src/HOL/OrderedGroup.thy
author haftmann
Fri Mar 10 15:33:48 2006 +0100 (2006-03-10)
changeset 19233 77ca20b0ed77
parent 17085 5b57f995a179
child 19404 9bf2cdc9e8e8
permissions -rw-r--r--
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
     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 Inductive LOrder
    11 uses "../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, Groups *}
    28  
    29 axclass semigroup_add \<subseteq> plus
    30   add_assoc: "(a + b) + c = a + (b + c)"
    31 
    32 axclass ab_semigroup_add \<subseteq> semigroup_add
    33   add_commute: "a + b = b + a"
    34 
    35 lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))"
    36   by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
    37 
    38 theorems add_ac = add_assoc add_commute add_left_commute
    39 
    40 axclass semigroup_mult \<subseteq> times
    41   mult_assoc: "(a * b) * c = a * (b * c)"
    42 
    43 axclass ab_semigroup_mult \<subseteq> semigroup_mult
    44   mult_commute: "a * b = b * a"
    45 
    46 lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))"
    47   by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
    48 
    49 theorems mult_ac = mult_assoc mult_commute mult_left_commute
    50 
    51 axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add
    52   add_0[simp]: "0 + a = a"
    53 
    54 axclass monoid_mult \<subseteq> one, semigroup_mult
    55   mult_1_left[simp]: "1 * a  = a"
    56   mult_1_right[simp]: "a * 1 = a"
    57 
    58 axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult
    59   mult_1: "1 * a = a"
    60 
    61 instance comm_monoid_mult \<subseteq> monoid_mult
    62 by (intro_classes, insert mult_1, simp_all add: mult_commute, auto)
    63 
    64 axclass cancel_semigroup_add \<subseteq> semigroup_add
    65   add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
    66   add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
    67 
    68 axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add
    69   add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
    70 
    71 instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add
    72 proof
    73   {
    74     fix a b c :: 'a
    75     assume "a + b = a + c"
    76     thus "b = c" by (rule add_imp_eq)
    77   }
    78   note f = this
    79   fix a b c :: 'a
    80   assume "b + a = c + a"
    81   hence "a + b = a + c" by (simp only: add_commute)
    82   thus "b = c" by (rule f)
    83 qed
    84 
    85 axclass ab_group_add \<subseteq> minus, comm_monoid_add
    86   left_minus[simp]: " - a + a = 0"
    87   diff_minus: "a - b = a + (-b)"
    88 
    89 instance ab_group_add \<subseteq> cancel_ab_semigroup_add
    90 proof 
    91   fix a b c :: 'a
    92   assume "a + b = a + c"
    93   hence "-a + a + b = -a + a + c" by (simp only: add_assoc)
    94   thus "b = c" by simp 
    95 qed
    96 
    97 lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)"
    98 proof -
    99   have "a + 0 = 0 + a" by (simp only: add_commute)
   100   also have "... = a" by simp
   101   finally show ?thesis .
   102 qed
   103 
   104 lemma add_left_cancel [simp]:
   105      "(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))"
   106 by (blast dest: add_left_imp_eq) 
   107 
   108 lemma add_right_cancel [simp]:
   109      "(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))"
   110   by (blast dest: add_right_imp_eq)
   111 
   112 lemma right_minus [simp]: "a + -(a::'a::ab_group_add) = 0"
   113 proof -
   114   have "a + -a = -a + a" by (simp add: add_ac)
   115   also have "... = 0" by simp
   116   finally show ?thesis .
   117 qed
   118 
   119 lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ab_group_add))"
   120 proof
   121   have "a = a - b + b" by (simp add: diff_minus add_ac)
   122   also assume "a - b = 0"
   123   finally show "a = b" by simp
   124 next
   125   assume "a = b"
   126   thus "a - b = 0" by (simp add: diff_minus)
   127 qed
   128 
   129 lemma minus_minus [simp]: "- (- (a::'a::ab_group_add)) = a"
   130 proof (rule add_left_cancel [of "-a", THEN iffD1])
   131   show "(-a + -(-a) = -a + a)"
   132   by simp
   133 qed
   134 
   135 lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ab_group_add)"
   136 apply (rule right_minus_eq [THEN iffD1, symmetric])
   137 apply (simp add: diff_minus add_commute) 
   138 done
   139 
   140 lemma minus_zero [simp]: "- 0 = (0::'a::ab_group_add)"
   141 by (simp add: equals_zero_I)
   142 
   143 lemma diff_self [simp]: "a - (a::'a::ab_group_add) = 0"
   144   by (simp add: diff_minus)
   145 
   146 lemma diff_0 [simp]: "(0::'a::ab_group_add) - a = -a"
   147 by (simp add: diff_minus)
   148 
   149 lemma diff_0_right [simp]: "a - (0::'a::ab_group_add) = a" 
   150 by (simp add: diff_minus)
   151 
   152 lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ab_group_add)"
   153 by (simp add: diff_minus)
   154 
   155 lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ab_group_add))" 
   156 proof 
   157   assume "- a = - b"
   158   hence "- (- a) = - (- b)"
   159     by simp
   160   thus "a=b" by simp
   161 next
   162   assume "a=b"
   163   thus "-a = -b" by simp
   164 qed
   165 
   166 lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))"
   167 by (subst neg_equal_iff_equal [symmetric], simp)
   168 
   169 lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))"
   170 by (subst neg_equal_iff_equal [symmetric], simp)
   171 
   172 text{*The next two equations can make the simplifier loop!*}
   173 
   174 lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))"
   175 proof -
   176   have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)
   177   thus ?thesis by (simp add: eq_commute)
   178 qed
   179 
   180 lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)"
   181 proof -
   182   have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)
   183   thus ?thesis by (simp add: eq_commute)
   184 qed
   185 
   186 lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)"
   187 apply (rule equals_zero_I)
   188 apply (simp add: add_ac) 
   189 done
   190 
   191 lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)"
   192 by (simp add: diff_minus add_commute)
   193 
   194 subsection {* (Partially) Ordered Groups *} 
   195 
   196 axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add
   197   add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
   198 
   199 axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add
   200 
   201 instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add ..
