src/HOL/OrderedGroup.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15093 49ede01e9ee6 child 15140 322485b816ac permissions -rw-r--r--
     1 (*  Title:   HOL/OrderedGroup.thy

     2     ID:      $Id$

     3     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel

     4 *)

     5

     6 header {* Ordered Groups *}

     7

     8 theory OrderedGroup

     9 import Inductive LOrder

    10 files "../Provers/Arith/abel_cancel.ML"

    11 begin

    12

    13 text {*

    14   The theory of partially ordered groups is taken from the books:

    15   \begin{itemize}

    16   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979

    17   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963

    18   \end{itemize}

    19   Most of the used notions can also be looked up in

    20   \begin{itemize}

    21   \item \url{http://www.mathworld.com} by Eric Weisstein et. al.

    22   \item \emph{Algebra I} by van der Waerden, Springer.

    23   \end{itemize}

    24 *}

    25

    26 subsection {* Semigroups, Groups *}

    27

    28 axclass semigroup_add \<subseteq> plus

    29   add_assoc: "(a + b) + c = a + (b + c)"

    30

    31 axclass ab_semigroup_add \<subseteq> semigroup_add

    32   add_commute: "a + b = b + a"

    33

    34 lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))"

    35   by (rule mk_left_commute [of "op +", OF add_assoc add_commute])

    36

    37 theorems add_ac = add_assoc add_commute add_left_commute

    38

    39 axclass semigroup_mult \<subseteq> times

    40   mult_assoc: "(a * b) * c = a * (b * c)"

    41

    42 axclass ab_semigroup_mult \<subseteq> semigroup_mult

    43   mult_commute: "a * b = b * a"

    44

    45 lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))"

    46   by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])

    47

    48 theorems mult_ac = mult_assoc mult_commute mult_left_commute

    49

    50 axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add

    51   add_0[simp]: "0 + a = a"

    52

    53 axclass monoid_mult \<subseteq> one, semigroup_mult

    54   mult_1_left[simp]: "1 * a  = a"

    55   mult_1_right[simp]: "a * 1 = a"

    56

    57 axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult

    58   mult_1: "1 * a = a"

    59

    60 instance comm_monoid_mult \<subseteq> monoid_mult

    61 by (intro_classes, insert mult_1, simp_all add: mult_commute, auto)

    62

    63 axclass cancel_semigroup_add \<subseteq> semigroup_add

    64   add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"

    65   add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"

    66

    67 axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add

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

    69

    70 instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add

    71 proof

    72   {

    73     fix a b c :: 'a

    74     assume "a + b = a + c"

    75     thus "b = c" by (rule add_imp_eq)

    76   }

    77   note f = this

    78   fix a b c :: 'a

    79   assume "b + a = c + a"

    80   hence "a + b = a + c" by (simp only: add_commute)

    81   thus "b = c" by (rule f)

    82 qed

    83

    84 axclass ab_group_add \<subseteq> minus, comm_monoid_add

    85   left_minus[simp]: " - a + a = 0"

    86   diff_minus: "a - b = a + (-b)"

    87

    88 instance ab_group_add \<subseteq> cancel_ab_semigroup_add

    89 proof

    90   fix a b c :: 'a

    91   assume "a + b = a + c"

    92   hence "-a + a + b = -a + a + c" by (simp only: add_assoc)

    93   thus "b = c" by simp

    94 qed

    95

    96 lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)"

    97 proof -

    98   have "a + 0 = 0 + a" by (simp only: add_commute)

    99   also have "... = a" by simp

   100   finally show ?thesis .

   101 qed

   102

   103 lemma add_left_cancel [simp]:

   104      "(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))"

   105 by (blast dest: add_left_imp_eq)

   106

   107 lemma add_right_cancel [simp]:

   108      "(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))"

   109   by (blast dest: add_right_imp_eq)

   110

   111 lemma right_minus [simp]: "a + -(a::'a::ab_group_add) = 0"

   112 proof -

   113   have "a + -a = -a + a" by (simp add: add_ac)

   114   also have "... = 0" by simp

   115   finally show ?thesis .

