src/HOL/OrderedGroup.thy
 author wenzelm Wed Jun 18 18:54:57 2008 +0200 (2008-06-18) changeset 27250 7eef2b183032 parent 26480 544cef16045b child 27474 a89d755b029d permissions -rw-r--r--
simplified Abel_Cancel setup;
1 (*  Title:   HOL/OrderedGroup.thy
2     ID:      $Id$
3     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel,
4              with contributions by Jeremy Avigad
5 *)
7 header {* Ordered Groups *}
9 theory OrderedGroup
10 imports Lattices
11 uses "~~/src/Provers/Arith/abel_cancel.ML"
12 begin
14 text {*
15   The theory of partially ordered groups is taken from the books:
16   \begin{itemize}
17   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
18   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
19   \end{itemize}
20   Most of the used notions can also be looked up in
21   \begin{itemize}
22   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
23   \item \emph{Algebra I} by van der Waerden, Springer.
24   \end{itemize}
25 *}
27 subsection {* Semigroups and Monoids *}
29 class semigroup_add = plus +
30   assumes add_assoc: "(a + b) + c = a + (b + c)"
33   assumes add_commute: "a + b = b + a"
34 begin
36 lemma add_left_commute: "a + (b + c) = b + (a + c)"
41 end
45 class semigroup_mult = times +
46   assumes mult_assoc: "(a * b) * c = a * (b * c)"
48 class ab_semigroup_mult = semigroup_mult +
49   assumes mult_commute: "a * b = b * a"
50 begin
52 lemma mult_left_commute: "a * (b * c) = b * (a * c)"
53   by (rule mk_left_commute [of "times", OF mult_assoc mult_commute])
55 theorems mult_ac = mult_assoc mult_commute mult_left_commute
57 end
59 theorems mult_ac = mult_assoc mult_commute mult_left_commute
61 class ab_semigroup_idem_mult = ab_semigroup_mult +
62   assumes mult_idem: "x * x = x"
63 begin
65 lemma mult_left_idem: "x * (x * y) = x * y"
66   unfolding mult_assoc [symmetric, of x] mult_idem ..
68 lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem
70 end
72 lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem
75   assumes add_0_left [simp]: "0 + a = a"
76     and add_0_right [simp]: "a + 0 = a"
78 lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
79   by (rule eq_commute)
82   assumes add_0: "0 + a = a"
83 begin
88 end
90 class monoid_mult = one + semigroup_mult +
91   assumes mult_1_left [simp]: "1 * a  = a"
92   assumes mult_1_right [simp]: "a * 1 = a"
94 lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
95   by (rule eq_commute)
97 class comm_monoid_mult = one + ab_semigroup_mult +
98   assumes mult_1: "1 * a = a"
99 begin
101 subclass monoid_mult
102   by unfold_locales (insert mult_1, simp_all add: mult_commute)
104 end
107   assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
108   assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
111   assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
112 begin
115 proof unfold_locales
116   fix a b c :: 'a
117   assume "a + b = a + c"
118   then show "b = c" by (rule add_imp_eq)
119 next
120   fix a b c :: 'a
121   assume "b + a = c + a"
122   then have "a + b = a + c" by (simp only: add_commute)
123   then show "b = c" by (rule add_imp_eq)
124 qed
126 end
129 begin
132   "a + b = a + c \<longleftrightarrow> b = c"
136   "b + a = c + a \<longleftrightarrow> b = c"
139 end
141 subsection {* Groups *}
144   assumes left_minus [simp]: "- a + a = 0"
145   assumes diff_minus: "a - b = a + (- b)"
146 begin
148 lemma minus_add_cancel: "- a + (a + b) = b"
151 lemma minus_zero [simp]: "- 0 = 0"
152 proof -
153   have "- 0 = - 0 + (0 + 0)" by (simp only: add_0_right)
154   also have "\<dots> = 0" by (rule minus_add_cancel)
155   finally show ?thesis .
156 qed
158 lemma minus_minus [simp]: "- (- a) = a"
159 proof -
160   have "- (- a) = - (- a) + (- a + a)" by simp
161   also have "\<dots> = a" by (rule minus_add_cancel)
162   finally show ?thesis .
163 qed
165 lemma right_minus [simp]: "a + - a = 0"
166 proof -
167   have "a + - a = - (- a) + - a" by simp
168   also have "\<dots> = 0" by (rule left_minus)
169   finally show ?thesis .