   202 
   203 axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add
   204   add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
   205 
   206 axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add
   207 
   208 instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le
   209 proof
   210   fix a b c :: 'a
   211   assume "c + a \<le> c + b"
   212   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
   213   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
   214   thus "a \<le> b" by simp
   215 qed
   216 
   217 axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder
   218 
   219 instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le
   220 proof
   221   fix a b c :: 'a
   222   assume le: "c + a <= c + b"  
   223   show "a <= b"
   224   proof (rule ccontr)
   225     assume w: "~ a \<le> b"
   226     hence "b <= a" by (simp add: linorder_not_le)
   227     hence le2: "c+b <= c+a" by (rule add_left_mono)
   228     have "a = b" 
   229       apply (insert le)
   230       apply (insert le2)
   231       apply (drule order_antisym, simp_all)
   232       done
   233     with w  show False 
   234       by (simp add: linorder_not_le [symmetric])
   235   qed
   236 qed
   237 
   238 lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c"
   239 by (simp add: add_commute[of _ c] add_left_mono)
   240 
   241 text {* non-strict, in both arguments *}
   242 lemma add_mono:
   243      "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)"
   244   apply (erule add_right_mono [THEN order_trans])
   245   apply (simp add: add_commute add_left_mono)
   246   done
   247 
   248 lemma add_strict_left_mono:
   249      "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)"
   250  by (simp add: order_less_le add_left_mono) 
   251 
   252 lemma add_strict_right_mono:
   253      "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)"
   254  by (simp add: add_commute [of _ c] add_strict_left_mono)
   255 
   256 text{*Strict monotonicity in both arguments*}
   257 lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
   258 apply (erule add_strict_right_mono [THEN order_less_trans])
   259 apply (erule add_strict_left_mono)
   260 done
   261 
   262 lemma add_less_le_mono:
   263      "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
   264 apply (erule add_strict_right_mono [THEN order_less_le_trans])
   265 apply (erule add_left_mono) 
   266 done
   267 
   268 lemma add_le_less_mono:
   269      "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
   270 apply (erule add_right_mono [THEN order_le_less_trans])
   271 apply (erule add_strict_left_mono) 
   272 done
   273 
   274 lemma add_less_imp_less_left:
   275       assumes less: "c + a < c + b"  shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)"
   276 proof -
   277   from less have le: "c + a <= c + b" by (simp add: order_le_less)
   278   have "a <= b" 
   279     apply (insert le)
   280     apply (drule add_le_imp_le_left)
   281     by (insert le, drule add_le_imp_le_left, assumption)
   282   moreover have "a \<noteq> b"
   283   proof (rule ccontr)
   284     assume "~(a \<noteq> b)"
   285     then have "a = b" by simp
   286     then have "c + a = c + b" by simp
   287     with less show "False"by simp
   288   qed
   289   ultimately show "a < b" by (simp add: order_le_less)
   290 qed
   291 
   292 lemma add_less_imp_less_right:
   293       "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)"
   294 apply (rule add_less_imp_less_left [of c])
   295 apply (simp add: add_commute)  
   296 done
   297 
   298 lemma add_less_cancel_left [simp]:
   299     "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
   300 by (blast intro: add_less_imp_less_left add_strict_left_mono) 
   301 
   302 lemma add_less_cancel_right [simp]:
   303     "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
   304 by (blast intro: add_less_imp_less_right add_strict_right_mono)
   305 
   306 lemma add_le_cancel_left [simp]:
   307     "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
   308 by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
   309 
   310 lemma add_le_cancel_right [simp]:
   311     "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
   312 by (simp add: add_commute[of a c] add_commute[of b c])
   313 
   314 lemma add_le_imp_le_right:
   315       "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)"
   316 by simp
   317 
   318 lemma add_increasing:
   319   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
   320   shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
   321 by (insert add_mono [of 0 a b c], simp)
   322 
   323 lemma add_increasing2:
   324   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
   325   shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
   326 by (simp add:add_increasing add_commute[of a])
   327 
   328 lemma add_strict_increasing:
   329   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
   330   shows "[|0<a; b\<le>c|] ==> b < a + c"
   331 by (insert add_less_le_mono [of 0 a b c], simp)
   332 
   333 lemma add_strict_increasing2:
   334   fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
   335   shows "[|0\<le>a; b<c|] ==> b < a + c"
   336 by (insert add_le_less_mono [of 0 a b c], simp)
   337 
   338 
   339 subsection {* Ordering Rules for Unary Minus *}
   340 
   341 lemma le_imp_neg_le:
   342       assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"
   343 proof -
   344   have "-a+a \<le> -a+b"
   345     by (rule add_left_mono) 
   346   hence "0 \<le> -a+b"
   347     by simp
   348   hence "0 + (-b) \<le> (-a + b) + (-b)"
   349     by (rule add_right_mono) 
   350   thus ?thesis
   351     by (simp add: add_assoc)
   352 qed
   353 
   354 lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"
   355 proof 
   356   assume "- b \<le> - a"
   357   hence "- (- a) \<le> - (- b)"
   358     by (rule le_imp_neg_le)
   359   thus "a\<le>b" by simp
   360 next
   361   assume "a\<le>b"
   362   thus "-b \<le> -a" by (rule le_imp_neg_le)
   363 qed
   364 
   365 lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"
   366 by (subst neg_le_iff_le [symmetric], simp)
   367 
   368 lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"
   369 by (subst neg_le_iff_le [symmetric], simp)
   370 
   371 lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"
   372 by (force simp add: order_less_le) 
   373 
   374 lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"
   375 by (subst neg_less_iff_less [symmetric], simp)
   376 
   377 lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"
   378 by (subst neg_less_iff_less [symmetric], simp)
   379 
   380 text{*The next several equations can make the simplifier loop!*}
   381 
   382 lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"
   383 proof -
   384   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
   385   thus ?thesis by simp
   386 qed
   387 
   388 lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"
   389 proof -
   390   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
   391   thus ?thesis by simp
   392 qed
   393 
   394 lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"
   395 proof -
   396   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
   397   have "(- (- a) <= -b) = (b <= - a)" 
   398     apply (auto simp only: order_le_less)
   399     apply (drule mm)
   400     apply (simp_all)
   401     apply (drule mm[simplified], assumption)
   402     done
   403   then show ?thesis by simp
   404 qed
   405 
   406 lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"
   407 by (auto simp add: order_le_less minus_less_iff)
   408 
   409 lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"
   410 by (simp add: diff_minus add_ac)
   411 
   412 lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"
   413 by (simp add: diff_minus add_ac)
   414 
   415 lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"
   416 by (auto simp add: diff_minus add_assoc)
   417 
   418 lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"
   419 by (auto simp add: diff_minus add_assoc)
   420 
   421 lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"
   422 by (simp add: diff_minus add_ac)
   423 
   424 lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"
   425 by (simp add: diff_minus add_ac)
   426 
   427 lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"
   428 by (simp add: diff_minus add_ac)
   429 
   430 lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"
   431 by (simp add: diff_minus add_ac)
   432 
   433 text{*Further subtraction laws*}
   434 
   435 lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"
   436 proof -
   437   have  "(a < b) = (a + (- b) < b + (-b))"  
   438     by (simp only: add_less_cancel_right)
   439   also have "... =  (a - b < 0)" by (simp add: diff_minus)
   440   finally show ?thesis .
   441 qed
   442 
   443 lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"
   444 apply (subst less_iff_diff_less_0 [of a])
   445 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
   446 apply (simp add: diff_minus add_ac)
   447 done
   448 
   449 lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"
   450 apply (subst less_iff_diff_less_0 [of "a+b"])
   451 apply (subst less_iff_diff_less_0 [of a])
   452 apply (simp add: diff_minus add_ac)
   453 done
   454 
   455 lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"
   456 by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)
   457 
   458 lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"
   459 by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)
   460 
   461 text{*This list of rewrites simplifies (in)equalities by bringing subtractions
   462   to the top and then moving negative terms to the other side.
