src/HOL/Groups.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62378 85ed00c1fe7c child 62608 19f87fa0cfcb permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
     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 begin

   757

   758 lemma pos_add_strict:

   759   shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

   760   using add_strict_mono [of 0 a b c] by simp

   761

   762 end

   763

   764 class ordered_cancel_comm_monoid_add = ordered_comm_monoid_add + cancel_ab_semigroup_add

   765 begin

   766

   767 subclass ordered_cancel_ab_semigroup_add ..

   768 subclass strict_ordered_comm_monoid_add ..

   769

   770 lemma add_strict_increasing:

   771   "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"

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

   773

   774 lemma add_strict_increasing2:

   775   "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"

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

   777

   778 end

   779

   780 class ordered_ab_group_add = ab_group_add + ordered_ab_semigroup_add

   781 begin

   782

   783 subclass ordered_cancel_ab_semigroup_add ..

   784

   785 subclass ordered_ab_semigroup_add_imp_le

   786 proof

   787   fix a b c :: 'a

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

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

   790   hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)

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

   792 qed

   793

   794 subclass ordered_cancel_comm_monoid_add ..

   795

   796 lemma add_less_same_cancel1 [simp]:

   797   "b + a < b \<longleftrightarrow> a < 0"

   798   using add_less_cancel_left [of _ _ 0] by simp

   799

   800 lemma add_less_same_cancel2 [simp]:

   801   "a + b < b \<longleftrightarrow> a < 0"

   802   using add_less_cancel_right [of _ _ 0] by simp

   803

   804 lemma less_add_same_cancel1 [simp]:

   805   "a < a + b \<longleftrightarrow> 0 < b"

   806   using add_less_cancel_left [of _ 0] by simp

   807

   808 lemma less_add_same_cancel2 [simp]:

   809   "a < b + a \<longleftrightarrow> 0 < b"

   810   using add_less_cancel_right [of 0] by simp

   811

   812 lemma add_le_same_cancel1 [simp]:

   813   "b + a \<le> b \<longleftrightarrow> a \<le> 0"

   814   using add_le_cancel_left [of _ _ 0] by simp

   815

   816 lemma add_le_same_cancel2 [simp]:

   817   "a + b \<le> b \<longleftrightarrow> a \<le> 0"

   818   using add_le_cancel_right [of _ _ 0] by simp

   819

   820 lemma le_add_same_cancel1 [simp]:

   821   "a \<le> a + b \<longleftrightarrow> 0 \<le> b"

   822   using add_le_cancel_left [of _ 0] by simp

   823

   824 lemma le_add_same_cancel2 [simp]:

   825   "a \<le> b + a \<longleftrightarrow> 0 \<le> b"

   826   using add_le_cancel_right [of 0] by simp

   827

   828 lemma max_diff_distrib_left:

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

   830   using max_add_distrib_left [of x y "- z"] by simp

   831

   832 lemma min_diff_distrib_left:

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

   834   using min_add_distrib_left [of x y "- z"] by simp

   835

   836 lemma le_imp_neg_le:

   837   assumes "a \<le> b" shows "-b \<le> -a"

   838 proof -

   839   have "-a+a \<le> -a+b" using \<open>a \<le> b\<close> by (rule add_left_mono)

   840   then have "0 \<le> -a+b" by simp

   841   then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono)

   842   then show ?thesis by (simp add: algebra_simps)

   843 qed

   844

   845 lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"

   846 proof

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

   848   hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)

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

   850 next

   851   assume "a\<le>b"

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

   853 qed

   854

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

   856 by (subst neg_le_iff_le [symmetric], simp)

   857

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

   859 by (subst neg_le_iff_le [symmetric], simp)

   860

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

   862 by (force simp add: less_le)

   863

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

   865 by (subst neg_less_iff_less [symmetric], simp)

   866

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

   868 by (subst neg_less_iff_less [symmetric], simp)

   869

   870 text\<open>The next several equations can make the simplifier loop!\<close>

   871

   872 lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"

   873 proof -

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

   875   thus ?thesis by simp

   876 qed

   877

   878 lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"

   879 proof -

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

   881   thus ?thesis by simp

   882 qed

   883

   884 lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"

   885 proof -

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

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

   888     apply (auto simp only: le_less)

   889     apply (drule mm)

   890     apply (simp_all)

   891     apply (drule mm[simplified], assumption)

   892     done

   893   then show ?thesis by simp

   894 qed

   895

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

   897 by (auto simp add: le_less minus_less_iff)

   898

   899 lemma diff_less_0_iff_less [simp]:

   900   "a - b < 0 \<longleftrightarrow> a < b"

   901 proof -

   902   have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp

   903   also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)

   904   finally show ?thesis .

