src/HOL/IntDiv.thy
 author nipkow Sun Mar 01 12:01:57 2009 +0100 (2009-03-01) changeset 30181 05629f28f0f7 parent 30180 6d29a873141f child 30242 aea5d7fa7ef5 permissions -rw-r--r--
removed redundant lemmas
1 (*  Title:      HOL/IntDiv.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1999  University of Cambridge
5 *)
7 header{* The Division Operators div and mod *}
9 theory IntDiv
10 imports Int Divides FunDef
11 begin
13 definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
14     --{*definition of quotient and remainder*}
15     [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
16                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
18 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
19     --{*for the division algorithm*}
20     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
21                          else (2 * q, r))"
23 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
24 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
25   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
26      else adjust b (posDivAlg a (2 * b)))"
27 by auto
28 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
30 text{*algorithm for the case @{text "a<0, b>0"}*}
31 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
32   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
33      else adjust b (negDivAlg a (2 * b)))"
34 by auto
35 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
37 text{*algorithm for the general case @{term "b\<noteq>0"}*}
38 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
39   [code inline]: "negateSnd = apsnd uminus"
41 definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
42     --{*The full division algorithm considers all possible signs for a, b
43        including the special case @{text "a=0, b<0"} because
44        @{term negDivAlg} requires @{term "a<0"}.*}
45   "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
46                   else if a = 0 then (0, 0)
47                        else negateSnd (negDivAlg (-a) (-b))
48                else
49                   if 0 < b then negDivAlg a b
50                   else negateSnd (posDivAlg (-a) (-b)))"
52 instantiation int :: Divides.div
53 begin
55 definition
56   div_def: "a div b = fst (divmod a b)"
58 definition
59   mod_def: "a mod b = snd (divmod a b)"
61 instance ..
63 end
65 lemma divmod_mod_div:
66   "divmod p q = (p div q, p mod q)"
67   by (auto simp add: div_def mod_def)
69 text{*
70 Here is the division algorithm in ML:
72 \begin{verbatim}
73     fun posDivAlg (a,b) =
74       if a<b then (0,a)
75       else let val (q,r) = posDivAlg(a, 2*b)
76 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
77 	   end
79     fun negDivAlg (a,b) =
80       if 0\<le>a+b then (~1,a+b)
81       else let val (q,r) = negDivAlg(a, 2*b)
82 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
83 	   end;
85     fun negateSnd (q,r:int) = (q,~r);
87     fun divmod (a,b) = if 0\<le>a then
88 			  if b>0 then posDivAlg (a,b)
89 			   else if a=0 then (0,0)
90 				else negateSnd (negDivAlg (~a,~b))
91 		       else
92 			  if 0<b then negDivAlg (a,b)
93 			  else        negateSnd (posDivAlg (~a,~b));
94 \end{verbatim}
95 *}
99 subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
101 lemma unique_quotient_lemma:
102      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]
103       ==> q' \<le> (q::int)"
104 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
105  prefer 2 apply (simp add: right_diff_distrib)
106 apply (subgoal_tac "0 < b * (1 + q - q') ")
107 apply (erule_tac [2] order_le_less_trans)
108  prefer 2 apply (simp add: right_diff_distrib right_distrib)
109 apply (subgoal_tac "b * q' < b * (1 + q) ")
110  prefer 2 apply (simp add: right_diff_distrib right_distrib)
112 done
114 lemma unique_quotient_lemma_neg:
115      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]
116       ==> q \<le> (q'::int)"
117 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
118     auto)
120 lemma unique_quotient:
121      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]
122       ==> q = q'"
123 apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)
124 apply (blast intro: order_antisym
125              dest: order_eq_refl [THEN unique_quotient_lemma]
126              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
127 done
130 lemma unique_remainder:
131      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]
132       ==> r = r'"
133 apply (subgoal_tac "q = q'")
135 apply (blast intro: unique_quotient)
136 done
139 subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
141 text{*And positive divisors*}
145       (let diff = r-b in
146 	if 0 \<le> diff then (2*q + 1, diff)
147                      else (2*q, r))"
150 declare posDivAlg.simps [simp del]
152 text{*use with a simproc to avoid repeatedly proving the premise*}
153 lemma posDivAlg_eqn:
154      "0 < b ==>
155       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
156 by (rule posDivAlg.simps [THEN trans], simp)
158 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
159 theorem posDivAlg_correct:
160   assumes "0 \<le> a" and "0 < b"
161   shows "divmod_rel a b (posDivAlg a b)"
162 using prems apply (induct a b rule: posDivAlg.induct)
163 apply auto
165 apply (subst posDivAlg_eqn, simp add: right_distrib)
166 apply (case_tac "a < b")
167 apply simp_all
168 apply (erule splitE)
169 apply (auto simp add: right_distrib Let_def)
170 done
173 subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
175 text{*And positive divisors*}
177 declare negDivAlg.simps [simp del]
179 text{*use with a simproc to avoid repeatedly proving the premise*}