   463   Use with @{text add_ac}*}
   464 lemmas compare_rls =
   465        diff_minus [symmetric]
   466        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
   467        diff_less_eq less_diff_eq diff_le_eq le_diff_eq
   468        diff_eq_eq eq_diff_eq
   469 
   470 subsection {* Support for reasoning about signs *}
   471 
   472 lemma add_pos_pos: "0 < 
   473     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   474       ==> 0 < y ==> 0 < x + y"
   475 apply (subgoal_tac "0 + 0 < x + y")
   476 apply simp
   477 apply (erule add_less_le_mono)
   478 apply (erule order_less_imp_le)
   479 done
   480 
   481 lemma add_pos_nonneg: "0 < 
   482     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   483       ==> 0 <= y ==> 0 < x + y"
   484 apply (subgoal_tac "0 + 0 < x + y")
   485 apply simp
   486 apply (erule add_less_le_mono, assumption)
   487 done
   488 
   489 lemma add_nonneg_pos: "0 <= 
   490     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   491       ==> 0 < y ==> 0 < x + y"
   492 apply (subgoal_tac "0 + 0 < x + y")
   493 apply simp
   494 apply (erule add_le_less_mono, assumption)
   495 done
   496 
   497 lemma add_nonneg_nonneg: "0 <= 
   498     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   499       ==> 0 <= y ==> 0 <= x + y"
   500 apply (subgoal_tac "0 + 0 <= x + y")
   501 apply simp
   502 apply (erule add_mono, assumption)
   503 done
   504 
   505 lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add})
   506     < 0 ==> y < 0 ==> x + y < 0"
   507 apply (subgoal_tac "x + y < 0 + 0")
   508 apply simp
   509 apply (erule add_less_le_mono)
   510 apply (erule order_less_imp_le)
   511 done
   512 
   513 lemma add_neg_nonpos: 
   514     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 
   515       ==> y <= 0 ==> x + y < 0"
   516 apply (subgoal_tac "x + y < 0 + 0")
   517 apply simp
   518 apply (erule add_less_le_mono, assumption)
   519 done
   520 
   521 lemma add_nonpos_neg: 
   522     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
   523       ==> y < 0 ==> x + y < 0"
   524 apply (subgoal_tac "x + y < 0 + 0")
   525 apply simp
   526 apply (erule add_le_less_mono, assumption)
   527 done
   528 
   529 lemma add_nonpos_nonpos: 
   530     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
   531       ==> y <= 0 ==> x + y <= 0"
   532 apply (subgoal_tac "x + y <= 0 + 0")
   533 apply simp
   534 apply (erule add_mono, assumption)
   535 done
   536 
   537 subsection{*Lemmas for the @{text cancel_numerals} simproc*}
   538 
   539 lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"
   540 by (simp add: compare_rls)
   541 
   542 lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"
   543 by (simp add: compare_rls)
   544 
   545 subsection {* Lattice Ordered (Abelian) Groups *}
   546 
   547 axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder
   548 
   549 axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder
   550 
   551 lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))"
   552 apply (rule order_antisym)
   553 apply (rule meet_imp_le, simp_all add: meet_join_le)
   554 apply (rule add_le_imp_le_left [of "-a"])
   555 apply (simp only: add_assoc[symmetric], simp)
   556 apply (rule meet_imp_le)
   557 apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
   558 done
   559 
   560 lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" 
   561 apply (rule order_antisym)
   562 apply (rule add_le_imp_le_left [of "-a"])
   563 apply (simp only: add_assoc[symmetric], simp)
   564 apply (rule join_imp_le)
   565 apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
   566 apply (rule join_imp_le)
   567 apply (simp_all add: meet_join_le)
   568 done
   569 
   570 lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))"
   571 apply (auto simp add: is_join_def)
   572 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
   573 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
   574 apply (subst neg_le_iff_le[symmetric]) 
   575 apply (simp add: meet_imp_le)
   576 done
   577 
   578 lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))"
   579 apply (auto simp add: is_meet_def)
   580 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
   581 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
   582 apply (subst neg_le_iff_le[symmetric]) 
   583 apply (simp add: join_imp_le)
   584 done
   585 
   586 axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder
   587 
   588 instance lordered_ab_group_meet \<subseteq> lordered_ab_group
   589 proof 
   590   show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet)
   591 qed
   592 
   593 instance lordered_ab_group_join \<subseteq> lordered_ab_group
   594 proof
   595   show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join)
   596 qed
   597 
   598 lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)"
   599 proof -
   600   have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left)
   601   thus ?thesis by (simp add: add_commute)
   602 qed
   603 
   604 lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)"
   605 proof -
   606   have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left)
   607   thus ?thesis by (simp add: add_commute)
   608 qed
   609 
   610 lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left
   611 
   612 lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)"
   613 by (simp add: is_join_unique[OF is_join_join is_join_neg_meet])
   614 
   615 lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)"
   616 by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join])
   617 
   618 lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))"
   619 proof -
   620   have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm)
   621   hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join)
   622   hence "0 = (-a + join a b) + (meet a b + (-b))"
   623     apply (simp add: add_join_distrib_left add_meet_distrib_right)
   624     by (simp add: diff_minus add_commute)
   625   thus ?thesis
   626     apply (simp add: compare_rls)
   627     apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"])
   628     apply (simp only: add_assoc, simp add: add_assoc[symmetric])
   629     done
   630 qed
   631 
   632 subsection {* Positive Part, Negative Part, Absolute Value *}
   633 
   634 constdefs
   635   pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
   636   "pprt x == join x 0"
   637   nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
   638   "nprt x == meet x 0"
   639 
   640 lemma prts: "a = pprt a + nprt a"
   641 by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric])
   642 
   643 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
   644 by (simp add: pprt_def meet_join_le)
   645 
   646 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
   647 by (simp add: nprt_def meet_join_le)
   648 
