src/HOL/Rings.thy
 author haftmann Thu Jun 25 15:01:42 2015 +0200 (2015-06-25) changeset 60570 7ed2cde6806d parent 60562 24af00b010cf child 60615 e5fa1d5d3952 permissions -rw-r--r--
more theorems
1 (*  Title:      HOL/Rings.thy
2     Author:     Gertrud Bauer
3     Author:     Steven Obua
4     Author:     Tobias Nipkow
5     Author:     Lawrence C Paulson
6     Author:     Markus Wenzel
8 *)
10 section {* Rings *}
12 theory Rings
13 imports Groups
14 begin
16 class semiring = ab_semigroup_add + semigroup_mult +
17   assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
18   assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
19 begin
21 text{*For the @{text combine_numerals} simproc*}
22 lemma combine_common_factor:
23   "a * e + (b * e + c) = (a + b) * e + c"
24 by (simp add: distrib_right ac_simps)
26 end
28 class mult_zero = times + zero +
29   assumes mult_zero_left [simp]: "0 * a = 0"
30   assumes mult_zero_right [simp]: "a * 0 = 0"
31 begin
33 lemma mult_not_zero:
34   "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
35   by auto
37 end
39 class semiring_0 = semiring + comm_monoid_add + mult_zero
41 class semiring_0_cancel = semiring + cancel_comm_monoid_add
42 begin
44 subclass semiring_0
45 proof
46   fix a :: 'a
47   have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
48   thus "0 * a = 0" by (simp only: add_left_cancel)
49 next
50   fix a :: 'a
51   have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
52   thus "a * 0 = 0" by (simp only: add_left_cancel)
53 qed
55 end
57 class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
58   assumes distrib: "(a + b) * c = a * c + b * c"
59 begin
61 subclass semiring
62 proof
63   fix a b c :: 'a
64   show "(a + b) * c = a * c + b * c" by (simp add: distrib)
65   have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
66   also have "... = b * a + c * a" by (simp only: distrib)
67   also have "... = a * b + a * c" by (simp add: ac_simps)
68   finally show "a * (b + c) = a * b + a * c" by blast
69 qed
71 end
73 class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
74 begin
76 subclass semiring_0 ..
78 end
80 class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
81 begin
83 subclass semiring_0_cancel ..
85 subclass comm_semiring_0 ..
87 end
89 class zero_neq_one = zero + one +
90   assumes zero_neq_one [simp]: "0 \<noteq> 1"
91 begin
93 lemma one_neq_zero [simp]: "1 \<noteq> 0"
94 by (rule not_sym) (rule zero_neq_one)
96 definition of_bool :: "bool \<Rightarrow> 'a"
97 where
98   "of_bool p = (if p then 1 else 0)"
100 lemma of_bool_eq [simp, code]:
101   "of_bool False = 0"
102   "of_bool True = 1"
105 lemma of_bool_eq_iff:
106   "of_bool p = of_bool q \<longleftrightarrow> p = q"
109 lemma split_of_bool [split]:
110   "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
111   by (cases p) simp_all
113 lemma split_of_bool_asm:
114   "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
115   by (cases p) simp_all
117 end
119 class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
121 text {* Abstract divisibility *}
123 class dvd = times
124 begin
126 definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
127   "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
129 lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
130   unfolding dvd_def ..
132 lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
133   unfolding dvd_def by blast
135 end
137 context comm_monoid_mult
138 begin
140 subclass dvd .
142 lemma dvd_refl [simp]:
143   "a dvd a"
144 proof
145   show "a = a * 1" by simp
146 qed
148 lemma dvd_trans:
149   assumes "a dvd b" and "b dvd c"
150   shows "a dvd c"
151 proof -
152   from assms obtain v where "b = a * v" by (auto elim!: dvdE)
153   moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
154   ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
155   then show ?thesis ..
156 qed
158 lemma one_dvd [simp]:
159   "1 dvd a"
160   by (auto intro!: dvdI)
162 lemma dvd_mult [simp]:
163   "a dvd c \<Longrightarrow> a dvd (b * c)"
164   by (auto intro!: mult.left_commute dvdI elim!: dvdE)
166 lemma dvd_mult2 [simp]:
167   "a dvd b \<Longrightarrow> a dvd (b * c)"
168   using dvd_mult [of a b c] by (simp add: ac_simps)
170 lemma dvd_triv_right [simp]:
171   "a dvd b * a"
172   by (rule dvd_mult) (rule dvd_refl)
174 lemma dvd_triv_left [simp]:
175   "a dvd a * b"
176   by (rule dvd_mult2) (rule dvd_refl)
178 lemma mult_dvd_mono:
179   assumes "a dvd b"
180     and "c dvd d"
181   shows "a * c dvd b * d"
182 proof -
183   from a dvd b obtain b' where "b = a * b'" ..
184   moreover from c dvd d obtain d' where "d = c * d'" ..
185   ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
186   then show ?thesis ..
187 qed
189 lemma dvd_mult_left:
190   "a * b dvd c \<Longrightarrow> a dvd c"
191   by (simp add: dvd_def mult.assoc) blast
193 lemma dvd_mult_right:
194   "a * b dvd c \<Longrightarrow> b dvd c"
195   using dvd_mult_left [of b a c] by (simp add: ac_simps)
197 end
199 class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
200 begin
202 subclass semiring_1 ..
