src/HOL/OrderedGroup.thy
author wenzelm
Tue Jun 06 20:42:28 2006 +0200 (2006-06-06)
changeset 19798 94f12468bbba
parent 19527 9b5c38e8e780
child 21245 23e6eb4d0975
permissions -rw-r--r--
tuned;
     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 "~~/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, 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 lemma max_add_distrib_left:
   339   fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
   340   shows  "(max x y) + z = max (x+z) (y+z)"
   341 by (rule max_of_mono [THEN sym], rule add_le_cancel_right)
   342 
   343 lemma min_add_distrib_left:
   344   fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
   345   shows  "(min x y) + z = min (x+z) (y+z)"
   346 by (rule min_of_mono [THEN sym], rule add_le_cancel_right)
   347 
   348 lemma max_diff_distrib_left:
   349   fixes z :: "'a::pordered_ab_group_add"
   350   shows  "(max x y) - z = max (x-z) (y-z)"
   351 by (simp add: diff_minus, rule max_add_distrib_left) 
   352 
   353 lemma min_diff_distrib_left:
   354   fixes z :: "'a::pordered_ab_group_add"
   355   shows  "(min x y) - z = min (x-z) (y-z)"
   356 by (simp add: diff_minus, rule min_add_distrib_left) 
   357 
   358 
   359 subsection {* Ordering Rules for Unary Minus *}
   360 
   361 lemma le_imp_neg_le:
   362       assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"
   363 proof -
   364   have "-a+a \<le> -a+b"
   365     by (rule add_left_mono) 
   366   hence "0 \<le> -a+b"
   367     by simp
   368   hence "0 + (-b) \<le> (-a + b) + (-b)"
   369     by (rule add_right_mono) 
   370   thus ?thesis
   371     by (simp add: add_assoc)
   372 qed
   373 
   374 lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"
   375 proof 
   376   assume "- b \<le> - a"
   377   hence "- (- a) \<le> - (- b)"
   378     by (rule le_imp_neg_le)
   379   thus "a\<le>b" by simp
   380 next
   381   assume "a\<le>b"
   382   thus "-b \<le> -a" by (rule le_imp_neg_le)
   383 qed
   384 
   385 lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"
   386 by (subst neg_le_iff_le [symmetric], simp)
   387 
   388 lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"
   389 by (subst neg_le_iff_le [symmetric], simp)
   390 
   391 lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"
   392 by (force simp add: order_less_le) 
   393 
   394 lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"
   395 by (subst neg_less_iff_less [symmetric], simp)
   396 
   397 lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"
   398 by (subst neg_less_iff_less [symmetric], simp)
   399 
   400 text{*The next several equations can make the simplifier loop!*}
   401 
   402 lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"
   403 proof -
   404   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
   405   thus ?thesis by simp
   406 qed
   407 
   408 lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"
   409 proof -
   410   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
   411   thus ?thesis by simp
   412 qed
   413 
   414 lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"
   415 proof -
   416   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
   417   have "(- (- a) <= -b) = (b <= - a)" 
   418     apply (auto simp only: order_le_less)
   419     apply (drule mm)
   420     apply (simp_all)
   421     apply (drule mm[simplified], assumption)
   422     done
   423   then show ?thesis by simp
   424 qed
   425 
   426 lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"
   427 by (auto simp add: order_le_less minus_less_iff)
   428 
   429 lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"
   430 by (simp add: diff_minus add_ac)
   431 
   432 lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"
   433 by (simp add: diff_minus add_ac)
   434 
   435 lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"
   436 by (auto simp add: diff_minus add_assoc)
   437 
   438 lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"
   439 by (auto simp add: diff_minus add_assoc)
   440 
   441 lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"
   442 by (simp add: diff_minus add_ac)
   443 
   444 lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"
   445 by (simp add: diff_minus add_ac)
   446 
   447 lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"
   448 by (simp add: diff_minus add_ac)
   449 
   450 lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"
   451 by (simp add: diff_minus add_ac)
   452 
   453 text{*Further subtraction laws*}
   454 
   455 lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"
   456 proof -
   457   have  "(a < b) = (a + (- b) < b + (-b))"  
   458     by (simp only: add_less_cancel_right)
   459   also have "... =  (a - b < 0)" by (simp add: diff_minus)
   460   finally show ?thesis .