   116 qed

   117

   118 lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ab_group_add))"

   119 proof

   120   have "a = a - b + b" by (simp add: diff_minus add_ac)

   121   also assume "a - b = 0"

   122   finally show "a = b" by simp

   123 next

   124   assume "a = b"

   125   thus "a - b = 0" by (simp add: diff_minus)

   126 qed

   127

   128 lemma minus_minus [simp]: "- (- (a::'a::ab_group_add)) = a"

   129 proof (rule add_left_cancel [of "-a", THEN iffD1])

   130   show "(-a + -(-a) = -a + a)"

   131   by simp

   132 qed

   133

   134 lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ab_group_add)"

   135 apply (rule right_minus_eq [THEN iffD1, symmetric])

   136 apply (simp add: diff_minus add_commute)

   137 done

   138

   139 lemma minus_zero [simp]: "- 0 = (0::'a::ab_group_add)"

   140 by (simp add: equals_zero_I)

   141

   142 lemma diff_self [simp]: "a - (a::'a::ab_group_add) = 0"

   143   by (simp add: diff_minus)

   144

   145 lemma diff_0 [simp]: "(0::'a::ab_group_add) - a = -a"

   146 by (simp add: diff_minus)

   147

   148 lemma diff_0_right [simp]: "a - (0::'a::ab_group_add) = a"

   149 by (simp add: diff_minus)

   150

   151 lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ab_group_add)"

   152 by (simp add: diff_minus)

   153

   154 lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ab_group_add))"

   155 proof

   156   assume "- a = - b"

   157   hence "- (- a) = - (- b)"

   158     by simp

   159   thus "a=b" by simp

   160 next

   161   assume "a=b"

   162   thus "-a = -b" by simp

   163 qed

   164

   165 lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))"

   166 by (subst neg_equal_iff_equal [symmetric], simp)

   167

   168 lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))"

   169 by (subst neg_equal_iff_equal [symmetric], simp)

   170

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

   172

   173 lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))"

   174 proof -

   175   have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)

   176   thus ?thesis by (simp add: eq_commute)

   177 qed

   178

   179 lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)"

   180 proof -

   181   have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)

   182   thus ?thesis by (simp add: eq_commute)

   183 qed

   184

   185 lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)"

   186 apply (rule equals_zero_I)

   187 apply (simp add: add_ac)

   188 done

   189

   190 lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)"

   191 by (simp add: diff_minus add_commute)

   192

   193 subsection {* (Partially) Ordered Groups *}

   194

   195 axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add

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

   197

   198 axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add

   199

   200 instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add ..

   201

   202 axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add

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

   204

   205 axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add

   206

   207 instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le

   208 proof

   209   fix a b c :: 'a

   210   assume "c + a \<le> c + b"

   211   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)

   212   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)

   213   thus "a \<le> b" by simp

   214 qed

   215

   216 axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder

   217

   218 instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le

   219 proof

   220   fix a b c :: 'a

   221   assume le: "c + a <= c + b"

   222   show "a <= b"

   223   proof (rule ccontr)

   224     assume w: "~ a \<le> b"

   225     hence "b <= a" by (simp add: linorder_not_le)

   226     hence le2: "c+b <= c+a" by (rule add_left_mono)

   227     have "a = b"

   228       apply (insert le)

   229       apply (insert le2)

   230       apply (drule order_antisym, simp_all)

   231       done

   232     with w  show False

   233       by (simp add: linorder_not_le [symmetric])

   234   qed

   235 qed

   236

   237 lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c"

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

   239

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

   241 lemma add_mono:

   242      "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)"

   243   apply (erule add_right_mono [THEN order_trans])

   244   apply (simp add: add_commute add_left_mono)

   245   done

   246

   247 lemma add_strict_left_mono:

   248      "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)"

   249  by (simp add: order_less_le add_left_mono)

   250

   251 lemma add_strict_right_mono:

   252      "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)"

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

   254

   255 text{*Strict monotonicity in both arguments*}

   256 lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"

   257 apply (erule add_strict_right_mono [THEN order_less_trans])

   258 apply (erule add_strict_left_mono)

   259 done

   260

   261 lemma add_less_le_mono:

   262      "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"

   263 apply (erule add_strict_right_mono [THEN order_less_le_trans])

   264 apply (erule add_left_mono)

   265 done

   266

   267 lemma add_le_less_mono:

   268      "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"

   269 apply (erule add_right_mono [THEN order_le_less_trans])

   270 apply (erule add_strict_left_mono)

   271 done

   272

   273 lemma add_less_imp_less_left:

   274       assumes less: "c + a < c + b"  shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)"

   275 proof -

   276   from less have le: "c + a <= c + b" by (simp add: order_le_less)

   277   have "a <= b"

   278     apply (insert le)

   279     apply (drule add_le_imp_le_left)

   280     by (insert le, drule add_le_imp_le_left, assumption)

   281   moreover have "a \<noteq> b"

   282   proof (rule ccontr)

   283     assume "~(a \<noteq> b)"

   284     then have "a = b" by simp

   285     then have "c + a = c + b" by simp

   286     with less show "False"by simp

   287   qed

   288   ultimately show "a < b" by (simp add: order_le_less)

   289 qed

   290

   291 lemma add_less_imp_less_right:

   292       "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)"

   293 apply (rule add_less_imp_less_left [of c])

   294 apply (simp add: add_commute)

   295 done

   296

   297 lemma add_less_cancel_left [simp]:

   298     "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"

   299 by (blast intro: add_less_imp_less_left add_strict_left_mono)

   300

   301 lemma add_less_cancel_right [simp]:

   302     "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"

   303 by (blast intro: add_less_imp_less_right add_strict_right_mono)

   304

   305 lemma add_le_cancel_left [simp]:

   306     "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"

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

   308

   309 lemma add_le_cancel_right [simp]:

   310     "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"

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

   312

   313 lemma add_le_imp_le_right:

   314       "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)"

   315 by simp

   316

   317 lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add})"

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

   319

   320 subsection {* Ordering Rules for Unary Minus *}

   321

   322 lemma le_imp_neg_le:

   323       assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"

   324 proof -

   325   have "-a+a \<le> -a+b"

   326     by (rule add_left_mono)

   327   hence "0 \<le> -a+b"

   328     by simp

   329   hence "0 + (-b) \<le> (-a + b) + (-b)"

   330     by (rule add_right_mono)

   331   thus ?thesis

   332     by (simp add: add_assoc)

   333 qed

   334

   335 lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"

   336 proof

   337   assume "- b \<le> - a"

   338   hence "- (- a) \<le> - (- b)"

   339     by (rule le_imp_neg_le)

   340   thus "a\<le>b" by simp

   341 next

   342   assume "a\<le>b"

   343   thus "-b \<le> -a" by (rule le_imp_neg_le)

   344 qed

   345

   346 lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"

   347 by (subst neg_le_iff_le [symmetric], simp)

   348

   349 lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"

   350 by (subst neg_le_iff_le [symmetric], simp)

   351

   352 lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"

   353 by (force simp add: order_less_le)

   354

   355 lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"

   356 by (subst neg_less_iff_less [symmetric], simp)

   357

   358 lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"

   359 by (subst neg_less_iff_less [symmetric], simp)

   360

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

   362

   363 lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"

   364 proof -

   365   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)

   366   thus ?thesis by simp

   367 qed

   368

   369 lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"

   370 proof -

   371   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)

   372   thus ?thesis by simp

   373 qed

   374

   375 lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"

   376 proof -

   377   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)

   378   have "(- (- a) <= -b) = (b <= - a)"

   379     apply (auto simp only: order_le_less)

   380     apply (drule mm)

   381     apply (simp_all)

   382     apply (drule mm[simplified], assumption)

   383     done

   384   then show ?thesis by simp

   385 qed

   386

   387 lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"

   388 by (auto simp add: order_le_less minus_less_iff)

   389

   390 lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"

   391 by (simp add: diff_minus add_ac)

   392

   393 lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"

   394 by (simp add: diff_minus add_ac)

   395

   396 lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"

   397 by (auto simp add: diff_minus add_assoc)

   398

   399 lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"

   400 by (auto simp add: diff_minus add_assoc)

   401

   402 lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"

   403 by (simp add: diff_minus add_ac)

   404

   405 lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"

   406 by (simp add: diff_minus add_ac)

   407

   408 lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"

   409 by (simp add: diff_minus add_ac)

   410

   411 lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"

   412 by (simp add: diff_minus add_ac)

   413

   414 text{*Further subtraction laws*}

   415

   416 lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"

   417 proof -

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

   419     by (simp only: add_less_cancel_right)

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

   421   finally show ?thesis .