   649 lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")
   650 proof -
   651   have a: "?l \<longrightarrow> ?r"
   652     apply (auto)
   653     apply (rule add_le_imp_le_right[of _ "-b" _])
   654     apply (simp add: add_assoc)
   655     done
   656   have b: "?r \<longrightarrow> ?l"
   657     apply (auto)
   658     apply (rule add_le_imp_le_right[of _ "b" _])
   659     apply (simp)
   660     done
   661   from a b show ?thesis by blast
   662 qed
   663 
   664 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
   665 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
   666 
   667 lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x"
   668   by (simp add: pprt_def le_def_join join_aci)
   669 
   670 lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x"
   671   by (simp add: nprt_def le_def_meet meet_aci)
   672 
   673 lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0"
   674   by (simp add: pprt_def le_def_join join_aci)
   675 
   676 lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0"
   677   by (simp add: nprt_def le_def_meet meet_aci)
   678 
   679 lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
   680 proof -
   681   {
   682     fix a::'a
   683     assume hyp: "join a (-a) = 0"
   684     hence "join a (-a) + a = a" by (simp)
   685     hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) 
   686     hence "join (a+a) 0 <= a" by (simp)
   687     hence "0 <= a" by (blast intro: order_trans meet_join_le)
   688   }
   689   note p = this
   690   assume hyp:"join a (-a) = 0"
   691   hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm)
   692   from p[OF hyp] p[OF hyp2] show "a = 0" by simp
   693 qed
   694 
   695 lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
   696 apply (simp add: meet_eq_neg_join)
   697 apply (simp add: join_comm)
   698 apply (erule join_0_imp_0)
   699 done
   700 
   701 lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
   702 by (auto, erule join_0_imp_0)
   703 
   704 lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
   705 by (auto, erule meet_0_imp_0)
   706 
   707 lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"
   708 proof
   709   assume "0 <= a + a"
   710   hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm)
   711   have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci)
   712   hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm)
   713   hence "meet a 0 = 0" by (simp only: add_right_cancel)
   714   then show "0 <= a" by (simp add: le_def_meet meet_comm)    
   715 next  
   716   assume a: "0 <= a"
   717   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
   718 qed
   719 
   720 lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 
   721 proof -
   722   have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)
   723   moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)
   724   ultimately show ?thesis by blast
   725 qed
   726 
   727 lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)
   728 proof cases
   729   assume a: "a < 0"
   730   thus ?s by (simp add:  add_strict_mono[OF a a, simplified])
   731 next
   732   assume "~(a < 0)" 
   733   hence a:"0 <= a" by (simp)
   734   hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified])
   735   hence "~(a+a < 0)" by simp
   736   with a show ?thesis by simp 
   737 qed
   738 
   739 axclass lordered_ab_group_abs \<subseteq> lordered_ab_group
   740   abs_lattice: "abs x = join x (-x)"
   741 
   742 lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"
   743 by (simp add: abs_lattice)
   744 
   745 lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"
   746 by (simp add: abs_lattice)
   747 
   748 lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"
   749 proof -
   750   have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
   751   thus ?thesis by simp
   752 qed
   753 
   754 lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)"
   755 by (simp add: meet_eq_neg_join)
   756 
   757 lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)"
   758 by (simp del: neg_meet_eq_join add: join_eq_neg_meet)
   759 
   760 lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"
   761 proof -
   762   note b = add_le_cancel_right[of a a "-a",symmetric,simplified]
   763   have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)
   764   show ?thesis by (auto simp add: join_max max_def b linorder_not_less)
   765 qed
   766 
   767 lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"
   768 proof -
   769   show ?thesis by (simp add: abs_lattice join_eq_if)
   770 qed
   771 
   772 lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"
   773 proof -
   774   have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le)
   775   show ?thesis by (rule add_mono[OF a b, simplified])
   776 qed
   777   
   778 lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 
   779 proof
   780   assume "abs a <= 0"
   781   hence "abs a = 0" by (auto dest: order_antisym)
   782   thus "a = 0" by simp
   783 next
   784   assume "a = 0"
   785   thus "abs a <= 0" by simp
   786 qed
   787 
   788 lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"
   789 by (simp add: order_less_le)
   790 
   791 lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"
   792 proof -
   793   have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto
   794   show ?thesis by (simp add: a)
   795 qed
   796 
   797 lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"
   798 by (simp add: abs_lattice meet_join_le)
   799 
   800 lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"
   801 by (simp add: abs_lattice meet_join_le)
   802 
   803 lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" 
   804 by (simp add: le_def_join)
   805 
   806 lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a"
   807 by (simp add: le_def_join join_aci)
   808 
   809 lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a"
   810 by (simp add: le_def_meet)
   811 
   812 lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b"
   813 by (simp add: le_def_meet meet_aci)
   814 
   815 lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"
   816 apply (simp add: pprt_def nprt_def diff_minus)
   817 apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric])
   818 apply (subst le_imp_join_eq, auto)
   819 done
   820 
   821 lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"
   822 by (simp add: abs_lattice join_comm)
   823 
   824 lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"
   825 apply (simp add: abs_lattice[of "abs a"])
   826 apply (subst ge_imp_join_eq)
   827 apply (rule order_trans[of _ 0])
   828 by auto
   829 
   830 lemma abs_minus_commute: 
   831   fixes a :: "'a::lordered_ab_group_abs"
   832   shows "abs (a-b) = abs(b-a)"
   833 proof -
   834   have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel)
   835   also have "... = abs(b-a)" by simp
   836   finally show ?thesis .
