src/HOL/Groups.thy
author wenzelm
Mon Dec 09 12:22:23 2013 +0100 (2013-12-09)
changeset 54703 499f92dc6e45
parent 54250 7d2544dd3988
child 54868 bab6cade3cc5
permissions -rw-r--r--
more antiquotations;
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(*  Title:   HOL/Groups.thy
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
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*)
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header {* Groups, also combined with orderings *}
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theory Groups
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imports Orderings
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begin
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subsection {* Fact collections *}
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ML {*
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structure Ac_Simps = Named_Thms
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(
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  val name = @{binding ac_simps}
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  val description = "associativity and commutativity simplification rules"
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)
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*}
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setup Ac_Simps.setup
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text{* The rewrites accumulated in @{text algebra_simps} deal with the
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classical algebraic structures of groups, rings and family. They simplify
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terms by multiplying everything out (in case of a ring) and bringing sums and
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products into a canonical form (by ordered rewriting). As a result it decides
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group and ring equalities but also helps with inequalities.
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Of course it also works for fields, but it knows nothing about multiplicative
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inverses or division. This is catered for by @{text field_simps}. *}
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ML {*
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structure Algebra_Simps = Named_Thms
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(
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  val name = @{binding algebra_simps}
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  val description = "algebra simplification rules"
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)
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*}
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setup Algebra_Simps.setup
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text{* Lemmas @{text field_simps} multiply with denominators in (in)equations
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if they can be proved to be non-zero (for equations) or positive/negative
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(for inequations). Can be too aggressive and is therefore separate from the
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more benign @{text algebra_simps}. *}
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ML {*
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structure Field_Simps = Named_Thms
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(
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  val name = @{binding field_simps}
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  val description = "algebra simplification rules for fields"
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)
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*}
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setup Field_Simps.setup
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subsection {* Abstract structures *}
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text {*
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  These locales provide basic structures for interpretation into
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  bigger structures;  extensions require careful thinking, otherwise
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  undesired effects may occur due to interpretation.
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*}
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locale semigroup =
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  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
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  assumes assoc [ac_simps]: "a * b * c = a * (b * c)"
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locale abel_semigroup = semigroup +
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  assumes commute [ac_simps]: "a * b = b * a"
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begin
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lemma left_commute [ac_simps]:
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  "b * (a * c) = a * (b * c)"
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proof -
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  have "(b * a) * c = (a * b) * c"
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    by (simp only: commute)
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  then show ?thesis
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    by (simp only: assoc)
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qed
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end
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locale monoid = semigroup +
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  fixes z :: 'a ("1")
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  assumes left_neutral [simp]: "1 * a = a"
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  assumes right_neutral [simp]: "a * 1 = a"
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locale comm_monoid = abel_semigroup +
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  fixes z :: 'a ("1")
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  assumes comm_neutral: "a * 1 = a"
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sublocale comm_monoid < monoid
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  by default (simp_all add: commute comm_neutral)
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subsection {* Generic operations *}
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class zero = 
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  fixes zero :: 'a  ("0")
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class one =
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  fixes one  :: 'a  ("1")
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hide_const (open) zero one
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lemma Let_0 [simp]: "Let 0 f = f 0"
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  unfolding Let_def ..
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lemma Let_1 [simp]: "Let 1 f = f 1"
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  unfolding Let_def ..
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setup {*
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  Reorient_Proc.add
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    (fn Const(@{const_name Groups.zero}, _) => true
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      | Const(@{const_name Groups.one}, _) => true
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      | _ => false)
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*}
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simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc
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simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc
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typed_print_translation {*
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  let
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    fun tr' c = (c, fn ctxt => fn T => fn ts =>
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      if null ts andalso Printer.type_emphasis ctxt T then
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        Syntax.const @{syntax_const "_constrain"} $ Syntax.const c $
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          Syntax_Phases.term_of_typ ctxt T
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      else raise Match);
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  in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end;
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*} -- {* show types that are presumably too general *}
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class plus =
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  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)
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class minus =
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  fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)
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class uminus =
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  fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)
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class times =
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  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)
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subsection {* Semigroups and Monoids *}
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class semigroup_add = plus +
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  assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
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sublocale semigroup_add < add!: semigroup plus
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  by default (fact add_assoc)
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class ab_semigroup_add = semigroup_add +
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  assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
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sublocale ab_semigroup_add < add!: abel_semigroup plus
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  by default (fact add_commute)
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context ab_semigroup_add
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begin
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lemmas add_left_commute [algebra_simps, field_simps] = add.left_commute
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theorems add_ac = add_assoc add_commute add_left_commute
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end
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theorems add_ac = add_assoc add_commute add_left_commute
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class semigroup_mult = times +
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  assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
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sublocale semigroup_mult < mult!: semigroup times
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  by default (fact mult_assoc)
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class ab_semigroup_mult = semigroup_mult +
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  assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
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sublocale ab_semigroup_mult < mult!: abel_semigroup times
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  by default (fact mult_commute)
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context ab_semigroup_mult
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begin
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lemmas mult_left_commute [algebra_simps, field_simps] = mult.left_commute
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theorems mult_ac = mult_assoc mult_commute mult_left_commute
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end
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theorems mult_ac = mult_assoc mult_commute mult_left_commute
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class monoid_add = zero + semigroup_add +
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  assumes add_0_left: "0 + a = a"
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    and add_0_right: "a + 0 = a"
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sublocale monoid_add < add!: monoid plus 0
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  by default (fact add_0_left add_0_right)+
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lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
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by (rule eq_commute)
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class comm_monoid_add = zero + ab_semigroup_add +
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  assumes add_0: "0 + a = a"
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sublocale comm_monoid_add < add!: comm_monoid plus 0
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  by default (insert add_0, simp add: ac_simps)
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subclass (in comm_monoid_add) monoid_add
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  by default (fact add.left_neutral add.right_neutral)+
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class comm_monoid_diff = comm_monoid_add + minus +
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  assumes diff_zero [simp]: "a - 0 = a"
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    and zero_diff [simp]: "0 - a = 0"
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    and add_diff_cancel_left [simp]: "(c + a) - (c + b) = a - b"
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    and diff_diff_add: "a - b - c = a - (b + c)"
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begin
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lemma add_diff_cancel_right [simp]:
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  "(a + c) - (b + c) = a - b"
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  using add_diff_cancel_left [symmetric] by (simp add: add.commute)
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lemma add_diff_cancel_left' [simp]:
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  "(b + a) - b = a"
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proof -
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  have "(b + a) - (b + 0) = a" by (simp only: add_diff_cancel_left diff_zero)
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  then show ?thesis by simp
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qed
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lemma add_diff_cancel_right' [simp]:
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  "(a + b) - b = a"
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  using add_diff_cancel_left' [symmetric] by (simp add: add.commute)
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lemma diff_add_zero [simp]:
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  "a - (a + b) = 0"
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proof -
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  have "a - (a + b) = (a + 0) - (a + b)" by simp
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  also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff)
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  finally show ?thesis .