   461 qed
   462 
   463 lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"
   464 apply (subst less_iff_diff_less_0 [of a])
   465 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
   466 apply (simp add: diff_minus add_ac)
   467 done
   468 
   469 lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"
   470 apply (subst less_iff_diff_less_0 [of "a+b"])
   471 apply (subst less_iff_diff_less_0 [of a])
   472 apply (simp add: diff_minus add_ac)
   473 done
   474 
   475 lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"
   476 by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)
   477 
   478 lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"
   479 by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)
   480 
   481 text{*This list of rewrites simplifies (in)equalities by bringing subtractions
   482   to the top and then moving negative terms to the other side.
   483   Use with @{text add_ac}*}
   484 lemmas compare_rls =
   485        diff_minus [symmetric]
   486        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
   487        diff_less_eq less_diff_eq diff_le_eq le_diff_eq
   488        diff_eq_eq eq_diff_eq
   489 
   490 subsection {* Support for reasoning about signs *}
   491 
   492 lemma add_pos_pos: "0 < 
   493     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   494       ==> 0 < y ==> 0 < x + y"
   495 apply (subgoal_tac "0 + 0 < x + y")
   496 apply simp
   497 apply (erule add_less_le_mono)
   498 apply (erule order_less_imp_le)
   499 done
   500 
   501 lemma add_pos_nonneg: "0 < 
   502     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   503       ==> 0 <= y ==> 0 < x + y"
   504 apply (subgoal_tac "0 + 0 < x + y")
   505 apply simp
   506 apply (erule add_less_le_mono, assumption)
   507 done
   508 
   509 lemma add_nonneg_pos: "0 <= 
   510     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   511       ==> 0 < y ==> 0 < x + y"
   512 apply (subgoal_tac "0 + 0 < x + y")
   513 apply simp
   514 apply (erule add_le_less_mono, assumption)
   515 done
   516 
   517 lemma add_nonneg_nonneg: "0 <= 
   518     (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
   519       ==> 0 <= y ==> 0 <= x + y"
   520 apply (subgoal_tac "0 + 0 <= x + y")
   521 apply simp
   522 apply (erule add_mono, assumption)
   523 done
   524 
   525 lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add})
   526     < 0 ==> y < 0 ==> x + y < 0"
   527 apply (subgoal_tac "x + y < 0 + 0")
   528 apply simp
   529 apply (erule add_less_le_mono)
   530 apply (erule order_less_imp_le)
   531 done
   532 
   533 lemma add_neg_nonpos: 
   534     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 
   535       ==> y <= 0 ==> x + y < 0"
   536 apply (subgoal_tac "x + y < 0 + 0")
   537 apply simp
   538 apply (erule add_less_le_mono, assumption)
   539 done
   540 
   541 lemma add_nonpos_neg: 
   542     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
   543       ==> y < 0 ==> x + y < 0"
   544 apply (subgoal_tac "x + y < 0 + 0")
   545 apply simp
   546 apply (erule add_le_less_mono, assumption)
   547 done
   548 
   549 lemma add_nonpos_nonpos: 
   550     "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
   551       ==> y <= 0 ==> x + y <= 0"
   552 apply (subgoal_tac "x + y <= 0 + 0")
   553 apply simp
   554 apply (erule add_mono, assumption)
   555 done
   556 
   557 subsection{*Lemmas for the @{text cancel_numerals} simproc*}
   558 
   559 lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"
   560 by (simp add: compare_rls)
   561 
   562 lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"
   563 by (simp add: compare_rls)
   564 
   565 subsection {* Lattice Ordered (Abelian) Groups *}
   566 
   567 axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder
   568 
   569 axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder
   570 
   571 lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))"
   572 apply (rule order_antisym)
   573 apply (rule meet_imp_le, simp_all add: meet_join_le)
   574 apply (rule add_le_imp_le_left [of "-a"])
   575 apply (simp only: add_assoc[symmetric], simp)
   576 apply (rule meet_imp_le)
   577 apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
   578 done
   579 
   580 lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" 
   581 apply (rule order_antisym)
   582 apply (rule add_le_imp_le_left [of "-a"])
   583 apply (simp only: add_assoc[symmetric], simp)
   584 apply (rule join_imp_le)
   585 apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
   586 apply (rule join_imp_le)
   587 apply (simp_all add: meet_join_le)
   588 done
   589 
   590 lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))"
   591 apply (auto simp add: is_join_def)
   592 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
   593 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