   837 qed
   838 
   839 lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"
   840 by (simp add: le_def_meet nprt_def meet_comm)
   841 
   842 lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"
   843 by (simp add: le_def_join pprt_def join_comm)
   844 
   845 lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"
   846 by (simp add: le_def_join pprt_def join_comm)
   847 
   848 lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"
   849 by (simp add: le_def_meet nprt_def meet_comm)
   850 
   851 lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b"
   852   by (simp add: le_def_join pprt_def join_aci)
   853 
   854 lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b"
   855   by (simp add: le_def_meet nprt_def meet_aci)
   856 
   857 lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"
   858 by (simp)
   859 
   860 lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"
   861 by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)
   862 
   863 lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x";
   864 by (rule abs_of_nonneg, rule order_less_imp_le);
   865 
   866 lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"
   867 by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)
   868 
   869 lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) <  0 ==> 
   870   abs x = - x"
   871 by (rule abs_of_nonpos, rule order_less_imp_le)
   872 
   873 lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"
   874 by (simp add: abs_lattice join_imp_le)
   875 
   876 lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"
   877 proof -
   878   from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" 
   879     by (simp add: add_assoc[symmetric])
   880   thus ?thesis by simp
   881 qed
   882 
   883 lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"
   884 proof -
   885   from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" 
   886     by (simp add: add_assoc[symmetric])
   887   thus ?thesis by simp
   888 qed
   889 
   890 lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"
   891 by (insert abs_ge_self, blast intro: order_trans)
   892 
   893 lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"
   894 by (insert abs_le_D1 [of "-a"], simp)
   895 
   896 lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"
   897 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
   898 
   899 lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)"
   900 proof -
   901   have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n")
   902     by (simp add: abs_lattice add_meet_join_distribs join_aci diff_minus)
   903   have a:"a+b <= join ?m ?n" by (simp add: meet_join_le)
   904   have b:"-a-b <= ?n" by (simp add: meet_join_le) 
   905   have c:"?n <= join ?m ?n" by (simp add: meet_join_le)
   906   from b c have d: "-a-b <= join ?m ?n" by simp
   907   have e:"-a-b = -(a+b)" by (simp add: diff_minus)
   908   from a d e have "abs(a+b) <= join ?m ?n" 
   909     by (drule_tac abs_leI, auto)
   910   with g[symmetric] show ?thesis by simp
   911 qed
   912 
   913 lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs) - 
   914     abs b <= abs (a - b)"
   915   apply (simp add: compare_rls)
   916   apply (subgoal_tac "abs a = abs (a - b + b)")
   917   apply (erule ssubst)
   918   apply (rule abs_triangle_ineq)
   919   apply (rule arg_cong);back;
   920   apply (simp add: compare_rls)
   921 done
   922 
   923 lemma abs_triangle_ineq3: 
   924     "abs(abs (a::'a::lordered_ab_group_abs) - abs b) <= abs (a - b)"
   925   apply (subst abs_le_iff)
   926   apply auto
   927   apply (rule abs_triangle_ineq2)
   928   apply (subst abs_minus_commute)
   929   apply (rule abs_triangle_ineq2)
   930 done
   931 
   932 lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs) - b) <= 
   933     abs a + abs b"
   934 proof -;
   935   have "abs(a - b) = abs(a + - b)"
   936     by (subst diff_minus, rule refl)
   937   also have "... <= abs a + abs (- b)"
   938     by (rule abs_triangle_ineq)
   939   finally show ?thesis
   940     by simp
   941 qed
   942 
   943 lemma abs_diff_triangle_ineq:
   944      "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"
   945 proof -
   946   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
   947   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
   948   finally show ?thesis .
   949 qed
   950 
   951 lemma abs_add_abs[simp]:
   952 fixes a:: "'a::{lordered_ab_group_abs}"
   953 shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R")
   954 proof (rule order_antisym)
   955   show "?L \<ge> ?R" by(rule abs_ge_self)
   956 next
   957   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
   958   also have "\<dots> = ?R" by simp
   959   finally show "?L \<le> ?R" .