170 qed
172 lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b"
173 proof
174   assume "a - b = 0"
175   have "a = (a - b) + b" by (simp add:diff_minus add_assoc)
176   also have "\<dots> = b" using a - b = 0 by simp
177   finally show "a = b" .
178 next
179   assume "a = b" thus "a - b = 0" by (simp add: diff_minus)
180 qed
182 lemma equals_zero_I:
183   assumes "a + b = 0"
184   shows "- a = b"
185 proof -
186   have "- a = - a + (a + b)" using assms by simp
188   finally show ?thesis .
189 qed
191 lemma diff_self [simp]: "a - a = 0"
194 lemma diff_0 [simp]: "0 - a = - a"
197 lemma diff_0_right [simp]: "a - 0 = a"
200 lemma diff_minus_eq_add [simp]: "a - - b = a + b"
203 lemma neg_equal_iff_equal [simp]:
204   "- a = - b \<longleftrightarrow> a = b"
205 proof
206   assume "- a = - b"
207   hence "- (- a) = - (- b)"
208     by simp
209   thus "a = b" by simp
210 next
211   assume "a = b"
212   thus "- a = - b" by simp
213 qed
215 lemma neg_equal_0_iff_equal [simp]:
216   "- a = 0 \<longleftrightarrow> a = 0"
217   by (subst neg_equal_iff_equal [symmetric], simp)
219 lemma neg_0_equal_iff_equal [simp]:
220   "0 = - a \<longleftrightarrow> 0 = a"
221   by (subst neg_equal_iff_equal [symmetric], simp)
223 text{*The next two equations can make the simplifier loop!*}
225 lemma equation_minus_iff:
226   "a = - b \<longleftrightarrow> b = - a"
227 proof -
228   have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
229   thus ?thesis by (simp add: eq_commute)
230 qed
232 lemma minus_equation_iff:
233   "- a = b \<longleftrightarrow> - b = a"
234 proof -
235   have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
236   thus ?thesis by (simp add: eq_commute)
237 qed
239 end
242   assumes ab_left_minus: "- a + a = 0"
243   assumes ab_diff_minus: "a - b = a + (- b)"
244 begin
247   by unfold_locales (simp_all add: ab_left_minus ab_diff_minus)
250 proof unfold_locales
251   fix a b c :: 'a
252   assume "a + b = a + c"
253   then have "- a + a + b = - a + a + c"
255   then show "b = c" by simp
256 qed
259   "- a + b = b - a"
263   "- (a + b) = - a + - b"
266 lemma minus_diff_eq [simp]:
267   "- (a - b) = b - a"
270 lemma add_diff_eq: "a + (b - c) = (a + b) - c"
273 lemma diff_add_eq: "(a - b) + c = (a + c) - b"
276 lemma diff_eq_eq: "a - b = c \<longleftrightarrow> a = c + b"
279 lemma eq_diff_eq: "a = c - b \<longleftrightarrow> a + b = c"
282 lemma diff_diff_eq: "(a - b) - c = a - (b + c)"
285 lemma diff_diff_eq2: "a - (b - c) = (a + c) - b"
288 lemma diff_add_cancel: "a - b + b = a"
291 lemma add_diff_cancel: "a + b - b = a"
294 lemmas compare_rls =
295        diff_minus [symmetric]
297        diff_eq_eq eq_diff_eq
299 lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
302 end
304 subsection {* (Partially) Ordered Groups *}
307   assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
308 begin
311   "a \<le> b \<Longrightarrow> a + c \<le> b + c"
314 text {* non-strict, in both arguments *}
316   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
317   apply (erule add_right_mono [THEN order_trans])
319   done
321 end
325 begin
328   "a < b \<Longrightarrow> c + a < c + b"
332   "a < b \<Longrightarrow> a + c < b + c"
335 text{*Strict monotonicity in both arguments*}
337   "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
338 apply (erule add_strict_right_mono [THEN less_trans])
340 done
343   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
344 apply (erule add_strict_right_mono [THEN less_le_trans])
346 done
349   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
350 apply (erule add_right_mono [THEN le_less_trans])
352 done
354 end
358   assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
359 begin
362    assumes less: "c + a < c + b"
363    shows "a < b"
364 proof -
365   from less have le: "c + a <= c + b" by (simp add: order_le_less)
366   have "a <= b"
367     apply (insert le)
369     by (insert le, drule add_le_imp_le_left, assumption)
370   moreover have "a \<noteq> b"
371   proof (rule ccontr)
372     assume "~(a \<noteq> b)"
373     then have "a = b" by simp
374     then have "c + a = c + b" by simp
375     with less show "False"by simp
376   qed
377   ultimately show "a < b" by (simp add: order_le_less)
378 qed
381   "a + c < b + c \<Longrightarrow> a < b"
382 apply (rule add_less_imp_less_left [of c])
384 done
387   "c + a < c + b \<longleftrightarrow> a < b"
391   "a + c < b + c \<longleftrightarrow> a < b"
