src/HOL/OrderedGroup.thy
 author haftmann Fri Feb 15 16:09:10 2008 +0100 (2008-02-15) changeset 26071 046fe7ddfc4b parent 26015 ad2756de580e child 26480 544cef16045b permissions -rw-r--r--
moved *_reorient lemmas here
     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 *)

     6

     7 header {* Ordered Groups *}

     8

     9 theory OrderedGroup

    10 imports Lattices

    11 uses "~~/src/Provers/Arith/abel_cancel.ML"

    12 begin

    13

    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 *}

    26

    27 subsection {* Semigroups and Monoids *}

    28

    29 class semigroup_add = plus +

    30   assumes add_assoc: "(a + b) + c = a + (b + c)"

    31

    32 class ab_semigroup_add = semigroup_add +

    33   assumes add_commute: "a + b = b + a"

    34 begin

    35

    36 lemma add_left_commute: "a + (b + c) = b + (a + c)"

    37   by (rule mk_left_commute [of "plus", OF add_assoc add_commute])

    38

    39 theorems add_ac = add_assoc add_commute add_left_commute

    40

    41 end

    42

    43 theorems add_ac = add_assoc add_commute add_left_commute

    44

    45 class semigroup_mult = times +

    46   assumes mult_assoc: "(a * b) * c = a * (b * c)"

    47

    48 class ab_semigroup_mult = semigroup_mult +

    49   assumes mult_commute: "a * b = b * a"

    50 begin

    51

    52 lemma mult_left_commute: "a * (b * c) = b * (a * c)"

    53   by (rule mk_left_commute [of "times", OF mult_assoc mult_commute])

    54

    55 theorems mult_ac = mult_assoc mult_commute mult_left_commute

    56

    57 end

    58

    59 theorems mult_ac = mult_assoc mult_commute mult_left_commute

    60

    61 class ab_semigroup_idem_mult = ab_semigroup_mult +

    62   assumes mult_idem: "x * x = x"

    63 begin

    64

    65 lemma mult_left_idem: "x * (x * y) = x * y"

    66   unfolding mult_assoc [symmetric, of x] mult_idem ..

    67

    68 lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem

    69

    70 end

    71

    72 lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem

    73

    74 class monoid_add = zero + semigroup_add +

    75   assumes add_0_left [simp]: "0 + a = a"

    76     and add_0_right [simp]: "a + 0 = a"

    77

    78 lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"

    79   by (rule eq_commute)

    80

    81 class comm_monoid_add = zero + ab_semigroup_add +

    82   assumes add_0: "0 + a = a"

    83 begin

    84

    85 subclass monoid_add

    86   by unfold_locales (insert add_0, simp_all add: add_commute)

    87

    88 end

    89

    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"

    93

    94 lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"

    95   by (rule eq_commute)

    96

    97 class comm_monoid_mult = one + ab_semigroup_mult +

    98   assumes mult_1: "1 * a = a"

    99 begin

   100

   101 subclass monoid_mult

   102   by unfold_locales (insert mult_1, simp_all add: mult_commute)

   103

   104 end

   105

   106 class cancel_semigroup_add = semigroup_add +

   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"

   109

   110 class cancel_ab_semigroup_add = ab_semigroup_add +

   111   assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"

   112 begin

   113

   114 subclass cancel_semigroup_add

   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

   125

   126 end

   127

   128 context cancel_ab_semigroup_add

   129 begin

   130

   131 lemma add_left_cancel [simp]:

   132   "a + b = a + c \<longleftrightarrow> b = c"

   133   by (blast dest: add_left_imp_eq)

   134

   135 lemma add_right_cancel [simp]:

   136   "b + a = c + a \<longleftrightarrow> b = c"

   137   by (blast dest: add_right_imp_eq)

   138

   139 end

   140

   141 subsection {* Groups *}

   142

   143 class group_add = minus + uminus + monoid_add +

   144   assumes left_minus [simp]: "- a + a = 0"

   145   assumes diff_minus: "a - b = a + (- b)"

   146 begin

   147

   148 lemma minus_add_cancel: "- a + (a + b) = b"

   149   by (simp add: add_assoc[symmetric])

   150

   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

   157

   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

   164

   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

   171

   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

   181

   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

   187   also have "\<dots> = b" by (simp add: add_assoc[symmetric])

   188   finally show ?thesis .

