src/HOL/Groups.thy
author hoelzl
Fri Feb 12 16:09:07 2016 +0100 (2016-02-12)
changeset 62377 ace69956d018
parent 62376 85f38d5f8807
child 62378 85ed00c1fe7c
permissions -rw-r--r--
moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
     1 (*  Title:   HOL/Groups.thy
     2     Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
     3 *)
     4 
     5 section \<open>Groups, also combined with orderings\<close>
     6 
     7 theory Groups
     8 imports Orderings
     9 begin
    10 
    11 subsection \<open>Dynamic facts\<close>
    12 
    13 named_theorems ac_simps "associativity and commutativity simplification rules"
    14 
    15 
    16 text\<open>The rewrites accumulated in \<open>algebra_simps\<close> deal with the
    17 classical algebraic structures of groups, rings and family. They simplify
    18 terms by multiplying everything out (in case of a ring) and bringing sums and
    19 products into a canonical form (by ordered rewriting). As a result it decides
    20 group and ring equalities but also helps with inequalities.
    21 
    22 Of course it also works for fields, but it knows nothing about multiplicative
    23 inverses or division. This is catered for by \<open>field_simps\<close>.\<close>
    24 
    25 named_theorems algebra_simps "algebra simplification rules"
    26 
    27 
    28 text\<open>Lemmas \<open>field_simps\<close> multiply with denominators in (in)equations
    29 if they can be proved to be non-zero (for equations) or positive/negative
    30 (for inequations). Can be too aggressive and is therefore separate from the
    31 more benign \<open>algebra_simps\<close>.\<close>
    32 
    33 named_theorems field_simps "algebra simplification rules for fields"
    34 
    35 
    36 subsection \<open>Abstract structures\<close>
    37 
    38 text \<open>
    39   These locales provide basic structures for interpretation into
    40   bigger structures;  extensions require careful thinking, otherwise
    41   undesired effects may occur due to interpretation.
    42 \<close>
    43 
    44 locale semigroup =
    45   fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
    46   assumes assoc [ac_simps]: "a * b * c = a * (b * c)"
    47 
    48 locale abel_semigroup = semigroup +
    49   assumes commute [ac_simps]: "a * b = b * a"
    50 begin
    51 
    52 lemma left_commute [ac_simps]:
    53   "b * (a * c) = a * (b * c)"
    54 proof -
    55   have "(b * a) * c = (a * b) * c"
    56     by (simp only: commute)
    57   then show ?thesis
    58     by (simp only: assoc)
    59 qed
    60 
    61 end
    62 
    63 locale monoid = semigroup +
    64   fixes z :: 'a ("1")
    65   assumes left_neutral [simp]: "1 * a = a"
    66   assumes right_neutral [simp]: "a * 1 = a"
    67 
    68 locale comm_monoid = abel_semigroup +
    69   fixes z :: 'a ("1")
    70   assumes comm_neutral: "a * 1 = a"
    71 begin
    72 
    73 sublocale monoid
    74   by standard (simp_all add: commute comm_neutral)
    75 
    76 end
    77 
    78 
    79 subsection \<open>Generic operations\<close>
    80 
    81 class zero =
    82   fixes zero :: 'a  ("0")
    83 
    84 class one =
    85   fixes one  :: 'a  ("1")
    86 
    87 hide_const (open) zero one
    88 
    89 lemma Let_0 [simp]: "Let 0 f = f 0"
    90   unfolding Let_def ..
    91 
    92 lemma Let_1 [simp]: "Let 1 f = f 1"
    93   unfolding Let_def ..
    94 
    95 setup \<open>
    96   Reorient_Proc.add
    97     (fn Const(@{const_name Groups.zero}, _) => true
    98       | Const(@{const_name Groups.one}, _) => true
    99       | _ => false)
   100 \<close>
   101 
   102 simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc
   103 simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc
   104 
   105 typed_print_translation \<open>
   106   let
   107     fun tr' c = (c, fn ctxt => fn T => fn ts =>
   108       if null ts andalso Printer.type_emphasis ctxt T then
   109         Syntax.const @{syntax_const "_constrain"} $ Syntax.const c $
   110           Syntax_Phases.term_of_typ ctxt T
   111       else raise Match);
   112   in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end;
   113 \<close> \<comment> \<open>show types that are presumably too general\<close>
   114 
   115 class plus =
   116   fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)
   117 
   118 class minus =
   119   fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)
   120 
   121 class uminus =
   122   fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)
   123 
   124 class times =
   125   fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)
   126 
   127 
   128 subsection \<open>Semigroups and Monoids\<close>
   129 
   130 class semigroup_add = plus +
   131   assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
   132 begin
   133 
   134 sublocale add: semigroup plus
   135   by standard (fact add_assoc)
   136 
   137 end
   138 
   139 hide_fact add_assoc
   140 
   141 class ab_semigroup_add = semigroup_add +
   142   assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
   143 begin
   144 
   145 sublocale add: abel_semigroup plus
   146   by standard (fact add_commute)
   147 
   148 declare add.left_commute [algebra_simps, field_simps]
   149 
   150 lemmas add_ac = add.assoc add.commute add.left_commute
   151 
   152 end
   153 
   154 hide_fact add_commute
   155 
   156 lemmas add_ac = add.assoc add.commute add.left_commute
   157 
   158 class semigroup_mult = times +
   159   assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
   160 begin
   161 
   162 sublocale mult: semigroup times
   163   by standard (fact mult_assoc)
   164 
   165 end
   166 
   167 hide_fact mult_assoc
   168 
   169 class ab_semigroup_mult = semigroup_mult +