   422 qed

   423

   424 lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"

   425 apply (subst less_iff_diff_less_0)

   426 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])

   427 apply (simp add: diff_minus add_ac)

   428 done

   429

   430 lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"

   431 apply (subst less_iff_diff_less_0)

   432 apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst])

   433 apply (simp add: diff_minus add_ac)

   434 done

   435

   436 lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"

   437 by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)

   438

   439 lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"

   440 by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)

   441

   442 text{*This list of rewrites simplifies (in)equalities by bringing subtractions

   443   to the top and then moving negative terms to the other side.

   444   Use with @{text add_ac}*}

   445 lemmas compare_rls =

   446        diff_minus [symmetric]

   447        add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   448        diff_less_eq less_diff_eq diff_le_eq le_diff_eq

   449        diff_eq_eq eq_diff_eq

   450

   451

   452 subsection{*Lemmas for the @{text cancel_numerals} simproc*}

   453

   454 lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"

   455 by (simp add: compare_rls)

   456

   457 lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"

   458 by (simp add: compare_rls)

   459

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

   461

   462 axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder

   463

   464 axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder

   465

   466 lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))"

   467 apply (rule order_antisym)

   468 apply (rule meet_imp_le, simp_all add: meet_join_le)

   469 apply (rule add_le_imp_le_left [of "-a"])

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

   471 apply (rule meet_imp_le)

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

   473 done

   474

   475 lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))"

   476 apply (rule order_antisym)

   477 apply (rule add_le_imp_le_left [of "-a"])

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

   479 apply (rule join_imp_le)

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

   481 apply (rule join_imp_le)

   482 apply (simp_all add: meet_join_le)

   483 done

   484

   485 lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))"

   486 apply (auto simp add: is_join_def)

   487 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)

   488 apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)

   489 apply (subst neg_le_iff_le[symmetric])

   490 apply (simp add: meet_imp_le)

   491 done

   492

   493 lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))"

   494 apply (auto simp add: is_meet_def)

   495 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)

   496 apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)

   497 apply (subst neg_le_iff_le[symmetric])

   498 apply (simp add: join_imp_le)

   499 done

   500

   501 axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder

   502

   503 instance lordered_ab_group_meet \<subseteq> lordered_ab_group

   504 proof

   505   show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet)

   506 qed

   507

   508 instance lordered_ab_group_join \<subseteq> lordered_ab_group

   509 proof

   510   show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join)

   511 qed

   512

   513 lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)"

   514 proof -

   515   have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left)

   516   thus ?thesis by (simp add: add_commute)

   517 qed

   518

   519 lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)"

   520 proof -

   521   have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left)

   522   thus ?thesis by (simp add: add_commute)

   523 qed

   524

   525 lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left

   526

   527 lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)"

   528 by (simp add: is_join_unique[OF is_join_join is_join_neg_meet])

   529

   530 lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)"

   531 by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join])

   532

   533 lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))"

   534 proof -

   535   have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm)

   536   hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join)

   537   hence "0 = (-a + join a b) + (meet a b + (-b))"

   538     apply (simp add: add_join_distrib_left add_meet_distrib_right)

   539     by (simp add: diff_minus add_commute)

   540   thus ?thesis

   541     apply (simp add: compare_rls)

   542     apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"])

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

   544     done

   545 qed

   546

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

   548

   549 constdefs

   550   pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"

   551   "pprt x == join x 0"

   552   nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"

   553   "nprt x == meet x 0"

   554

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

   556 by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric])

   557

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

   559 by (simp add: pprt_def meet_join_le)

   560

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

   562 by (simp add: nprt_def meet_join_le)