   905 qed

   906

   907 lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]

   908

   909 lemma diff_less_eq [algebra_simps, field_simps]:

   910   "a - b < c \<longleftrightarrow> a < c + b"

   911 apply (subst less_iff_diff_less_0 [of a])

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

   913 apply (simp add: algebra_simps)

   914 done

   915

   916 lemma less_diff_eq[algebra_simps, field_simps]:

   917   "a < c - b \<longleftrightarrow> a + b < c"

   918 apply (subst less_iff_diff_less_0 [of "a + b"])

   919 apply (subst less_iff_diff_less_0 [of a])

   920 apply (simp add: algebra_simps)

   921 done

   922

   923 lemma diff_gt_0_iff_gt [simp]:

   924   "a - b > 0 \<longleftrightarrow> a > b"

   925   by (simp add: less_diff_eq)

   926

   927 lemma diff_le_eq [algebra_simps, field_simps]:

   928   "a - b \<le> c \<longleftrightarrow> a \<le> c + b"

   929   by (auto simp add: le_less diff_less_eq )

   930

   931 lemma le_diff_eq [algebra_simps, field_simps]:

   932   "a \<le> c - b \<longleftrightarrow> a + b \<le> c"

   933   by (auto simp add: le_less less_diff_eq)

   934

   935 lemma diff_le_0_iff_le [simp]:

   936   "a - b \<le> 0 \<longleftrightarrow> a \<le> b"

   937   by (simp add: algebra_simps)

   938

   939 lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]

   940

   941 lemma diff_ge_0_iff_ge [simp]:

   942   "a - b \<ge> 0 \<longleftrightarrow> a \<ge> b"

   943   by (simp add: le_diff_eq)

   944

   945 lemma diff_eq_diff_less:

   946   "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"

   947   by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])

   948

   949 lemma diff_eq_diff_less_eq:

   950   "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"

   951   by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])

   952

   953 lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d"

   954   by (simp add: field_simps add_mono)

   955

   956 lemma diff_left_mono: "b \<le> a \<Longrightarrow> c - a \<le> c - b"

   957   by (simp add: field_simps)

   958

   959 lemma diff_right_mono: "a \<le> b \<Longrightarrow> a - c \<le> b - c"

   960   by (simp add: field_simps)

   961

   962 lemma diff_strict_mono: "a < b \<Longrightarrow> d < c \<Longrightarrow> a - c < b - d"

   963   by (simp add: field_simps add_strict_mono)

   964

   965 lemma diff_strict_left_mono: "b < a \<Longrightarrow> c - a < c - b"

   966   by (simp add: field_simps)

   967

   968 lemma diff_strict_right_mono: "a < b \<Longrightarrow> a - c < b - c"

   969   by (simp add: field_simps)

   970

   971 end

   972

   973 ML_file "Tools/group_cancel.ML"

   974

   975 simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =

   976   \<open>fn phi => fn ss => try Group_Cancel.cancel_add_conv\<close>

   977

   978 simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =

   979   \<open>fn phi => fn ss => try Group_Cancel.cancel_diff_conv\<close>

   980

   981 simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =

   982   \<open>fn phi => fn ss => try Group_Cancel.cancel_eq_conv\<close>

   983

   984 simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =

   985   \<open>fn phi => fn ss => try Group_Cancel.cancel_le_conv\<close>

   986

   987 simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =

   988   \<open>fn phi => fn ss => try Group_Cancel.cancel_less_conv\<close>

   989

   990 class linordered_ab_semigroup_add =

   991   linorder + ordered_ab_semigroup_add

   992

   993 class linordered_cancel_ab_semigroup_add =

   994   linorder + ordered_cancel_ab_semigroup_add

   995 begin

   996

   997 subclass linordered_ab_semigroup_add ..