   170   assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
   171 begin
   172 
   173 sublocale mult: abel_semigroup times
   174   by standard (fact mult_commute)
   175 
   176 declare mult.left_commute [algebra_simps, field_simps]
   177 
   178 lemmas mult_ac = mult.assoc mult.commute mult.left_commute
   179 
   180 end
   181 
   182 hide_fact mult_commute
   183 
   184 lemmas mult_ac = mult.assoc mult.commute mult.left_commute
   185 
   186 class monoid_add = zero + semigroup_add +
   187   assumes add_0_left: "0 + a = a"
   188     and add_0_right: "a + 0 = a"
   189 begin
   190 
   191 sublocale add: monoid plus 0
   192   by standard (fact add_0_left add_0_right)+
   193 
   194 end
   195 
   196 lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
   197   by (fact eq_commute)
   198 
   199 class comm_monoid_add = zero + ab_semigroup_add +
   200   assumes add_0: "0 + a = a"
   201 begin
   202 
   203 subclass monoid_add
   204   by standard (simp_all add: add_0 add.commute [of _ 0])
   205 
   206 sublocale add: comm_monoid plus 0
   207   by standard (simp add: ac_simps)
   208 
   209 end
   210 
   211 class monoid_mult = one + semigroup_mult +
   212   assumes mult_1_left: "1 * a  = a"
   213     and mult_1_right: "a * 1 = a"
   214 begin
   215 
   216 sublocale mult: monoid times 1
   217   by standard (fact mult_1_left mult_1_right)+
   218 
   219 end
   220 
   221 lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
   222   by (fact eq_commute)
   223 
   224 class comm_monoid_mult = one + ab_semigroup_mult +
   225   assumes mult_1: "1 * a = a"
   226 begin
   227 
   228 subclass monoid_mult
   229   by standard (simp_all add: mult_1 mult.commute [of _ 1])
   230 
   231 sublocale mult: comm_monoid times 1
   232   by standard (simp add: ac_simps)
   233 
   234 end
   235 
   236 class cancel_semigroup_add = semigroup_add +
   237   assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
   238   assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
   239 begin
   240 
   241 lemma add_left_cancel [simp]:
   242   "a + b = a + c \<longleftrightarrow> b = c"
   243 by (blast dest: add_left_imp_eq)
   244 
   245 lemma add_right_cancel [simp]:
   246   "b + a = c + a \<longleftrightarrow> b = c"
   247 by (blast dest: add_right_imp_eq)
   248 
   249 end
   250 
   251 class cancel_ab_semigroup_add = ab_semigroup_add + minus +
   252   assumes add_diff_cancel_left' [simp]: "(a + b) - a = b"
   253   assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)"
   254 begin
   255 
   256 lemma add_diff_cancel_right' [simp]:
   257   "(a + b) - b = a"
   258   using add_diff_cancel_left' [of b a] by (simp add: ac_simps)
   259 
   260 subclass cancel_semigroup_add
   261 proof
   262   fix a b c :: 'a
   263   assume "a + b = a + c"
   264   then have "a + b - a = a + c - a"
   265     by simp
   266   then show "b = c"
   267     by simp
   268 next
   269   fix a b c :: 'a
   270   assume "b + a = c + a"
   271   then have "b + a - a = c + a - a"
   272     by simp
   273   then show "b = c"
   274     by simp
   275 qed
   276 
   277 lemma add_diff_cancel_left [simp]:
   278   "(c + a) - (c + b) = a - b"
   279   unfolding diff_diff_add [symmetric] by simp
   280 
   281 lemma add_diff_cancel_right [simp]:
   282   "(a + c) - (b + c) = a - b"
   283   using add_diff_cancel_left [symmetric] by (simp add: ac_simps)
   284 
   285 lemma diff_right_commute:
   286   "a - c - b = a - b - c"
   287   by (simp add: diff_diff_add add.commute)
   288 
   289 end
   290 
   291 class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
   292 begin
   293 
   294 lemma diff_zero [simp]:
   295   "a - 0 = a"
   296   using add_diff_cancel_right' [of a 0] by simp
   297 
   298 lemma diff_cancel [simp]:
   299   "a - a = 0"
   300 proof -
   301   have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero)
   302   then show ?thesis by simp
   303 qed
   304 
   305 lemma add_implies_diff:
   306   assumes "c + b = a"
   307   shows "c = a - b"
   308 proof -
   309   from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute)
   310   then show "c = a - b" by simp
   311 qed
   312 
   313 end
   314 
   315 class comm_monoid_diff = cancel_comm_monoid_add +
   316   assumes zero_diff [simp]: "0 - a = 0"
   317 begin
   318 
   319 lemma diff_add_zero [simp]:
   320   "a - (a + b) = 0"
   321 proof -
   322   have "a - (a + b) = (a + 0) - (a + b)" by simp
   323   also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff)
   324   finally show ?thesis .
   325 qed
   326 
   327 end
   328 
   329 
   330 subsection \<open>Groups\<close>
   331 
   332 class group_add = minus + uminus + monoid_add +
   333   assumes left_minus [simp]: "- a + a = 0"
   334   assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
   335 begin
   336 
   337 lemma diff_conv_add_uminus:
   338   "a - b = a + (- b)"
   339   by simp
   340 
   341 lemma minus_unique:
   342   assumes "a + b = 0" shows "- a = b"
   343 proof -
   344   have "- a = - a + (a + b)" using assms by simp
   345   also have "\<dots> = b" by (simp add: add.assoc [symmetric])
   346   finally show ?thesis .
   347 qed
   348 
   349 lemma minus_zero [simp]: "- 0 = 0"
   350 proof -
   351   have "0 + 0 = 0" by (rule add_0_right)
   352   thus "- 0 = 0" by (rule minus_unique)
   353 qed
   354 
   355 lemma minus_minus [simp]: "- (- a) = a"
   356 proof -
   357   have "- a + a = 0" by (rule left_minus)
   358   thus "- (- a) = a" by (rule minus_unique)
   359 qed
   360 
   361 lemma right_minus: "a + - a = 0"
   362 proof -
   363   have "a + - a = - (- a) + - a" by simp
   364   also have "\<dots> = 0" by (rule left_minus)
   365   finally show ?thesis .