180 lemma negDivAlg_eqn:
181      "0 < b ==>
182       negDivAlg a b =
183        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
184 by (rule negDivAlg.simps [THEN trans], simp)
186 (*Correctness of negDivAlg: it computes quotients correctly
187   It doesn't work if a=0 because the 0/b equals 0, not -1*)
188 lemma negDivAlg_correct:
189   assumes "a < 0" and "b > 0"
190   shows "divmod_rel a b (negDivAlg a b)"
191 using prems apply (induct a b rule: negDivAlg.induct)
192 apply (auto simp add: linorder_not_le)
194 apply (subst negDivAlg_eqn, assumption)
195 apply (case_tac "a + b < (0\<Colon>int)")
196 apply simp_all
197 apply (erule splitE)
198 apply (auto simp add: right_distrib Let_def)
199 done
202 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
204 (*the case a=0*)
205 lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
206 by (auto simp add: divmod_rel_def linorder_neq_iff)
208 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
209 by (subst posDivAlg.simps, auto)
211 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
212 by (subst negDivAlg.simps, auto)
214 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
217 lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
218 by (auto simp add: split_ifs divmod_rel_def)
220 lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
221 by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg
222                     posDivAlg_correct negDivAlg_correct)
224 text{*Arbitrary definitions for division by zero.  Useful to simplify
225     certain equations.*}
227 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
228 by (simp add: div_def mod_def divmod_def posDivAlg.simps)
231 text{*Basic laws about division and remainder*}
233 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
234 apply (case_tac "b = 0", simp)
235 apply (cut_tac a = a and b = b in divmod_correct)
236 apply (auto simp add: divmod_rel_def div_def mod_def)
237 done
239 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
242 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
245 text {* Tool setup *}
247 ML {*
248 local
250 structure CancelDivMod = CancelDivModFun(
251 struct
252   val div_name = @{const_name Divides.div};
253   val mod_name = @{const_name Divides.mod};
254   val mk_binop = HOLogic.mk_binop;
255   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
256   val dest_sum = Int_Numeral_Simprocs.dest_sum;
257   val div_mod_eqs =
258     map mk_meta_eq [@{thm zdiv_zmod_equality},
259       @{thm zdiv_zmod_equality2}];
260   val trans = trans;
261   val prove_eq_sums =
262     let
264     in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
265 end)
267 in
269 val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
270   "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
272 end;
275 *}
277 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
278 apply (cut_tac a = a and b = b in divmod_correct)
279 apply (auto simp add: divmod_rel_def mod_def)
280 done
282 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
283    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
285 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
286 apply (cut_tac a = a and b = b in divmod_correct)
287 apply (auto simp add: divmod_rel_def div_def mod_def)
288 done
290 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
291    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
295 subsection{*General Properties of div and mod*}
297 lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"
298 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
299 apply (force simp add: divmod_rel_def linorder_neq_iff)
300 done
302 lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
303 by (simp add: divmod_rel_div_mod [THEN unique_quotient])
305 lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
306 by (simp add: divmod_rel_div_mod [THEN unique_remainder])
308 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
309 apply (rule divmod_rel_div)
310 apply (auto simp add: divmod_rel_def)
311 done
313 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
314 apply (rule divmod_rel_div)
315 apply (auto simp add: divmod_rel_def)
316 done
318 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
319 apply (rule divmod_rel_div)
320 apply (auto simp add: divmod_rel_def)
321 done
323 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
325 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
326 apply (rule_tac q = 0 in divmod_rel_mod)
327 apply (auto simp add: divmod_rel_def)
328 done
330 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
331 apply (rule_tac q = 0 in divmod_rel_mod)
332 apply (auto simp add: divmod_rel_def)
333 done
335 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
336 apply (rule_tac q = "-1" in divmod_rel_mod)
337 apply (auto simp add: divmod_rel_def)
338 done
340 text{*There is no @{text mod_neg_pos_trivial}.*}
343 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
344 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
345 apply (case_tac "b = 0", simp)
346 apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified,
347                                  THEN divmod_rel_div, THEN sym])
349 done
351 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
352 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
353 apply (case_tac "b = 0", simp)
354 apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
355        auto)
356 done
359 subsection{*Laws for div and mod with Unary Minus*}
361 lemma zminus1_lemma:
362      "divmod_rel a b (q, r)
363       ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,
364                           if r=0 then 0 else b-r)"