   960 qed
   961 
   962 text {* Needed for abelian cancellation simprocs: *}
   963 
   964 lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
   965 apply (subst add_left_commute)
   966 apply (subst add_left_cancel)
   967 apply simp
   968 done
   969 
   970 lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
   971 apply (subst add_cancel_21[of _ _ _ 0, simplified])
   972 apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
   973 done
   974 
   975 lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
   976 by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
   977 
   978 lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
   979 apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
   980 apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
   981 done
   982 
   983 lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
   984 by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
   985 
   986 lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
   987 by (simp add: diff_minus)
   988 
   989 lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"
   990 by (simp add: add_assoc[symmetric])
   991 
   992 lemma minus_add_cancel: "-(a::'a::ab_group_add) + (a + b) = b"
   993 by (simp add: add_assoc[symmetric])
   994 
   995 lemma  le_add_right_mono: 
   996   assumes 
   997   "a <= b + (c::'a::pordered_ab_group_add)"
   998   "c <= d"    
   999   shows "a <= b + d"
  1000   apply (rule_tac order_trans[where y = "b+c"])
  1001   apply (simp_all add: prems)
  1002   done
  1003 
  1004 lemmas group_eq_simps =
  1005   mult_ac
  1006   add_ac
  1007   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
  1008   diff_eq_eq eq_diff_eq
  1009 
  1010 lemma estimate_by_abs:
  1011 "a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 
  1012 proof -
  1013   assume 1: "a+b <= c"
  1014   have 2: "a <= c+(-b)"
  1015     apply (insert 1)
  1016     apply (drule_tac add_right_mono[where c="-b"])
  1017     apply (simp add: group_eq_simps)
  1018     done
  1019   have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
  1020   show ?thesis by (rule le_add_right_mono[OF 2 3])
  1021 qed
  1022 
  1023 text{*Simplification of @{term "x-y < 0"}, etc.*}
  1024 lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric]
  1025 lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric]
  1026 lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric]
  1027 declare diff_less_0_iff_less [simp]
  1028 declare diff_eq_0_iff_eq [simp]
  1029 declare diff_le_0_iff_le [simp]
  1030 
  1031 
  1032 ML {*
  1033 val add_zero_left = thm"add_0";
  1034 val add_zero_right = thm"add_0_right";
  1035 *}
  1036 
  1037 ML {*
  1038 val add_assoc = thm "add_assoc";
  1039 val add_commute = thm "add_commute";
  1040 val add_left_commute = thm "add_left_commute";
  1041 val add_ac = thms "add_ac";
  1042 val mult_assoc = thm "mult_assoc";
  1043 val mult_commute = thm "mult_commute";
  1044 val mult_left_commute = thm "mult_left_commute";
  1045 val mult_ac = thms "mult_ac";
  1046 val add_0 = thm "add_0";
  1047 val mult_1_left = thm "mult_1_left";
  1048 val mult_1_right = thm "mult_1_right";
  1049 val mult_1 = thm "mult_1";
  1050 val add_left_imp_eq = thm "add_left_imp_eq";
  1051 val add_right_imp_eq = thm "add_right_imp_eq";
  1052 val add_imp_eq = thm "add_imp_eq";
  1053 val left_minus = thm "left_minus";
  1054 val diff_minus = thm "diff_minus";
  1055 val add_0_right = thm "add_0_right";
  1056 val add_left_cancel = thm "add_left_cancel";
  1057 val add_right_cancel = thm "add_right_cancel";
  1058 val right_minus = thm "right_minus";
  1059 val right_minus_eq = thm "right_minus_eq";
  1060 val minus_minus = thm "minus_minus";
  1061 val equals_zero_I = thm "equals_zero_I";
  1062 val minus_zero = thm "minus_zero";
  1063 val diff_self = thm "diff_self";
  1064 val diff_0 = thm "diff_0";
  1065 val diff_0_right = thm "diff_0_right";
  1066 val diff_minus_eq_add = thm "diff_minus_eq_add";
  1067 val neg_equal_iff_equal = thm "neg_equal_iff_equal";
  1068 val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal";
  1069 val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal";
  1070 val equation_minus_iff = thm "equation_minus_iff";
  1071 val minus_equation_iff = thm "minus_equation_iff";
  1072 val minus_add_distrib = thm "minus_add_distrib";
  1073 val minus_diff_eq = thm "minus_diff_eq";
  1074 val add_left_mono = thm "add_left_mono";
  1075 val add_le_imp_le_left = thm "add_le_imp_le_left";
  1076 val add_right_mono = thm "add_right_mono";
  1077 val add_mono = thm "add_mono";
  1078 val add_strict_left_mono = thm "add_strict_left_mono";
  1079 val add_strict_right_mono = thm "add_strict_right_mono";
  1080 val add_strict_mono = thm "add_strict_mono";
  1081 val add_less_le_mono = thm "add_less_le_mono";