   366 qed
   367 
   368 lemma diff_self [simp]:
   369   "a - a = 0"
   370   using right_minus [of a] by simp
   371 
   372 subclass cancel_semigroup_add
   373 proof
   374   fix a b c :: 'a
   375   assume "a + b = a + c"
   376   then have "- a + a + b = - a + a + c"
   377     unfolding add.assoc by simp
   378   then show "b = c" by simp
   379 next
   380   fix a b c :: 'a
   381   assume "b + a = c + a"
   382   then have "b + a + - a = c + a  + - a" by simp
   383   then show "b = c" unfolding add.assoc by simp
   384 qed
   385 
   386 lemma minus_add_cancel [simp]:
   387   "- a + (a + b) = b"
   388   by (simp add: add.assoc [symmetric])
   389 
   390 lemma add_minus_cancel [simp]:
   391   "a + (- a + b) = b"
   392   by (simp add: add.assoc [symmetric])
   393 
   394 lemma diff_add_cancel [simp]:
   395   "a - b + b = a"
   396   by (simp only: diff_conv_add_uminus add.assoc) simp
   397 
   398 lemma add_diff_cancel [simp]:
   399   "a + b - b = a"
   400   by (simp only: diff_conv_add_uminus add.assoc) simp
   401 
   402 lemma minus_add:
   403   "- (a + b) = - b + - a"
   404 proof -
   405   have "(a + b) + (- b + - a) = 0"
   406     by (simp only: add.assoc add_minus_cancel) simp
   407   then show "- (a + b) = - b + - a"
   408     by (rule minus_unique)
   409 qed
   410 
   411 lemma right_minus_eq [simp]:
   412   "a - b = 0 \<longleftrightarrow> a = b"
   413 proof
   414   assume "a - b = 0"
   415   have "a = (a - b) + b" by (simp add: add.assoc)
   416   also have "\<dots> = b" using \<open>a - b = 0\<close> by simp
   417   finally show "a = b" .
   418 next
   419   assume "a = b" thus "a - b = 0" by simp
   420 qed
   421 
   422 lemma eq_iff_diff_eq_0:
   423   "a = b \<longleftrightarrow> a - b = 0"
   424   by (fact right_minus_eq [symmetric])
   425 
   426 lemma diff_0 [simp]:
   427   "0 - a = - a"
   428   by (simp only: diff_conv_add_uminus add_0_left)
   429 
   430 lemma diff_0_right [simp]:
   431   "a - 0 = a"
   432   by (simp only: diff_conv_add_uminus minus_zero add_0_right)
   433 
   434 lemma diff_minus_eq_add [simp]:
   435   "a - - b = a + b"
   436   by (simp only: diff_conv_add_uminus minus_minus)
   437 
   438 lemma neg_equal_iff_equal [simp]:
   439   "- a = - b \<longleftrightarrow> a = b"
   440 proof
   441   assume "- a = - b"
   442   hence "- (- a) = - (- b)" by simp
   443   thus "a = b" by simp
   444 next
   445   assume "a = b"
   446   thus "- a = - b" by simp
   447 qed
   448 
   449 lemma neg_equal_0_iff_equal [simp]:
   450   "- a = 0 \<longleftrightarrow> a = 0"
   451   by (subst neg_equal_iff_equal [symmetric]) simp
   452 
   453 lemma neg_0_equal_iff_equal [simp]:
   454   "0 = - a \<longleftrightarrow> 0 = a"
   455   by (subst neg_equal_iff_equal [symmetric]) simp
   456 
   457 text\<open>The next two equations can make the simplifier loop!\<close>
   458 
   459 lemma equation_minus_iff:
   460   "a = - b \<longleftrightarrow> b = - a"
   461 proof -
   462   have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
   463   thus ?thesis by (simp add: eq_commute)
   464 qed
   465 
   466 lemma minus_equation_iff:
   467   "- a = b \<longleftrightarrow> - b = a"
   468 proof -
   469   have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
   470   thus ?thesis by (simp add: eq_commute)
   471 qed
   472 
   473 lemma eq_neg_iff_add_eq_0:
   474   "a = - b \<longleftrightarrow> a + b = 0"
   475 proof
   476   assume "a = - b" then show "a + b = 0" by simp
   477 next
   478   assume "a + b = 0"
   479   moreover have "a + (b + - b) = (a + b) + - b"
   480     by (simp only: add.assoc)
   481   ultimately show "a = - b" by simp
   482 qed
   483 
   484 lemma add_eq_0_iff2:
   485   "a + b = 0 \<longleftrightarrow> a = - b"
   486   by (fact eq_neg_iff_add_eq_0 [symmetric])
   487 
   488 lemma neg_eq_iff_add_eq_0:
   489   "- a = b \<longleftrightarrow> a + b = 0"
   490   by (auto simp add: add_eq_0_iff2)
   491 
   492 lemma add_eq_0_iff:
   493   "a + b = 0 \<longleftrightarrow> b = - a"
   494   by (auto simp add: neg_eq_iff_add_eq_0 [symmetric])
   495 
   496 lemma minus_diff_eq [simp]:
   497   "- (a - b) = b - a"
   498   by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add.assoc minus_add_cancel) simp
   499 
   500 lemma add_diff_eq [algebra_simps, field_simps]:
   501   "a + (b - c) = (a + b) - c"
   502   by (simp only: diff_conv_add_uminus add.assoc)
   503 
   504 lemma diff_add_eq_diff_diff_swap:
   505   "a - (b + c) = a - c - b"
   506   by (simp only: diff_conv_add_uminus add.assoc minus_add)
   507 
   508 lemma diff_eq_eq [algebra_simps, field_simps]:
   509   "a - b = c \<longleftrightarrow> a = c + b"
   510   by auto
   511 
   512 lemma eq_diff_eq [algebra_simps, field_simps]:
   513   "a = c - b \<longleftrightarrow> a + b = c"
   514   by auto
   515 
   516 lemma diff_diff_eq2 [algebra_simps, field_simps]:
   517   "a - (b - c) = (a + c) - b"
   518   by (simp only: diff_conv_add_uminus add.assoc) simp
   519 
   520 lemma diff_eq_diff_eq:
   521   "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
   522   by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
   523 
   524 end
   525 
   526 class ab_group_add = minus + uminus + comm_monoid_add +
   527   assumes ab_left_minus: "- a + a = 0"
   528   assumes ab_diff_conv_add_uminus: "a - b = a + (- b)"
   529 begin
   530 
   531 subclass group_add
   532   proof qed (simp_all add: ab_left_minus ab_diff_conv_add_uminus)
   533 
   534 subclass cancel_comm_monoid_add
   535 proof
   536   fix a b c :: 'a
   537   have "b + a - a = b"
   538     by simp
   539   then show "a + b - a = b"
   540     by (simp add: ac_simps)
   541   show "a - b - c = a - (b + c)"
   542     by (simp add: algebra_simps)
   543 qed
   544 
   545 lemma uminus_add_conv_diff [simp]:
   546   "- a + b = b - a"
   547   by (simp add: add.commute)
   548 
   549 lemma minus_add_distrib [simp]:
   550   "- (a + b) = - a + - b"
   551   by (simp add: algebra_simps)
   552 
   553 lemma diff_add_eq [algebra_simps, field_simps]:
   554   "(a - b) + c = (a + c) - b"
   555   by (simp add: algebra_simps)
   556 
   557 end
   558 
   559 
   560 subsection \<open>(Partially) Ordered Groups\<close>
   561 
   562 text \<open>
   563   The theory of partially ordered groups is taken from the books:
   564   \begin{itemize}
   565   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
   566   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
   567   \end{itemize}
   568   Most of the used notions can also be looked up in
   569   \begin{itemize}
   570   \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
   571   \item \emph{Algebra I} by van der Waerden, Springer.