   189 qed

   190

   191 lemma diff_self [simp]: "a - a = 0"

   192   by (simp add: diff_minus)

   193

   194 lemma diff_0 [simp]: "0 - a = - a"

   195   by (simp add: diff_minus)

   196

   197 lemma diff_0_right [simp]: "a - 0 = a"

   198   by (simp add: diff_minus)

   199

   200 lemma diff_minus_eq_add [simp]: "a - - b = a + b"

   201   by (simp add: diff_minus)

   202

   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

   214

   215 lemma neg_equal_0_iff_equal [simp]:

   216   "- a = 0 \<longleftrightarrow> a = 0"

   217   by (subst neg_equal_iff_equal [symmetric], simp)

   218

   219 lemma neg_0_equal_iff_equal [simp]:

   220   "0 = - a \<longleftrightarrow> 0 = a"

   221   by (subst neg_equal_iff_equal [symmetric], simp)

   222

   223 text{*The next two equations can make the simplifier loop!*}

   224

   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

   231

   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

   238

   239 end

   240

   241 class ab_group_add = minus + uminus + comm_monoid_add +

   242   assumes ab_left_minus: "- a + a = 0"

   243   assumes ab_diff_minus: "a - b = a + (- b)"

   244 begin

   245

   246 subclass group_add

   247   by unfold_locales (simp_all add: ab_left_minus ab_diff_minus)

   248

   249 subclass cancel_ab_semigroup_add

   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"

   254     unfolding add_assoc by simp

   255   then show "b = c" by simp

   256 qed

   257

   258 lemma uminus_add_conv_diff:

   259   "- a + b = b - a"

   260   by (simp add:diff_minus add_commute)

   261

   262 lemma minus_add_distrib [simp]:

   263   "- (a + b) = - a + - b"

   264   by (rule equals_zero_I) (simp add: add_ac)

   265

   266 lemma minus_diff_eq [simp]:

   267   "- (a - b) = b - a"

   268   by (simp add: diff_minus add_commute)

   269

   270 lemma add_diff_eq: "a + (b - c) = (a + b) - c"

   271   by (simp add: diff_minus add_ac)

   272

   273 lemma diff_add_eq: "(a - b) + c = (a + c) - b"

   274   by (simp add: diff_minus add_ac)

   275

   276 lemma diff_eq_eq: "a - b = c \<longleftrightarrow> a = c + b"

   277   by (auto simp add: diff_minus add_assoc)

   278

   279 lemma eq_diff_eq: "a = c - b \<longleftrightarrow> a + b = c"

   280   by (auto simp add: diff_minus add_assoc)

   281

   282 lemma diff_diff_eq: "(a - b) - c = a - (b + c)"

   283   by (simp add: diff_minus add_ac)

   284

   285 lemma diff_diff_eq2: "a - (b - c) = (a + c) - b"

   286   by (simp add: diff_minus add_ac)

   287

   288 lemma diff_add_cancel: "a - b + b = a"

   289   by (simp add: diff_minus add_ac)

   290

   291 lemma add_diff_cancel: "a + b - b = a"

   292   by (simp add: diff_minus add_ac)

   293

   294 lemmas compare_rls =

   295        diff_minus [symmetric]

   296        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   297        diff_eq_eq eq_diff_eq

   298

   299 lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"

   300   by (simp add: compare_rls)

   301

   302 end

   303

   304 subsection {* (Partially) Ordered Groups *}

   305

   306 class pordered_ab_semigroup_add = order + ab_semigroup_add +

   307   assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"

   308 begin

   309

   310 lemma add_right_mono:

   311   "a \<le> b \<Longrightarrow> a + c \<le> b + c"

   312   by (simp add: add_commute [of _ c] add_left_mono)

   313

   314 text {* non-strict, in both arguments *}

   315 lemma add_mono:

   316   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"

   317   apply (erule add_right_mono [THEN order_trans])

   318   apply (simp add: add_commute add_left_mono)

   319   done

   320

   321 end

   322

   323 class pordered_cancel_ab_semigroup_add =

   324   pordered_ab_semigroup_add + cancel_ab_semigroup_add

   325 begin

   326

   327 lemma add_strict_left_mono:

   328   "a < b \<Longrightarrow> c + a < c + b"

   329   by (auto simp add: less_le add_left_mono)

   330

   331 lemma add_strict_right_mono:

   332   "a < b \<Longrightarrow> a + c < b + c"

   333   by (simp add: add_commute [of _ c] add_strict_left_mono)

   334

   335 text{*Strict monotonicity in both arguments*}

   336 lemma add_strict_mono:

   337   "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"

   338 apply (erule add_strict_right_mono [THEN less_trans])

   339 apply (erule add_strict_left_mono)

   340 done

   341

   342 lemma add_less_le_mono:

   343   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"

   344 apply (erule add_strict_right_mono [THEN less_le_trans])

   345 apply (erule add_left_mono)

   346 done

   347

   348 lemma add_le_less_mono:

   349   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"

   350 apply (erule add_right_mono [THEN le_less_trans])

   351 apply (erule add_strict_left_mono)

   352 done

   353

   354 end

   355

   356 class pordered_ab_semigroup_add_imp_le =

   357   pordered_cancel_ab_semigroup_add +

   358   assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"

   359 begin

   360

   361 lemma add_less_imp_less_left:

   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)

   368     apply (drule add_le_imp_le_left)

   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

   379

   380 lemma add_less_imp_less_right:

   381   "a + c < b + c \<Longrightarrow> a < b"

   382 apply (rule add_less_imp_less_left [of c])

   383 apply (simp add: add_commute)

   384 done

   385

   386 lemma add_less_cancel_left [simp]:

   387   "c + a < c + b \<longleftrightarrow> a < b"

   388   by (blast intro: add_less_imp_less_left add_strict_left_mono)

   389

   390 lemma add_less_cancel_right [simp]:

   391   "a + c < b + c \<longleftrightarrow> a < b"

   392   by (blast intro: add_less_imp_less_right add_strict_right_mono)

   393

   394 lemma add_le_cancel_left [simp]:

   395   "c + a \<le> c + b \<longleftrightarrow> a \<le> b"

   396   by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)

   397

   398 lemma add_le_cancel_right [simp]:

   399   "a + c \<le> b + c \<longleftrightarrow> a \<le> b"

   400   by (simp add: add_commute [of a c] add_commute [of b c])

   401

   402 lemma add_le_imp_le_right:

   403   "a + c \<le> b + c \<Longrightarrow> a \<le> b"

   404   by simp

   405

   406 lemma max_add_distrib_left:

   407   "max x y + z = max (x + z) (y + z)"

   408   unfolding max_def by auto

   409

   410 lemma min_add_distrib_left:

   411   "min x y + z = min (x + z) (y + z)"

   412   unfolding min_def by auto

   413

   414 end

   415

   416 subsection {* Support for reasoning about signs *}

   417

   418 class pordered_comm_monoid_add =

   419   pordered_cancel_ab_semigroup_add + comm_monoid_add

   420 begin

   421

   422 lemma add_pos_nonneg:

   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

   430

   431 lemma add_pos_pos:

   432   assumes "0 < a" and "0 < b"

   433     shows "0 < a + b"

   434   by (rule add_pos_nonneg) (insert assms, auto)

   435

   436 lemma add_nonneg_pos:

   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

   444

   445 lemma add_nonneg_nonneg:

   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

   453

   454 lemma add_neg_nonpos:

   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

   462

   463 lemma add_neg_neg:

   464   assumes "a < 0" and "b < 0"

   465   shows "a + b < 0"

   466   by (rule add_neg_nonpos) (insert assms, auto)

   467

   468 lemma add_nonpos_neg:

   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

   476

   477 lemma add_nonpos_nonpos:

   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

   485

   486 end

   487

   488 class pordered_ab_group_add =

   489   ab_group_add + pordered_ab_semigroup_add

   490 begin

   491

   492 subclass pordered_cancel_ab_semigroup_add

   493   by intro_locales

   494

   495 subclass pordered_ab_semigroup_add_imp_le

   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

   503

   504 subclass pordered_comm_monoid_add

   505   by intro_locales

   506

   507 lemma max_diff_distrib_left:

   508   shows "max x y - z = max (x - z) (y - z)"

   509   by (simp add: diff_minus, rule max_add_distrib_left)

   510

   511 lemma min_diff_distrib_left:

   512   shows "min x y - z = min (x - z) (y - z)"

   513   by (simp add: diff_minus, rule min_add_distrib_left)

   514

   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)"

   524     by (rule add_right_mono)