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qed
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lemma diff_cancel [simp]:
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  "a - a = 0"
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proof -
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  have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero)
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  then show ?thesis by simp
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qed
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lemma diff_right_commute:
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  "a - c - b = a - b - c"
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  by (simp add: diff_diff_add add.commute)
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lemma add_implies_diff:
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  assumes "c + b = a"
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  shows "c = a - b"
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proof -
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  from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute)
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  then show "c = a - b" by simp
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qed
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end
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class monoid_mult = one + semigroup_mult +
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  assumes mult_1_left: "1 * a  = a"
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    and mult_1_right: "a * 1 = a"
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sublocale monoid_mult < mult!: monoid times 1
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  by default (fact mult_1_left mult_1_right)+
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lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
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by (rule eq_commute)
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class comm_monoid_mult = one + ab_semigroup_mult +
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  assumes mult_1: "1 * a = a"
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sublocale comm_monoid_mult < mult!: comm_monoid times 1
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  by default (insert mult_1, simp add: ac_simps)
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subclass (in comm_monoid_mult) monoid_mult
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  by default (fact mult.left_neutral mult.right_neutral)+
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class cancel_semigroup_add = semigroup_add +
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  assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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  assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
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begin
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lemma add_left_cancel [simp]:
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  "a + b = a + c \<longleftrightarrow> b = c"
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by (blast dest: add_left_imp_eq)
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lemma add_right_cancel [simp]:
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  "b + a = c + a \<longleftrightarrow> b = c"
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by (blast dest: add_right_imp_eq)
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end
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class cancel_ab_semigroup_add = ab_semigroup_add +
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  assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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begin
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subclass cancel_semigroup_add
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proof
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  fix a b c :: 'a
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  assume "a + b = a + c" 
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  then show "b = c" by (rule add_imp_eq)
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next
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  fix a b c :: 'a
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  assume "b + a = c + a"
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  then have "a + b = a + c" by (simp only: add_commute)
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  then show "b = c" by (rule add_imp_eq)
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qed
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end
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class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
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subsection {* Groups *}
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class group_add = minus + uminus + monoid_add +
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  assumes left_minus [simp]: "- a + a = 0"
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  assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
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begin
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lemma diff_conv_add_uminus:
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  "a - b = a + (- b)"
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  by simp
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lemma minus_unique:
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  assumes "a + b = 0" shows "- a = b"
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proof -
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  have "- a = - a + (a + b)" using assms by simp
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  also have "\<dots> = b" by (simp add: add_assoc [symmetric])
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  finally show ?thesis .
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qed
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lemma minus_zero [simp]: "- 0 = 0"
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proof -
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  have "0 + 0 = 0" by (rule add_0_right)
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  thus "- 0 = 0" by (rule minus_unique)
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qed
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lemma minus_minus [simp]: "- (- a) = a"
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proof -
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  have "- a + a = 0" by (rule left_minus)
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  thus "- (- a) = a" by (rule minus_unique)
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qed
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lemma right_minus: "a + - a = 0"
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proof -
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  have "a + - a = - (- a) + - a" by simp
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  also have "\<dots> = 0" by (rule left_minus)
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  finally show ?thesis .
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qed
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lemma diff_self [simp]:
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  "a - a = 0"
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  using right_minus [of a] by simp
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subclass cancel_semigroup_add
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proof
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  fix a b c :: 'a
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   365
  assume "a + b = a + c"
haftmann@40368
   366
  then have "- a + a + b = - a + a + c"
haftmann@40368
   367
    unfolding add_assoc by simp
haftmann@40368
   368
  then show "b = c" by simp
haftmann@40368
   369
next
haftmann@40368
   370
  fix a b c :: 'a
haftmann@40368
   371
  assume "b + a = c + a"
haftmann@40368
   372
  then have "b + a + - a = c + a  + - a" by simp
haftmann@40368
   373
  then show "b = c" unfolding add_assoc by simp
haftmann@40368
   374
qed
haftmann@40368
   375
haftmann@54230
   376
lemma minus_add_cancel [simp]:
haftmann@54230
   377
  "- a + (a + b) = b"
haftmann@54230
   378
  by (simp add: add_assoc [symmetric])
haftmann@54230
   379
haftmann@54230
   380
lemma add_minus_cancel [simp]:
haftmann@54230
   381
  "a + (- a + b) = b"
haftmann@54230
   382
  by (simp add: add_assoc [symmetric])
huffman@34147
   383
haftmann@54230
   384
lemma diff_add_cancel [simp]:
haftmann@54230
   385
  "a - b + b = a"
haftmann@54230
   386
  by (simp only: diff_conv_add_uminus add_assoc) simp
huffman@34147
   387
haftmann@54230
   388
lemma add_diff_cancel [simp]:
haftmann@54230
   389
  "a + b - b = a"
haftmann@54230
   390
  by (simp only: diff_conv_add_uminus add_assoc) simp
haftmann@54230
   391
haftmann@54230
   392
lemma minus_add:
haftmann@54230
   393
  "- (a + b) = - b + - a"
huffman@34147
   394
proof -
huffman@34147
   395
  have "(a + b) + (- b + - a) = 0"
haftmann@54230
   396
    by (simp only: add_assoc add_minus_cancel) simp
haftmann@54230
   397
  then show "- (a + b) = - b + - a"
huffman@34147
   398
    by (rule minus_unique)
huffman@34147
   399
qed
huffman@34147
   400
haftmann@54230
   401
lemma right_minus_eq [simp]:
haftmann@54230
   402
  "a - b = 0 \<longleftrightarrow> a = b"
obua@14738
   403
proof
nipkow@23085
   404
  assume "a - b = 0"
haftmann@54230
   405
  have "a = (a - b) + b" by (simp add: add_assoc)
nipkow@23085
   406
  also have "\<dots> = b" using `a - b = 0` by simp
nipkow@23085
   407
  finally show "a = b" .