395   "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
399   "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
403   "a + c \<le> b + c \<Longrightarrow> a \<le> b"
404   by simp
407   "max x y + z = max (x + z) (y + z)"
408   unfolding max_def by auto
411   "min x y + z = min (x + z) (y + z)"
412   unfolding min_def by auto
414 end
416 subsection {* Support for reasoning about signs *}
420 begin
423   assumes "0 < a" and "0 \<le> b"
424     shows "0 < a + b"
425 proof -
426   have "0 + 0 < a + b"
427     using assms by (rule add_less_le_mono)
428   then show ?thesis by simp
429 qed
432   assumes "0 < a" and "0 < b"
433     shows "0 < a + b"
434   by (rule add_pos_nonneg) (insert assms, auto)
437   assumes "0 \<le> a" and "0 < b"
438     shows "0 < a + b"
439 proof -
440   have "0 + 0 < a + b"
441     using assms by (rule add_le_less_mono)
442   then show ?thesis by simp
443 qed
446   assumes "0 \<le> a" and "0 \<le> b"
447     shows "0 \<le> a + b"
448 proof -
449   have "0 + 0 \<le> a + b"
450     using assms by (rule add_mono)
451   then show ?thesis by simp
452 qed
455   assumes "a < 0" and "b \<le> 0"
456   shows "a + b < 0"
457 proof -
458   have "a + b < 0 + 0"
459     using assms by (rule add_less_le_mono)
460   then show ?thesis by simp
461 qed
464   assumes "a < 0" and "b < 0"
465   shows "a + b < 0"
466   by (rule add_neg_nonpos) (insert assms, auto)
469   assumes "a \<le> 0" and "b < 0"
470   shows "a + b < 0"
471 proof -
472   have "a + b < 0 + 0"
473     using assms by (rule add_le_less_mono)
474   then show ?thesis by simp
475 qed
478   assumes "a \<le> 0" and "b \<le> 0"
479   shows "a + b \<le> 0"
480 proof -
481   have "a + b \<le> 0 + 0"
482     using assms by (rule add_mono)
483   then show ?thesis by simp
484 qed
486 end
490 begin
493   by intro_locales
496 proof unfold_locales
497   fix a b c :: 'a
498   assume "c + a \<le> c + b"
499   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
500   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
501   thus "a \<le> b" by simp
502 qed
505   by intro_locales
507 lemma max_diff_distrib_left:
508   shows "max x y - z = max (x - z) (y - z)"
511 lemma min_diff_distrib_left:
512   shows "min x y - z = min (x - z) (y - z)"
515 lemma le_imp_neg_le:
516   assumes "a \<le> b"
517   shows "-b \<le> -a"
518 proof -
519   have "-a+a \<le> -a+b"
520     using a \<le> b by (rule add_left_mono)
521   hence "0 \<le> -a+b"
522     by simp
523   hence "0 + (-b) \<le> (-a + b) + (-b)"
525   thus ?thesis
527 qed
529 lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
530 proof
531   assume "- b \<le> - a"
532   hence "- (- a) \<le> - (- b)"
533     by (rule le_imp_neg_le)
534   thus "a\<le>b" by simp
535 next
536   assume "a\<le>b"
537   thus "-b \<le> -a" by (rule le_imp_neg_le)
538 qed
540 lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
541   by (subst neg_le_iff_le [symmetric], simp)
543 lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
544   by (subst neg_le_iff_le [symmetric], simp)
546 lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
547   by (force simp add: less_le)
549 lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
550   by (subst neg_less_iff_less [symmetric], simp)
552 lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
553   by (subst neg_less_iff_less [symmetric], simp)
555 text{*The next several equations can make the simplifier loop!*}
557 lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
558 proof -
559   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
560   thus ?thesis by simp
561 qed
563 lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
564 proof -
565   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
566   thus ?thesis by simp
567 qed
569 lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
570 proof -
571   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
572   have "(- (- a) <= -b) = (b <= - a)"
573     apply (auto simp only: le_less)
574     apply (drule mm)
575     apply (simp_all)
576     apply (drule mm[simplified], assumption)
577     done
578   then show ?thesis by simp
579 qed
581 lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
582   by (auto simp add: le_less minus_less_iff)
584 lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
585 proof -
586   have  "(a < b) = (a + (- b) < b + (-b))"
588   also have "... =  (a - b < 0)" by (simp add: diff_minus)
589   finally show ?thesis .