204 lemma dvd_0_left_iff [simp]:
205   "0 dvd a \<longleftrightarrow> a = 0"
206   by (auto intro: dvd_refl elim!: dvdE)
208 lemma dvd_0_right [iff]:
209   "a dvd 0"
210 proof
211   show "0 = a * 0" by simp
212 qed
214 lemma dvd_0_left:
215   "0 dvd a \<Longrightarrow> a = 0"
216   by simp
219   assumes "a dvd b" and "a dvd c"
220   shows "a dvd (b + c)"
221 proof -
222   from a dvd b obtain b' where "b = a * b'" ..
223   moreover from a dvd c obtain c' where "c = a * c'" ..
224   ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
225   then show ?thesis ..
226 qed
228 end
230 class semiring_1_cancel = semiring + cancel_comm_monoid_add
231   + zero_neq_one + monoid_mult
232 begin
234 subclass semiring_0_cancel ..
236 subclass semiring_1 ..
238 end
240 class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
241                                zero_neq_one + comm_monoid_mult +
242   assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
243 begin
245 subclass semiring_1_cancel ..
246 subclass comm_semiring_0_cancel ..
247 subclass comm_semiring_1 ..
249 lemma left_diff_distrib' [algebra_simps]:
250   "(b - c) * a = b * a - c * a"
254   "a dvd c * a + b \<longleftrightarrow> a dvd b"
255 proof -
256   have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
257   proof
258     assume ?Q then show ?P by simp
259   next
260     assume ?P
261     then obtain d where "a * c + b = a * d" ..
262     then have "a * c + b - a * c = a * d - a * c" by simp
263     then have "b = a * d - a * c" by simp
264     then have "b = a * (d - c)" by (simp add: algebra_simps)
265     then show ?Q ..
266   qed
267   then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
268 qed
271   "a dvd b + c * a \<longleftrightarrow> a dvd b"
275   "a dvd a + b \<longleftrightarrow> a dvd b"
276   using dvd_add_times_triv_left_iff [of a 1 b] by simp
279   "a dvd b + a \<longleftrightarrow> a dvd b"
280   using dvd_add_times_triv_right_iff [of a b 1] by simp
283   assumes "a dvd b"
284   shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
285 proof
286   assume ?P then obtain d where "b + c = a * d" ..
287   moreover from a dvd b obtain e where "b = a * e" ..
288   ultimately have "a * e + c = a * d" by simp
289   then have "a * e + c - a * e = a * d - a * e" by simp
290   then have "c = a * d - a * e" by simp
291   then have "c = a * (d - e)" by (simp add: algebra_simps)
292   then show ?Q ..
293 next
294   assume ?Q with assms show ?P by simp
295 qed
298   assumes "a dvd c"
299   shows "a dvd b + c \<longleftrightarrow> a dvd b"
300   using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
302 end
304 class ring = semiring + ab_group_add
305 begin
307 subclass semiring_0_cancel ..
309 text {* Distribution rules *}
311 lemma minus_mult_left: "- (a * b) = - a * b"
312 by (rule minus_unique) (simp add: distrib_right [symmetric])
314 lemma minus_mult_right: "- (a * b) = a * - b"
315 by (rule minus_unique) (simp add: distrib_left [symmetric])
317 text{*Extract signs from products*}
318 lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
319 lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
321 lemma minus_mult_minus [simp]: "- a * - b = a * b"
322 by simp
324 lemma minus_mult_commute: "- a * b = a * - b"
325 by simp
327 lemma right_diff_distrib [algebra_simps]:
328   "a * (b - c) = a * b - a * c"
329   using distrib_left [of a b "-c "] by simp
331 lemma left_diff_distrib [algebra_simps]:
332   "(a - b) * c = a * c - b * c"
333   using distrib_right [of a "- b" c] by simp
335 lemmas ring_distribs =
336   distrib_left distrib_right left_diff_distrib right_diff_distrib
339   "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
343   "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
346 end
348 lemmas ring_distribs =
349   distrib_left distrib_right left_diff_distrib right_diff_distrib
351 class comm_ring = comm_semiring + ab_group_add
352 begin
354 subclass ring ..
355 subclass comm_semiring_0_cancel ..
357 lemma square_diff_square_factored:
358   "x * x - y * y = (x + y) * (x - y)"
361 end
363 class ring_1 = ring + zero_neq_one + monoid_mult
364 begin
366 subclass semiring_1_cancel ..
368 lemma square_diff_one_factored:
369   "x * x - 1 = (x + 1) * (x - 1)"
372 end
374 class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
375 begin
377 subclass ring_1 ..
378 subclass comm_semiring_1_cancel
379   by unfold_locales (simp add: algebra_simps)
381 lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
382 proof
383   assume "x dvd - y"
384   then have "x dvd - 1 * - y" by (rule dvd_mult)
385   then show "x dvd y" by simp
386 next
387   assume "x dvd y"
388   then have "x dvd - 1 * y" by (rule dvd_mult)
389   then show "x dvd - y" by simp
390 qed
392 lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
393 proof
394   assume "- x dvd y"
395   then obtain k where "y = - x * k" ..
396   then have "y = x * - k" by simp
397   then show "x dvd y" ..
398 next
399   assume "x dvd y"
400   then obtain k where "y = x * k" ..
401   then have "y = - x * - k" by simp
402   then show "- x dvd y" ..