obua@14738
   408
next
haftmann@54230
   409
  assume "a = b" thus "a - b = 0" by simp
obua@14738
   410
qed
obua@14738
   411
haftmann@54230
   412
lemma eq_iff_diff_eq_0:
haftmann@54230
   413
  "a = b \<longleftrightarrow> a - b = 0"
haftmann@54230
   414
  by (fact right_minus_eq [symmetric])
obua@14738
   415
haftmann@54230
   416
lemma diff_0 [simp]:
haftmann@54230
   417
  "0 - a = - a"
haftmann@54230
   418
  by (simp only: diff_conv_add_uminus add_0_left)
obua@14738
   419
haftmann@54230
   420
lemma diff_0_right [simp]:
haftmann@54230
   421
  "a - 0 = a" 
haftmann@54230
   422
  by (simp only: diff_conv_add_uminus minus_zero add_0_right)
obua@14738
   423
haftmann@54230
   424
lemma diff_minus_eq_add [simp]:
haftmann@54230
   425
  "a - - b = a + b"
haftmann@54230
   426
  by (simp only: diff_conv_add_uminus minus_minus)
obua@14738
   427
haftmann@25062
   428
lemma neg_equal_iff_equal [simp]:
haftmann@25062
   429
  "- a = - b \<longleftrightarrow> a = b" 
obua@14738
   430
proof 
obua@14738
   431
  assume "- a = - b"
nipkow@29667
   432
  hence "- (- a) = - (- b)" by simp
haftmann@25062
   433
  thus "a = b" by simp
obua@14738
   434
next
haftmann@25062
   435
  assume "a = b"
haftmann@25062
   436
  thus "- a = - b" by simp
obua@14738
   437
qed
obua@14738
   438
haftmann@25062
   439
lemma neg_equal_0_iff_equal [simp]:
haftmann@25062
   440
  "- a = 0 \<longleftrightarrow> a = 0"
haftmann@54230
   441
  by (subst neg_equal_iff_equal [symmetric]) simp
obua@14738
   442
haftmann@25062
   443
lemma neg_0_equal_iff_equal [simp]:
haftmann@25062
   444
  "0 = - a \<longleftrightarrow> 0 = a"
haftmann@54230
   445
  by (subst neg_equal_iff_equal [symmetric]) simp
obua@14738
   446
obua@14738
   447
text{*The next two equations can make the simplifier loop!*}
obua@14738
   448
haftmann@25062
   449
lemma equation_minus_iff:
haftmann@25062
   450
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   451
proof -
haftmann@25062
   452
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   453
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   454
qed
haftmann@25062
   455
haftmann@25062
   456
lemma minus_equation_iff:
haftmann@25062
   457
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   458
proof -
haftmann@25062
   459
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   460
  thus ?thesis by (simp add: eq_commute)
obua@14738
   461
qed
obua@14738
   462
haftmann@54230
   463
lemma eq_neg_iff_add_eq_0:
haftmann@54230
   464
  "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   465
proof
huffman@29914
   466
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   467
next
huffman@29914
   468
  assume "a + b = 0"
huffman@29914
   469
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   470
    by (simp only: add_assoc)
huffman@29914
   471
  ultimately show "a = - b" by simp
huffman@29914
   472
qed
huffman@29914
   473
haftmann@54230
   474
lemma add_eq_0_iff2:
haftmann@54230
   475
  "a + b = 0 \<longleftrightarrow> a = - b"
haftmann@54230
   476
  by (fact eq_neg_iff_add_eq_0 [symmetric])
haftmann@54230
   477
haftmann@54230
   478
lemma neg_eq_iff_add_eq_0:
haftmann@54230
   479
  "- a = b \<longleftrightarrow> a + b = 0"
haftmann@54230
   480
  by (auto simp add: add_eq_0_iff2)
huffman@44348
   481
haftmann@54230
   482
lemma add_eq_0_iff:
haftmann@54230
   483
  "a + b = 0 \<longleftrightarrow> b = - a"
haftmann@54230
   484
  by (auto simp add: neg_eq_iff_add_eq_0 [symmetric])
huffman@45548
   485
haftmann@54230
   486
lemma minus_diff_eq [simp]:
haftmann@54230
   487
  "- (a - b) = b - a"
haftmann@54230
   488
  by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add_assoc minus_add_cancel) simp
huffman@45548
   489
haftmann@54230
   490
lemma add_diff_eq [algebra_simps, field_simps]:
haftmann@54230
   491
  "a + (b - c) = (a + b) - c"
haftmann@54230
   492
  by (simp only: diff_conv_add_uminus add_assoc)
huffman@45548
   493
haftmann@54230
   494
lemma diff_add_eq_diff_diff_swap:
haftmann@54230
   495
  "a - (b + c) = a - c - b"
haftmann@54230
   496
  by (simp only: diff_conv_add_uminus add_assoc minus_add)
huffman@45548
   497
haftmann@54230
   498
lemma diff_eq_eq [algebra_simps, field_simps]:
haftmann@54230
   499
  "a - b = c \<longleftrightarrow> a = c + b"
haftmann@54230
   500
  by auto
huffman@45548
   501
haftmann@54230
   502
lemma eq_diff_eq [algebra_simps, field_simps]:
haftmann@54230
   503
  "a = c - b \<longleftrightarrow> a + b = c"
haftmann@54230
   504
  by auto
haftmann@54230
   505
haftmann@54230
   506
lemma diff_diff_eq2 [algebra_simps, field_simps]:
haftmann@54230
   507
  "a - (b - c) = (a + c) - b"
haftmann@54230
   508
  by (simp only: diff_conv_add_uminus add_assoc) simp
huffman@45548
   509
huffman@45548
   510
lemma diff_eq_diff_eq:
huffman@45548
   511
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
haftmann@54230
   512
  by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
huffman@45548
   513
haftmann@25062
   514
end
haftmann@25062
   515
haftmann@25762
   516
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   517
  assumes ab_left_minus: "- a + a = 0"
haftmann@54230
   518
  assumes ab_add_uminus_conv_diff: "a - b = a + (- b)"
haftmann@25267
   519
begin
haftmann@25062
   520
haftmann@25267
   521