590 qed
592 lemma diff_less_eq: "a - b < c \<longleftrightarrow> a < c + b"
593 apply (subst less_iff_diff_less_0 [of a])
594 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
596 done
598 lemma less_diff_eq: "a < c - b \<longleftrightarrow> a + b < c"
599 apply (subst less_iff_diff_less_0 [of "plus a b"])
600 apply (subst less_iff_diff_less_0 [of a])
602 done
604 lemma diff_le_eq: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
607 lemma le_diff_eq: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
610 lemmas compare_rls =
611        diff_minus [symmetric]
613        diff_less_eq less_diff_eq diff_le_eq le_diff_eq
614        diff_eq_eq eq_diff_eq
616 text{*This list of rewrites simplifies (in)equalities by bringing subtractions
617   to the top and then moving negative terms to the other side.
619 lemmas (in -) compare_rls =
620        diff_minus [symmetric]
622        diff_less_eq less_diff_eq diff_le_eq le_diff_eq
623        diff_eq_eq eq_diff_eq
625 lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
628 lemmas group_simps =
631   diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
632   diff_less_eq less_diff_eq diff_le_eq le_diff_eq
634 end
636 lemmas group_simps =
637   mult_ac
640   diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
641   diff_less_eq less_diff_eq diff_le_eq le_diff_eq
648 begin
651   by intro_locales
654 proof unfold_locales
655   fix a b c :: 'a
656   assume le: "c + a <= c + b"
657   show "a <= b"
658   proof (rule ccontr)
659     assume w: "~ a \<le> b"
660     hence "b <= a" by (simp add: linorder_not_le)
661     hence le2: "c + b <= c + a" by (rule add_left_mono)
662     have "a = b"
663       apply (insert le)
664       apply (insert le2)
665       apply (drule antisym, simp_all)
666       done
667     with w show False
668       by (simp add: linorder_not_le [symmetric])
669   qed
670 qed
672 end
676 begin
679   by intro_locales
681 lemma neg_less_eq_nonneg:
682   "- a \<le> a \<longleftrightarrow> 0 \<le> a"
683 proof
684   assume A: "- a \<le> a" show "0 \<le> a"
685   proof (rule classical)
686     assume "\<not> 0 \<le> a"
687     then have "a < 0" by auto
688     with A have "- a < 0" by (rule le_less_trans)
689     then show ?thesis by auto
690   qed
691 next
692   assume A: "0 \<le> a" show "- a \<le> a"
693   proof (rule order_trans)
694     show "- a \<le> 0" using A by (simp add: minus_le_iff)
695   next
696     show "0 \<le> a" using A .
697   qed
698 qed
700 lemma less_eq_neg_nonpos:
701   "a \<le> - a \<longleftrightarrow> a \<le> 0"
702 proof
703   assume A: "a \<le> - a" show "a \<le> 0"
704   proof (rule classical)
705     assume "\<not> a \<le> 0"
706     then have "0 < a" by auto
707     then have "0 < - a" using A by (rule less_le_trans)
708     then show ?thesis by auto
709   qed
710 next
711   assume A: "a \<le> 0" show "a \<le> - a"
712   proof (rule order_trans)
713     show "0 \<le> - a" using A by (simp add: minus_le_iff)
714   next
715     show "a \<le> 0" using A .