   563

   564 lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")

   565 proof -

   566   have a: "?l \<longrightarrow> ?r"

   567     apply (auto)

   568     apply (rule add_le_imp_le_right[of _ "-b" _])

   569     apply (simp add: add_assoc)

   570     done

   571   have b: "?r \<longrightarrow> ?l"

   572     apply (auto)

   573     apply (rule add_le_imp_le_right[of _ "b" _])

   574     apply (simp)

   575     done

   576   from a b show ?thesis by blast

   577 qed

   578

   579 lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"

   580 proof -

   581   {

   582     fix a::'a

   583     assume hyp: "join a (-a) = 0"

   584     hence "join a (-a) + a = a" by (simp)

   585     hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right)

   586     hence "join (a+a) 0 <= a" by (simp)

   587     hence "0 <= a" by (blast intro: order_trans meet_join_le)

   588   }

   589   note p = this

   590   assume hyp:"join a (-a) = 0"

   591   hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm)

   592   from p[OF hyp] p[OF hyp2] show "a = 0" by simp

   593 qed

   594

   595 lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"

   596 apply (simp add: meet_eq_neg_join)

   597 apply (simp add: join_comm)

   598 apply (subst join_0_imp_0)

   599 by auto

   600

   601 lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))"

   602 by (auto, erule join_0_imp_0)

   603

   604 lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))"

   605 by (auto, erule meet_0_imp_0)

   606

   607 lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"

   608 proof

   609   assume "0 <= a + a"

   610   hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm)

   611   have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci)

   612   hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm)

   613   hence "meet a 0 = 0" by (simp only: add_right_cancel)

   614   then show "0 <= a" by (simp add: le_def_meet meet_comm)

   615 next

   616   assume a: "0 <= a"

   617   show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])

   618 qed

   619

   620 lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)"

   621 proof -

   622   have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)

   623   moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)

   624   ultimately show ?thesis by blast

   625 qed

   626

   627 lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)

   628 proof cases

   629   assume a: "a < 0"

   630   thus ?s by (simp add:  add_strict_mono[OF a a, simplified])

   631 next

   632   assume "~(a < 0)"

   633   hence a:"0 <= a" by (simp)

   634   hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified])

   635   hence "~(a+a < 0)" by simp

   636   with a show ?thesis by simp

   637 qed

   638

   639 axclass lordered_ab_group_abs \<subseteq> lordered_ab_group

   640   abs_lattice: "abs x = join x (-x)"

   641

   642 lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"

   643 by (simp add: abs_lattice)

   644

   645 lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"

   646 by (simp add: abs_lattice)

   647

   648 lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"

   649 proof -

   650   have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)

   651   thus ?thesis by simp

   652 qed

   653

   654 lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)"

   655 by (simp add: meet_eq_neg_join)

   656

   657 lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)"

   658 by (simp del: neg_meet_eq_join add: join_eq_neg_meet)

   659

   660 lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"

   661 proof -

   662   note b = add_le_cancel_right[of a a "-a",symmetric,simplified]

   663   have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)

   664   show ?thesis

   665     apply (auto simp add: join_max max_def b linorder_not_less)

   666     apply (drule order_antisym, auto)

   667     done

   668 qed

   669

   670 lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"

   671 proof -

   672   show ?thesis by (simp add: abs_lattice join_eq_if)

   673 qed

   674

   675 lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"

   676 proof -

   677   have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le)

   678   show ?thesis by (rule add_mono[OF a b, simplified])

   679 qed

   680

   681 lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)"

   682 proof

   683   assume "abs a <= 0"

   684   hence "abs a = 0" by (auto dest: order_antisym)

   685   thus "a = 0" by simp

   686 next

   687   assume "a = 0"

   688   thus "abs a <= 0" by simp

   689 qed

   690

   691 lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"

   692 by (simp add: order_less_le)

   693

   694 lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"

   695 proof -

   696   have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto

   697   show ?thesis by (simp add: a)

   698 qed

   699

   700 lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"

   701 by (simp add: abs_lattice meet_join_le)

   702

   703 lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"

   704 by (simp add: abs_lattice meet_join_le)