subclass group_add
haftmann@54230
   522
  proof qed (simp_all add: ab_left_minus ab_add_uminus_conv_diff)
haftmann@25062
   523
huffman@29904
   524
subclass cancel_comm_monoid_add
haftmann@28823
   525
proof
haftmann@25062
   526
  fix a b c :: 'a
haftmann@25062
   527
  assume "a + b = a + c"
haftmann@25062
   528
  then have "- a + a + b = - a + a + c"
haftmann@54230
   529
    by (simp only: add_assoc)
haftmann@25062
   530
  then show "b = c" by simp
haftmann@25062
   531
qed
haftmann@25062
   532
haftmann@54230
   533
lemma uminus_add_conv_diff [simp]:
haftmann@25062
   534
  "- a + b = b - a"
haftmann@54230
   535
  by (simp add: add_commute)
haftmann@25062
   536
haftmann@25062
   537
lemma minus_add_distrib [simp]:
haftmann@25062
   538
  "- (a + b) = - a + - b"
haftmann@54230
   539
  by (simp add: algebra_simps)
haftmann@25062
   540
haftmann@54230
   541
lemma diff_add_eq [algebra_simps, field_simps]:
haftmann@54230
   542
  "(a - b) + c = (a + c) - b"
haftmann@54230
   543
  by (simp add: algebra_simps)
haftmann@25077
   544
haftmann@54230
   545
lemma diff_diff_eq [algebra_simps, field_simps]:
haftmann@54230
   546
  "(a - b) - c = a - (b + c)"
haftmann@54230
   547
  by (simp add: algebra_simps)
huffman@30629
   548
haftmann@54230
   549
lemma diff_add_eq_diff_diff:
haftmann@54230
   550
  "a - (b + c) = a - b - c"
haftmann@54230
   551
  using diff_add_eq_diff_diff_swap [of a c b] by (simp add: add.commute)
haftmann@54230
   552
haftmann@54230
   553
lemma add_diff_cancel_left [simp]:
haftmann@54230
   554
  "(c + a) - (c + b) = a - b"
haftmann@54230
   555
  by (simp add: algebra_simps)
huffman@48556
   556
haftmann@25062
   557
end
obua@14738
   558
haftmann@37884
   559
obua@14738
   560
subsection {* (Partially) Ordered Groups *} 
obua@14738
   561
haftmann@35301
   562
text {*
haftmann@35301
   563
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   564
  \begin{itemize}
haftmann@35301
   565
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   566
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   567
  \end{itemize}
haftmann@35301
   568
  Most of the used notions can also be looked up in 
haftmann@35301
   569
  \begin{itemize}
wenzelm@54703
   570
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35301
   571
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   572
  \end{itemize}
haftmann@35301
   573
*}
haftmann@35301
   574
haftmann@35028
   575
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   576
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   577
begin
haftmann@24380
   578
haftmann@25062
   579
lemma add_right_mono:
haftmann@25062
   580
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   581
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   582
obua@14738
   583
text {* non-strict, in both arguments *}
obua@14738
   584
lemma add_mono:
haftmann@25062
   585
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   586
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   587
  apply (simp add: add_commute add_left_mono)
obua@14738
   588
  done
obua@14738
   589
haftmann@25062
   590
end
haftmann@25062
   591
haftmann@35028
   592
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   593
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   594
begin
haftmann@25062
   595
obua@14738
   596
lemma add_strict_left_mono:
haftmann@25062
   597
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   598
by (auto simp add: less_le add_left_mono)
obua@14738
   599
obua@14738
   600
lemma add_strict_right_mono:
haftmann@25062
   601
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   602
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   603
obua@14738
   604
text{*Strict monotonicity in both arguments*}
haftmann@25062
   605
lemma add_strict_mono:
haftmann@25062
   606
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   607
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   608
apply (erule add_strict_left_mono)
obua@14738
   609
done
obua@14738
   610
obua@14738
   611
lemma add_less_le_mono:
haftmann@25062
   612
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   613
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   614
apply (erule add_left_mono)
obua@14738
   615
done
obua@14738
   616
obua@14738
   617
lemma add_le_less_mono:
haftmann@25062
   618
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   619
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   620
apply (erule add_strict_left_mono) 
obua@14738
   621
done
obua@14738
   622
haftmann@25062
   623
end
haftmann@25062
   624
haftmann@35028
   625
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   626
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   627
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   628
begin
haftmann@25062
   629
obua@14738
   630
lemma add_less_imp_less_left:
nipkow@29667
   631
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   632
proof -
obua@14738
   633
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   634
  have "a <= b" 
obua@14738
   635
    apply (insert le)
obua@14738
   636
    apply (drule add_le_imp_le_left)
obua@14738
   637
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   638
  moreover have "a \<noteq> b"
obua@14738
   639
  proof (rule ccontr)
obua@14738
   640