   525   thus ?thesis

   526     by (simp add: add_assoc)

   527 qed

   528

   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

   539

   540 lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"

   541   by (subst neg_le_iff_le [symmetric], simp)

   542

   543 lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"

   544   by (subst neg_le_iff_le [symmetric], simp)

   545

   546 lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"

   547   by (force simp add: less_le)

   548

   549 lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"

   550   by (subst neg_less_iff_less [symmetric], simp)

   551

   552 lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"

   553   by (subst neg_less_iff_less [symmetric], simp)

   554

   555 text{*The next several equations can make the simplifier loop!*}

   556

   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

   562

   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

   568

   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

   580

   581 lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"

   582   by (auto simp add: le_less minus_less_iff)

   583

   584 lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"

   585 proof -

   586   have  "(a < b) = (a + (- b) < b + (-b))"

   587     by (simp only: add_less_cancel_right)

   588   also have "... =  (a - b < 0)" by (simp add: diff_minus)

   589   finally show ?thesis .

   590 qed

   591

   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])

   595 apply (simp add: diff_minus add_ac)

   596 done

   597

   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])

   601 apply (simp add: diff_minus add_ac)

   602 done

   603

   604 lemma diff_le_eq: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"

   605   by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)

   606

   607 lemma le_diff_eq: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"

   608   by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)

   609

   610 lemmas compare_rls =

   611        diff_minus [symmetric]

   612        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   613        diff_less_eq less_diff_eq diff_le_eq le_diff_eq

   614        diff_eq_eq eq_diff_eq

   615

   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.

   618   Use with @{text add_ac}*}

   619 lemmas (in -) compare_rls =

   620        diff_minus [symmetric]

   621        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   622        diff_less_eq less_diff_eq diff_le_eq le_diff_eq

   623        diff_eq_eq eq_diff_eq

   624

   625 lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"

   626   by (simp add: compare_rls)

   627

   628 lemmas group_simps =

   629   add_ac

   630   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   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

   633

   634 end

   635

   636 lemmas group_simps =

   637   mult_ac

   638   add_ac

   639   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   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

   642

   643 class ordered_ab_semigroup_add =

   644   linorder + pordered_ab_semigroup_add

   645

   646 class ordered_cancel_ab_semigroup_add =

   647   linorder + pordered_cancel_ab_semigroup_add

   648 begin

   649

   650 subclass ordered_ab_semigroup_add

   651   by intro_locales

   652

   653 subclass pordered_ab_semigroup_add_imp_le

   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

   671

   672 end

   673

   674 class ordered_ab_group_add =

   675   linorder + pordered_ab_group_add

   676 begin

   677

   678 subclass ordered_cancel_ab_semigroup_add

   679   by intro_locales

   680

   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

   699

   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

   718

   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

   735

   736 lemma neg_equal_zero:

   737   "- a = a \<longleftrightarrow> a = 0"

   738   unfolding equal_neg_zero [symmetric] by auto

   739

   740 end

   741

   742 -- {* FIXME localize the following *}

   743

   744 lemma add_increasing:

   745   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"

   746   shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"

   747 by (insert add_mono [of 0 a b c], simp)

   748

   749 lemma add_increasing2:

   750   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"

   751   shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"

   752 by (simp add:add_increasing add_commute[of a])

   753

   754 lemma add_strict_increasing:

   755   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"

   756   shows "[|0<a; b\<le>c|] ==> b < a + c"

   757 by (insert add_less_le_mono [of 0 a b c], simp)

   758

   759 lemma add_strict_increasing2:

   760   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"

   761   shows "[|0\<le>a; b<c|] ==> b < a + c"

   762 by (insert add_le_less_mono [of 0 a b c], simp)

   763

   764

   765 class pordered_ab_group_add_abs = pordered_ab_group_add + abs +

   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

   772

   773 lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"

   774   unfolding neg_le_0_iff_le by simp

   775

   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)

   784

   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"])

   788

   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

   801

   802 lemma abs_zero [simp]: "\<bar>0\<bar> = 0"

   803   by simp

   804

   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

   810

   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

   820

   821 lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"

   822   by (simp add: less_le)

   823

   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

   829

   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

   835

   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

   843

   844 lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"

   845   by (rule abs_of_nonneg, rule less_imp_le)

   846

   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

   858

   859 lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"