   705

   706 lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b"

   707 by (simp add: le_def_join)

   708

   709 lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a"

   710 by (simp add: le_def_join join_aci)

   711

   712 lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a"

   713 by (simp add: le_def_meet)

   714

   715 lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b"

   716 by (simp add: le_def_meet meet_aci)

   717

   718 lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"

   719 apply (simp add: pprt_def nprt_def diff_minus)

   720 apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric])

   721 apply (subst le_imp_join_eq, auto)

   722 done

   723

   724 lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"

   725 by (simp add: abs_lattice join_comm)

   726

   727 lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"

   728 apply (simp add: abs_lattice[of "abs a"])

   729 apply (subst ge_imp_join_eq)

   730 apply (rule order_trans[of _ 0])

   731 by auto

   732

   733 lemma abs_minus_commute:

   734   fixes a :: "'a::lordered_ab_group_abs"

   735   shows "abs (a-b) = abs(b-a)"

   736 proof -

   737   have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel)

   738   also have "... = abs(b-a)" by simp

   739   finally show ?thesis .

   740 qed

   741

   742 lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"

   743 by (simp add: le_def_meet nprt_def meet_comm)

   744

   745 lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"

   746 by (simp add: le_def_join pprt_def join_comm)

   747

   748 lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"

   749 by (simp add: le_def_join pprt_def join_comm)

   750

   751 lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"

   752 by (simp add: le_def_meet nprt_def meet_comm)

   753

   754 lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"

   755 by (simp)

   756

   757 lemma imp_abs_id: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"

   758 by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)

   759

   760 lemma imp_abs_neg_id: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"

   761 by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)

   762

   763 lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"

   764 by (simp add: abs_lattice join_imp_le)

   765

   766 lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"

   767 proof -

   768   from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)"

   769     by (simp add: add_assoc[symmetric])

   770   thus ?thesis by simp

   771 qed

   772

   773 lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"

   774 proof -

   775   from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)"

   776     by (simp add: add_assoc[symmetric])

   777   thus ?thesis by simp

   778 qed

   779

   780 lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"

   781 by (insert abs_ge_self, blast intro: order_trans)

   782

   783 lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"

   784 by (insert abs_le_D1 [of "-a"], simp)

   785

   786 lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"

   787 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)

   788

   789 lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::lordered_ab_group_abs)"

   790 proof -

   791   have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n")

   792     apply (simp add: abs_lattice add_meet_join_distribs join_aci)

   793     by (simp only: diff_minus)

   794   have a:"a+b <= join ?m ?n" by (simp add: meet_join_le)

   795   have b:"-a-b <= ?n" by (simp add: meet_join_le)

   796   have c:"?n <= join ?m ?n" by (simp add: meet_join_le)

   797   from b c have d: "-a-b <= join ?m ?n" by simp

   798   have e:"-a-b = -(a+b)" by (simp add: diff_minus)

   799   from a d e have "abs(a+b) <= join ?m ?n"

   800     by (drule_tac abs_leI, auto)

   801   with g[symmetric] show ?thesis by simp

   802 qed

   803

   804 lemma abs_diff_triangle_ineq:

   805      "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"

   806 proof -

   807   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)

   808   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)

   809   finally show ?thesis .

   810 qed

   811

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

   813

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

   815 apply (subst add_left_commute)

   816 apply (subst add_left_cancel)

   817 apply simp

   818 done

   819

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

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

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

   823 done

   824

   825 lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"

   826 by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])

   827

   828 lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"

   829 apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])

   830 apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])

   831 done

   832

   833 lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"

   834 by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])

   835

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

   837 by (simp add: diff_minus)

   838

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

   840 by (simp add: add_assoc[symmetric])

   841

   842 lemma minus_add_cancel: "-(a::'a::ab_group_add) + (a + b) = b"

   843 by (simp add: add_assoc[symmetric])

   844

   845 ML {*

   846 val add_zero_left = thm"add_0";

   847 val add_zero_right = thm"add_0_right";

   848 *}

   849

   850 ML {*

   851 val add_assoc = thm "add_assoc";

   852 val add_commute = thm "add_commute";