    assume "~(a \<noteq> b)"
obua@14738
   641
    then have "a = b" by simp
obua@14738
   642
    then have "c + a = c + b" by simp
obua@14738
   643
    with less show "False"by simp
obua@14738
   644
  qed
obua@14738
   645
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   646
qed
obua@14738
   647
obua@14738
   648
lemma add_less_imp_less_right:
haftmann@25062
   649
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   650
apply (rule add_less_imp_less_left [of c])
obua@14738
   651
apply (simp add: add_commute)  
obua@14738
   652
done
obua@14738
   653
obua@14738
   654
lemma add_less_cancel_left [simp]:
haftmann@25062
   655
  "c + a < c + b \<longleftrightarrow> a < b"
haftmann@54230
   656
  by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   657
obua@14738
   658
lemma add_less_cancel_right [simp]:
haftmann@25062
   659
  "a + c < b + c \<longleftrightarrow> a < b"
haftmann@54230
   660
  by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   661
obua@14738
   662
lemma add_le_cancel_left [simp]:
haftmann@25062
   663
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
haftmann@54230
   664
  by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   665
obua@14738
   666
lemma add_le_cancel_right [simp]:
haftmann@25062
   667
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
haftmann@54230
   668
  by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   669
obua@14738
   670
lemma add_le_imp_le_right:
haftmann@25062
   671
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   672
by simp
haftmann@25062
   673
haftmann@25077
   674
lemma max_add_distrib_left:
haftmann@25077
   675
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   676
  unfolding max_def by auto
haftmann@25077
   677
haftmann@25077
   678
lemma min_add_distrib_left:
haftmann@25077
   679
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   680
  unfolding min_def by auto
haftmann@25077
   681
huffman@44848
   682
lemma max_add_distrib_right:
huffman@44848
   683
  "x + max y z = max (x + y) (x + z)"
huffman@44848
   684
  unfolding max_def by auto
huffman@44848
   685
huffman@44848
   686
lemma min_add_distrib_right:
huffman@44848
   687
  "x + min y z = min (x + y) (x + z)"
huffman@44848
   688
  unfolding min_def by auto
huffman@44848
   689
haftmann@25062
   690
end
haftmann@25062
   691
haftmann@52289
   692
class ordered_cancel_comm_monoid_diff = comm_monoid_diff + ordered_ab_semigroup_add_imp_le +
haftmann@52289
   693
  assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"
haftmann@52289
   694
begin
haftmann@52289
   695
haftmann@52289
   696
context
haftmann@52289
   697
  fixes a b
haftmann@52289
   698
  assumes "a \<le> b"
haftmann@52289
   699
begin
haftmann@52289
   700
haftmann@52289
   701
lemma add_diff_inverse:
haftmann@52289
   702
  "a + (b - a) = b"
haftmann@52289
   703
  using `a \<le> b` by (auto simp add: le_iff_add)
haftmann@52289
   704
haftmann@52289
   705
lemma add_diff_assoc:
haftmann@52289
   706
  "c + (b - a) = c + b - a"
haftmann@52289
   707
  using `a \<le> b` by (auto simp add: le_iff_add add_left_commute [of c])
haftmann@52289
   708
haftmann@52289
   709
lemma add_diff_assoc2:
haftmann@52289
   710
  "b - a + c = b + c - a"
haftmann@52289
   711
  using `a \<le> b` by (auto simp add: le_iff_add add_assoc)
haftmann@52289
   712
haftmann@52289
   713
lemma diff_add_assoc:
haftmann@52289
   714
  "c + b - a = c + (b - a)"
haftmann@52289
   715
  using `a \<le> b` by (simp add: add_commute add_diff_assoc)
haftmann@52289
   716
haftmann@52289
   717
lemma diff_add_assoc2:
haftmann@52289
   718
  "b + c - a = b - a + c"
haftmann@52289
   719
  using `a \<le> b`by (simp add: add_commute add_diff_assoc)
haftmann@52289
   720
haftmann@52289
   721
lemma diff_diff_right:
haftmann@52289
   722
  "c - (b - a) = c + a - b"
haftmann@52289
   723
  by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add_commute)
haftmann@52289
   724
haftmann@52289
   725
lemma diff_add:
haftmann@52289
   726
  "b - a + a = b"
haftmann@52289
   727
  by (simp add: add_commute add_diff_inverse)
haftmann@52289
   728
haftmann@52289
   729
lemma le_add_diff:
haftmann@52289
   730
  "c \<le> b + c - a"
haftmann@52289
   731
  by (auto simp add: add_commute diff_add_assoc2 le_iff_add)
haftmann@52289
   732
haftmann@52289
   733
lemma le_imp_diff_is_add:
haftmann@52289
   734
  "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
haftmann@52289
   735
  by (auto simp add: add_commute add_diff_inverse)
haftmann@52289
   736
haftmann@52289
   737
lemma le_diff_conv2:
haftmann@52289
   738
  "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
haftmann@52289
   739
proof
haftmann@52289
   740
  assume ?P
haftmann@52289
   741
  then have "c + a \<le> b - a + a" by (rule add_right_mono)
haftmann@52289
   742
  then show ?Q by (simp add: add_diff_inverse add_commute)
haftmann@52289
   743
next
haftmann@52289
   744
  assume ?Q
haftmann@52289
   745
  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add_commute)
haftmann@52289
   746
  then show ?P by simp
haftmann@52289
   747
qed
haftmann@52289
   748
haftmann@52289
   749
end
haftmann@52289
   750
haftmann@52289
   751
end
haftmann@52289
   752
haftmann@52289
   753
haftmann@25303
   754
subsection {* Support for reasoning about signs *}