403 qed
405 lemma dvd_diff [simp]:
406   "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
407   using dvd_add [of x y "- z"] by simp
409 end
411 class semiring_no_zero_divisors = semiring_0 +
412   assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
413 begin
415 lemma divisors_zero:
416   assumes "a * b = 0"
417   shows "a = 0 \<or> b = 0"
418 proof (rule classical)
419   assume "\<not> (a = 0 \<or> b = 0)"
420   then have "a \<noteq> 0" and "b \<noteq> 0" by auto
421   with no_zero_divisors have "a * b \<noteq> 0" by blast
422   with assms show ?thesis by simp
423 qed
425 lemma mult_eq_0_iff [simp]:
426   shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
427 proof (cases "a = 0 \<or> b = 0")
428   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
429     then show ?thesis using no_zero_divisors by simp
430 next
431   case True then show ?thesis by auto
432 qed
434 end
436 class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
437   assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
438     and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
439 begin
441 lemma mult_left_cancel:
442   "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
443   by simp
445 lemma mult_right_cancel:
446   "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
447   by simp
449 end
451 class ring_no_zero_divisors = ring + semiring_no_zero_divisors
452 begin
454 subclass semiring_no_zero_divisors_cancel
455 proof
456   fix a b c
457   have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
459   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
460     by auto
461   finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
462   have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
464   also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
465     by auto
466   finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
467 qed
469 end
471 class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
472 begin
474 lemma square_eq_1_iff:
475   "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
476 proof -
477   have "(x - 1) * (x + 1) = x * x - 1"
479   hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
480     by simp
481   thus ?thesis
483 qed
485 lemma mult_cancel_right1 [simp]:
486   "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
487 by (insert mult_cancel_right [of 1 c b], force)
489 lemma mult_cancel_right2 [simp]:
490   "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
491 by (insert mult_cancel_right [of a c 1], simp)
493 lemma mult_cancel_left1 [simp]:
494   "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
495 by (insert mult_cancel_left [of c 1 b], force)
497 lemma mult_cancel_left2 [simp]:
498   "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
499 by (insert mult_cancel_left [of c a 1], simp)
501 end
503 class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
505 class idom = comm_ring_1 + semiring_no_zero_divisors
506 begin
508 subclass semidom ..
510 subclass ring_1_no_zero_divisors ..
512 lemma dvd_mult_cancel_right [simp]:
513   "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
514 proof -
515   have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
516     unfolding dvd_def by (simp add: ac_simps)
517   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
518     unfolding dvd_def by simp
519   finally show ?thesis .
520 qed
522 lemma dvd_mult_cancel_left [simp]:
523   "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
524 proof -
525   have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
526     unfolding dvd_def by (simp add: ac_simps)
527   also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
528     unfolding dvd_def by simp
529   finally show ?thesis .
530 qed
532 lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
533 proof
534   assume "a * a = b * b"
535   then have "(a - b) * (a + b) = 0"
537   then show "a = b \<or> a = - b"
539 next
540   assume "a = b \<or> a = - b"
541   then show "a * a = b * b" by auto
542 qed
544 end
546 text {*
547   The theory of partially ordered rings is taken from the books:
548   \begin{itemize}
549   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
550   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
551   \end{itemize}
552   Most of the used notions can also be looked up in
553   \begin{itemize}
554   \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
555   \item \emph{Algebra I} by van der Waerden, Springer.
556   \end{itemize}
557 *}
559 class divide =
560   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
562 setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
564 context semiring
565 begin
567 lemma [field_simps]:
568   shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
569     and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
570   by (rule distrib_left distrib_right)+
572 end
574 context ring
575 begin
577 lemma [field_simps]:
578   shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
579     and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
580   by (rule left_diff_distrib right_diff_distrib)+
582 end
584 setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
586 class semidom_divide = semidom + divide +
587   assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
588   assumes divide_zero [simp]: "a div 0 = 0"
589 begin
591 lemma nonzero_mult_divide_cancel_left [simp]:
592   "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
593   using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
595 subclass semiring_no_zero_divisors_cancel
596 proof
597   fix a b c
598   { fix a b c
599     show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
600     proof (cases "c = 0")
601       case True then show ?thesis by simp
602     next
603       case False
604       { assume "a * c = b * c"
605         then have "a * c div c = b * c div c"
606           by simp
607         with False have "a = b"
608           by simp
609       } then show ?thesis by auto
610     qed
611   }
612   from this [of a c b]
613   show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
615 qed
617 lemma div_self [simp]:
618   assumes "a \<noteq> 0"
619   shows "a div a = 1"
620   using assms nonzero_mult_divide_cancel_left [of a 1] by simp
622 lemma divide_zero_left [simp]:
623   "0 div a = 0"
624 proof (cases "a = 0")
625   case True then show ?thesis by simp
626 next
627   case False then have "a * 0 div a = 0"
628     by (rule nonzero_mult_divide_cancel_left)
629   then show ?thesis by simp
630 qed
632 end
634 class idom_divide = idom + semidom_divide
636 class algebraic_semidom = semidom_divide
637 begin
639 lemma dvd_div_mult_self [simp]:
640   "a dvd b \<Longrightarrow> b div a * a = b"
641   by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
643 lemma dvd_mult_div_cancel [simp]:
644   "a dvd b \<Longrightarrow> a * (b div a) = b"
645   using dvd_div_mult_self [of a b] by (simp add: ac_simps)
647 lemma div_mult_swap:
648   assumes "c dvd b"
649   shows "a * (b div c) = (a * b) div c"
650 proof (cases "c = 0")
651   case True then show ?thesis by simp
652 next
653   case False from assms obtain d where "b = c * d" ..