   594 apply (subst neg_le_iff_le[symmetric]) 
   595 apply (simp add: meet_imp_le)
   596 done
   597 
   598 lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))"
   599 apply (auto simp add: is_meet_def)
   600 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
   601 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
   602 apply (subst neg_le_iff_le[symmetric]) 
   603 apply (simp add: join_imp_le)
   604 done
   605 
   606 axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder
   607 
   608 instance lordered_ab_group_meet \<subseteq> lordered_ab_group
   609 proof 
   610   show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet)
   611 qed
   612 
   613 instance lordered_ab_group_join \<subseteq> lordered_ab_group
   614 proof
   615   show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join)
   616 qed
   617 
   618 lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)"
   619 proof -
   620   have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left)
   621   thus ?thesis by (simp add: add_commute)
   622 qed
   623 
   624 lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)"
   625 proof -
   626   have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left)
   627   thus ?thesis by (simp add: add_commute)
   628 qed
   629 
   630 lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left
   631 
   632 lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)"
   633 by (simp add: is_join_unique[OF is_join_join is_join_neg_meet])
   634 
   635 lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)"
   636 by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join])
   637 
   638 lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))"
   639 proof -
   640   have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm)
   641   hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join)
   642   hence "0 = (-a + join a b) + (meet a b + (-b))"
   643     apply (simp add: add_join_distrib_left add_meet_distrib_right)
   644     by (simp add: diff_minus add_commute)
   645   thus ?thesis
   646     apply (simp add: compare_rls)
   647     apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"])
   648     apply (simp only: add_assoc, simp add: add_assoc[symmetric])
   649     done
   650 qed
   651 
   652 subsection {* Positive Part, Negative Part, Absolute Value *}
   653 
   654 constdefs
   655   pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
   656   "pprt x == join x 0"
   657   nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
   658   "nprt x == meet x 0"
   659 
   660 lemma prts: "a = pprt a + nprt a"
   661 by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric])
   662 
   663 lemma zero_le_pprt[simp]: "0 \<le> pprt a"
   664 by (simp add: pprt_def meet_join_le)
   665 
   666 lemma nprt_le_zero[simp]: "nprt a \<le> 0"
   667 by (simp add: nprt_def meet_join_le)
   668 
   669 lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")
   670 proof -
   671   have a: "?l \<longrightarrow> ?r"
   672     apply (auto)
   673     apply (rule add_le_imp_le_right[of _ "-b" _])
   674     apply (simp add: add_assoc)
   675     done
   676   have b: "?r \<longrightarrow> ?l"
   677     apply (auto)
   678     apply (rule add_le_imp_le_right[of _ "b" _])
   679     apply (simp)
   680     done
   681   from a b show ?thesis by blast
   682 qed
   683 
   684 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
   685 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
   686 
   687 lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x"
   688   by (simp add: pprt_def le_def_join join_aci)
   689 
   690 lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x"
   691   by (simp add: nprt_def le_def_meet meet_aci)
   692 
   693 lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0"
   694   by (simp add: pprt_def le_def_join join_aci)
   695 
   696 lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0"
   697   by (simp add: nprt_def le_def_meet meet_aci)
   698 
   699 lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
   700 proof -
   701   {
   702     fix a::'a
   703     assume hyp: "join a (-a) = 0"
   704     hence "join a (-a) + a = a" by (simp)
   705     hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) 
   706     hence "join (a+a) 0 <= a" by (simp)
   707     hence "0 <= a" by (blast intro: order_trans meet_join_le)
   708   }
   709   note p = this
   710   assume hyp:"join a (-a) = 0"
   711   hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm)
   712   from p[OF hyp] p[OF hyp2] show "a = 0" by simp
   713 qed
   714 
   715 lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
   716 apply (simp add: meet_eq_neg_join)
   717 apply (simp add: join_comm)
   718 apply (erule join_0_imp_0)
   719 done
   720 
   721 lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
   722 by (auto, erule join_0_imp_0)
   723 
   724 lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