   860   by (rule abs_of_nonpos, rule less_imp_le)

   861

   862 lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"

   863   by (insert abs_ge_self, blast intro: order_trans)

   864

   865 lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"

   866   by (insert abs_le_D1 [of "uminus a"], simp)

   867

   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)

   870

   871 lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"

   872   apply (simp add: compare_rls)

   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

   877   apply (simp add: compare_rls)

   878 done

   879

   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

   887

   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

   897

   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

   904

   905 lemma abs_add_abs [simp]:

   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

   914

   915 end

   916

   917

   918 subsection {* Lattice Ordered (Abelian) Groups *}

   919

   920 class lordered_ab_group_add_meet = pordered_ab_group_add + lower_semilattice

   921 begin

   922

   923 lemma add_inf_distrib_left:

   924   "a + inf b c = inf (a + b) (a + c)"

   925 apply (rule antisym)

   926 apply (simp_all add: le_infI)

   927 apply (rule add_le_imp_le_left [of "uminus a"])

   928 apply (simp only: add_assoc [symmetric], simp)

   929 apply rule

   930 apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+

   931 done

   932

   933 lemma add_inf_distrib_right:

   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)

   937   thus ?thesis by (simp add: add_commute)

   938 qed

   939

   940 end

   941

   942 class lordered_ab_group_add_join = pordered_ab_group_add + upper_semilattice

   943 begin

   944

   945 lemma add_sup_distrib_left:

   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

   951 apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+

   952 apply (rule le_supI)

   953 apply (simp_all)

   954 done

   955

   956 lemma add_sup_distrib_right:

   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)

   960   thus ?thesis by (simp add: add_commute)

   961 qed

   962

   963 end

   964

   965 class lordered_ab_group_add = pordered_ab_group_add + lattice

   966 begin

   967

   968 subclass lordered_ab_group_add_meet by intro_locales

   969 subclass lordered_ab_group_add_join by intro_locales

   970

   971 lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left

   972

   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)"])

   978       (simp, simp add: add_sup_distrib_left)

   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)"])

   983       (simp, simp add: add_sup_distrib_left)

   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])

   988     (simp add: le_supI)

   989 qed

   990

   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)"])

   996       (simp, simp add: add_inf_distrib_left)

   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)"])

  1001       (simp, simp add: add_inf_distrib_left)

  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])

  1006     (simp add: le_infI)

  1007 qed

  1008

  1009 lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"

  1010   by (simp add: inf_eq_neg_sup)

  1011

  1012 lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"

  1013   by (simp add: sup_eq_neg_inf)

  1014

  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))"

  1020     apply (simp add: add_sup_distrib_left add_inf_distrib_right)

  1021     by (simp add: diff_minus add_commute)

  1022   thus ?thesis

  1023     apply (simp add: compare_rls)

  1024     apply (subst add_left_cancel [symmetric, of "plus a b" "plus (sup a b) (inf a b)" "uminus a"])

  1025     apply (simp only: add_assoc, simp add: add_assoc[symmetric])

  1026     done

  1027 qed

  1028

  1029 subsection {* Positive Part, Negative Part, Absolute Value *}

  1030

  1031 definition

  1032   nprt :: "'a \<Rightarrow> 'a" where

  1033   "nprt x = inf x 0"

  1034

  1035 definition

  1036   pprt :: "'a \<Rightarrow> 'a" where

  1037   "pprt x = sup x 0"

  1038

  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

  1046

  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

  1053

  1054 lemma prts: "a = pprt a + nprt a"

  1055   by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])

  1056

  1057 lemma zero_le_pprt[simp]: "0 \<le> pprt a"

  1058   by (simp add: pprt_def)

  1059

  1060 lemma nprt_le_zero[simp]: "nprt a \<le> 0"

  1061   by (simp add: nprt_def)

  1062

  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" _])

  1068     apply (simp add: add_assoc)

  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

  1077

  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)

  1080

  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)

  1083

  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)

  1086

  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)

  1089

  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)

  1092

  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)

  1099     hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)

  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

  1108

  1109 lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"

  1110 apply (simp add: inf_eq_neg_sup)

  1111 apply (simp add: sup_commute)

  1112 apply (erule sup_0_imp_0)

  1113 done

  1114

  1115 lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"

  1116   by (rule, erule inf_0_imp_0) simp

  1117

  1118 lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"