365 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)
368 lemma zdiv_zminus1_eq_if:
369      "b \<noteq> (0::int)
370       ==> (-a) div b =
371           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
372 by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])
374 lemma zmod_zminus1_eq_if:
375      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
376 apply (case_tac "b = 0", simp)
377 apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
378 done
380 lemma zmod_zminus1_not_zero:
381   fixes k l :: int
382   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
383   unfolding zmod_zminus1_eq_if by auto
385 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
386 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
388 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
389 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
391 lemma zdiv_zminus2_eq_if:
392      "b \<noteq> (0::int)
393       ==> a div (-b) =
394           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
395 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
397 lemma zmod_zminus2_eq_if:
398      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
399 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
401 lemma zmod_zminus2_not_zero:
402   fixes k l :: int
403   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
404   unfolding zmod_zminus2_eq_if by auto
407 subsection{*Division of a Number by Itself*}
409 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
410 apply (subgoal_tac "0 < a*q")
411  apply (simp add: zero_less_mult_iff, arith)
412 done
414 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
415 apply (subgoal_tac "0 \<le> a* (1-q) ")
418 done
420 lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
421 apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
422 apply (rule order_antisym, safe, simp_all)
423 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
424 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
426 done
428 lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
429 apply (frule self_quotient, assumption)
431 done
433 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
434 by (simp add: divmod_rel_div_mod [THEN self_quotient])
436 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
437 lemma zmod_self [simp]: "a mod a = (0::int)"
438 apply (case_tac "a = 0", simp)
439 apply (simp add: divmod_rel_div_mod [THEN self_remainder])
440 done
443 subsection{*Computation of Division and Remainder*}
445 lemma zdiv_zero [simp]: "(0::int) div b = 0"
446 by (simp add: div_def divmod_def)
448 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
449 by (simp add: div_def divmod_def)
451 lemma zmod_zero [simp]: "(0::int) mod b = 0"
452 by (simp add: mod_def divmod_def)
454 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
455 by (simp add: mod_def divmod_def)
457 text{*a positive, b positive *}
459 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
460 by (simp add: div_def divmod_def)
462 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
463 by (simp add: mod_def divmod_def)
465 text{*a negative, b positive *}
467 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
468 by (simp add: div_def divmod_def)
470 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
471 by (simp add: mod_def divmod_def)
473 text{*a positive, b negative *}
475 lemma div_pos_neg:
476      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
477 by (simp add: div_def divmod_def)
479 lemma mod_pos_neg:
480      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
481 by (simp add: mod_def divmod_def)
483 text{*a negative, b negative *}
485 lemma div_neg_neg:
486      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
487 by (simp add: div_def divmod_def)
489 lemma mod_neg_neg:
490      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
491 by (simp add: mod_def divmod_def)
493 text {*Simplify expresions in which div and mod combine numerical constants*}
495 lemma divmod_relI:
496   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
497     \<Longrightarrow> divmod_rel a b (q, r)"
498   unfolding divmod_rel_def by simp
500 lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
501 lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
502 lemmas arithmetic_simps =
503   arith_simps
507   mult_zero_left
508   mult_zero_right
509   mult_1_left
510   mult_1_right
512 (* simprocs adapted from HOL/ex/Binary.thy *)
513 ML {*
514 local
515   infix ==;
516   val op == = Logic.mk_equals;
517   fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $m$ n;
518   fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $m$ n;
520   val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};
521   fun prove ctxt prop =
522     Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
524   fun binary_proc proc ss ct =
525     (case Thm.term_of ct of
526       _ $t$ u =>
527       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of
528         SOME args => proc (Simplifier.the_context ss) args
529       | NONE => NONE)
530     | _ => NONE);
531 in
533 fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
534   if n = 0 then NONE
535   else
536     let val (k, l) = Integer.div_mod m n;
537         fun mk_num x = HOLogic.mk_number HOLogic.intT x;
538     in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])
539     end);
541 end;
542 *}
544 simproc_setup binary_int_div ("number_of m div number_of n :: int") =
545   {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
547 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
548   {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
550 lemmas posDivAlg_eqn_number_of [simp] =