   853 val add_left_commute = thm "add_left_commute";

   854 val add_ac = thms "add_ac";

   855 val mult_assoc = thm "mult_assoc";

   856 val mult_commute = thm "mult_commute";

   857 val mult_left_commute = thm "mult_left_commute";

   858 val mult_ac = thms "mult_ac";

   859 val add_0 = thm "add_0";

   860 val mult_1_left = thm "mult_1_left";

   861 val mult_1_right = thm "mult_1_right";

   862 val mult_1 = thm "mult_1";

   863 val add_left_imp_eq = thm "add_left_imp_eq";

   864 val add_right_imp_eq = thm "add_right_imp_eq";

   865 val add_imp_eq = thm "add_imp_eq";

   866 val left_minus = thm "left_minus";

   867 val diff_minus = thm "diff_minus";

   868 val add_0_right = thm "add_0_right";

   869 val add_left_cancel = thm "add_left_cancel";

   870 val add_right_cancel = thm "add_right_cancel";

   871 val right_minus = thm "right_minus";

   872 val right_minus_eq = thm "right_minus_eq";

   873 val minus_minus = thm "minus_minus";

   874 val equals_zero_I = thm "equals_zero_I";

   875 val minus_zero = thm "minus_zero";

   876 val diff_self = thm "diff_self";

   877 val diff_0 = thm "diff_0";

   878 val diff_0_right = thm "diff_0_right";

   879 val diff_minus_eq_add = thm "diff_minus_eq_add";

   880 val neg_equal_iff_equal = thm "neg_equal_iff_equal";

   881 val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal";

   882 val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal";

   883 val equation_minus_iff = thm "equation_minus_iff";

   884 val minus_equation_iff = thm "minus_equation_iff";

   885 val minus_add_distrib = thm "minus_add_distrib";

   886 val minus_diff_eq = thm "minus_diff_eq";

   887 val add_left_mono = thm "add_left_mono";

   888 val add_le_imp_le_left = thm "add_le_imp_le_left";

   889 val add_right_mono = thm "add_right_mono";

   890 val add_mono = thm "add_mono";

   891 val add_strict_left_mono = thm "add_strict_left_mono";

   892 val add_strict_right_mono = thm "add_strict_right_mono";

   893 val add_strict_mono = thm "add_strict_mono";

   894 val add_less_le_mono = thm "add_less_le_mono";

   895 val add_le_less_mono = thm "add_le_less_mono";

   896 val add_less_imp_less_left = thm "add_less_imp_less_left";

   897 val add_less_imp_less_right = thm "add_less_imp_less_right";

   898 val add_less_cancel_left = thm "add_less_cancel_left";

   899 val add_less_cancel_right = thm "add_less_cancel_right";

   900 val add_le_cancel_left = thm "add_le_cancel_left";

   901 val add_le_cancel_right = thm "add_le_cancel_right";

   902 val add_le_imp_le_right = thm "add_le_imp_le_right";

   903 val add_increasing = thm "add_increasing";

   904 val le_imp_neg_le = thm "le_imp_neg_le";

   905 val neg_le_iff_le = thm "neg_le_iff_le";

   906 val neg_le_0_iff_le = thm "neg_le_0_iff_le";

   907 val neg_0_le_iff_le = thm "neg_0_le_iff_le";

   908 val neg_less_iff_less = thm "neg_less_iff_less";

   909 val neg_less_0_iff_less = thm "neg_less_0_iff_less";

   910 val neg_0_less_iff_less = thm "neg_0_less_iff_less";

   911 val less_minus_iff = thm "less_minus_iff";

   912 val minus_less_iff = thm "minus_less_iff";

   913 val le_minus_iff = thm "le_minus_iff";

   914 val minus_le_iff = thm "minus_le_iff";

   915 val add_diff_eq = thm "add_diff_eq";

   916 val diff_add_eq = thm "diff_add_eq";

   917 val diff_eq_eq = thm "diff_eq_eq";

   918 val eq_diff_eq = thm "eq_diff_eq";

   919 val diff_diff_eq = thm "diff_diff_eq";