   998

   999 subclass ordered_ab_semigroup_add_imp_le

  1000 proof

  1001   fix a b c :: 'a

  1002   assume le: "c + a <= c + b"

  1003   show "a <= b"

  1004   proof (rule ccontr)

  1005     assume w: "~ a \<le> b"

  1006     hence "b <= a" by (simp add: linorder_not_le)

  1007     hence le2: "c + b <= c + a" by (rule add_left_mono)

  1008     have "a = b"

  1009       apply (insert le)

  1010       apply (insert le2)

  1011       apply (drule antisym, simp_all)

  1012       done

  1013     with w show False

  1014       by (simp add: linorder_not_le [symmetric])

  1015   qed

  1016 qed

  1017

  1018 end

  1019

  1020 class linordered_ab_group_add = linorder + ordered_ab_group_add

  1021 begin

  1022

  1023 subclass linordered_cancel_ab_semigroup_add ..

  1024

  1025 lemma equal_neg_zero [simp]:

  1026   "a = - a \<longleftrightarrow> a = 0"

  1027 proof

  1028   assume "a = 0" then show "a = - a" by simp

  1029 next

  1030   assume A: "a = - a" show "a = 0"

  1031   proof (cases "0 \<le> a")

  1032     case True with A have "0 \<le> - a" by auto

  1033     with le_minus_iff have "a \<le> 0" by simp

  1034     with True show ?thesis by (auto intro: order_trans)

  1035   next

  1036     case False then have B: "a \<le> 0" by auto

  1037     with A have "- a \<le> 0" by auto

  1038     with B show ?thesis by (auto intro: order_trans)

  1039   qed

  1040 qed

  1041

  1042 lemma neg_equal_zero [simp]:

  1043   "- a = a \<longleftrightarrow> a = 0"

  1044   by (auto dest: sym)

  1045

  1046 lemma neg_less_eq_nonneg [simp]:

  1047   "- a \<le> a \<longleftrightarrow> 0 \<le> a"

  1048 proof

  1049   assume A: "- a \<le> a" show "0 \<le> a"

  1050   proof (rule classical)

  1051     assume "\<not> 0 \<le> a"

  1052     then have "a < 0" by auto

  1053     with A have "- a < 0" by (rule le_less_trans)

  1054     then show ?thesis by auto

  1055   qed

  1056 next

  1057   assume A: "0 \<le> a" show "- a \<le> a"

  1058   proof (rule order_trans)

  1059     show "- a \<le> 0" using A by (simp add: minus_le_iff)

  1060   next

  1061     show "0 \<le> a" using A .

  1062   qed

  1063 qed

  1064

  1065 lemma neg_less_pos [simp]:

  1066   "- a < a \<longleftrightarrow> 0 < a"

  1067   by (auto simp add: less_le)

  1068

  1069 lemma less_eq_neg_nonpos [simp]:

  1070   "a \<le> - a \<longleftrightarrow> a \<le> 0"

  1071   using neg_less_eq_nonneg [of "- a"] by simp

  1072

  1073 lemma less_neg_neg [simp]:

  1074   "a < - a \<longleftrightarrow> a < 0"

  1075   using neg_less_pos [of "- a"] by simp

  1076

  1077 lemma double_zero [simp]:

  1078   "a + a = 0 \<longleftrightarrow> a = 0"

  1079 proof

  1080   assume assm: "a + a = 0"

  1081   then have a: "- a = a" by (rule minus_unique)

  1082   then show "a = 0" by (simp only: neg_equal_zero)

  1083 qed simp

  1084

  1085 lemma double_zero_sym [simp]:

  1086   "0 = a + a \<longleftrightarrow> a = 0"

  1087   by (rule, drule sym) simp_all

  1088

  1089 lemma zero_less_double_add_iff_zero_less_single_add [simp]:

  1090   "0 < a + a \<longleftrightarrow> 0 < a"

  1091 proof

  1092   assume "0 < a + a"

  1093   then have "0 - a < a" by (simp only: diff_less_eq)

  1094   then have "- a < a" by simp

  1095   then show "0 < a" by simp

  1096 next

  1097   assume "0 < a"

  1098   with this have "0 + 0 < a + a"

  1099     by (rule add_strict_mono)