551     posDivAlg_eqn [of "number_of v" "number_of w", standard]
553 lemmas negDivAlg_eqn_number_of [simp] =
554     negDivAlg_eqn [of "number_of v" "number_of w", standard]
557 text{*Special-case simplification *}
559 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
560 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
561 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
562 apply (auto simp del: neg_mod_sign neg_mod_bound)
563 done
565 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
566 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
568 (** The last remaining special cases for constant arithmetic:
569     1 div z and 1 mod z **)
571 lemmas div_pos_pos_1_number_of [simp] =
572     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
574 lemmas div_pos_neg_1_number_of [simp] =
575     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
577 lemmas mod_pos_pos_1_number_of [simp] =
578     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
580 lemmas mod_pos_neg_1_number_of [simp] =
581     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
584 lemmas posDivAlg_eqn_1_number_of [simp] =
585     posDivAlg_eqn [of concl: 1 "number_of w", standard]
587 lemmas negDivAlg_eqn_1_number_of [simp] =
588     negDivAlg_eqn [of concl: 1 "number_of w", standard]
592 subsection{*Monotonicity in the First Argument (Dividend)*}
594 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
595 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
596 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
597 apply (rule unique_quotient_lemma)
598 apply (erule subst)
599 apply (erule subst, simp_all)
600 done
602 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
603 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
604 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
605 apply (rule unique_quotient_lemma_neg)
606 apply (erule subst)
607 apply (erule subst, simp_all)
608 done
611 subsection{*Monotonicity in the Second Argument (Divisor)*}
613 lemma q_pos_lemma:
614      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
615 apply (subgoal_tac "0 < b'* (q' + 1) ")
618 done
620 lemma zdiv_mono2_lemma:
621      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
622          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
623       ==> q \<le> (q'::int)"
624 apply (frule q_pos_lemma, assumption+)
625 apply (subgoal_tac "b*q < b* (q' + 1) ")
627 apply (subgoal_tac "b*q = r' - r + b'*q'")
628  prefer 2 apply simp
629 apply (simp (no_asm_simp) add: right_distrib)
631 apply (rule mult_right_mono, auto)
632 done
634 lemma zdiv_mono2:
635      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
636 apply (subgoal_tac "b \<noteq> 0")
637  prefer 2 apply arith
638 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
639 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
640 apply (rule zdiv_mono2_lemma)
641 apply (erule subst)
642 apply (erule subst, simp_all)
643 done
645 lemma q_neg_lemma:
646      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
647 apply (subgoal_tac "b'*q' < 0")
648  apply (simp add: mult_less_0_iff, arith)
649 done
651 lemma zdiv_mono2_neg_lemma:
652      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
653          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
654       ==> q' \<le> (q::int)"
655 apply (frule q_neg_lemma, assumption+)
656 apply (subgoal_tac "b*q' < b* (q + 1) ")
659 apply (subgoal_tac "b*q' \<le> b'*q'")
660  prefer 2 apply (simp add: mult_right_mono_neg, arith)
661 done
663 lemma zdiv_mono2_neg:
664      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
665 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
666 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
667 apply (rule zdiv_mono2_neg_lemma)
668 apply (erule subst)
669 apply (erule subst, simp_all)
670 done
673 subsection{*More Algebraic Laws for div and mod*}
675 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
677 lemma zmult1_lemma:
678      "[| divmod_rel b c (q, r);  c \<noteq> 0 |]
679       ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
680 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
682 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
683 apply (case_tac "c = 0", simp)
684 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
685 done
687 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
688 apply (case_tac "c = 0", simp)
689 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
690 done
692 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
693 apply (case_tac "b = 0", simp)
694 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
695 done
697 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
700      "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]
701       ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
702 by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
704 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
706      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
707 apply (case_tac "c = 0", simp)
708 apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
709 done
711 instance int :: ring_div
712 proof
713   fix a b c :: int
714   assume not0: "b \<noteq> 0"
715   show "(a + c * b) div b = c + a div b"
716     unfolding zdiv_zadd1_eq [of a "c * b"] using not0
717       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
718 qed auto
720 lemma posDivAlg_div_mod:
721   assumes "k \<ge> 0"
722   and "l \<ge> 0"
723   shows "posDivAlg k l = (k div l, k mod l)"
724 proof (cases "l = 0")
725   case True then show ?thesis by (simp add: posDivAlg.simps)
726 next
727   case False with assms posDivAlg_correct