  1119   by (rule, erule sup_0_imp_0) simp

  1120

  1121 lemma zero_le_double_add_iff_zero_le_single_add [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=_")

  1127     by (simp add: add_sup_inf_distribs inf_ACI)

  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

  1135

  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

  1145   unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]

  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

  1150

  1151 lemma zero_less_double_add_iff_zero_less_single_add:

  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

  1165

  1166 lemma double_add_le_zero_iff_single_add_le_zero [simp]:

  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)

  1170   moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by (simp add: zero_le_double_add_iff_zero_le_single_add)

  1171   ultimately show ?thesis by blast

  1172 qed

  1173

  1174 lemma double_add_less_zero_iff_single_less_zero [simp]:

  1175   "a + a < 0 \<longleftrightarrow> a < 0"

  1176 proof -

  1177   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)

  1178   moreover have "\<dots> \<longleftrightarrow> a < 0" by (simp add: zero_less_double_add_iff_zero_less_single_add)

  1179   ultimately show ?thesis by blast

  1180 qed

  1181

  1182 declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]

  1183

  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)"

  1188     by (simp add: add_assoc[symmetric])

  1189   thus ?thesis by simp

  1190 qed

  1191

  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)"

  1196     by (simp add: add_assoc[symmetric])

  1197   thus ?thesis by simp

  1198 qed

  1199

  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)

  1202

  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)

  1205

  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)

  1208

  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)

  1211

  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])

  1214

  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])

  1217

  1218 end

  1219

  1220 lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left

  1221

  1222

  1223 class lordered_ab_group_add_abs = lordered_ab_group_add + abs +

  1224   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"

  1225 begin

  1226

  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

  1237     by (simp add: add_sup_inf_distribs sup_ACI

  1238       pprt_def nprt_def diff_minus abs_lattice)

  1239 qed

  1240

  1241 subclass pordered_ab_group_add_abs

  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")

  1271         by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)

  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

  1283

  1284 end

  1285

  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

  1294

  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

  1299

  1300

  1301 text {* Needed for abelian cancellation simprocs: *}

  1302

  1303 lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"

  1304 apply (subst add_left_commute)

  1305 apply (subst add_left_cancel)

  1306 apply simp

  1307 done

  1308

  1309 lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"

  1310 apply (subst add_cancel_21[of _ _ _ 0, simplified])

  1311 apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])

  1312 done

  1313

  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'])

  1316

  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

  1321

  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'])

  1324

  1325 lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"

  1326 by (simp add: diff_minus)

  1327

  1328 lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"

  1329 by (simp add: add_assoc[symmetric])

  1330

  1331 lemma le_add_right_mono:

  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"])

  1337   apply (simp_all add: prems)

  1338   done

  1339

  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

  1348

  1349 subsection {* Tools setup *}

  1350

  1351 lemma add_mono_thms_ordered_semiring [noatp]:

  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"

  1357 by (rule add_mono, clarify+)+

  1358

  1359 lemma add_mono_thms_ordered_field [noatp]:

  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"

  1366 by (auto intro: add_strict_right_mono add_strict_left_mono

  1367   add_less_le_mono add_le_less_mono add_strict_mono)

  1368

  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]

  1373

  1374 ML {*

  1375 structure ab_group_add_cancel = Abel_Cancel(

  1376 struct

  1377

  1378 (* term order for abelian groups *)

  1379

  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;

  1384

  1385 fun termless_agrp (a, b) = (Term.term_lpo agrp_ord (a, b) = LESS);

  1386

  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}

  1400     "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;

  1401 end;

  1402

  1403 val cancel_ss = HOL_basic_ss settermless termless_agrp

  1404   addsimprocs [add_ac_proc] addsimps

  1405   [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},

  1406    @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},

  1407    @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},

  1408    @{thm minus_add_cancel}];

  1409

  1410 val eq_reflection = @{thm eq_reflection};

  1411

  1412 val thy_ref = Theory.check_thy @{theory};

  1413

  1414 val T = @{typ "'a\<Colon>ab_group_add"};

  1415

  1416 val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];

  1417

  1418 val dest_eqI =

  1419   fst o HOLogic.dest_bin "op =" HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;

  1420

  1421 end);

  1422 *}

  1423

  1424 ML_setup {*

  1425   Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];

  1426 *}

  1427

  1428 end