  1082 val add_le_less_mono = thm "add_le_less_mono";
  1083 val add_less_imp_less_left = thm "add_less_imp_less_left";
  1084 val add_less_imp_less_right = thm "add_less_imp_less_right";
  1085 val add_less_cancel_left = thm "add_less_cancel_left";
  1086 val add_less_cancel_right = thm "add_less_cancel_right";
  1087 val add_le_cancel_left = thm "add_le_cancel_left";
  1088 val add_le_cancel_right = thm "add_le_cancel_right";
  1089 val add_le_imp_le_right = thm "add_le_imp_le_right";
  1090 val add_increasing = thm "add_increasing";
  1091 val le_imp_neg_le = thm "le_imp_neg_le";
  1092 val neg_le_iff_le = thm "neg_le_iff_le";
  1093 val neg_le_0_iff_le = thm "neg_le_0_iff_le";
  1094 val neg_0_le_iff_le = thm "neg_0_le_iff_le";
  1095 val neg_less_iff_less = thm "neg_less_iff_less";
  1096 val neg_less_0_iff_less = thm "neg_less_0_iff_less";
  1097 val neg_0_less_iff_less = thm "neg_0_less_iff_less";
  1098 val less_minus_iff = thm "less_minus_iff";
  1099 val minus_less_iff = thm "minus_less_iff";
  1100 val le_minus_iff = thm "le_minus_iff";
  1101 val minus_le_iff = thm "minus_le_iff";
  1102 val add_diff_eq = thm "add_diff_eq";
  1103 val diff_add_eq = thm "diff_add_eq";
  1104 val diff_eq_eq = thm "diff_eq_eq";
  1105 val eq_diff_eq = thm "eq_diff_eq";
  1106 val diff_diff_eq = thm "diff_diff_eq";
  1107 val diff_diff_eq2 = thm "diff_diff_eq2";
  1108 val diff_add_cancel = thm "diff_add_cancel";
  1109 val add_diff_cancel = thm "add_diff_cancel";
  1110 val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
  1111 val diff_less_eq = thm "diff_less_eq";
  1112 val less_diff_eq = thm "less_diff_eq";
  1113 val diff_le_eq = thm "diff_le_eq";
  1114 val le_diff_eq = thm "le_diff_eq";
  1115 val compare_rls = thms "compare_rls";
  1116 val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
  1117 val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
  1118 val add_meet_distrib_left = thm "add_meet_distrib_left";
  1119 val add_join_distrib_left = thm "add_join_distrib_left";
  1120 val is_join_neg_meet = thm "is_join_neg_meet";
  1121 val is_meet_neg_join = thm "is_meet_neg_join";
  1122 val add_join_distrib_right = thm "add_join_distrib_right";
  1123 val add_meet_distrib_right = thm "add_meet_distrib_right";
  1124 val add_meet_join_distribs = thms "add_meet_join_distribs";
  1125 val join_eq_neg_meet = thm "join_eq_neg_meet";
  1126 val meet_eq_neg_join = thm "meet_eq_neg_join";
  1127 val add_eq_meet_join = thm "add_eq_meet_join";
  1128 val prts = thm "prts";
  1129 val zero_le_pprt = thm "zero_le_pprt";
  1130 val nprt_le_zero = thm "nprt_le_zero";
  1131 val le_eq_neg = thm "le_eq_neg";
  1132 val join_0_imp_0 = thm "join_0_imp_0";
  1133 val meet_0_imp_0 = thm "meet_0_imp_0";
  1134 val join_0_eq_0 = thm "join_0_eq_0";
  1135 val meet_0_eq_0 = thm "meet_0_eq_0";
  1136 val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add";
  1137 val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero";
  1138 val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero";
  1139 val abs_lattice = thm "abs_lattice";
  1140 val abs_zero = thm "abs_zero";
  1141 val abs_eq_0 = thm "abs_eq_0";
  1142 val abs_0_eq = thm "abs_0_eq";
  1143 val neg_meet_eq_join = thm "neg_meet_eq_join";
  1144 val neg_join_eq_meet = thm "neg_join_eq_meet";
  1145 val join_eq_if = thm "join_eq_if";
  1146 val abs_if_lattice = thm "abs_if_lattice";
  1147 val abs_ge_zero = thm "abs_ge_zero";
  1148 val abs_le_zero_iff = thm "abs_le_zero_iff";
  1149 val zero_less_abs_iff = thm "zero_less_abs_iff";
  1150 val abs_not_less_zero = thm "abs_not_less_zero";
  1151 val abs_ge_self = thm "abs_ge_self";
  1152 val abs_ge_minus_self = thm "abs_ge_minus_self";
  1153 val le_imp_join_eq = thm "le_imp_join_eq";
  1154 val ge_imp_join_eq = thm "ge_imp_join_eq";
  1155 val le_imp_meet_eq = thm "le_imp_meet_eq";
  1156 val ge_imp_meet_eq = thm "ge_imp_meet_eq";
  1157 val abs_prts = thm "abs_prts";
  1158 val abs_minus_cancel = thm "abs_minus_cancel";
  1159 val abs_idempotent = thm "abs_idempotent";
  1160 val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt";
  1161 val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt";
  1162 val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id";
  1163 val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id";
  1164 val iff2imp = thm "iff2imp";
  1165 (* val imp_abs_id = thm "imp_abs_id";
  1166 val imp_abs_neg_id = thm "imp_abs_neg_id"; *)
  1167 val abs_leI = thm "abs_leI";
  1168 val le_minus_self_iff = thm "le_minus_self_iff";
  1169 val minus_le_self_iff = thm "minus_le_self_iff";
  1170 val abs_le_D1 = thm "abs_le_D1";
  1171 val abs_le_D2 = thm "abs_le_D2";
  1172 val abs_le_iff = thm "abs_le_iff";
  1173 val abs_triangle_ineq = thm "abs_triangle_ineq";
  1174 val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
  1175 *}
  1176 
  1177 end