728     have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
729     by simp
730   from divmod_rel_div [OF this l \<noteq> 0] divmod_rel_mod [OF this l \<noteq> 0]
731   show ?thesis by simp
732 qed
734 lemma negDivAlg_div_mod:
735   assumes "k < 0"
736   and "l > 0"
737   shows "negDivAlg k l = (k div l, k mod l)"
738 proof -
739   from assms have "l \<noteq> 0" by simp
740   from assms negDivAlg_correct
741     have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
742     by simp
743   from divmod_rel_div [OF this l \<noteq> 0] divmod_rel_mod [OF this l \<noteq> 0]
744   show ?thesis by simp
745 qed
747 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
748 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
750 (* REVISIT: should this be generalized to all semiring_div types? *)
751 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
754 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
756 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
757   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
758   to cause particular problems.*)
760 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
762 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
763 apply (subgoal_tac "b * (c - q mod c) < r * 1")
765 apply (rule order_le_less_trans)
766  apply (erule_tac [2] mult_strict_right_mono)
767  apply (rule mult_left_mono_neg)
769  apply (simp)
770 apply (simp)
771 done
773 lemma zmult2_lemma_aux2:
774      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
775 apply (subgoal_tac "b * (q mod c) \<le> 0")
776  apply arith
778 done
780 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
781 apply (subgoal_tac "0 \<le> b * (q mod c) ")
782 apply arith
784 done
786 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
787 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
789 apply (rule order_less_le_trans)
790  apply (erule mult_strict_right_mono)
791  apply (rule_tac [2] mult_left_mono)
792   apply simp
794 apply simp
795 done
797 lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]
798       ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
799 by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
800                    zero_less_mult_iff right_distrib [symmetric]
801                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
803 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
804 apply (case_tac "b = 0", simp)
805 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
806 done
808 lemma zmod_zmult2_eq:
809      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
810 apply (case_tac "b = 0", simp)
811 apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
812 done
815 subsection{*Cancellation of Common Factors in div*}
817 lemma zdiv_zmult_zmult1_aux1:
818      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
819 by (subst zdiv_zmult2_eq, auto)
821 lemma zdiv_zmult_zmult1_aux2:
822      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
823 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
824 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
825 done
827 lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
828 apply (case_tac "b = 0", simp)
829 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
830 done
832 lemma zdiv_zmult_zmult1_if[simp]:
833   "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
837 subsection{*Distribution of Factors over mod*}
839 lemma zmod_zmult_zmult1_aux1:
840      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
841 by (subst zmod_zmult2_eq, auto)
843 lemma zmod_zmult_zmult1_aux2:
844      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
845 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
846 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
847 done
849 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
850 apply (case_tac "b = 0", simp)
851 apply (case_tac "c = 0", simp)
852 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
853 done
855 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
856 apply (cut_tac c = c in zmod_zmult_zmult1)
857 apply (auto simp add: mult_commute)
858 done
861 subsection {*Splitting Rules for div and mod*}
863 text{*The proofs of the two lemmas below are essentially identical*}
865 lemma split_pos_lemma:
866  "0<k ==>
867     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
868 apply (rule iffI, clarify)
869  apply (erule_tac P="P ?x ?y" in rev_mp)
872  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
873 txt{*converse direction*}
874 apply (drule_tac x = "n div k" in spec)
875 apply (drule_tac x = "n mod k" in spec, simp)
876 done
878 lemma split_neg_lemma:
879  "k<0 ==>
880     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
881 apply (rule iffI, clarify)
882  apply (erule_tac P="P ?x ?y" in rev_mp)
885  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
886 txt{*converse direction*}
887 apply (drule_tac x = "n div k" in spec)
888 apply (drule_tac x = "n mod k" in spec, simp)
889 done
891 lemma split_zdiv:
892  "P(n div k :: int) =
893   ((k = 0 --> P 0) &
894    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
895    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
896 apply (case_tac "k=0", simp)
897 apply (simp only: linorder_neq_iff)
898 apply (erule disjE)
899  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
900                       split_neg_lemma [of concl: "%x y. P x"])
901 done
903 lemma split_zmod:
904  "P(n mod k :: int) =
905   ((k = 0 --> P n) &
906    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
907    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
908 apply (case_tac "k=0", simp)
909 apply (simp only: linorder_neq_iff)
910 apply (erule disjE)