   725 by (auto, erule meet_0_imp_0)
   726 
   727 lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"
   728 proof
   729   assume "0 <= a + a"
   730   hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm)
   731   have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci)
   732   hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm)
   733   hence "meet a 0 = 0" by (simp only: add_right_cancel)
   734   then show "0 <= a" by (simp add: le_def_meet meet_comm)    
   735 next  
   736   assume a: "0 <= a"
   737   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
   738 qed
   739 
   740 lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 
   741 proof -
   742   have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)
   743   moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)
   744   ultimately show ?thesis by blast
   745 qed
   746 
   747 lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)
   748 proof cases
   749   assume a: "a < 0"
   750   thus ?s by (simp add:  add_strict_mono[OF a a, simplified])
   751 next
   752   assume "~(a < 0)" 
   753   hence a:"0 <= a" by (simp)
   754   hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified])
   755   hence "~(a+a < 0)" by simp
   756   with a show ?thesis by simp 
   757 qed
   758 
   759 axclass lordered_ab_group_abs \<subseteq> lordered_ab_group
   760   abs_lattice: "abs x = join x (-x)"
   761 
   762 lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"
   763 by (simp add: abs_lattice)
   764 
   765 lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"
   766 by (simp add: abs_lattice)
   767 
   768 lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"
   769 proof -
   770   have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
   771   thus ?thesis by simp
   772 qed
   773 
   774 lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)"
   775 by (simp add: meet_eq_neg_join)
   776 
   777 lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)"
   778 by (simp del: neg_meet_eq_join add: join_eq_neg_meet)
   779 
   780 lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"
   781 proof -
   782   note b = add_le_cancel_right[of a a "-a",symmetric,simplified]
   783   have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)
   784   show ?thesis by (auto simp add: join_max max_def b linorder_not_less)
   785 qed
   786 
   787 lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"
   788 proof -
   789   show ?thesis by (simp add: abs_lattice join_eq_if)
   790 qed
   791 
   792 lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"
   793 proof -
   794   have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le)
   795   show ?thesis by (rule add_mono[OF a b, simplified])
   796 qed
   797   
   798 lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 
   799 proof
   800   assume "abs a <= 0"
   801   hence "abs a = 0" by (auto dest: order_antisym)
   802   thus "a = 0" by simp
   803 next
   804   assume "a = 0"
   805   thus "abs a <= 0" by simp
   806 qed
   807 
   808 lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"
   809 by (simp add: order_less_le)
   810 
   811 lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"
   812 proof -
   813   have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto
   814   show ?thesis by (simp add: a)
   815 qed
   816 
   817 lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"
   818 by (simp add: abs_lattice meet_join_le)
   819 
   820 lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"
   821 by (simp add: abs_lattice meet_join_le)
   822 
   823 lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" 
   824 by (simp add: le_def_join)
   825 
   826 lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a"
   827 by (simp add: le_def_join join_aci)
   828 
   829 lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a"
   830 by (simp add: le_def_meet)
   831 
   832 lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b"
   833 by (simp add: le_def_meet meet_aci)
   834 
   835 lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"
   836 apply (simp add: pprt_def nprt_def diff_minus)
   837 apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric])
   838 apply (subst le_imp_join_eq, auto)
   839 done
   840 
   841 lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"
   842 by (simp add: abs_lattice join_comm)
   843 
   844 lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"
   845 apply (simp add: abs_lattice[of "abs a"])
   846 apply (subst ge_imp_join_eq)
   847 apply (rule order_trans[of _ 0])
   848 by auto
   849 
   850 lemma abs_minus_commute: 
   851   fixes a :: "'a::lordered_ab_group_abs"
   852   shows "abs (a-b) = abs(b-a)"
   853 proof -
   854   have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel)
   855   also have "... = abs(b-a)" by simp
   856   finally show ?thesis .