   572   \end{itemize}
   573 \<close>
   574 
   575 class ordered_ab_semigroup_add = order + ab_semigroup_add +
   576   assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
   577 begin
   578 
   579 lemma add_right_mono:
   580   "a \<le> b \<Longrightarrow> a + c \<le> b + c"
   581 by (simp add: add.commute [of _ c] add_left_mono)
   582 
   583 text \<open>non-strict, in both arguments\<close>
   584 lemma add_mono:
   585   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
   586   apply (erule add_right_mono [THEN order_trans])
   587   apply (simp add: add.commute add_left_mono)
   588   done
   589 
   590 end
   591 
   592 text\<open>Strict monotonicity in both arguments\<close>
   593 class strict_ordered_ab_semigroup_add = ordered_ab_semigroup_add +
   594   assumes add_strict_mono: "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
   595 
   596 class ordered_cancel_ab_semigroup_add =
   597   ordered_ab_semigroup_add + cancel_ab_semigroup_add
   598 begin
   599 
   600 lemma add_strict_left_mono:
   601   "a < b \<Longrightarrow> c + a < c + b"
   602 by (auto simp add: less_le add_left_mono)
   603 
   604 lemma add_strict_right_mono:
   605   "a < b \<Longrightarrow> a + c < b + c"
   606 by (simp add: add.commute [of _ c] add_strict_left_mono)
   607 
   608 subclass strict_ordered_ab_semigroup_add
   609   apply standard
   610   apply (erule add_strict_right_mono [THEN less_trans])
   611   apply (erule add_strict_left_mono)
   612   done
   613 
   614 lemma add_less_le_mono:
   615   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
   616 apply (erule add_strict_right_mono [THEN less_le_trans])
   617 apply (erule add_left_mono)
   618 done
   619 
   620 lemma add_le_less_mono:
   621   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
   622 apply (erule add_right_mono [THEN le_less_trans])
   623 apply (erule add_strict_left_mono)
   624 done
   625 
   626 end
   627 
   628 class ordered_ab_semigroup_add_imp_le = ordered_cancel_ab_semigroup_add +
   629   assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
   630 begin
   631 
   632 lemma add_less_imp_less_left:
   633   assumes less: "c + a < c + b" shows "a < b"
   634 proof -
   635   from less have le: "c + a <= c + b" by (simp add: order_le_less)
   636   have "a <= b"
   637     apply (insert le)
   638     apply (drule add_le_imp_le_left)
   639     by (insert le, drule add_le_imp_le_left, assumption)
   640   moreover have "a \<noteq> b"
   641   proof (rule ccontr)
   642     assume "~(a \<noteq> b)"
   643     then have "a = b" by simp
   644     then have "c + a = c + b" by simp
   645     with less show "False"by simp
   646   qed
   647   ultimately show "a < b" by (simp add: order_le_less)
   648 qed
   649 
   650 lemma add_less_imp_less_right:
   651   "a + c < b + c \<Longrightarrow> a < b"
   652 apply (rule add_less_imp_less_left [of c])
   653 apply (simp add: add.commute)
   654 done
   655 
   656 lemma add_less_cancel_left [simp]:
   657   "c + a < c + b \<longleftrightarrow> a < b"
   658   by (blast intro: add_less_imp_less_left add_strict_left_mono)
   659 
   660 lemma add_less_cancel_right [simp]:
   661   "a + c < b + c \<longleftrightarrow> a < b"
   662   by (blast intro: add_less_imp_less_right add_strict_right_mono)
   663 
   664 lemma add_le_cancel_left [simp]:
   665   "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
   666   by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)
   667 
   668 lemma add_le_cancel_right [simp]:
   669   "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
   670   by (simp add: add.commute [of a c] add.commute [of b c])
   671 
   672 lemma add_le_imp_le_right:
   673   "a + c \<le> b + c \<Longrightarrow> a \<le> b"
   674 by simp
   675 
   676 lemma max_add_distrib_left:
   677   "max x y + z = max (x + z) (y + z)"
   678   unfolding max_def by auto
   679 
   680 lemma min_add_distrib_left:
   681   "min x y + z = min (x + z) (y + z)"
   682   unfolding min_def by auto
   683 
   684 lemma max_add_distrib_right:
   685   "x + max y z = max (x + y) (x + z)"
   686   unfolding max_def by auto
   687 
   688 lemma min_add_distrib_right:
   689   "x + min y z = min (x + y) (x + z)"
   690   unfolding min_def by auto
   691 
   692 end
   693 
   694 subsection \<open>Support for reasoning about signs\<close>
   695 
   696 class ordered_comm_monoid_add = comm_monoid_add + ordered_ab_semigroup_add
   697 begin
   698 
   699 lemma add_nonneg_nonneg [simp]:
   700   "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
   701   using add_mono[of 0 a 0 b] by simp
   702 
   703 lemma add_nonpos_nonpos:
   704   "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b \<le> 0"
   705   using add_mono[of a 0 b 0] by simp
   706 
   707 lemma add_nonneg_eq_0_iff:
   708   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   709   using add_left_mono[of 0 y x] add_right_mono[of 0 x y] by auto
   710 
   711 lemma add_nonpos_eq_0_iff:
   712   "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   713   using add_left_mono[of y 0 x] add_right_mono[of x 0 y] by auto
   714 
   715 lemma add_increasing:
   716   "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
   717   by (insert add_mono [of 0 a b c], simp)
   718 
   719 lemma add_increasing2:
   720   "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
   721   by (simp add: add_increasing add.commute [of a])
   722 
   723 lemma add_decreasing:
   724   "a \<le> 0 \<Longrightarrow> c \<le> b \<Longrightarrow> a + c \<le> b"
   725   using add_mono[of a 0 c b] by simp
   726 
   727 lemma add_decreasing2:
   728   "c \<le> 0 \<Longrightarrow> a \<le> b \<Longrightarrow> a + c \<le> b"
   729   using add_mono[of a b c 0] by simp
   730 
   731 lemma add_pos_nonneg: "0 < a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < a + b"
   732   using less_le_trans[of 0 a "a + b"] by (simp add: add_increasing2)
   733 
   734 lemma add_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a + b"
   735   by (intro add_pos_nonneg less_imp_le)
   736 