911  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
912                       split_neg_lemma [of concl: "%x y. P y"])
913 done
915 (* Enable arith to deal with div 2 and mod 2: *)
916 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
917 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
920 subsection{*Speeding up the Division Algorithm with Shifting*}
922 text{*computing div by shifting *}
924 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
925 proof cases
926   assume "a=0"
927     thus ?thesis by simp
928 next
929   assume "a\<noteq>0" and le_a: "0\<le>a"
930   hence a_pos: "1 \<le> a" by arith
931   hence one_less_a2: "1 < 2*a" by arith
932   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
934   with a_pos have "0 \<le> b mod a" by simp
935   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
936     by (simp add: mod_pos_pos_trivial one_less_a2)
937   with  le_2a
938   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
940                   right_distrib)
941   thus ?thesis
943         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
944                   div_pos_pos_trivial)
945 qed
947 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
948 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
949 apply (rule_tac [2] pos_zdiv_mult_2)
950 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
951 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
952 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
953        simp)
954 done
956 lemma zdiv_number_of_Bit0 [simp]:
957      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =
958           number_of v div (number_of w :: int)"
959 by (simp only: number_of_eq numeral_simps) simp
961 lemma zdiv_number_of_Bit1 [simp]:
962      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =
963           (if (0::int) \<le> number_of w
964            then number_of v div (number_of w)
965            else (number_of v + (1::int)) div (number_of w))"
966 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)
968 done
971 subsection{*Computing mod by Shifting (proofs resemble those for div)*}
973 lemma pos_zmod_mult_2:
974      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
975 apply (case_tac "a = 0", simp)
976 apply (subgoal_tac "1 < a * 2")
977  prefer 2 apply arith
978 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
979  apply (rule_tac [2] mult_left_mono)
981                       pos_mod_bound)
983 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
984 apply (rule mod_pos_pos_trivial)
985 apply (auto simp add: mod_pos_pos_trivial ring_distribs)
986 apply (subgoal_tac "0 \<le> b mod a", arith, simp)
987 done
989 lemma neg_zmod_mult_2:
990      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
991 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =
992                     1 + 2* ((-b - 1) mod (-a))")
993 apply (rule_tac [2] pos_zmod_mult_2)
994 apply (auto simp add: right_diff_distrib)
995 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
996  prefer 2 apply simp
997 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
998 done
1000 lemma zmod_number_of_Bit0 [simp]:
1001      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =
1002       (2::int) * (number_of v mod number_of w)"
1003 apply (simp only: number_of_eq numeral_simps)
1004 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
1006 done
1008 lemma zmod_number_of_Bit1 [simp]:
1009      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =
1010       (if (0::int) \<le> number_of w
1011                 then 2 * (number_of v mod number_of w) + 1
1012                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
1013 apply (simp only: number_of_eq numeral_simps)
1014 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
1016 done
1019 subsection{*Quotients of Signs*}
1021 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
1022 apply (subgoal_tac "a div b \<le> -1", force)
1023 apply (rule order_trans)
1024 apply (rule_tac a' = "-1" in zdiv_mono1)
1025 apply (auto simp add: div_eq_minus1)
1026 done
1028 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
1029 by (drule zdiv_mono1_neg, auto)
1031 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
1032 apply auto
1033 apply (drule_tac [2] zdiv_mono1)
1034 apply (auto simp add: linorder_neq_iff)
1035 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
1036 apply (blast intro: div_neg_pos_less0)
1037 done
1039 lemma neg_imp_zdiv_nonneg_iff:
1040      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
1041 apply (subst zdiv_zminus_zminus [symmetric])
1042 apply (subst pos_imp_zdiv_nonneg_iff, auto)
1043 done
1045 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
1046 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
1047 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
1049 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
1050 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
1051 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
1054 subsection {* The Divides Relation *}
1056 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
1057   dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
1059 lemma zdvd_anti_sym:
1060     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
1061   apply (simp add: dvd_def, auto)
1062   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
1063   done
1065 lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a"
1066   shows "\<bar>a\<bar> = \<bar>b\<bar>"
1067 proof-
1068   from a dvd b obtain k where k:"b = a*k" unfolding dvd_def by blast
1069   from b dvd a obtain k' where k':"a = b*k'" unfolding dvd_def by blast
1070   from k k' have "a = a*k*k'" by simp