   857 qed
   858 
   859 lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"
   860 by (simp add: le_def_meet nprt_def meet_comm)
   861 
   862 lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"
   863 by (simp add: le_def_join pprt_def join_comm)
   864 
   865 lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"
   866 by (simp add: le_def_join pprt_def join_comm)
   867 
   868 lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"
   869 by (simp add: le_def_meet nprt_def meet_comm)
   870 
   871 lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b"
   872   by (simp add: le_def_join pprt_def join_aci)
   873 
   874 lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b"
   875   by (simp add: le_def_meet nprt_def meet_aci)
   876 
   877 lemma pprt_neg: "pprt (-x) = - nprt x"
   878   by (simp add: pprt_def nprt_def)
   879 
   880 lemma nprt_neg: "nprt (-x) = - pprt x"
   881   by (simp add: pprt_def nprt_def)
   882 
   883 lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"
   884 by (simp)
   885 
   886 lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"
   887 by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)
   888 
   889 lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x";
   890 by (rule abs_of_nonneg, rule order_less_imp_le);
   891 
   892 lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"
   893 by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)
   894 
   895 lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) <  0 ==> 
   896   abs x = - x"
   897 by (rule abs_of_nonpos, rule order_less_imp_le)
   898 
   899 lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"
   900 by (simp add: abs_lattice join_imp_le)
   901 
   902 lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"
   903 proof -
   904   from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" 
   905     by (simp add: add_assoc[symmetric])
   906   thus ?thesis by simp
   907 qed
   908 
   909 lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"
   910 proof -
   911   from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" 
   912     by (simp add: add_assoc[symmetric])
   913   thus ?thesis by simp
   914 qed
   915 
   916 lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"
   917 by (insert abs_ge_self, blast intro: order_trans)
   918 
   919 lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"
   920 by (insert abs_le_D1 [of "-a"], simp)
   921 
   922 lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"
   923 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
   924 
   925 lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)"
   926 proof -
   927   have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n")
   928     by (simp add: abs_lattice add_meet_join_distribs join_aci diff_minus)
   929   have a:"a+b <= join ?m ?n" by (simp add: meet_join_le)
   930   have b:"-a-b <= ?n" by (simp add: meet_join_le) 
   931   have c:"?n <= join ?m ?n" by (simp add: meet_join_le)
   932   from b c have d: "-a-b <= join ?m ?n" by simp
   933   have e:"-a-b = -(a+b)" by (simp add: diff_minus)
   934   from a d e have "abs(a+b) <= join ?m ?n" 
   935     by (drule_tac abs_leI, auto)
   936   with g[symmetric] show ?thesis by simp
   937 qed
   938 
   939 lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs) - 
   940     abs b <= abs (a - b)"
   941   apply (simp add: compare_rls)
   942   apply (subgoal_tac "abs a = abs (a - b + b)")
   943   apply (erule ssubst)
   944   apply (rule abs_triangle_ineq)
   945   apply (rule arg_cong);back;
   946   apply (simp add: compare_rls)
   947 done
   948 
   949 lemma abs_triangle_ineq3: 
   950     "abs(abs (a::'a::lordered_ab_group_abs) - abs b) <= abs (a - b)"
   951   apply (subst abs_le_iff)
   952   apply auto
   953   apply (rule abs_triangle_ineq2)
   954   apply (subst abs_minus_commute)
   955   apply (rule abs_triangle_ineq2)
   956 done
   957 
   958 lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs) - b) <= 
   959     abs a + abs b"
   960 proof -;
   961   have "abs(a - b) = abs(a + - b)"
   962     by (subst diff_minus, rule refl)
   963   also have "... <= abs a + abs (- b)"
   964     by (rule abs_triangle_ineq)
   965   finally show ?thesis
   966     by simp
   967 qed
   968 
   969 lemma abs_diff_triangle_ineq:
   970      "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"
   971 proof -
   972   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
   973   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
   974   finally show ?thesis .