haftmann@25303
   755
haftmann@35028
   756
class ordered_comm_monoid_add =
haftmann@35028
   757
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   758
begin
haftmann@25303
   759
haftmann@25303
   760
lemma add_pos_nonneg:
nipkow@29667
   761
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   762
proof -
haftmann@25303
   763
  have "0 + 0 < a + b" 
haftmann@25303
   764
    using assms by (rule add_less_le_mono)
haftmann@25303
   765
  then show ?thesis by simp
haftmann@25303
   766
qed
haftmann@25303
   767
haftmann@25303
   768
lemma add_pos_pos:
nipkow@29667
   769
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   770
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   771
haftmann@25303
   772
lemma add_nonneg_pos:
nipkow@29667
   773
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   774
proof -
haftmann@25303
   775
  have "0 + 0 < a + b" 
haftmann@25303
   776
    using assms by (rule add_le_less_mono)
haftmann@25303
   777
  then show ?thesis by simp
haftmann@25303
   778
qed
haftmann@25303
   779
huffman@36977
   780
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   781
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   782
proof -
haftmann@25303
   783
  have "0 + 0 \<le> a + b" 
haftmann@25303
   784
    using assms by (rule add_mono)
haftmann@25303
   785
  then show ?thesis by simp
haftmann@25303
   786
qed
haftmann@25303
   787
huffman@30691
   788
lemma add_neg_nonpos:
nipkow@29667
   789
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   790
proof -
haftmann@25303
   791
  have "a + b < 0 + 0"
haftmann@25303
   792
    using assms by (rule add_less_le_mono)
haftmann@25303
   793
  then show ?thesis by simp
haftmann@25303
   794
qed
haftmann@25303
   795
haftmann@25303
   796
lemma add_neg_neg: 
nipkow@29667
   797
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   798
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   799
haftmann@25303
   800
lemma add_nonpos_neg:
nipkow@29667
   801
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   802
proof -
haftmann@25303
   803
  have "a + b < 0 + 0"
haftmann@25303
   804
    using assms by (rule add_le_less_mono)
haftmann@25303
   805
  then show ?thesis by simp
haftmann@25303
   806
qed
haftmann@25303
   807
haftmann@25303
   808
lemma add_nonpos_nonpos:
nipkow@29667
   809
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   810
proof -
haftmann@25303
   811
  have "a + b \<le> 0 + 0"
haftmann@25303
   812
    using assms by (rule add_mono)
haftmann@25303
   813
  then show ?thesis by simp
haftmann@25303
   814
qed
haftmann@25303
   815
huffman@30691
   816
lemmas add_sign_intros =
huffman@30691
   817
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   818
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   819
huffman@29886
   820
lemma add_nonneg_eq_0_iff:
huffman@29886
   821
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   822
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   823
proof (intro iffI conjI)
huffman@29886
   824
  have "x = x + 0" by simp
huffman@29886
   825
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   826
  also assume "x + y = 0"
huffman@29886
   827
  also have "0 \<le> x" using x .
huffman@29886
   828
  finally show "x = 0" .
huffman@29886
   829
next
huffman@29886
   830
  have "y = 0 + y" by simp
huffman@29886
   831
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   832
  also assume "x + y = 0"
huffman@29886
   833
  also have "0 \<le> y" using y .
huffman@29886
   834
  finally show "y = 0" .
huffman@29886
   835
next
huffman@29886
   836
  assume "x = 0 \<and> y = 0"
huffman@29886
   837
  then show "x + y = 0" by simp
huffman@29886
   838
qed
huffman@29886
   839
haftmann@54230
   840
lemma add_increasing:
haftmann@54230
   841
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@54230
   842
  by (insert add_mono [of 0 a b c], simp)
haftmann@54230
   843
haftmann@54230
   844
lemma add_increasing2:
haftmann@54230
   845
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@54230
   846
  by (simp add: add_increasing add_commute [of a])
haftmann@54230
   847
haftmann@54230
   848
lemma add_strict_increasing:
haftmann@54230
   849
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@54230
   850
  by (insert add_less_le_mono [of 0 a b c], simp)
haftmann@54230
   851
haftmann@54230
   852
lemma add_strict_increasing2:
haftmann@54230
   853
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@54230
   854
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@54230
   855
haftmann@25303
   856
end
haftmann@25303
   857
haftmann@35028
   858
class ordered_ab_group_add =
haftmann@35028
   859
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   860
begin
haftmann@25062
   861
haftmann@35028
   862
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   863
haftmann@35028
   864
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   865
proof
haftmann@25062
   866
  fix a b c :: 'a
haftmann@25062
   867
  assume "c + a \<le> c + b"
haftmann@25062
   868
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   869
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   870
  thus "a \<le> b" by simp
haftmann@25062
   871
qed
haftmann@25062
   872
haftmann@35028
   873
subclass ordered_comm_monoid_add ..