654   moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
655     by simp
656   ultimately show ?thesis by (simp add: ac_simps)
657 qed
659 lemma dvd_div_mult:
660   assumes "c dvd b"
661   shows "b div c * a = (b * a) div c"
662   using assms div_mult_swap [of c b a] by (simp add: ac_simps)
664 lemma dvd_div_mult2_eq:
665   assumes "b * c dvd a"
666   shows "a div (b * c) = a div b div c"
667 using assms proof
668   fix k
669   assume "a = b * c * k"
670   then show ?thesis
671     by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
672 qed
675 text \<open>Units: invertible elements in a ring\<close>
677 abbreviation is_unit :: "'a \<Rightarrow> bool"
678 where
679   "is_unit a \<equiv> a dvd 1"
681 lemma not_is_unit_0 [simp]:
682   "\<not> is_unit 0"
683   by simp
685 lemma unit_imp_dvd [dest]:
686   "is_unit b \<Longrightarrow> b dvd a"
687   by (rule dvd_trans [of _ 1]) simp_all
689 lemma unit_dvdE:
690   assumes "is_unit a"
691   obtains c where "a \<noteq> 0" and "b = a * c"
692 proof -
693   from assms have "a dvd b" by auto
694   then obtain c where "b = a * c" ..
695   moreover from assms have "a \<noteq> 0" by auto
696   ultimately show thesis using that by blast
697 qed
699 lemma dvd_unit_imp_unit:
700   "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
701   by (rule dvd_trans)
703 lemma unit_div_1_unit [simp, intro]:
704   assumes "is_unit a"
705   shows "is_unit (1 div a)"
706 proof -
707   from assms have "1 = 1 div a * a" by simp
708   then show "is_unit (1 div a)" by (rule dvdI)
709 qed
711 lemma is_unitE [elim?]:
712   assumes "is_unit a"
713   obtains b where "a \<noteq> 0" and "b \<noteq> 0"
714     and "is_unit b" and "1 div a = b" and "1 div b = a"
715     and "a * b = 1" and "c div a = c * b"
716 proof (rule that)
717   def b \<equiv> "1 div a"
718   then show "1 div a = b" by simp
719   from b_def is_unit a show "is_unit b" by simp
720   from is_unit a and is_unit b show "a \<noteq> 0" and "b \<noteq> 0" by auto
721   from b_def is_unit a show "a * b = 1" by simp
722   then have "1 = a * b" ..
723   with b_def b \<noteq> 0 show "1 div b = a" by simp
724   from is_unit a have "a dvd c" ..
725   then obtain d where "c = a * d" ..
726   with a \<noteq> 0 a * b = 1 show "c div a = c * b"
727     by (simp add: mult.assoc mult.left_commute [of a])
728 qed
730 lemma unit_prod [intro]:
731   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
732   by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
734 lemma unit_div [intro]:
735   "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
736   by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
738 lemma mult_unit_dvd_iff:
739   assumes "is_unit b"
740   shows "a * b dvd c \<longleftrightarrow> a dvd c"
741 proof
742   assume "a * b dvd c"
743   with assms show "a dvd c"
745 next
746   assume "a dvd c"
747   then obtain k where "c = a * k" ..
748   with assms have "c = (a * b) * (1 div b * k)"
750   then show "a * b dvd c" by (rule dvdI)
751 qed
753 lemma dvd_mult_unit_iff:
754   assumes "is_unit b"
755   shows "a dvd c * b \<longleftrightarrow> a dvd c"
756 proof
757   assume "a dvd c * b"
758   with assms have "c * b dvd c * (b * (1 div b))"
759     by (subst mult_assoc [symmetric]) simp
760   also from is_unit b have "b * (1 div b) = 1" by (rule is_unitE) simp
761   finally have "c * b dvd c" by simp
762   with a dvd c * b show "a dvd c" by (rule dvd_trans)
763 next
764   assume "a dvd c"
765   then show "a dvd c * b" by simp
766 qed
768 lemma div_unit_dvd_iff:
769   "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
770   by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
772 lemma dvd_div_unit_iff:
773   "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
774   by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
776 lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
777   dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
779 lemma unit_mult_div_div [simp]:
780   "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
781   by (erule is_unitE [of _ b]) simp
783 lemma unit_div_mult_self [simp]:
784   "is_unit a \<Longrightarrow> b div a * a = b"
785   by (rule dvd_div_mult_self) auto
787 lemma unit_div_1_div_1 [simp]:
788   "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
789   by (erule is_unitE) simp
791 lemma unit_div_mult_swap:
792   "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
793   by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
795 lemma unit_div_commute:
796   "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
797   using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
799 lemma unit_eq_div1:
800   "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
801   by (auto elim: is_unitE)
803 lemma unit_eq_div2:
804   "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
805   using unit_eq_div1 [of b c a] by auto
807 lemma unit_mult_left_cancel:
808   assumes "is_unit a"
809   shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
810   using assms mult_cancel_left [of a b c] by auto
812 lemma unit_mult_right_cancel:
813   "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
814   using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
816 lemma unit_div_cancel:
817   assumes "is_unit a"
818   shows "b div a = c div a \<longleftrightarrow> b = c"
819 proof -
820   from assms have "is_unit (1 div a)" by simp
821   then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
822     by (rule unit_mult_right_cancel)
823   with assms show ?thesis by simp
824 qed
826 lemma is_unit_div_mult2_eq:
827   assumes "is_unit b" and "is_unit c"
828   shows "a div (b * c) = a div b div c"
829 proof -
830   from assms have "is_unit (b * c)" by (simp add: unit_prod)
831   then have "b * c dvd a"
832     by (rule unit_imp_dvd)
833   then show ?thesis
834     by (rule dvd_div_mult2_eq)
835 qed
838 text \<open>Associated elements in a ring --- an equivalence relation induced
839   by the quasi-order divisibility.\<close>
841 definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
842 where
843   "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
845 lemma associatedI:
846   "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
849 lemma associatedD1:
850   "associated a b \<Longrightarrow> a dvd b"
853 lemma associatedD2:
854   "associated a b \<Longrightarrow> b dvd a"
857 lemma associated_refl [simp]:
858   "associated a a"
859   by (auto intro: associatedI)
861 lemma associated_sym:
862   "associated b a \<longleftrightarrow> associated a b"
863   by (auto intro: associatedI dest: associatedD1 associatedD2)
865 lemma associated_trans:
866   "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
867   by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
869 lemma associated_0 [simp]:
870   "associated 0 b \<longleftrightarrow> b = 0"