   975 qed
   976 
   977 lemma abs_add_abs[simp]:
   978 fixes a:: "'a::{lordered_ab_group_abs}"
   979 shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R")
   980 proof (rule order_antisym)
   981   show "?L \<ge> ?R" by(rule abs_ge_self)
   982 next
   983   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
   984   also have "\<dots> = ?R" by simp
   985   finally show "?L \<le> ?R" .
   986 qed
   987 
   988 text {* Needed for abelian cancellation simprocs: *}
   989 
   990 lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
   991 apply (subst add_left_commute)
   992 apply (subst add_left_cancel)
   993 apply simp
   994 done
   995 
   996 lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
   997 apply (subst add_cancel_21[of _ _ _ 0, simplified])
   998 apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
   999 done
  1000 
  1001 lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
  1002 by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
  1003 
  1004 lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
  1005 apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
  1006 apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
  1007 done
  1008 
  1009 lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
  1010 by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
  1011 
  1012 lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
  1013 by (simp add: diff_minus)
  1014 
  1015 lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"
  1016 by (simp add: add_assoc[symmetric])
  1017 
  1018 lemma minus_add_cancel: "-(a::'a::ab_group_add) + (a + b) = b"
  1019 by (simp add: add_assoc[symmetric])
  1020 
  1021 lemma  le_add_right_mono: 
  1022   assumes 
  1023   "a <= b + (c::'a::pordered_ab_group_add)"
  1024   "c <= d"    
  1025   shows "a <= b + d"
  1026   apply (rule_tac order_trans[where y = "b+c"])
  1027   apply (simp_all add: prems)
  1028   done
  1029 
  1030 lemmas group_eq_simps =
  1031   mult_ac
  1032   add_ac
  1033   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
  1034   diff_eq_eq eq_diff_eq
  1035 
  1036 lemma estimate_by_abs:
  1037 "a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 
  1038 proof -
  1039   assume 1: "a+b <= c"
  1040   have 2: "a <= c+(-b)"
  1041     apply (insert 1)
  1042     apply (drule_tac add_right_mono[where c="-b"])
  1043     apply (simp add: group_eq_simps)
  1044     done
  1045   have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
  1046   show ?thesis by (rule le_add_right_mono[OF 2 3])
  1047 qed
  1048 
  1049 text{*Simplification of @{term "x-y < 0"}, etc.*}
  1050 lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric]
  1051 lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric]
  1052 lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric]
  1053 declare diff_less_0_iff_less [simp]
  1054 declare diff_eq_0_iff_eq [simp]
  1055 declare diff_le_0_iff_le [simp]
  1056 
  1057 
  1058 
  1059 
  1060 ML {*
  1061 val add_zero_left = thm"add_0";
  1062 val add_zero_right = thm"add_0_right";
  1063 *}
  1064 
  1065 ML {*
  1066 val add_assoc = thm "add_assoc";
  1067 val add_commute = thm "add_commute";
  1068 val add_left_commute = thm "add_left_commute";
  1069 val add_ac = thms "add_ac";
  1070 val mult_assoc = thm "mult_assoc";
  1071 val mult_commute = thm "mult_commute";
