 0)"
780   from i have p: "0<p" by simp
781   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
782   proof
783     fix k
784     show "?A k"
785     proof (induct k)
786       show "?A 0" by simp  -- "by contradiction"
787     next
788       fix n
789       assume ih: "?A n"
790       show "?A (Suc n)"
791       proof (clarsimp)
792         assume y: "P (p - Suc n)"
793         have n: "Suc n < p"
794         proof (rule ccontr)
795           assume "\<not>(Suc n < p)"
796           hence "p - Suc n = 0"
797             by simp
798           with y contra show "False"
799             by simp
800         qed
801         hence n2: "Suc (p - Suc n) = p-n" by arith
802         from p have "p - Suc n < p" by arith
803         with y step have z: "P ((Suc (p - Suc n)) mod p)"
804           by blast
805         show "False"
806         proof (cases "n=0")
807           case True
808           with z n2 contra show ?thesis by simp
809         next
810           case False
811           with p have "p-n < p" by arith
812           with z n2 False ih show ?thesis by simp
813         qed
814       qed
815     qed
816   qed
817   moreover
818   from i obtain k where "0<k \<and> i+k=p"
820   hence "0<k \<and> i=p-k" by auto
821   moreover
822   note base
823   ultimately
824   show "False" by blast
825 qed
827 lemma mod_induct:
828   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
829   and base: "P i" and i: "i<p" and j: "j<p"
830   shows "P j"
831 proof -
832   have "\<forall>j<p. P j"
833   proof
834     fix j
835     show "j<p \<longrightarrow> P j" (is "?A j")
836     proof (induct j)
837       from step base i show "?A 0"
838         by (auto elim: mod_induct_0)
839     next
840       fix k
841       assume ih: "?A k"
842       show "?A (Suc k)"
843       proof
844         assume suc: "Suc k < p"
845         hence k: "k<p" by simp
846         with ih have "P k" ..
847         with step k have "P (Suc k mod p)"
848           by blast
849         moreover
850         from suc have "Suc k mod p = Suc k"
851           by simp
852         ultimately
853         show "P (Suc k)" by simp
854       qed
855     qed
856   qed
857   with j show ?thesis by blast
858 qed
861 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
862   apply (rule trans [symmetric])
865   done
867 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
868   apply (rule trans [symmetric])
871   done
874 subsection {* Code generation for div and mod *}
876 definition
877   "divmod (m\<Colon>nat) n = (m div n, m mod n)"
879 lemma divmod_zero [code]: "divmod m 0 = (0, m)"
880   unfolding divmod_def by simp
882 lemma divmod_succ [code]:
883   "divmod m (Suc k) = (if m < Suc k then (0, m) else
884     let
885       (p, q) = divmod (m - Suc k) (Suc k)
886     in (Suc p, q))"
887   unfolding divmod_def Let_def split_def
888   by (auto intro: div_geq mod_geq)
890 lemma div_divmod [code]: "m div n = fst (divmod m n)"
891   unfolding divmod_def by simp
893 lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
894   unfolding divmod_def by simp
896 code_modulename SML
897   Divides Integer
899 code_modulename OCaml
900   Divides Integer
902 hide (open) const divmod
904 subsection {* Legacy bindings *}
906 ML
907 {*
908 val div_def = thm "div_def"
909 val mod_def = thm "mod_def"
910 val dvd_def = thm "dvd_def"
911 val quorem_def = thm "quorem_def"
913 val wf_less_trans = thm "wf_less_trans";
914 val mod_eq = thm "mod_eq";
915 val div_eq = thm "div_eq";
916 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
917 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
918 val mod_less = thm "mod_less";
919 val mod_geq = thm "mod_geq";
920 val le_mod_geq = thm "le_mod_geq";
921 val mod_if = thm "mod_if";
922 val mod_1 = thm "mod_1";
923 val mod_self = thm "mod_self";
926 val mod_mult_self1 = thm "mod_mult_self1";
927 val mod_mult_self2 = thm "mod_mult_self2";
928 val mod_mult_distrib = thm "mod_mult_distrib";
929 val mod_mult_distrib2 = thm "mod_mult_distrib2";