871   "associated a 0 \<longleftrightarrow> a = 0"
872   by (auto dest: associatedD1 associatedD2)
874 lemma associated_unit:
875   "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
876   using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
878 lemma is_unit_associatedI:
879   assumes "is_unit c" and "a = c * b"
880   shows "associated a b"
881 proof (rule associatedI)
882   from a = c * b show "b dvd a" by auto
883   from is_unit c obtain d where "c * d = 1" by (rule is_unitE)
884   moreover from a = c * b have "d * a = d * (c * b)" by simp
885   ultimately have "b = a * d" by (simp add: ac_simps)
886   then show "a dvd b" ..
887 qed
889 lemma associated_is_unitE:
890   assumes "associated a b"
891   obtains c where "is_unit c" and "a = c * b"
892 proof (cases "b = 0")
893   case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
894   with that show thesis .
895 next
896   case False
897   from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
898   then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
899   then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
900   with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
901   then have "is_unit c" by auto
902   with a = c * b that show thesis by blast
903 qed
905 lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
906   dvd_div_unit_iff unit_div_mult_swap unit_div_commute
907   unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
908   unit_eq_div1 unit_eq_div2
910 end
913   assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
914   assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
915 begin
917 lemma mult_mono:
918   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
919 apply (erule mult_right_mono [THEN order_trans], assumption)
920 apply (erule mult_left_mono, assumption)
921 done
923 lemma mult_mono':
924   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
925 apply (rule mult_mono)
926 apply (fast intro: order_trans)+
927 done
929 end
931 class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
932 begin
934 subclass semiring_0_cancel ..
936 lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
937 using mult_left_mono [of 0 b a] by simp
939 lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
940 using mult_left_mono [of b 0 a] by simp
942 lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
943 using mult_right_mono [of a 0 b] by simp
945 text {* Legacy - use @{text mult_nonpos_nonneg} *}
946 lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
947 by (drule mult_right_mono [of b 0], auto)
949 lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
950 by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
952 end
954 class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
955 begin
957 subclass ordered_cancel_semiring ..
961 lemma mult_left_less_imp_less:
962   "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
963 by (force simp add: mult_left_mono not_le [symmetric])
965 lemma mult_right_less_imp_less:
966   "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
967 by (force simp add: mult_right_mono not_le [symmetric])
969 end
971 class linordered_semiring_1 = linordered_semiring + semiring_1
972 begin
974 lemma convex_bound_le:
975   assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
976   shows "u * x + v * y \<le> a"
977 proof-
978   from assms have "u * x + v * y \<le> u * a + v * a"
980   thus ?thesis using assms unfolding distrib_right[symmetric] by simp
981 qed
983 end
986   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
987   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
988 begin
990 subclass semiring_0_cancel ..
992 subclass linordered_semiring
993 proof
994   fix a b c :: 'a
995   assume A: "a \<le> b" "0 \<le> c"
996   from A show "c * a \<le> c * b"
997     unfolding le_less
998     using mult_strict_left_mono by (cases "c = 0") auto
999   from A show "a * c \<le> b * c"
1000     unfolding le_less
1001     using mult_strict_right_mono by (cases "c = 0") auto
1002 qed
1004 lemma mult_left_le_imp_le:
1005   "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
1006 by (force simp add: mult_strict_left_mono _not_less [symmetric])
1008 lemma mult_right_le_imp_le:
1009   "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
1010 by (force simp add: mult_strict_right_mono not_less [symmetric])
1012 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
1013 using mult_strict_left_mono [of 0 b a] by simp
1015 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
1016 using mult_strict_left_mono [of b 0 a] by simp
1018 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
1019 using mult_strict_right_mono [of a 0 b] by simp
1021 text {* Legacy - use @{text mult_neg_pos} *}
1022 lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
1023 by (drule mult_strict_right_mono [of b 0], auto)
1025 lemma zero_less_mult_pos:
1026   "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
1027 apply (cases "b\<le>0")
1028  apply (auto simp add: le_less not_less)
1029 apply (drule_tac mult_pos_neg [of a b])
1030  apply (auto dest: less_not_sym)
1031 done
1033 lemma zero_less_mult_pos2:
1034   "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
1035 apply (cases "b\<le>0")
1036  apply (auto simp add: le_less not_less)
1037 apply (drule_tac mult_pos_neg2 [of a b])
1038  apply (auto dest: less_not_sym)
1039 done
1041 text{*Strict monotonicity in both arguments*}
1042 lemma mult_strict_mono:
1043   assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
1044   shows "a * c < b * d"
1045   using assms apply (cases "c=0")
1046   apply (simp)
1047   apply (erule mult_strict_right_mono [THEN less_trans])
1048   apply (force simp add: le_less)
1049   apply (erule mult_strict_left_mono, assumption)
1050   done
1052 text{*This weaker variant has more natural premises*}
1053 lemma mult_strict_mono':
1054   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
1055   shows "a * c < b * d"
1056 by (rule mult_strict_mono) (insert assms, auto)
1058 lemma mult_less_le_imp_less:
1059   assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
1060   shows "a * c < b * d"
1061   using assms apply (subgoal_tac "a * c < b * c")
1062   apply (erule less_le_trans)
1063   apply (erule mult_left_mono)
1064   apply simp
1065   apply (erule mult_strict_right_mono)
1066   apply assumption
1067   done
1069 lemma mult_le_less_imp_less:
1070   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
1071   shows "a * c < b * d"
1072   using assms apply (subgoal_tac "a * c \<le> b * c")
1073   apply (erule le_less_trans)
1074   apply (erule mult_strict_left_mono)
1075   apply simp
1076   apply (erule mult_right_mono)
1077   apply simp
1078   done
1080 end
1082 class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
1083 begin
1085 subclass linordered_semiring_1 ..