  1072 val mult_left_commute = thm "mult_left_commute";
  1073 val mult_ac = thms "mult_ac";
  1074 val add_0 = thm "add_0";
  1075 val mult_1_left = thm "mult_1_left";
  1076 val mult_1_right = thm "mult_1_right";
  1077 val mult_1 = thm "mult_1";
  1078 val add_left_imp_eq = thm "add_left_imp_eq";
  1079 val add_right_imp_eq = thm "add_right_imp_eq";
  1080 val add_imp_eq = thm "add_imp_eq";
  1081 val left_minus = thm "left_minus";
  1082 val diff_minus = thm "diff_minus";
  1083 val add_0_right = thm "add_0_right";
  1084 val add_left_cancel = thm "add_left_cancel";
  1085 val add_right_cancel = thm "add_right_cancel";
  1086 val right_minus = thm "right_minus";
  1087 val right_minus_eq = thm "right_minus_eq";
  1088 val minus_minus = thm "minus_minus";
  1089 val equals_zero_I = thm "equals_zero_I";
  1090 val minus_zero = thm "minus_zero";
  1091 val diff_self = thm "diff_self";
  1092 val diff_0 = thm "diff_0";
  1093 val diff_0_right = thm "diff_0_right";
  1094 val diff_minus_eq_add = thm "diff_minus_eq_add";
  1095 val neg_equal_iff_equal = thm "neg_equal_iff_equal";
  1096 val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal";
  1097 val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal";
  1098 val equation_minus_iff = thm "equation_minus_iff";
  1099 val minus_equation_iff = thm "minus_equation_iff";
  1100 val minus_add_distrib = thm "minus_add_distrib";
  1101 val minus_diff_eq = thm "minus_diff_eq";
  1102 val add_left_mono = thm "add_left_mono";
  1103 val add_le_imp_le_left = thm "add_le_imp_le_left";
  1104 val add_right_mono = thm "add_right_mono";
  1105 val add_mono = thm "add_mono";
  1106 val add_strict_left_mono = thm "add_strict_left_mono";
  1107 val add_strict_right_mono = thm "add_strict_right_mono";
  1108 val add_strict_mono = thm "add_strict_mono";
  1109 val add_less_le_mono = thm "add_less_le_mono";
  1110 val add_le_less_mono = thm "add_le_less_mono";
  1111 val add_less_imp_less_left = thm "add_less_imp_less_left";
  1112 val add_less_imp_less_right = thm "add_less_imp_less_right";
  1113 val add_less_cancel_left = thm "add_less_cancel_left";
  1114 val add_less_cancel_right = thm "add_less_cancel_right";
  1115 val add_le_cancel_left = thm "add_le_cancel_left";
  1116 val add_le_cancel_right = thm "add_le_cancel_right";
  1117 val add_le_imp_le_right = thm "add_le_imp_le_right";
  1118 val add_increasing = thm "add_increasing";
  1119 val le_imp_neg_le = thm "le_imp_neg_le";
  1120 val neg_le_iff_le = thm "neg_le_iff_le";
  1121 val neg_le_0_iff_le = thm "neg_le_0_iff_le";
  1122 val neg_0_le_iff_le = thm "neg_0_le_iff_le";
  1123 val neg_less_iff_less = thm "neg_less_iff_less";
  1124 val neg_less_0_iff_less = thm "neg_less_0_iff_less";
  1125 val neg_0_less_iff_less = thm "neg_0_less_iff_less";
  1126 val less_minus_iff = thm "less_minus_iff";
  1127 val minus_less_iff = thm "minus_less_iff";
  1128 val le_minus_iff = thm "le_minus_iff";
  1129 val minus_le_iff = thm "minus_le_iff";
  1130 val add_diff_eq = thm "add_diff_eq";
  1131 val diff_add_eq = thm "diff_add_eq";
  1132 val diff_eq_eq = thm "diff_eq_eq";
  1133 val eq_diff_eq = thm "eq_diff_eq";
  1134 val diff_diff_eq = thm "diff_diff_eq";
  1135 val diff_diff_eq2 = thm "diff_diff_eq2";
  1136 val diff_add_cancel = thm "diff_add_cancel";
  1137 val add_diff_cancel = thm "add_diff_cancel";