  1100   then show "0 < a + a" by simp

  1101 qed

  1102

  1103 lemma zero_le_double_add_iff_zero_le_single_add [simp]:

  1104   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"

  1105   by (auto simp add: le_less)

  1106

  1107 lemma double_add_less_zero_iff_single_add_less_zero [simp]:

  1108   "a + a < 0 \<longleftrightarrow> a < 0"

  1109 proof -

  1110   have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"

  1111     by (simp add: not_less)

  1112   then show ?thesis by simp

  1113 qed

  1114

  1115 lemma double_add_le_zero_iff_single_add_le_zero [simp]:

  1116   "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"

  1117 proof -

  1118   have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"

  1119     by (simp add: not_le)

  1120   then show ?thesis by simp

  1121 qed

  1122

  1123 lemma minus_max_eq_min:

  1124   "- max x y = min (-x) (-y)"

  1125   by (auto simp add: max_def min_def)

  1126

  1127 lemma minus_min_eq_max:

  1128   "- min x y = max (-x) (-y)"

  1129   by (auto simp add: max_def min_def)

  1130

  1131 end

  1132

  1133 class abs =

  1134   fixes abs :: "'a \<Rightarrow> 'a"  ("\<bar>_\<bar>")

  1135

  1136 class sgn =

  1137   fixes sgn :: "'a \<Rightarrow> 'a"

  1138

  1139 class abs_if = minus + uminus + ord + zero + abs +

  1140   assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"

  1141

  1142 class sgn_if = minus + uminus + zero + one + ord + sgn +

  1143   assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

  1144 begin

  1145

  1146 lemma sgn0 [simp]: "sgn 0 = 0"

  1147   by (simp add:sgn_if)

  1148

  1149 end

  1150

  1151 class ordered_ab_group_add_abs = ordered_ab_group_add + abs +

  1152   assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"

  1153     and abs_ge_self: "a \<le> \<bar>a\<bar>"

  1154     and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"

  1155     and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"

  1156     and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"

  1157 begin

  1158

  1159 lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"

  1160   unfolding neg_le_0_iff_le by simp

  1161

  1162 lemma abs_of_nonneg [simp]:

  1163   assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"

  1164 proof (rule antisym)

  1165   from nonneg le_imp_neg_le have "- a \<le> 0" by simp

  1166   from this nonneg have "- a \<le> a" by (rule order_trans)

  1167   then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)

  1168 qed (rule abs_ge_self)

  1169

  1170 lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"

  1171 by (rule antisym)

  1172    (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])

  1173

  1174 lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"

  1175 proof -

  1176   have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"

  1177   proof (rule antisym)

  1178     assume zero: "\<bar>a\<bar> = 0"

  1179     with abs_ge_self show "a \<le> 0" by auto

  1180     from zero have "\<bar>-a\<bar> = 0" by simp

  1181     with abs_ge_self [of "- a"] have "- a \<le> 0" by auto

  1182     with neg_le_0_iff_le show "0 \<le> a" by auto

  1183   qed

  1184   then show ?thesis by auto

  1185 qed

  1186

  1187 lemma abs_zero [simp]: "\<bar>0\<bar> = 0"

  1188 by simp

  1189

  1190 lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"

  1191 proof -

  1192   have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)

  1193   thus ?thesis by simp

  1194 qed

  1195

  1196 lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"

  1197 proof

  1198   assume "\<bar>a\<bar> \<le> 0"

  1199   then have "\<bar>a\<bar> = 0" by (rule antisym) simp

  1200   thus "a = 0" by simp

  1201 next

  1202   assume "a = 0"

  1203   thus "\<bar>a\<bar> \<le> 0" by simp

  1204 qed

  1205

  1206 lemma abs_le_self_iff [simp]: "\<bar>a\<bar> \<le> a \<longleftrightarrow> 0 \<le> a"

  1207 proof -

  1208   have "\<forall>a. (0::'a) \<le> \<bar>a\<bar>"

  1209     using abs_ge_zero by blast

  1210   then have "\<bar>a\<bar> \<le> a \<Longrightarrow> 0 \<le> a"

  1211     using order.trans by blast

  1212   then show ?thesis

  1213     using abs_of_nonneg eq_refl by blast

  1214 qed

  1215

  1216 lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"

  1217 by (simp add: less_le)

  1218

  1219 lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"

  1220 proof -

  1221   have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto

  1222   show ?thesis by (simp add: a)

  1223 qed

  1224

  1225 lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"

  1226 proof -

  1227   have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)

  1228   then show ?thesis by simp

  1229 qed

  1230

  1231 lemma abs_minus_commute:

  1232   "\<bar>a - b\<bar> = \<bar>b - a\<bar>"

  1233 proof -

  1234   have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)

  1235   also have "... = \<bar>b - a\<bar>" by simp

  1236   finally show ?thesis .