1087 lemma convex_bound_lt:
1088   assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
1089   shows "u * x + v * y < a"
1090 proof -
1091   from assms have "u * x + v * y < u * a + v * a"
1092     by (cases "u = 0")
1093        (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
1094   thus ?thesis using assms unfolding distrib_right[symmetric] by simp
1095 qed
1097 end
1099 class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
1100   assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
1101 begin
1103 subclass ordered_semiring
1104 proof
1105   fix a b c :: 'a
1106   assume "a \<le> b" "0 \<le> c"
1107   thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
1108   thus "a * c \<le> b * c" by (simp only: mult.commute)
1109 qed
1111 end
1113 class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
1114 begin
1116 subclass comm_semiring_0_cancel ..
1117 subclass ordered_comm_semiring ..
1118 subclass ordered_cancel_semiring ..
1120 end
1122 class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
1123   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
1124 begin
1126 subclass linordered_semiring_strict
1127 proof
1128   fix a b c :: 'a
1129   assume "a < b" "0 < c"
1130   thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
1131   thus "a * c < b * c" by (simp only: mult.commute)
1132 qed
1134 subclass ordered_cancel_comm_semiring
1135 proof
1136   fix a b c :: 'a
1137   assume "a \<le> b" "0 \<le> c"
1138   thus "c * a \<le> c * b"
1139     unfolding le_less
1140     using mult_strict_left_mono by (cases "c = 0") auto
1141 qed
1143 end
1145 class ordered_ring = ring + ordered_cancel_semiring
1146 begin
1151   "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
1155   "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
1159   "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
1163   "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
1166 lemma mult_left_mono_neg:
1167   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
1168   apply (drule mult_left_mono [of _ _ "- c"])
1169   apply simp_all
1170   done
1172 lemma mult_right_mono_neg:
1173   "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
1174   apply (drule mult_right_mono [of _ _ "- c"])
1175   apply simp_all
1176   done
1178 lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
1179 using mult_right_mono_neg [of a 0 b] by simp
1181 lemma split_mult_pos_le:
1182   "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
1183 by (auto simp add: mult_nonpos_nonpos)
1185 end
1187 class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
1188 begin
1190 subclass ordered_ring ..
1193 proof
1194   fix a b
1195   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
1197 qed (auto simp add: abs_if)
1199 lemma zero_le_square [simp]: "0 \<le> a * a"
1200   using linear [of 0 a]
1201   by (auto simp add: mult_nonpos_nonpos)
1203 lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
1206 end
1208 class linordered_ring_strict = ring + linordered_semiring_strict
1210 begin
1212 subclass linordered_ring ..