   920 val diff_diff_eq2 = thm "diff_diff_eq2";

   921 val diff_add_cancel = thm "diff_add_cancel";

   922 val add_diff_cancel = thm "add_diff_cancel";

   923 val less_iff_diff_less_0 = thm "less_iff_diff_less_0";

   924 val diff_less_eq = thm "diff_less_eq";

   925 val less_diff_eq = thm "less_diff_eq";

   926 val diff_le_eq = thm "diff_le_eq";

   927 val le_diff_eq = thm "le_diff_eq";

   928 val compare_rls = thms "compare_rls";

   929 val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";

   930 val le_iff_diff_le_0 = thm "le_iff_diff_le_0";

   931 val add_meet_distrib_left = thm "add_meet_distrib_left";

   932 val add_join_distrib_left = thm "add_join_distrib_left";

   933 val is_join_neg_meet = thm "is_join_neg_meet";

   934 val is_meet_neg_join = thm "is_meet_neg_join";

   935 val add_join_distrib_right = thm "add_join_distrib_right";

   936 val add_meet_distrib_right = thm "add_meet_distrib_right";

   937 val add_meet_join_distribs = thms "add_meet_join_distribs";

   938 val join_eq_neg_meet = thm "join_eq_neg_meet";

   939 val meet_eq_neg_join = thm "meet_eq_neg_join";

   940 val add_eq_meet_join = thm "add_eq_meet_join";

   941 val prts = thm "prts";

   942 val zero_le_pprt = thm "zero_le_pprt";

   943 val nprt_le_zero = thm "nprt_le_zero";

   944 val le_eq_neg = thm "le_eq_neg";

   945 val join_0_imp_0 = thm "join_0_imp_0";

   946 val meet_0_imp_0 = thm "meet_0_imp_0";

   947 val join_0_eq_0 = thm "join_0_eq_0";

   948 val meet_0_eq_0 = thm "meet_0_eq_0";

   949 val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add";

   950 val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero";

   951 val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero";

   952 val abs_lattice = thm "abs_lattice";

   953 val abs_zero = thm "abs_zero";

   954 val abs_eq_0 = thm "abs_eq_0";

   955 val abs_0_eq = thm "abs_0_eq";

   956 val neg_meet_eq_join = thm "neg_meet_eq_join";

   957 val neg_join_eq_meet = thm "neg_join_eq_meet";

   958 val join_eq_if = thm "join_eq_if";

   959 val abs_if_lattice = thm "abs_if_lattice";

   960 val abs_ge_zero = thm "abs_ge_zero";

   961 val abs_le_zero_iff = thm "abs_le_zero_iff";

   962 val zero_less_abs_iff = thm "zero_less_abs_iff";

   963 val abs_not_less_zero = thm "abs_not_less_zero";

   964 val abs_ge_self = thm "abs_ge_self";

   965 val abs_ge_minus_self = thm "abs_ge_minus_self";

   966 val le_imp_join_eq = thm "le_imp_join_eq";

   967 val ge_imp_join_eq = thm "ge_imp_join_eq";

   968 val le_imp_meet_eq = thm "le_imp_meet_eq";

   969 val ge_imp_meet_eq = thm "ge_imp_meet_eq";

   970 val abs_prts = thm "abs_prts";

   971 val abs_minus_cancel = thm "abs_minus_cancel";

   972 val abs_idempotent = thm "abs_idempotent";

   973 val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt";

   974 val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt";

   975 val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id";

   976 val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id";

   977 val iff2imp = thm "iff2imp";

   978 val imp_abs_id = thm "imp_abs_id";

   979 val imp_abs_neg_id = thm "imp_abs_neg_id";

   980 val abs_leI = thm "abs_leI";

   981 val le_minus_self_iff = thm "le_minus_self_iff";

   982 val minus_le_self_iff = thm "minus_le_self_iff";

   983 val abs_le_D1 = thm "abs_le_D1";

   984 val abs_le_D2 = thm "abs_le_D2";

   985 val abs_le_iff = thm "abs_le_iff";

   986 val abs_triangle_ineq = thm "abs_triangle_ineq";

   987 val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";

   988 *}

   989

   990 end