haftmann@25303
   874
haftmann@54230
   875
lemma add_less_same_cancel1 [simp]:
haftmann@54230
   876
  "b + a < b \<longleftrightarrow> a < 0"
haftmann@54230
   877
  using add_less_cancel_left [of _ _ 0] by simp
haftmann@54230
   878
haftmann@54230
   879
lemma add_less_same_cancel2 [simp]:
haftmann@54230
   880
  "a + b < b \<longleftrightarrow> a < 0"
haftmann@54230
   881
  using add_less_cancel_right [of _ _ 0] by simp
haftmann@54230
   882
haftmann@54230
   883
lemma less_add_same_cancel1 [simp]:
haftmann@54230
   884
  "a < a + b \<longleftrightarrow> 0 < b"
haftmann@54230
   885
  using add_less_cancel_left [of _ 0] by simp
haftmann@54230
   886
haftmann@54230
   887
lemma less_add_same_cancel2 [simp]:
haftmann@54230
   888
  "a < b + a \<longleftrightarrow> 0 < b"
haftmann@54230
   889
  using add_less_cancel_right [of 0] by simp
haftmann@54230
   890
haftmann@54230
   891
lemma add_le_same_cancel1 [simp]:
haftmann@54230
   892
  "b + a \<le> b \<longleftrightarrow> a \<le> 0"
haftmann@54230
   893
  using add_le_cancel_left [of _ _ 0] by simp
haftmann@54230
   894
haftmann@54230
   895
lemma add_le_same_cancel2 [simp]:
haftmann@54230
   896
  "a + b \<le> b \<longleftrightarrow> a \<le> 0"
haftmann@54230
   897
  using add_le_cancel_right [of _ _ 0] by simp
haftmann@54230
   898
haftmann@54230
   899
lemma le_add_same_cancel1 [simp]:
haftmann@54230
   900
  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
haftmann@54230
   901
  using add_le_cancel_left [of _ 0] by simp
haftmann@54230
   902
haftmann@54230
   903
lemma le_add_same_cancel2 [simp]:
haftmann@54230
   904
  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
haftmann@54230
   905
  using add_le_cancel_right [of 0] by simp
haftmann@54230
   906
haftmann@25077
   907
lemma max_diff_distrib_left:
haftmann@25077
   908
  shows "max x y - z = max (x - z) (y - z)"
haftmann@54230
   909
  using max_add_distrib_left [of x y "- z"] by simp
haftmann@25077
   910
haftmann@25077
   911
lemma min_diff_distrib_left:
haftmann@25077
   912
  shows "min x y - z = min (x - z) (y - z)"
haftmann@54230
   913
  using min_add_distrib_left [of x y "- z"] by simp
haftmann@25077
   914
haftmann@25077
   915
lemma le_imp_neg_le:
nipkow@29667
   916
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   917
proof -
nipkow@29667
   918
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
haftmann@54230
   919
  then have "0 \<le> -a+b" by simp
haftmann@54230
   920
  then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
haftmann@54230
   921
  then show ?thesis by (simp add: algebra_simps)
haftmann@25077
   922
qed
haftmann@25077
   923
haftmann@25077
   924
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   925
proof 
haftmann@25077
   926
  assume "- b \<le> - a"
nipkow@29667
   927
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   928
  thus "a\<le>b" by simp
haftmann@25077
   929
next
haftmann@25077
   930
  assume "a\<le>b"
haftmann@25077
   931
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   932
qed
haftmann@25077
   933
haftmann@25077
   934
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   935
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   936
haftmann@25077
   937
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   938
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   939
haftmann@25077
   940
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   941
by (force simp add: less_le) 
haftmann@25077
   942
haftmann@25077
   943
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   944
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   945
haftmann@25077
   946
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   947
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   948
haftmann@25077
   949
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   950
haftmann@25077
   951
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   952
proof -
haftmann@25077
   953
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   954
  thus ?thesis by simp
haftmann@25077
   955
qed
haftmann@25077
   956
haftmann@25077
   957
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   958
proof -
haftmann@25077
   959
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   960
  thus ?thesis by simp
haftmann@25077
   961
qed
haftmann@25077
   962
haftmann@25077
   963
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   964
proof -
haftmann@25077
   965
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   966
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   967
    apply (auto simp only: le_less)
haftmann@25077
   968
    apply (drule mm)
haftmann@25077
   969
    apply (simp_all)
haftmann@25077
   970
    apply (drule mm[simplified], assumption)
haftmann@25077
   971
    done
haftmann@25077
   972
  then show ?thesis by simp
haftmann@25077
   973
qed
haftmann@25077
   974
haftmann@25077
   975
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   976
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   977
blanchet@54148
   978
lemma diff_less_0_iff_less [simp]:
haftmann@37884
   979
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   980
proof -
haftmann@54230
   981
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
haftmann@37884
   982
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   983
  finally show ?thesis .
haftmann@25077
   984
qed
haftmann@25077
   985
haftmann@37884
   986
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   987
haftmann@54230
   988
lemma diff_less_eq [algebra_simps, field_simps]:
haftmann@54230
   989
  "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   990
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   991
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@54230
   992
apply (simp add: algebra_simps)
haftmann@25077
   993
done
haftmann@25077
   994
haftmann@54230
   995
lemma less_diff_eq[algebra_simps, field_simps]:
haftmann@54230
   996
  "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   997
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   998
apply (subst less_iff_diff_less_0 [of a])
haftmann@54230
   999
apply (simp add: algebra_simps)
haftmann@25077
  1000
done
haftmann@25077
  1001
haftmann@36348
  1002
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
haftmann@54230
  1003
by (auto simp add: le_less diff_less_eq )
haftmann@25077
  1004
haftmann@36348
  1005
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
haftmann@54230
  1006
by (auto simp add: le_less less_diff_eq)
haftmann@25077
  1007
blanchet@54148
  1008
lemma diff_le_0_iff_le [simp]:
haftmann@37884
  1009
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
  1010
  by (simp add: algebra_simps)
haftmann@37884
  1011
haftmann@37884
  1012
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
  1013
haftmann@37884
  1014
lemma diff_eq_diff_less:
haftmann@37884
  1015
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
  1016
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
  1017
haftmann@37889
  1018
lemma diff_eq_diff_less_eq:
haftmann@37889
  1019
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
  1020
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
  1021
haftmann@25077
  1022
end
haftmann@25077
  1023
wenzelm@48891
  1024
ML_file "Tools/group_cancel.ML"
huffman@48556
  1025
huffman@48556
  1026
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =
huffman@48556
  1027
  {* fn phi => fn ss => try Group_Cancel.cancel_add_conv *}
huffman@48556
  1028
huffman@48556
  1029
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
huffman@48556
  1030
  {* fn phi => fn ss => try Group_Cancel.cancel_diff_conv *}
haftmann@37884
  1031
huffman@48556
  1032
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
huffman@48556
  1033
  {* fn phi => fn ss => try Group_Cancel.cancel_eq_conv *}
haftmann@37889
  1034
huffman@48556
  1035
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
huffman@48556
  1036
  {* fn phi => fn ss => try Group_Cancel.cancel_le_conv *}
huffman@48556
  1037
huffman@48556
  1038
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
huffman@48556
  1039
  {* fn phi => fn ss => try Group_Cancel.cancel_less_conv *}
haftmann@37884
  1040
haftmann@35028
  1041
class linordered_ab_semigroup_add =
haftmann@35028
  1042
  linorder + ordered_ab_semigroup_add
haftmann@25062
  1043
haftmann@35028
  1044
class linordered_cancel_ab_semigroup_add =
haftmann@35028
  1045
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
  1046
begin
haftmann@25062
  1047
haftmann@35028
  1048
subclass linordered_ab_semigroup_add ..