1071   with mult_cancel_left1[where c="a" and b="k*k'"]
1072   have kk':"k*k' = 1" using a\<noteq>0` by (simp add: mult_assoc)
1073   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
1074   thus ?thesis using k k' by auto
1075 qed
1077 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
1078   apply (subgoal_tac "m = n + (m - n)")
1079    apply (erule ssubst)
1080    apply (blast intro: dvd_add, simp)
1081   done
1083 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
1084 apply (rule iffI)
1086  apply (subgoal_tac "n = (n + k * m) - k * m")
1087   apply (erule ssubst)
1088   apply (erule dvd_diff)
1089   apply(simp_all)
1090 done
1092 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
1094   apply (auto simp add: zmod_zmult_zmult1)
1095   done
1097 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
1098   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
1099    apply (simp add: zmod_zdiv_equality [symmetric])
1100   apply (simp only: dvd_add dvd_mult2)
1101   done
1103 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
1104   apply (auto elim!: dvdE)
1105   apply (subgoal_tac "0 < n")
1106    prefer 2
1107    apply (blast intro: order_less_trans)
1109   apply (subgoal_tac "n * k < n * 1")
1110    apply (drule mult_less_cancel_left [THEN iffD1], auto)
1111   done
1113 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
1114   using zmod_zdiv_equality[where a="m" and b="n"]
1117 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
1118 apply (subgoal_tac "m mod n = 0")
1120 apply (simp only: dvd_eq_mod_eq_0)
1121 done
1123 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
1124   shows "m dvd n"
1125 proof-
1126   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
1127   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
1128     with h have False by (simp add: mult_assoc)}
1129   hence "n = m * h" by blast
1130   thus ?thesis by simp
1131 qed
1134 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
1135 apply (simp split add: split_nat)
1136 apply (rule iffI)
1137 apply (erule exE)
1138 apply (rule_tac x = "int x" in exI)
1139 apply simp
1140 apply (erule exE)
1141 apply (rule_tac x = "nat x" in exI)
1142 apply (erule conjE)
1143 apply (erule_tac x = "nat x" in allE)
1144 apply simp
1145 done
1147 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
1148 proof -
1149   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
1150   proof -
1151     fix k
1152     assume A: "int y = int x * k"
1153     then show "x dvd y" proof (cases k)
1154       case (1 n) with A have "y = x * n" by (simp add: zmult_int)
1155       then show ?thesis ..
1156     next
1157       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
1158       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
1159       also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
1160       finally have "- int (x * Suc n) = int y" ..
1161       then show ?thesis by (simp only: negative_eq_positive) auto
1162     qed
1163   qed
1164   then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)
1165 qed
1167 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
1168 proof
1169   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
1170   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
1171   hence "nat \<bar>x\<bar> = 1"  by simp
1172   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
1173 next
1174   assume "\<bar>x\<bar>=1" thus "x dvd 1"
1175     by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)
1176 qed
1177 lemma zdvd_mult_cancel1:
1178   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
1179 proof
1180   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
1181     by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)
1182 next
1183   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
1184   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
1185 qed
1187 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
1188   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
1190 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
1191   unfolding zdvd_int by (cases "z \<ge> 0") simp_all
1193 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
1194   by (auto simp add: dvd_int_iff)
1196 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
1197   apply (rule_tac z=n in int_cases)
1198   apply (auto simp add: dvd_int_iff)
1199   apply (rule_tac z=z in int_cases)
1200   apply (auto simp add: dvd_imp_le)
1201   done
1203 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
1204 apply (induct "y", auto)
1205 apply (rule zmod_zmult1_eq [THEN trans])
1206 apply (simp (no_asm_simp))
1207 apply (rule mod_mult_eq [symmetric])
1208 done
1210 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
1211 apply (subst split_div, auto)
1212 apply (subst split_zdiv, auto)
1213 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
1214 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
1215 done
1217 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
1218 apply (subst split_mod, auto)
1219 apply (subst split_zmod, auto)
1220 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia
1221        in unique_remainder)
1222 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
1223 done
1225 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
1226 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
1228 text{*Suggested by Matthias Daum*}
1229 lemma int_power_div_base:
1230      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
1231 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
1232  apply (erule ssubst)
1234  apply simp_all
1235 done
1237 text {* by Brian Huffman *}
1238 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