716   qed
717 qed
719 lemma equal_neg_zero:
720   "a = - a \<longleftrightarrow> a = 0"
721 proof
722   assume "a = 0" then show "a = - a" by simp
723 next
724   assume A: "a = - a" show "a = 0"
725   proof (cases "0 \<le> a")
726     case True with A have "0 \<le> - a" by auto
727     with le_minus_iff have "a \<le> 0" by simp
728     with True show ?thesis by (auto intro: order_trans)
729   next
730     case False then have B: "a \<le> 0" by auto
731     with A have "- a \<le> 0" by auto
732     with B show ?thesis by (auto intro: order_trans)
733   qed
734 qed
736 lemma neg_equal_zero:
737   "- a = a \<longleftrightarrow> a = 0"
738   unfolding equal_neg_zero [symmetric] by auto
740 end
742 -- {* FIXME localize the following *}
746   shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
747 by (insert add_mono [of 0 a b c], simp)
751   shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
756   shows "[|0<a; b\<le>c|] ==> b < a + c"
757 by (insert add_less_le_mono [of 0 a b c], simp)
761   shows "[|0\<le>a; b<c|] ==> b < a + c"
762 by (insert add_le_less_mono [of 0 a b c], simp)
766   assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
767     and abs_ge_self: "a \<le> \<bar>a\<bar>"
768     and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
769     and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
770     and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
771 begin
773 lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
774   unfolding neg_le_0_iff_le by simp
776 lemma abs_of_nonneg [simp]:
777   assumes nonneg: "0 \<le> a"
778   shows "\<bar>a\<bar> = a"
779 proof (rule antisym)
780   from nonneg le_imp_neg_le have "- a \<le> 0" by simp
781   from this nonneg have "- a \<le> a" by (rule order_trans)
782   then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
783 qed (rule abs_ge_self)
785 lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
786   by (rule antisym)
787     (auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"])
789 lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
790 proof -
791   have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
792   proof (rule antisym)
793     assume zero: "\<bar>a\<bar> = 0"
794     with abs_ge_self show "a \<le> 0" by auto
795     from zero have "\<bar>-a\<bar> = 0" by simp
796     with abs_ge_self [of "uminus a"] have "- a \<le> 0" by auto
797     with neg_le_0_iff_le show "0 \<le> a" by auto
798   qed
799   then show ?thesis by auto
800 qed
802 lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
803   by simp
805 lemma abs_0_eq [simp, noatp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
806 proof -
807   have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
808   thus ?thesis by simp
809 qed
811 lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"
812 proof
813   assume "\<bar>a\<bar> \<le> 0"
814   then have "\<bar>a\<bar> = 0" by (rule antisym) simp
815   thus "a = 0" by simp
816 next
817   assume "a = 0"
818   thus "\<bar>a\<bar> \<le> 0" by simp
819 qed
821 lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
824 lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
825 proof -
826   have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
827   show ?thesis by (simp add: a)
828 qed
830 lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
831 proof -
832   have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
833   then show ?thesis by simp
834 qed
836 lemma abs_minus_commute:
837   "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
838 proof -
839   have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
840   also have "... = \<bar>b - a\<bar>" by simp
841   finally show ?thesis .
842 qed
844 lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
845   by (rule abs_of_nonneg, rule less_imp_le)
847 lemma abs_of_nonpos [simp]:
848   assumes "a \<le> 0"
849   shows "\<bar>a\<bar> = - a"
850 proof -
851   let ?b = "- a"
852   have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
853   unfolding abs_minus_cancel [of "?b"]
854   unfolding neg_le_0_iff_le [of "?b"]
855   unfolding minus_minus by (erule abs_of_nonneg)
856   then show ?thesis using assms by auto
857 qed
859 lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
860   by (rule abs_of_nonpos, rule less_imp_le)
862 lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
863   by (insert abs_ge_self, blast intro: order_trans)
865 lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
866   by (insert abs_le_D1 [of "uminus a"], simp)
868 lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
869   by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
871 lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
873   apply (subgoal_tac "abs a = abs (plus (minus a b) b)")
874   apply (erule ssubst)
875   apply (rule abs_triangle_ineq)
876   apply (rule arg_cong) back
878 done
880 lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
881   apply (subst abs_le_iff)
882   apply auto
883   apply (rule abs_triangle_ineq2)
884   apply (subst abs_minus_commute)
885   apply (rule abs_triangle_ineq2)
886 done
888 lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
889 proof -
890   have "abs(a - b) = abs(a + - b)"
891     by (subst diff_minus, rule refl)
892   also have "... <= abs a + abs (- b)"
893     by (rule abs_triangle_ineq)
894   finally show ?thesis
895     by simp
896 qed
898 lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
899 proof -
900   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
901   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
902   finally show ?thesis .