930 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
931 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
932 val div_less = thm "div_less";
933 val div_geq = thm "div_geq";
934 val le_div_geq = thm "le_div_geq";
935 val div_if = thm "div_if";
936 val mod_div_equality = thm "mod_div_equality";
937 val mod_div_equality2 = thm "mod_div_equality2";
938 val div_mod_equality = thm "div_mod_equality";
939 val div_mod_equality2 = thm "div_mod_equality2";
940 val mult_div_cancel = thm "mult_div_cancel";
941 val mod_less_divisor = thm "mod_less_divisor";
942 val div_mult_self_is_m = thm "div_mult_self_is_m";
943 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
944 val unique_quotient_lemma = thm "unique_quotient_lemma";
945 val unique_quotient = thm "unique_quotient";
946 val unique_remainder = thm "unique_remainder";
947 val div_0 = thm "div_0";
948 val mod_0 = thm "mod_0";
949 val div_mult1_eq = thm "div_mult1_eq";
950 val mod_mult1_eq = thm "mod_mult1_eq";
951 val mod_mult1_eq' = thm "mod_mult1_eq'";
952 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
957 val mod_lemma = thm "mod_lemma";
958 val div_mult2_eq = thm "div_mult2_eq";
959 val mod_mult2_eq = thm "mod_mult2_eq";
960 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
961 val div_mult_mult1 = thm "div_mult_mult1";
962 val div_mult_mult2 = thm "div_mult_mult2";
963 val div_1 = thm "div_1";
964 val div_self = thm "div_self";
967 val div_mult_self1 = thm "div_mult_self1";
968 val div_mult_self2 = thm "div_mult_self2";
969 val div_le_mono = thm "div_le_mono";
970 val div_le_mono2 = thm "div_le_mono2";
971 val div_le_dividend = thm "div_le_dividend";
972 val div_less_dividend = thm "div_less_dividend";
973 val mod_Suc = thm "mod_Suc";
974 val dvdI = thm "dvdI";
975 val dvdE = thm "dvdE";
976 val dvd_0_right = thm "dvd_0_right";
977 val dvd_0_left = thm "dvd_0_left";
978 val dvd_0_left_iff = thm "dvd_0_left_iff";
979 val dvd_1_left = thm "dvd_1_left";
980 val dvd_1_iff_1 = thm "dvd_1_iff_1";
981 val dvd_refl = thm "dvd_refl";
982 val dvd_trans = thm "dvd_trans";
983 val dvd_anti_sym = thm "dvd_anti_sym";
985 val dvd_diff = thm "dvd_diff";
986 val dvd_diffD = thm "dvd_diffD";
987 val dvd_diffD1 = thm "dvd_diffD1";
988 val dvd_mult = thm "dvd_mult";
989 val dvd_mult2 = thm "dvd_mult2";
990 val dvd_reduce = thm "dvd_reduce";
991 val dvd_mod = thm "dvd_mod";
992 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
993 val dvd_mod_iff = thm "dvd_mod_iff";
994 val dvd_mult_cancel = thm "dvd_mult_cancel";
995 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
996 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
997 val mult_dvd_mono = thm "mult_dvd_mono";
998 val dvd_mult_left = thm "dvd_mult_left";
999 val dvd_mult_right = thm "dvd_mult_right";
1000 val dvd_imp_le = thm "dvd_imp_le";
1001 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
1002 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
1003 val mod_eq_0_iff = thm "mod_eq_0_iff";
1004 val mod_eqD = thm "mod_eqD";
1005 *}
1007 (*
1008 lemma split_div:
1009 assumes m: "m \<noteq> 0"
1010 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
1011        (is "?P = ?Q")
1012 proof
1013   assume P: ?P
1014   show ?Q
1015   proof (intro allI impI)
1016     fix i j
1017     assume n: "n = m*i + j" and j: "j < m"
1018     show "P i"
1019     proof (cases)
1020       assume "i = 0"
1021       with n j P show "P i" by simp
1022     next
1023       assume "i \<noteq> 0"
1024       with n j P show "P i" by (simp add:add_ac div_mult_self1)
1025     qed
1026   qed
1027 next
1028   assume Q: ?Q
1029   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
1030   show ?P by simp
1031 qed
1033 lemma split_mod:
1034 assumes m: "m \<noteq> 0"
1035 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
1036        (is "?P = ?Q")
1037 proof
1038   assume P: ?P
1039   show ?Q
1040   proof (intro allI impI)
1041     fix i j
1042     assume "n = m*i + j" "j < m"