1239 by (rule mod_minus_eq [symmetric])
1241 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
1242 by (rule mod_diff_left_eq [symmetric])
1244 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
1245 by (rule mod_diff_right_eq [symmetric])
1247 lemmas zmod_simps =
1250   IntDiv.zmod_zmult1_eq     [symmetric]
1251   mod_mult_left_eq          [symmetric]
1252   IntDiv.zpower_zmod
1253   zminus_zmod zdiff_zmod_left zdiff_zmod_right
1255 text {* Distributive laws for function @{text nat}. *}
1257 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
1258 apply (rule linorder_cases [of y 0])
1260 apply simp
1261 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
1262 done
1264 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
1265 lemma nat_mod_distrib:
1266   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
1267 apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
1268 apply (simp add: nat_eq_iff zmod_int)
1269 done
1271 text{*Suggested by Matthias Daum*}
1272 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
1273 apply (subgoal_tac "nat x div nat k < nat x")
1274  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
1275 apply (rule Divides.div_less_dividend, simp_all)
1276 done
1278 text {* code generator setup *}
1280 context ring_1
1281 begin
1283 lemma of_int_num [code]:
1284   "of_int k = (if k = 0 then 0 else if k < 0 then
1285      - of_int (- k) else let
1286        (l, m) = divmod k 2;
1287        l' = of_int l
1288      in if m = 0 then l' + l' else l' + l' + 1)"
1289 proof -
1290   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>
1291     of_int k = of_int (k div 2 * 2 + 1)"
1292   proof -
1293     have "k mod 2 < 2" by (auto intro: pos_mod_bound)
1294     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
1295     moreover assume "k mod 2 \<noteq> 0"
1296     ultimately have "k mod 2 = 1" by arith
1297     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
1298     ultimately show ?thesis by auto
1299   qed
1300   have aux2: "\<And>x. of_int 2 * x = x + x"
1301   proof -
1302     fix x
1303     have int2: "(2::int) = 1 + 1" by arith
1304     show "of_int 2 * x = x + x"
1305     unfolding int2 of_int_add left_distrib by simp
1306   qed
1307   have aux3: "\<And>x. x * of_int 2 = x + x"
1308   proof -
1309     fix x
1310     have int2: "(2::int) = 1 + 1" by arith
1311     show "x * of_int 2 = x + x"
1312     unfolding int2 of_int_add right_distrib by simp
1313   qed
1314   from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
1315 qed
1317 end
1319 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
1320 proof
1321   assume H: "x mod n = y mod n"
1322   hence "x mod n - y mod n = 0" by simp
1323   hence "(x mod n - y mod n) mod n = 0" by simp
1324   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
1325   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
1326 next
1327   assume H: "n dvd x - y"
1328   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
1329   hence "x = n*k + y" by simp
1330   hence "x mod n = (n*k + y) mod n" by simp
1331   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
1332 qed
1334 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
1335   shows "\<exists>q. x = y + n * q"
1336 proof-
1337   from xy have th: "int x - int y = int (x - y)" by simp
1338   from xyn have "int x mod int n = int y mod int n"
1340   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
1341   hence "n dvd x - y" by (simp add: th zdvd_int)
1342   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
1343 qed
1345 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
1346   (is "?lhs = ?rhs")
1347 proof
1348   assume H: "x mod n = y mod n"
1349   {assume xy: "x \<le> y"
1350     from H have th: "y mod n = x mod n" by simp
1351     from nat_mod_eq_lemma[OF th xy] have ?rhs
1352       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
1353   moreover
1354   {assume xy: "y \<le> x"
1355     from nat_mod_eq_lemma[OF H xy] have ?rhs
1356       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
1357   ultimately  show ?rhs using linear[of x y] by blast
1358 next
1359   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
1360   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
1361   thus  ?lhs by simp
1362 qed
1365 subsection {* Code generation *}
1367 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1368   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
1370 lemma pdivmod_posDivAlg [code]:
1371   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
1372 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
1374 lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
1375   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
1376     then pdivmod k l
1377     else (let (r, s) = pdivmod k l in
1378       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
1379 proof -
1380   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
1381   show ?thesis
1382     by (simp add: divmod_mod_div pdivmod_def)
1383       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
1384       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
1385 qed
1387 lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
1388   apsnd ((op *) (sgn l)) (if sgn k = sgn l
1389     then pdivmod k l
1390     else (let (r, s) = pdivmod k l in
1391       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
1392 proof -
1393   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"
1394     by (auto simp add: not_less sgn_if)
1395   then show ?thesis by (simp add: divmod_pdivmod)
1396 qed
1398 code_modulename SML
1399   IntDiv Integer
1401 code_modulename OCaml
1402   IntDiv Integer