1214 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
1215 using mult_strict_left_mono [of b a "- c"] by simp
1217 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
1218 using mult_strict_right_mono [of b a "- c"] by simp
1220 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
1221 using mult_strict_right_mono_neg [of a 0 b] by simp
1223 subclass ring_no_zero_divisors
1224 proof
1225   fix a b
1226   assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
1227   assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
1228   have "a * b < 0 \<or> 0 < a * b"
1229   proof (cases "a < 0")
1230     case True note A' = this
1231     show ?thesis proof (cases "b < 0")
1232       case True with A'
1233       show ?thesis by (auto dest: mult_neg_neg)
1234     next
1235       case False with B have "0 < b" by auto
1236       with A' show ?thesis by (auto dest: mult_strict_right_mono)
1237     qed
1238   next
1239     case False with A have A': "0 < a" by auto
1240     show ?thesis proof (cases "b < 0")
1241       case True with A'
1242       show ?thesis by (auto dest: mult_strict_right_mono_neg)
1243     next
1244       case False with B have "0 < b" by auto
1245       with A' show ?thesis by auto
1246     qed
1247   qed
1248   then show "a * b \<noteq> 0" by (simp add: neq_iff)
1249 qed
1251 lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
1252   by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
1253      (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
1255 lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
1256   by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
1258 lemma mult_less_0_iff:
1259   "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
1260   apply (insert zero_less_mult_iff [of "-a" b])
1261   apply force
1262   done
1264 lemma mult_le_0_iff:
1265   "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
1266   apply (insert zero_le_mult_iff [of "-a" b])
1267   apply force
1268   done
1270 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
1271    also with the relations @{text "\<le>"} and equality.*}
1273 text{*These disjunction'' versions produce two cases when the comparison is
1274  an assumption, but effectively four when the comparison is a goal.*}
1276 lemma mult_less_cancel_right_disj:
1277   "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
1278   apply (cases "c = 0")
1279   apply (auto simp add: neq_iff mult_strict_right_mono
1280                       mult_strict_right_mono_neg)
1281   apply (auto simp add: not_less
1282                       not_le [symmetric, of "a*c"]
1283                       not_le [symmetric, of a])
1284   apply (erule_tac [!] notE)
1285   apply (auto simp add: less_imp_le mult_right_mono
1286                       mult_right_mono_neg)
1287   done
1289 lemma mult_less_cancel_left_disj:
1290   "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
1291   apply (cases "c = 0")
1292   apply (auto simp add: neq_iff mult_strict_left_mono
1293                       mult_strict_left_mono_neg)
1294   apply (auto simp add: not_less
1295                       not_le [symmetric, of "c*a"]
1296                       not_le [symmetric, of a])
1297   apply (erule_tac [!] notE)
1298   apply (auto simp add: less_imp_le mult_left_mono
1299                       mult_left_mono_neg)
1300   done
1302 text{*The conjunction of implication'' lemmas produce two cases when the
1303 comparison is a goal, but give four when the comparison is an assumption.*}
1305 lemma mult_less_cancel_right:
1306   "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
1307   using mult_less_cancel_right_disj [of a c b] by auto
1309 lemma mult_less_cancel_left:
1310   "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
1311   using mult_less_cancel_left_disj [of c a b] by auto
1313 lemma mult_le_cancel_right:
1314    "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
1315 by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
1317 lemma mult_le_cancel_left:
1318   "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
1319 by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
1321 lemma mult_le_cancel_left_pos:
1322   "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
1323 by (auto simp: mult_le_cancel_left)
1325 lemma mult_le_cancel_left_neg:
1326   "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
1327 by (auto simp: mult_le_cancel_left)
1329 lemma mult_less_cancel_left_pos:
1330   "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
1331 by (auto simp: mult_less_cancel_left)
1333 lemma mult_less_cancel_left_neg:
1334   "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
1335 by (auto simp: mult_less_cancel_left)
1337 end
1339 lemmas mult_sign_intros =
1340   mult_nonneg_nonneg mult_nonneg_nonpos
1341   mult_nonpos_nonneg mult_nonpos_nonpos
1342   mult_pos_pos mult_pos_neg
1343   mult_neg_pos mult_neg_neg
1345 class ordered_comm_ring = comm_ring + ordered_comm_semiring
1346 begin
1348 subclass ordered_ring ..
1349 subclass ordered_cancel_comm_semiring ..
1351 end
1353 class linordered_semidom = semidom + linordered_comm_semiring_strict +
1354   assumes zero_less_one [simp]: "0 < 1"
1355   assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
1356 begin
1358 text {* Addition is the inverse of subtraction. *}
1360 lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
1363 lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
1364   by simp
1367   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
1368   using add_strict_mono [of 0 a b c] by simp
1370 lemma zero_le_one [simp]: "0 \<le> 1"
1371 by (rule zero_less_one [THEN less_imp_le])
1373 lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
1376 lemma not_one_less_zero [simp]: "\<not> 1 < 0"
1379 lemma less_1_mult:
1380   assumes "1 < m" and "1 < n"
1381   shows "1 < m * n"
1382   using assms mult_strict_mono [of 1 m 1 n]
1383     by (simp add:  less_trans [OF zero_less_one])
1385 lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
1386   using mult_left_mono[of c 1 a] by simp
1388 lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
1389   using mult_mono[of a 1 b 1] by simp
1391 end
1393 class linordered_idom = comm_ring_1 +
1395   abs_if + sgn_if
1396 begin
1398 subclass linordered_semiring_1_strict ..
1399 subclass linordered_ring_strict ..
1400 subclass ordered_comm_ring ..
1401 subclass idom ..
1403 subclass linordered_semidom
1404 proof
1405   have "0 \<le> 1 * 1" by (rule zero_le_square)
1406   thus "0 < 1" by (simp add: le_less)
1407   show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
1408     by simp
1409 qed
1411 lemma linorder_neqE_linordered_idom:
1412   assumes "x \<noteq> y" obtains "x < y" | "y < x"
1413   using assms by (rule neqE)
1415 text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
1417 lemma mult_le_cancel_right1:
1418   "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
1419 by (insert mult_le_cancel_right [of 1 c b], simp)
1421 lemma mult_le_cancel_right2:
1422   "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
1423 by (insert mult_le_cancel_right [of a c 1], simp)
1425 lemma mult_le_cancel_left1:
1426   "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
1427 by (insert mult_le_cancel_left [of c 1 b], simp)
1429 lemma mult_le_cancel_left2:
1430   "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
1431 by (insert mult_le_cancel_left [of c a 1], simp)
1433 lemma mult_less_cancel_right1:
1434   "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
1435 by (insert mult_less_cancel_right [of 1 c b], simp)
1437 lemma mult_less_cancel_right2:
1438   "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
1439 by (insert mult_less_cancel_right [of a c 1], simp)
1441 lemma mult_less_cancel_left1:
1442   "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
1443 by (insert mult_less_cancel_left [of c 1 b], simp)
1445 lemma mult_less_cancel_left2:
1446   "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
1447 by (insert mult_less_cancel_left [of c a 1], simp)
1449 lemma sgn_sgn [simp]:
1450   "sgn (sgn a) = sgn a"
1451 unfolding sgn_if by simp
1453 lemma sgn_0_0:
1454   "sgn a = 0 \<longleftrightarrow> a = 0"
1455 unfolding sgn_if by simp
1457 lemma sgn_1_pos:
1458   "sgn a = 1 \<longleftrightarrow> a > 0"
1459 unfolding sgn_if by simp
1461 lemma sgn_1_neg:
1462   "sgn a = - 1 \<longleftrightarrow> a < 0"
1463 unfolding sgn_if by auto
1465 lemma sgn_pos [simp]:
1466   "0 < a \<Longrightarrow> sgn a = 1"
1467 unfolding sgn_1_pos .