903 qed
906   "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
907 proof (rule antisym)
908   show "?L \<ge> ?R" by(rule abs_ge_self)
909 next
910   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
911   also have "\<dots> = ?R" by simp
912   finally show "?L \<le> ?R" .
913 qed
915 end
918 subsection {* Lattice Ordered (Abelian) Groups *}
921 begin
924   "a + inf b c = inf (a + b) (a + c)"
925 apply (rule antisym)
927 apply (rule add_le_imp_le_left [of "uminus a"])
928 apply (simp only: add_assoc [symmetric], simp)
929 apply rule
931 done
934   "inf a b + c = inf (a + c) (b + c)"
935 proof -
936   have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
938 qed
940 end
943 begin
946   "a + sup b c = sup (a + b) (a + c)"
947 apply (rule antisym)
948 apply (rule add_le_imp_le_left [of "uminus a"])
949 apply (simp only: add_assoc[symmetric], simp)
950 apply rule
952 apply (rule le_supI)
953 apply (simp_all)
954 done
957   "sup a b + c = sup (a+c) (b+c)"
958 proof -
959   have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
961 qed
963 end
966 begin
973 lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
974 proof (rule inf_unique)
975   fix a b :: 'a
976   show "- sup (-a) (-b) \<le> a"
977     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
979 next
980   fix a b :: 'a
981   show "- sup (-a) (-b) \<le> b"
982     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
984 next
985   fix a b c :: 'a
986   assume "a \<le> b" "a \<le> c"
987   then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
989 qed
991 lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
992 proof (rule sup_unique)
993   fix a b :: 'a
994   show "a \<le> - inf (-a) (-b)"
995     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
997 next
998   fix a b :: 'a
999   show "b \<le> - inf (-a) (-b)"
1000     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
1002 next
1003   fix a b c :: 'a
1004   assume "a \<le> c" "b \<le> c"
1005   then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
1007 qed
1009 lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
1012 lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
1015 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
1016 proof -
1017   have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
1018   hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
1019   hence "0 = (-a + sup a b) + (inf a b + (-b))"
1022   thus ?thesis
1024     apply (subst add_left_cancel [symmetric, of "plus a b" "plus (sup a b) (inf a b)" "uminus a"])
1026     done
1027 qed
1029 subsection {* Positive Part, Negative Part, Absolute Value *}
1031 definition
1032   nprt :: "'a \<Rightarrow> 'a" where
1033   "nprt x = inf x 0"
1035 definition
1036   pprt :: "'a \<Rightarrow> 'a" where
1037   "pprt x = sup x 0"
1039 lemma pprt_neg: "pprt (- x) = - nprt x"
1040 proof -
1041   have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
1042   also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
1043   finally have "sup (- x) 0 = - inf x 0" .
1044   then show ?thesis unfolding pprt_def nprt_def .