  1138 val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
  1139 val diff_less_eq = thm "diff_less_eq";
  1140 val less_diff_eq = thm "less_diff_eq";
  1141 val diff_le_eq = thm "diff_le_eq";
  1142 val le_diff_eq = thm "le_diff_eq";
  1143 val compare_rls = thms "compare_rls";
  1144 val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
  1145 val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
  1146 val add_meet_distrib_left = thm "add_meet_distrib_left";
  1147 val add_join_distrib_left = thm "add_join_distrib_left";
  1148 val is_join_neg_meet = thm "is_join_neg_meet";
  1149 val is_meet_neg_join = thm "is_meet_neg_join";
  1150 val add_join_distrib_right = thm "add_join_distrib_right";
  1151 val add_meet_distrib_right = thm "add_meet_distrib_right";
  1152 val add_meet_join_distribs = thms "add_meet_join_distribs";
  1153 val join_eq_neg_meet = thm "join_eq_neg_meet";
  1154 val meet_eq_neg_join = thm "meet_eq_neg_join";
  1155 val add_eq_meet_join = thm "add_eq_meet_join";
  1156 val prts = thm "prts";
  1157 val zero_le_pprt = thm "zero_le_pprt";
  1158 val nprt_le_zero = thm "nprt_le_zero";
  1159 val le_eq_neg = thm "le_eq_neg";
  1160 val join_0_imp_0 = thm "join_0_imp_0";
  1161 val meet_0_imp_0 = thm "meet_0_imp_0";
  1162 val join_0_eq_0 = thm "join_0_eq_0";
  1163 val meet_0_eq_0 = thm "meet_0_eq_0";
  1164 val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add";
  1165 val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero";
  1166 val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero";
  1167 val abs_lattice = thm "abs_lattice";
  1168 val abs_zero = thm "abs_zero";
  1169 val abs_eq_0 = thm "abs_eq_0";
  1170 val abs_0_eq = thm "abs_0_eq";
  1171 val neg_meet_eq_join = thm "neg_meet_eq_join";
  1172 val neg_join_eq_meet = thm "neg_join_eq_meet";
  1173 val join_eq_if = thm "join_eq_if";
  1174 val abs_if_lattice = thm "abs_if_lattice";
  1175 val abs_ge_zero = thm "abs_ge_zero";
  1176 val abs_le_zero_iff = thm "abs_le_zero_iff";
  1177 val zero_less_abs_iff = thm "zero_less_abs_iff";
  1178 val abs_not_less_zero = thm "abs_not_less_zero";
  1179 val abs_ge_self = thm "abs_ge_self";
  1180 val abs_ge_minus_self = thm "abs_ge_minus_self";
  1181 val le_imp_join_eq = thm "le_imp_join_eq";
  1182 val ge_imp_join_eq = thm "ge_imp_join_eq";
  1183 val le_imp_meet_eq = thm "le_imp_meet_eq";
  1184 val ge_imp_meet_eq = thm "ge_imp_meet_eq";
  1185 val abs_prts = thm "abs_prts";
  1186 val abs_minus_cancel = thm "abs_minus_cancel";
  1187 val abs_idempotent = thm "abs_idempotent";
  1188 val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt";
  1189 val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt";
  1190 val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id";
  1191 val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id";
  1192 val iff2imp = thm "iff2imp";
  1193 (* val imp_abs_id = thm "imp_abs_id";
  1194 val imp_abs_neg_id = thm "imp_abs_neg_id"; *)
  1195 val abs_leI = thm "abs_leI";
  1196 val le_minus_self_iff = thm "le_minus_self_iff";
  1197 val minus_le_self_iff = thm "minus_le_self_iff";
  1198 val abs_le_D1 = thm "abs_le_D1";
  1199 val abs_le_D2 = thm "abs_le_D2";
  1200 val abs_le_iff = thm "abs_le_iff";
  1201 val abs_triangle_ineq = thm "abs_triangle_ineq";
  1202 val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
  1203 *}
  1204 
  1205 end