   737 lemma add_nonneg_pos: "0 \<le> a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a + b"
   738   using add_pos_nonneg[of b a] by (simp add: add_commute)
   739 
   740 lemma add_neg_nonpos: "a < 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> a + b < 0"
   741   using le_less_trans[of "a + b" a 0] by (simp add: add_decreasing2)
   742 
   743 lemma add_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> a + b < 0"
   744   by (intro add_neg_nonpos less_imp_le)
   745 
   746 lemma add_nonpos_neg: "a \<le> 0 \<Longrightarrow> b < 0 \<Longrightarrow> a + b < 0"
   747   using add_neg_nonpos[of b a] by (simp add: add_commute)
   748 
   749 lemmas add_sign_intros =
   750   add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
   751   add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
   752 
   753 end
   754 
   755 class strict_ordered_comm_monoid_add = comm_monoid_add + strict_ordered_ab_semigroup_add
   756 
   757 class ordered_cancel_comm_monoid_add = ordered_comm_monoid_add + cancel_ab_semigroup_add
   758 begin
   759 
   760 subclass ordered_cancel_ab_semigroup_add ..
   761 subclass strict_ordered_comm_monoid_add ..
   762 
   763 lemma add_strict_increasing:
   764   "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
   765   by (insert add_less_le_mono [of 0 a b c], simp)
   766 
   767 lemma add_strict_increasing2:
   768   "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
   769   by (insert add_le_less_mono [of 0 a b c], simp)
   770 
   771 end
   772 
   773 class ordered_ab_group_add = ab_group_add + ordered_ab_semigroup_add
   774 begin
   775 
   776 subclass ordered_cancel_ab_semigroup_add ..
   777 
   778 subclass ordered_ab_semigroup_add_imp_le
   779 proof
   780   fix a b c :: 'a
   781   assume "c + a \<le> c + b"
   782   hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
   783   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)
   784   thus "a \<le> b" by simp
   785 qed
   786 
   787 subclass ordered_cancel_comm_monoid_add ..
   788 
   789 lemma add_less_same_cancel1 [simp]:
   790   "b + a < b \<longleftrightarrow> a < 0"
   791   using add_less_cancel_left [of _ _ 0] by simp
   792 
   793 lemma add_less_same_cancel2 [simp]:
   794   "a + b < b \<longleftrightarrow> a < 0"
   795   using add_less_cancel_right [of _ _ 0] by simp
   796 
   797 lemma less_add_same_cancel1 [simp]:
   798   "a < a + b \<longleftrightarrow> 0 < b"
   799   using add_less_cancel_left [of _ 0] by simp
   800 
   801 lemma less_add_same_cancel2 [simp]:
   802   "a < b + a \<longleftrightarrow> 0 < b"
   803   using add_less_cancel_right [of 0] by simp
   804 
   805 lemma add_le_same_cancel1 [simp]:
   806   "b + a \<le> b \<longleftrightarrow> a \<le> 0"
   807   using add_le_cancel_left [of _ _ 0] by simp
   808 
   809 lemma add_le_same_cancel2 [simp]:
   810   "a + b \<le> b \<longleftrightarrow> a \<le> 0"
   811   using add_le_cancel_right [of _ _ 0] by simp
   812 
   813 lemma le_add_same_cancel1 [simp]:
   814   "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
   815   using add_le_cancel_left [of _ 0] by simp
   816 
   817 lemma le_add_same_cancel2 [simp]:
   818   "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
   819   using add_le_cancel_right [of 0] by simp
   820 
   821 lemma max_diff_distrib_left:
   822   shows "max x y - z = max (x - z) (y - z)"
   823   using max_add_distrib_left [of x y "- z"] by simp
   824 
   825 lemma min_diff_distrib_left:
   826   shows "min x y - z = min (x - z) (y - z)"
   827   using min_add_distrib_left [of x y "- z"] by simp
   828 
   829 lemma le_imp_neg_le:
   830   assumes "a \<le> b" shows "-b \<le> -a"
   831 proof -
   832   have "-a+a \<le> -a+b" using \<open>a \<le> b\<close> by (rule add_left_mono)
   833   then have "0 \<le> -a+b" by simp
   834   then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono)
   835   then show ?thesis by (simp add: algebra_simps)
   836 qed
   837 
   838 lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
   839 proof
   840   assume "- b \<le> - a"
   841   hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
   842   thus "a\<le>b" by simp
   843 next
   844   assume "a\<le>b"
   845   thus "-b \<le> -a" by (rule le_imp_neg_le)
   846 qed
   847 
   848 lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
   849 by (subst neg_le_iff_le [symmetric], simp)
   850 
   851 lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
   852 by (subst neg_le_iff_le [symmetric], simp)
   853 
   854 lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
   855 by (force simp add: less_le)
   856 
   857 lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
   858 by (subst neg_less_iff_less [symmetric], simp)
   859 
   860 lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
   861 by (subst neg_less_iff_less [symmetric], simp)
   862 
   863 text\<open>The next several equations can make the simplifier loop!\<close>
   864 
   865 lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
   866 proof -
   867   have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
   868   thus ?thesis by simp
   869 qed
   870 
   871 lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
   872 proof -
   873   have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
   874   thus ?thesis by simp
   875 qed
   876 
   877 lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
   878 proof -
   879   have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
   880   have "(- (- a) <= -b) = (b <= - a)"
   881     apply (auto simp only: le_less)
   882     apply (drule mm)
   883     apply (simp_all)
   884     apply (drule mm[simplified], assumption)
   885     done
   886   then show ?thesis by simp
   887 qed
   888 
   889 lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
   890 by (auto simp add: le_less minus_less_iff)
   891 
   892 lemma diff_less_0_iff_less [simp]:
   893   "a - b < 0 \<longleftrightarrow> a < b"
   894 proof -
   895   have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
   896   also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
   897   finally show ?thesis .