1469 lemma sgn_neg [simp]:
1470   "a < 0 \<Longrightarrow> sgn a = - 1"
1471 unfolding sgn_1_neg .
1473 lemma sgn_times:
1474   "sgn (a * b) = sgn a * sgn b"
1475 by (auto simp add: sgn_if zero_less_mult_iff)
1477 lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
1478 unfolding sgn_if abs_if by auto
1480 lemma sgn_greater [simp]:
1481   "0 < sgn a \<longleftrightarrow> 0 < a"
1482   unfolding sgn_if by auto
1484 lemma sgn_less [simp]:
1485   "sgn a < 0 \<longleftrightarrow> a < 0"
1486   unfolding sgn_if by auto
1488 lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
1491 lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
1494 lemma dvd_if_abs_eq:
1495   "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
1496 by(subst abs_dvd_iff[symmetric]) simp
1498 text {* The following lemmas can be proven in more general structures, but
1499 are dangerous as simp rules in absence of @{thm neg_equal_zero},
1500 @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
1502 lemma equation_minus_iff_1 [simp, no_atp]:
1503   "1 = - a \<longleftrightarrow> a = - 1"
1504   by (fact equation_minus_iff)
1506 lemma minus_equation_iff_1 [simp, no_atp]:
1507   "- a = 1 \<longleftrightarrow> a = - 1"
1508   by (subst minus_equation_iff, auto)
1510 lemma le_minus_iff_1 [simp, no_atp]:
1511   "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
1512   by (fact le_minus_iff)
1514 lemma minus_le_iff_1 [simp, no_atp]:
1515   "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
1516   by (fact minus_le_iff)
1518 lemma less_minus_iff_1 [simp, no_atp]:
1519   "1 < - b \<longleftrightarrow> b < - 1"
1520   by (fact less_minus_iff)
1522 lemma minus_less_iff_1 [simp, no_atp]:
1523   "- a < 1 \<longleftrightarrow> - 1 < a"
1524   by (fact minus_less_iff)
1526 end
1528 text {* Simprules for comparisons where common factors can be cancelled. *}
1530 lemmas mult_compare_simps =
1531     mult_le_cancel_right mult_le_cancel_left
1532     mult_le_cancel_right1 mult_le_cancel_right2
1533     mult_le_cancel_left1 mult_le_cancel_left2
1534     mult_less_cancel_right mult_less_cancel_left
1535     mult_less_cancel_right1 mult_less_cancel_right2
1536     mult_less_cancel_left1 mult_less_cancel_left2
1537     mult_cancel_right mult_cancel_left
1538     mult_cancel_right1 mult_cancel_right2
1539     mult_cancel_left1 mult_cancel_left2
1541 text {* Reasoning about inequalities with division *}
1543 context linordered_semidom
1544 begin
1546 lemma less_add_one: "a < a + 1"
1547 proof -
1548   have "a + 0 < a + 1"
1549     by (blast intro: zero_less_one add_strict_left_mono)
1550   thus ?thesis by simp
1551 qed
1553 lemma zero_less_two: "0 < 1 + 1"
1554 by (blast intro: less_trans zero_less_one less_add_one)
1556 end
1558 context linordered_idom
1559 begin
1561 lemma mult_right_le_one_le:
1562   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
1563   by (rule mult_left_le)
1565 lemma mult_left_le_one_le:
1566   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
1567   by (auto simp add: mult_le_cancel_right2)
1569 end
1571 text {* Absolute Value *}
1573 context linordered_idom
1574 begin
1576 lemma mult_sgn_abs:
1577   "sgn x * \<bar>x\<bar> = x"
1578   unfolding abs_if sgn_if by auto
1580 lemma abs_one [simp]:
1581   "\<bar>1\<bar> = 1"
1584 end
1586 class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
1587   assumes abs_eq_mult:
1588     "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
1590 context linordered_idom
1591 begin
1593 subclass ordered_ring_abs proof
1594 qed (auto simp add: abs_if not_less mult_less_0_iff)
1596 lemma abs_mult:
1597   "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
1598   by (rule abs_eq_mult) auto
1600 lemma abs_mult_self:
1601   "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
1604 lemma abs_mult_less:
1605   "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
1606 proof -
1607   assume ac: "\<bar>a\<bar> < c"
1608   hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
1609   assume "\<bar>b\<bar> < d"
1610   thus ?thesis by (simp add: ac cpos mult_strict_mono)
1611 qed
1613 lemma abs_less_iff:
1614   "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
1617 lemma abs_mult_pos:
1618   "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"