1045 qed
1047 lemma nprt_neg: "nprt (- x) = - pprt x"
1048 proof -
1049   from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
1050   then have "pprt x = - nprt (- x)" by simp
1051   then show ?thesis by simp
1052 qed
1054 lemma prts: "a = pprt a + nprt a"
1057 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
1060 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
1063 lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
1064 proof -
1065   have a: "?l \<longrightarrow> ?r"
1066     apply (auto)
1067     apply (rule add_le_imp_le_right[of _ "uminus b" _])
1069     done
1070   have b: "?r \<longrightarrow> ?l"
1071     apply (auto)
1072     apply (rule add_le_imp_le_right[of _ "b" _])
1073     apply (simp)
1074     done
1075   from a b show ?thesis by blast
1076 qed
1078 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
1079 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
1081 lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
1082   by (simp add: pprt_def le_iff_sup sup_ACI)
1084 lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
1085   by (simp add: nprt_def le_iff_inf inf_ACI)
1087 lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
1088   by (simp add: pprt_def le_iff_sup sup_ACI)
1090 lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
1091   by (simp add: nprt_def le_iff_inf inf_ACI)
1093 lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
1094 proof -
1095   {
1096     fix a::'a
1097     assume hyp: "sup a (-a) = 0"
1098     hence "sup a (-a) + a = a" by (simp)
1100     hence "sup (a+a) 0 <= a" by (simp)
1101     hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
1102   }
1103   note p = this
1104   assume hyp:"sup a (-a) = 0"
1105   hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
1106   from p[OF hyp] p[OF hyp2] show "a = 0" by simp
1107 qed
1109 lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
1112 apply (erule sup_0_imp_0)
1113 done
1115 lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
1116   by (rule, erule inf_0_imp_0) simp
1118 lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
1119   by (rule, erule sup_0_imp_0) simp
1122   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
1123 proof
1124   assume "0 <= a + a"
1125   hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute)
1126   have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
1128   hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
1129   hence "inf a 0 = 0" by (simp only: add_right_cancel)
1130   then show "0 <= a" by (simp add: le_iff_inf inf_commute)
1131 next
1132   assume a: "0 <= a"
1133   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
1134 qed
1136 lemma double_zero: "a + a = 0 \<longleftrightarrow> a = 0"
1137 proof
1138   assume assm: "a + a = 0"
1139   then have "a + a + - a = - a" by simp
1140   then have "a + (a + - a) = - a" by (simp only: add_assoc)
1141   then have a: "- a = a" by simp (*FIXME tune proof*)
1142   show "a = 0" apply (rule antisym)
1143   apply (unfold neg_le_iff_le [symmetric, of a])
1144   unfolding a apply simp
1146   unfolding assm unfolding le_less apply simp_all done
1147 next
1148   assume "a = 0" then show "a + a = 0" by simp
1149 qed
1152   "0 < a + a \<longleftrightarrow> 0 < a"
1153 proof (cases "a = 0")
1154   case True then show ?thesis by auto
1155 next
1156   case False then show ?thesis (*FIXME tune proof*)
1157   unfolding less_le apply simp apply rule
1158   apply clarify
1159   apply rule
1160   apply assumption
1161   apply (rule notI)
1162   unfolding double_zero [symmetric, of a] apply simp
1163   done
1164 qed
1167   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
1168 proof -
1169   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
1171   ultimately show ?thesis by blast
1172 qed
1175   "a + a < 0 \<longleftrightarrow> a < 0"
1176 proof -
1177   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
1179   ultimately show ?thesis by blast
1180 qed
1182 declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
1184 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
1185 proof -
1186   from add_le_cancel_left [of "uminus a" "plus a a" zero]
1187   have "(a <= -a) = (a+a <= 0)"
1189   thus ?thesis by simp
1190 qed
1192 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
1193 proof -
1194   from add_le_cancel_left [of "uminus a" zero "plus a a"]
1195   have "(-a <= a) = (0 <= a+a)"
1197   thus ?thesis by simp
1198 qed
1200 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
1201   by (simp add: le_iff_inf nprt_def inf_commute)
1203 lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
1204   by (simp add: le_iff_sup pprt_def sup_commute)
1206 lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
1207   by (simp add: le_iff_sup pprt_def sup_commute)
1209 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
1210   by (simp add: le_iff_inf nprt_def inf_commute)
1212 lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
1213   by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
1215 lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
1216   by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
1218 end
1224   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
1225 begin
1227 lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
1228 proof -
1229   have "0 \<le> \<bar>a\<bar>"
1230   proof -
1231     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
1232     show ?thesis by (rule add_mono [OF a b, simplified])
1233   qed
1234   then have "0 \<le> sup a (- a)" unfolding abs_lattice .