   898 qed
   899 
   900 lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
   901 
   902 lemma diff_less_eq [algebra_simps, field_simps]:
   903   "a - b < c \<longleftrightarrow> a < c + b"
   904 apply (subst less_iff_diff_less_0 [of a])
   905 apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
   906 apply (simp add: algebra_simps)
   907 done
   908 
   909 lemma less_diff_eq[algebra_simps, field_simps]:
   910   "a < c - b \<longleftrightarrow> a + b < c"
   911 apply (subst less_iff_diff_less_0 [of "a + b"])
   912 apply (subst less_iff_diff_less_0 [of a])
   913 apply (simp add: algebra_simps)
   914 done
   915 
   916 lemma diff_gt_0_iff_gt [simp]:
   917   "a - b > 0 \<longleftrightarrow> a > b"
   918   by (simp add: less_diff_eq)
   919 
   920 lemma diff_le_eq [algebra_simps, field_simps]:
   921   "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
   922   by (auto simp add: le_less diff_less_eq )
   923 
   924 lemma le_diff_eq [algebra_simps, field_simps]:
   925   "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
   926   by (auto simp add: le_less less_diff_eq)
   927 
   928 lemma diff_le_0_iff_le [simp]:
   929   "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
   930   by (simp add: algebra_simps)
   931 
   932 lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
   933 
   934 lemma diff_ge_0_iff_ge [simp]:
   935   "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
   936   by (simp add: le_diff_eq)
   937 
   938 lemma diff_eq_diff_less:
   939   "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
   940   by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
   941 
   942 lemma diff_eq_diff_less_eq:
   943   "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
   944   by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
   945 
   946 lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d"
   947   by (simp add: field_simps add_mono)
   948 
   949 lemma diff_left_mono: "b \<le> a \<Longrightarrow> c - a \<le> c - b"
   950   by (simp add: field_simps)
   951 
   952 lemma diff_right_mono: "a \<le> b \<Longrightarrow> a - c \<le> b - c"
   953   by (simp add: field_simps)
   954 
   955 lemma diff_strict_mono: "a < b \<Longrightarrow> d < c \<Longrightarrow> a - c < b - d"
   956   by (simp add: field_simps add_strict_mono)
   957 
   958 lemma diff_strict_left_mono: "b < a \<Longrightarrow> c - a < c - b"
   959   by (simp add: field_simps)
   960 
   961 lemma diff_strict_right_mono: "a < b \<Longrightarrow> a - c < b - c"
   962   by (simp add: field_simps)
   963 
   964 end
   965 
   966 ML_file "Tools/group_cancel.ML"
   967 
   968 simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =
   969   \<open>fn phi => fn ss => try Group_Cancel.cancel_add_conv\<close>
   970 
   971 simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
   972   \<open>fn phi => fn ss => try Group_Cancel.cancel_diff_conv\<close>
   973 
   974 simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
   975   \<open>fn phi => fn ss => try Group_Cancel.cancel_eq_conv\<close>
   976 
   977 simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
   978   \<open>fn phi => fn ss => try Group_Cancel.cancel_le_conv\<close>
   979 
   980 simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
   981   \<open>fn phi => fn ss => try Group_Cancel.cancel_less_conv\<close>
   982 
   983 class linordered_ab_semigroup_add =
   984   linorder + ordered_ab_semigroup_add
   985 
   986 class linordered_cancel_ab_semigroup_add =
   987   linorder + ordered_cancel_ab_semigroup_add
   988 begin
   989 
   990 subclass linordered_ab_semigroup_add ..
   991 
   992 subclass ordered_ab_semigroup_add_imp_le
   993 proof
   994   fix a b c :: 'a
   995   assume le: "c + a <= c + b"
   996   show "a <= b"
   997   proof (rule ccontr)
   998     assume w: "~ a \<le> b"
   999     hence "b <= a" by (simp add: linorder_not_le)
  1000     hence le2: "c + b <= c + a" by (rule add_left_mono)
  1001     have "a = b"
  1002       apply (insert le)
  1003       apply (insert le2)
  1004       apply (drule antisym, simp_all)
  1005       done
  1006     with w show False
  1007       by (simp add: linorder_not_le [symmetric])
  1008   qed
  1009 qed
  1010 
  1011 end
  1012 
  1013 class linordered_ab_group_add = linorder + ordered_ab_group_add
  1014 begin
  1015 
  1016 subclass linordered_cancel_ab_semigroup_add ..
  1017 
  1018 lemma equal_neg_zero [simp]:
  1019   "a = - a \<longleftrightarrow> a = 0"
  1020 proof
  1021   assume "a = 0" then show "a = - a" by simp
  1022 next
  1023   assume A: "a = - a" show "a = 0"
  1024   proof (cases "0 \<le> a")
  1025     case True with A have "0 \<le> - a" by auto
  1026     with le_minus_iff have "a \<le> 0" by simp
  1027     with True show ?thesis by (auto intro: order_trans)
  1028   next
  1029     case False then have B: "a \<le> 0" by auto
  1030     with A have "- a \<le> 0" by auto
  1031     with B show ?thesis by (auto intro: order_trans)
  1032   qed
  1033 qed
  1034 
  1035 lemma neg_equal_zero [simp]:
  1036   "- a = a \<longleftrightarrow> a = 0"
  1037   by (auto dest: sym)
  1038 
  1039 lemma neg_less_eq_nonneg [simp]:
  1040   "- a \<le> a \<longleftrightarrow> 0 \<le> a"
  1041 proof
  1042   assume A: "- a \<le> a" show "0 \<le> a"
  1043   proof (rule classical)
  1044     assume "\<not> 0 \<le> a"
  1045     then have "a < 0" by auto
  1046     with A have "- a < 0" by (rule le_less_trans)
  1047     then show ?thesis by auto
  1048   qed
  1049 next
  1050   assume A: "0 \<le> a" show "- a \<le> a"
  1051   proof (rule order_trans)
  1052     show "- a \<le> 0" using A by (simp add: minus_le_iff)
  1053   next
  1054     show "0 \<le> a" using A .