  1237 qed

  1238

  1239 lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"

  1240 by (rule abs_of_nonneg, rule less_imp_le)

  1241

  1242 lemma abs_of_nonpos [simp]:

  1243   assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"

  1244 proof -

  1245   let ?b = "- a"

  1246   have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"

  1247   unfolding abs_minus_cancel [of "?b"]

  1248   unfolding neg_le_0_iff_le [of "?b"]

  1249   unfolding minus_minus by (erule abs_of_nonneg)

  1250   then show ?thesis using assms by auto

  1251 qed

  1252

  1253 lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"

  1254 by (rule abs_of_nonpos, rule less_imp_le)

  1255

  1256 lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"

  1257 by (insert abs_ge_self, blast intro: order_trans)

  1258

  1259 lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"

  1260 by (insert abs_le_D1 [of "- a"], simp)

  1261

  1262 lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"

  1263 by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)

  1264

  1265 lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"

  1266 proof -

  1267   have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"

  1268     by (simp add: algebra_simps)

  1269   then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"

  1270     by (simp add: abs_triangle_ineq)

  1271   then show ?thesis

  1272     by (simp add: algebra_simps)

  1273 qed

  1274

  1275 lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"

  1276   by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)

  1277

  1278 lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"

  1279   by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)

  1280

  1281 lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"

  1282 proof -

  1283   have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)

  1284   also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)

  1285   finally show ?thesis by simp

  1286 qed

  1287

  1288 lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"

  1289 proof -

  1290   have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)

  1291   also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)

  1292   finally show ?thesis .

  1293 qed

  1294

  1295 lemma abs_add_abs [simp]:

  1296   "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")

  1297 proof (rule antisym)

  1298   show "?L \<ge> ?R" by(rule abs_ge_self)

  1299 next

  1300   have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)

  1301   also have "\<dots> = ?R" by simp

  1302   finally show "?L \<le> ?R" .

  1303 qed

  1304

  1305 end

  1306

  1307 lemma dense_eq0_I:

  1308   fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}"

  1309   shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) ==> x = 0"

  1310   apply (cases "\<bar>x\<bar> = 0", simp)

  1311   apply (simp only: zero_less_abs_iff [symmetric])

  1312   apply (drule dense)

  1313   apply (auto simp add: not_less [symmetric])

  1314   done

  1315

  1316 hide_fact (open) ab_diff_conv_add_uminus add_0 mult_1 ab_left_minus

  1317

  1318 lemmas add_0 = add_0_left \<comment> \<open>FIXME duplicate\<close>

  1319 lemmas mult_1 = mult_1_left \<comment> \<open>FIXME duplicate\<close>

  1320 lemmas ab_left_minus = left_minus \<comment> \<open>FIXME duplicate\<close>

  1321 lemmas diff_diff_eq = diff_diff_add \<comment> \<open>FIXME duplicate\<close>

  1322

  1323 subsection \<open>Canonically ordered monoids\<close>

  1324

  1325 text \<open>Canonically ordered monoids are never groups.\<close>

  1326

  1327 class canonically_ordered_monoid_add = comm_monoid_add + order +

  1328   assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"

  1329 begin

  1330

  1331 lemma zero_le[simp]: "0 \<le> x"

  1332   by (auto simp: le_iff_add)

  1333

  1334 lemma le_zero_eq[simp]: "n \<le> 0 \<longleftrightarrow> n = 0"

  1335   by (auto intro: antisym)

  1336

  1337 lemma not_less_zero[simp]: "\<not> n < 0"

  1338   by (auto simp: less_le)

  1339

  1340 lemma zero_less_iff_neq_zero: "(0 < n) \<longleftrightarrow> (n \<noteq> 0)"

  1341   by (auto simp: less_le)