1235   then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
1236   then show ?thesis
1238       pprt_def nprt_def diff_minus abs_lattice)
1239 qed
1242 proof -
1243   have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
1244   proof -
1245     fix a b
1246     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
1247     show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
1248   qed
1249   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
1250     by (simp add: abs_lattice le_supI)
1251   show ?thesis
1252   proof unfold_locales
1253     fix a
1254     show "0 \<le> \<bar>a\<bar>" by simp
1255   next
1256     fix a
1257     show "a \<le> \<bar>a\<bar>"
1258       by (auto simp add: abs_lattice)
1259   next
1260     fix a
1261     show "\<bar>-a\<bar> = \<bar>a\<bar>"
1262       by (simp add: abs_lattice sup_commute)
1263   next
1264     fix a b
1265     show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (erule abs_leI)
1266   next
1267     fix a b
1268     show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
1269     proof -
1270       have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
1272       have a:"a+b <= sup ?m ?n" by (simp)
1273       have b:"-a-b <= ?n" by (simp)
1274       have c:"?n <= sup ?m ?n" by (simp)
1275       from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
1276       have e:"-a-b = -(a+b)" by (simp add: diff_minus)
1277       from a d e have "abs(a+b) <= sup ?m ?n"
1278         by (drule_tac abs_leI, auto)
1279       with g[symmetric] show ?thesis by simp
1280     qed
1281   qed auto
1282 qed
1284 end
1286 lemma sup_eq_if:
1287   fixes a :: "'a\<Colon>{lordered_ab_group_add, linorder}"
1288   shows "sup a (- a) = (if a < 0 then - a else a)"
1289 proof -
1290   note add_le_cancel_right [of a a "- a", symmetric, simplified]
1291   moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
1292   then show ?thesis by (auto simp: sup_max max_def)
1293 qed
1295 lemma abs_if_lattice:
1296   fixes a :: "'a\<Colon>{lordered_ab_group_add_abs, linorder}"
1297   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
1298   by auto
1301 text {* Needed for abelian cancellation simprocs: *}
1303 lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
1306 apply simp
1307 done
1309 lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
1310 apply (subst add_cancel_21[of _ _ _ 0, simplified])
1312 done
1314 lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
1315 by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
1317 lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
1318 apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
1319 apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
1320 done
1322 lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
1323 by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
1325 lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
1332   assumes
1333   "a <= b + (c::'a::pordered_ab_group_add)"
1334   "c <= d"
1335   shows "a <= b + d"
1336   apply (rule_tac order_trans[where y = "b+c"])
1338   done
1340 lemma estimate_by_abs:
1341   "a + b <= (c::'a::lordered_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
1342 proof -
1343   assume "a+b <= c"
1344   hence 2: "a <= c+(-b)" by (simp add: group_simps)
1345   have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
1346   show ?thesis by (rule le_add_right_mono[OF 2 3])
1347 qed
1349 subsection {* Tools setup *}
1352   fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
1353   shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
1354     and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
1355     and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
1356     and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
1360   fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
1361   shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
1362     and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
1363     and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
1364     and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
1365     and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
1369 text{*Simplification of @{term "x-y < 0"}, etc.*}
1370 lemmas diff_less_0_iff_less [simp] = less_iff_diff_less_0 [symmetric]
1371 lemmas diff_eq_0_iff_eq [simp, noatp] = eq_iff_diff_eq_0 [symmetric]
1372 lemmas diff_le_0_iff_le [simp] = le_iff_diff_le_0 [symmetric]
1374 ML {*
1376 (
1378 (* term order for abelian groups *)
1380 fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
1381       [@{const_name HOL.zero}, @{const_name HOL.plus},
1382         @{const_name HOL.uminus}, @{const_name HOL.minus}]
1383   | agrp_ord _ = ~1;
1385 fun termless_agrp (a, b) = (Term.term_lpo agrp_ord (a, b) = LESS);
1387 local
1388   val ac1 = mk_meta_eq @{thm add_assoc};
1389   val ac2 = mk_meta_eq @{thm add_commute};
1390   val ac3 = mk_meta_eq @{thm add_left_commute};
1391   fun solve_add_ac thy _ (_ $(Const (@{const_name HOL.plus},_)$ _ $_)$ _) =
1392         SOME ac1
1393     | solve_add_ac thy _ (_ $x$ (Const (@{const_name HOL.plus},_) $y$ z)) =
1394         if termless_agrp (y, x) then SOME ac3 else NONE
1395     | solve_add_ac thy _ (_ $x$ y) =
1396         if termless_agrp (y, x) then SOME ac2 else NONE
1397     | solve_add_ac thy _ _ = NONE
1398 in
1399   val add_ac_proc = Simplifier.simproc @{theory}
1401 end;
1403 val eq_reflection = @{thm eq_reflection};
1405 val T = @{typ "'a::ab_group_add"};
1407 val cancel_ss = HOL_basic_ss settermless termless_agrp