  1055   qed
  1056 qed
  1057 
  1058 lemma neg_less_pos [simp]:
  1059   "- a < a \<longleftrightarrow> 0 < a"
  1060   by (auto simp add: less_le)
  1061 
  1062 lemma less_eq_neg_nonpos [simp]:
  1063   "a \<le> - a \<longleftrightarrow> a \<le> 0"
  1064   using neg_less_eq_nonneg [of "- a"] by simp
  1065 
  1066 lemma less_neg_neg [simp]:
  1067   "a < - a \<longleftrightarrow> a < 0"
  1068   using neg_less_pos [of "- a"] by simp
  1069 
  1070 lemma double_zero [simp]:
  1071   "a + a = 0 \<longleftrightarrow> a = 0"
  1072 proof
  1073   assume assm: "a + a = 0"
  1074   then have a: "- a = a" by (rule minus_unique)
  1075   then show "a = 0" by (simp only: neg_equal_zero)
  1076 qed simp
  1077 
  1078 lemma double_zero_sym [simp]:
  1079   "0 = a + a \<longleftrightarrow> a = 0"
  1080   by (rule, drule sym) simp_all
  1081 
  1082 lemma zero_less_double_add_iff_zero_less_single_add [simp]:
  1083   "0 < a + a \<longleftrightarrow> 0 < a"
  1084 proof
  1085   assume "0 < a + a"
  1086   then have "0 - a < a" by (simp only: diff_less_eq)
  1087   then have "- a < a" by simp
  1088   then show "0 < a" by simp
  1089 next
  1090   assume "0 < a"
  1091   with this have "0 + 0 < a + a"
  1092     by (rule add_strict_mono)
  1093   then show "0 < a + a" by simp
  1094 qed
  1095 
  1096 lemma zero_le_double_add_iff_zero_le_single_add [simp]:
  1097   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
  1098   by (auto simp add: le_less)
  1099 
  1100 lemma double_add_less_zero_iff_single_add_less_zero [simp]:
  1101   "a + a < 0 \<longleftrightarrow> a < 0"
  1102 proof -
  1103   have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
  1104     by (simp add: not_less)
  1105   then show ?thesis by simp
  1106 qed
  1107 
  1108 lemma double_add_le_zero_iff_single_add_le_zero [simp]:
  1109   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
  1110 proof -
  1111   have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
  1112     by (simp add: not_le)
  1113   then show ?thesis by simp
  1114 qed
  1115 
  1116 lemma minus_max_eq_min:
  1117   "- max x y = min (-x) (-y)"
  1118   by (auto simp add: max_def min_def)
  1119 
  1120 lemma minus_min_eq_max:
  1121   "- min x y = max (-x) (-y)"
  1122   by (auto simp add: max_def min_def)
  1123 
  1124 end
  1125 
  1126 class abs =
  1127   fixes abs :: "'a \<Rightarrow> 'a"  ("\<bar>_\<bar>")
  1128 
  1129 class sgn =
  1130   fixes sgn :: "'a \<Rightarrow> 'a"
  1131 
  1132 class abs_if = minus + uminus + ord + zero + abs +
  1133   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
  1134 
  1135 class sgn_if = minus + uminus + zero + one + ord + sgn +
  1136   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
  1137 begin
  1138 
  1139 lemma sgn0 [simp]: "sgn 0 = 0"
  1140   by (simp add:sgn_if)
  1141 
  1142 end
  1143 
  1144 class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
  1145   assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
  1146     and abs_ge_self: "a \<le> \<bar>a\<bar>"
  1147     and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
  1148     and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
  1149     and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
  1150 begin
  1151 
  1152 lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
  1153   unfolding neg_le_0_iff_le by simp
  1154 
  1155 lemma abs_of_nonneg [simp]:
  1156   assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
  1157 proof (rule antisym)
  1158   from nonneg le_imp_neg_le have "- a \<le> 0" by simp
  1159   from this nonneg have "- a \<le> a" by (rule order_trans)
  1160   then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
  1161 qed (rule abs_ge_self)
  1162 
  1163 lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
  1164 by (rule antisym)
  1165    (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
  1166 
  1167 lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
  1168 proof -
  1169   have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
  1170   proof (rule antisym)
  1171     assume zero: "\<bar>a\<bar> = 0"
  1172     with abs_ge_self show "a \<le> 0" by auto
  1173     from zero have "\<bar>-a\<bar> = 0" by simp
  1174     with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
  1175     with neg_le_0_iff_le show "0 \<le> a" by auto
  1176   qed
  1177   then show ?thesis by auto
  1178 qed
  1179 
  1180 lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
  1181 by simp
  1182 
  1183 lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
  1184 proof -
  1185   have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
  1186   thus ?thesis by simp
  1187 qed
  1188 
  1189 lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"
  1190 proof
  1191   assume "\<bar>a\<bar> \<le> 0"
  1192   then have "\<bar>a\<bar> = 0" by (rule antisym) simp
  1193   thus "a = 0" by simp
  1194 next
  1195   assume "a = 0"
  1196   thus "\<bar>a\<bar> \<le> 0" by simp
  1197 qed
  1198 
  1199 lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
  1200 by (simp add: less_le)
  1201 
  1202 lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
  1203 proof -
  1204   have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
  1205   show ?thesis by (simp add: a)
  1206 qed
  1207 
  1208 lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
  1209 proof -
  1210   have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
  1211   then show ?thesis by simp
  1212 qed
  1213 
  1214 lemma abs_minus_commute:
  1215   "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
  1216 proof -
  1217   have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
  1218   also have "... = \<bar>b - a\<bar>" by simp
  1219   finally show ?thesis .