  1342

  1343 text \<open>This theorem is useful with \<open>blast\<close>\<close>

  1344 lemma gr_zeroI: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"

  1345   by (rule zero_less_iff_neq_zero[THEN iffD2]) iprover

  1346

  1347 lemma not_gr_zero[simp]: "(\<not> (0 < n)) \<longleftrightarrow> (n = 0)"

  1348   by (simp add: zero_less_iff_neq_zero)

  1349

  1350 subclass ordered_comm_monoid_add

  1351   proof qed (auto simp: le_iff_add add_ac)

  1352

  1353 lemma add_eq_0_iff_both_eq_0: "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"

  1354   by (intro add_nonneg_eq_0_iff zero_le)

  1355

  1356 lemma gr_implies_not_zero: "m < n \<Longrightarrow> n \<noteq> 0"

  1357   using add_eq_0_iff_both_eq_0[of m] by (auto simp: le_iff_add less_le)

  1358

  1359 lemmas zero_order = zero_le le_zero_eq not_less_zero zero_less_iff_neq_zero not_gr_zero

  1360   -- \<open>This should be attributed with \<open>[iff]\<close>, but then \<open>blast\<close> fails in \<open>Set\<close>.\<close>

  1361

  1362 end

  1363

  1364 class ordered_cancel_comm_monoid_diff =

  1365   canonically_ordered_monoid_add + comm_monoid_diff + ordered_ab_semigroup_add_imp_le

  1366 begin

  1367

  1368 context

  1369   fixes a b

  1370   assumes "a \<le> b"

  1371 begin

  1372

  1373 lemma add_diff_inverse:

  1374   "a + (b - a) = b"

  1375   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add)

  1376

  1377 lemma add_diff_assoc:

  1378   "c + (b - a) = c + b - a"

  1379   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.left_commute [of c])

  1380

  1381 lemma add_diff_assoc2:

  1382   "b - a + c = b + c - a"

  1383   using \<open>a \<le> b\<close> by (auto simp add: le_iff_add add.assoc)

  1384

  1385 lemma diff_add_assoc:

  1386   "c + b - a = c + (b - a)"

  1387   using \<open>a \<le> b\<close> by (simp add: add.commute add_diff_assoc)

  1388

  1389 lemma diff_add_assoc2:

  1390   "b + c - a = b - a + c"

  1391   using \<open>a \<le> b\<close>by (simp add: add.commute add_diff_assoc)

  1392

  1393 lemma diff_diff_right:

  1394   "c - (b - a) = c + a - b"

  1395   by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add.commute)

  1396

  1397 lemma diff_add:

  1398   "b - a + a = b"

  1399   by (simp add: add.commute add_diff_inverse)

  1400

  1401 lemma le_add_diff:

  1402   "c \<le> b + c - a"

  1403   by (auto simp add: add.commute diff_add_assoc2 le_iff_add)

  1404

  1405 lemma le_imp_diff_is_add:

  1406   "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"

  1407   by (auto simp add: add.commute add_diff_inverse)

  1408

  1409 lemma le_diff_conv2:

  1410   "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")

  1411 proof

  1412   assume ?P

  1413   then have "c + a \<le> b - a + a" by (rule add_right_mono)

  1414   then show ?Q by (simp add: add_diff_inverse add.commute)

  1415 next

  1416   assume ?Q

  1417   then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add.commute)

  1418   then show ?P by simp

  1419 qed

  1420

  1421 end

  1422

  1423 end

  1424

  1425 subsection \<open>Tools setup\<close>

  1426

  1427 lemma add_mono_thms_linordered_semiring:

  1428   fixes i j k :: "'a::ordered_ab_semigroup_add"

  1429   shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"

  1430     and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"

  1431     and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"

  1432     and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"

  1433 by (rule add_mono, clarify+)+

  1434

  1435 lemma add_mono_thms_linordered_field:

  1436   fixes i j k :: "'a::ordered_cancel_ab_semigroup_add"

  1437   shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"

  1438     and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"

  1439     and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"

  1440     and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"

  1441     and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"

  1442 by (auto intro: add_strict_right_mono add_strict_left_mono

  1443   add_less_le_mono add_le_less_mono add_strict_mono)

  1444

  1445 code_identifier

  1446   code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1447

  1448 end