  1220 qed
  1221 
  1222 lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
  1223 by (rule abs_of_nonneg, rule less_imp_le)
  1224 
  1225 lemma abs_of_nonpos [simp]:
  1226   assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
  1227 proof -
  1228   let ?b = "- a"
  1229   have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
  1230   unfolding abs_minus_cancel [of "?b"]
  1231   unfolding neg_le_0_iff_le [of "?b"]
  1232   unfolding minus_minus by (erule abs_of_nonneg)
  1233   then show ?thesis using assms by auto
  1234 qed
  1235 
  1236 lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
  1237 by (rule abs_of_nonpos, rule less_imp_le)
  1238 
  1239 lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
  1240 by (insert abs_ge_self, blast intro: order_trans)
  1241 
  1242 lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
  1243 by (insert abs_le_D1 [of "- a"], simp)
  1244 
  1245 lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
  1246 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
  1247 
  1248 lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
  1249 proof -
  1250   have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
  1251     by (simp add: algebra_simps)
  1252   then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
  1253     by (simp add: abs_triangle_ineq)
  1254   then show ?thesis
  1255     by (simp add: algebra_simps)
  1256 qed
  1257 
  1258 lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
  1259   by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
  1260 
  1261 lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
  1262   by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
  1263 
  1264 lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
  1265 proof -
  1266   have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)
  1267   also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
  1268   finally show ?thesis by simp
  1269 qed
  1270 
  1271 lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
  1272 proof -
  1273   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)
  1274   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
  1275   finally show ?thesis .
  1276 qed
  1277 
  1278 lemma abs_add_abs [simp]:
  1279   "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
  1280 proof (rule antisym)
  1281   show "?L \<ge> ?R" by(rule abs_ge_self)
  1282 next
  1283   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
  1284   also have "\<dots> = ?R" by simp
  1285   finally show "?L \<le> ?R" .
  1286 qed
  1287 
  1288 end
  1289 
  1290 lemma dense_eq0_I:
  1291   fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}"
  1292   shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) ==> x = 0"
  1293   apply (cases "\<bar>x\<bar> = 0", simp)
  1294   apply (simp only: zero_less_abs_iff [symmetric])
  1295   apply (drule dense)
  1296   apply (auto simp add: not_less [symmetric])
  1297   done
  1298 
  1299 hide_fact (open) ab_diff_conv_add_uminus add_0 mult_1 ab_left_minus
  1300 
  1301 lemmas add_0 = add_0_left \<comment> \<open>FIXME duplicate\<close>
  1302 lemmas mult_1 = mult_1_left \<comment> \<open>FIXME duplicate\<close>
  1303 lemmas ab_left_minus = left_minus \<comment> \<open>FIXME duplicate\<close>
  1304 lemmas diff_diff_eq = diff_diff_add \<comment> \<open>FIXME duplicate\<close>
  1305 
  1306 subsection \<open>Canonically ordered monoids\<close>
  1307 
  1308 text \<open>Canonically ordered monoids are never groups.\<close>
  1309 
  1310 class canonically_ordered_monoid_add = comm_monoid_add + order +
  1311   assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"
  1312 begin
  1313 
  1314 lemma zero_le: "0 \<le> x"
  1315   by (auto simp: le_iff_add)
  1316 
  1317 subclass ordered_comm_monoid_add
  1318   proof qed (auto simp: le_iff_add add_ac)
  1319 
  1320 lemma add_eq_0_iff_both_eq_0: "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  1321   by (intro add_nonneg_eq_0_iff zero_le)
  1322 
  1323 end
  1324 
  1325 class ordered_cancel_comm_monoid_diff =
  1326   canonically_ordered_monoid_add + comm_monoid_diff + ordered_ab_semigroup_add_imp_le
  1327 begin
  1328 
  1329 context
  1330   fixes a b
  1331   assumes "a \<le> b"
  1332 begin
  1333 
  1334 lemma add_diff_inverse:
  1335   "a + (b - a) = b"
  1336   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add)
  1337 
  1338 lemma add_diff_assoc:
  1339   "c + (b - a) = c + b - a"
  1340   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.left_commute [of c])
  1341 
  1342 lemma add_diff_assoc2:
  1343   "b - a + c = b + c - a"
  1344   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.assoc)
  1345 
  1346 lemma diff_add_assoc:
  1347   "c + b - a = c + (b - a)"
  1348   using \<open>a \<le> b\<close> by (simp add: add.commute add_diff_assoc)
  1349 
  1350 lemma diff_add_assoc2:
  1351   "b + c - a = b - a + c"
  1352   using \<open>a \<le> b\<close>by (simp add: add.commute add_diff_assoc)
  1353 
  1354 lemma diff_diff_right:
  1355   "c - (b - a) = c + a - b"
  1356   by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute)
  1357 
  1358 lemma diff_add:
  1359   "b - a + a = b"
  1360   by (simp add: add.commute add_diff_inverse)
  1361 
  1362 lemma le_add_diff:
  1363   "c \<le> b + c - a"
  1364   by (auto simp add: add.commute diff_add_assoc2 le_iff_add)
  1365 
  1366 lemma le_imp_diff_is_add:
  1367   "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
  1368   by (auto simp add: add.commute add_diff_inverse)
  1369 
  1370 lemma le_diff_conv2:
  1371   "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
  1372 proof
  1373   assume ?P
  1374   then have "c + a \<le> b - a + a" by (rule add_right_mono)
  1375   then show ?Q by (simp add: add_diff_inverse add.commute)
  1376 next
  1377   assume ?Q
  1378   then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add.commute)
  1379   then show ?P by simp
  1380 qed
  1381 
  1382 end
  1383 
  1384 end
  1385 
  1386 subsection \<open>Tools setup\<close>
  1387 
  1388 lemma add_mono_thms_linordered_semiring:
  1389   fixes i j k :: "'a::ordered_ab_semigroup_add"
  1390   shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
  1391     and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
  1392     and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
  1393     and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
  1394 by (rule add_mono, clarify+)+
  1395 
  1396 lemma add_mono_thms_linordered_field:
  1397   fixes i j k :: "'a::ordered_cancel_ab_semigroup_add"
  1398   shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
  1399     and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
  1400     and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
  1401     and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
  1402     and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
  1403 by (auto intro: add_strict_right_mono add_strict_left_mono
  1404   add_less_le_mono add_le_less_mono add_strict_mono)
  1405 
  1406 code_identifier
  1407   code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
  1408 
  1409 end