haftmann@25062
  1049
haftmann@35028
  1050
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
  1051
proof
haftmann@25062
  1052
  fix a b c :: 'a
haftmann@25062
  1053
  assume le: "c + a <= c + b"  
haftmann@25062
  1054
  show "a <= b"
haftmann@25062
  1055
  proof (rule ccontr)
haftmann@25062
  1056
    assume w: "~ a \<le> b"
haftmann@25062
  1057
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
  1058
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
  1059
    have "a = b" 
haftmann@25062
  1060
      apply (insert le)
haftmann@25062
  1061
      apply (insert le2)
haftmann@25062
  1062
      apply (drule antisym, simp_all)
haftmann@25062
  1063
      done
haftmann@25062
  1064
    with w show False 
haftmann@25062
  1065
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
  1066
  qed
haftmann@25062
  1067
qed
haftmann@25062
  1068
haftmann@25267
  1069
end
haftmann@25267
  1070
haftmann@35028
  1071
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
  1072
begin
haftmann@25230
  1073
haftmann@35028
  1074
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
  1075
haftmann@35036
  1076
lemma equal_neg_zero [simp]:
haftmann@25303
  1077
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
  1078
proof
haftmann@25303
  1079
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
  1080
next
haftmann@25303
  1081
  assume A: "a = - a" show "a = 0"
haftmann@25303
  1082
  proof (cases "0 \<le> a")
haftmann@25303
  1083
    case True with A have "0 \<le> - a" by auto
haftmann@25303
  1084
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
  1085
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
  1086
  next
haftmann@25303
  1087
    case False then have B: "a \<le> 0" by auto
haftmann@25303
  1088
    with A have "- a \<le> 0" by auto
haftmann@25303
  1089
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
  1090
  qed
haftmann@25303
  1091
qed
haftmann@25303
  1092
haftmann@35036
  1093
lemma neg_equal_zero [simp]:
haftmann@25303
  1094
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
  1095
  by (auto dest: sym)
haftmann@35036
  1096
haftmann@54250
  1097
lemma neg_less_eq_nonneg [simp]:
haftmann@54250
  1098
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@54250
  1099
proof
haftmann@54250
  1100
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@54250
  1101
  proof (rule classical)
haftmann@54250
  1102
    assume "\<not> 0 \<le> a"
haftmann@54250
  1103
    then have "a < 0" by auto
haftmann@54250
  1104
    with A have "- a < 0" by (rule le_less_trans)
haftmann@54250
  1105
    then show ?thesis by auto
haftmann@54250
  1106
  qed
haftmann@54250
  1107
next
haftmann@54250
  1108
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@54250
  1109
  proof (rule order_trans)
haftmann@54250
  1110
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@54250
  1111
  next
haftmann@54250
  1112
    show "0 \<le> a" using A .
haftmann@54250
  1113
  qed
haftmann@54250
  1114
qed
haftmann@54250
  1115
haftmann@54250
  1116
lemma neg_less_pos [simp]:
haftmann@54250
  1117
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@54250
  1118
  by (auto simp add: less_le)
haftmann@54250
  1119
haftmann@54250
  1120
lemma less_eq_neg_nonpos [simp]:
haftmann@54250
  1121
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@54250
  1122
  using neg_less_eq_nonneg [of "- a"] by simp
haftmann@54250
  1123
haftmann@54250
  1124
lemma less_neg_neg [simp]:
haftmann@54250
  1125
  "a < - a \<longleftrightarrow> a < 0"
haftmann@54250
  1126
  using neg_less_pos [of "- a"] by simp
haftmann@54250
  1127
haftmann@35036
  1128
lemma double_zero [simp]:
haftmann@35036
  1129
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
  1130
proof
haftmann@35036
  1131
  assume assm: "a + a = 0"
haftmann@35036
  1132
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
  1133
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
  1134
qed simp
haftmann@35036
  1135
haftmann@35036
  1136
lemma double_zero_sym [simp]:
haftmann@35036
  1137
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
  1138
  by (rule, drule sym) simp_all
haftmann@35036
  1139
haftmann@35036
  1140
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
  1141
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
  1142
proof
haftmann@35036
  1143
  assume "0 < a + a"
haftmann@35036
  1144
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
  1145
  then have "- a < a" by simp
haftmann@54250
  1146
  then show "0 < a" by simp
haftmann@35036
  1147
next
haftmann@35036
  1148
  assume "0 < a"
haftmann@35036
  1149
  with this have "0 + 0 < a + a"
haftmann@35036
  1150
    by (rule add_strict_mono)
haftmann@35036
  1151
  then show "0 < a + a" by simp
haftmann@35036
  1152
qed
haftmann@35036
  1153
haftmann@35036
  1154
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
  1155
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
  1156
  by (auto simp add: le_less)
haftmann@35036
  1157
haftmann@35036
  1158
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
  1159
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
  1160
proof -
haftmann@35036
  1161
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
  1162
    by (simp add: not_less)
haftmann@35036
  1163
  then show ?thesis by simp
haftmann@35036
  1164
qed
haftmann@35036
  1165
haftmann@35036
  1166
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
  1167
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
  1168
proof -
haftmann@35036
  1169
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
  1170
    by (simp add: not_le)
haftmann@35036
  1171
  then show ?thesis by simp
haftmann@35036
  1172
qed
haftmann@35036
  1173
haftmann@35036
  1174
lemma minus_max_eq_min:
haftmann@35036
  1175
  "- max x y = min (-x) (-y)"
haftmann@35036
  1176
  by (auto simp add: max_def min_def)
haftmann@35036
  1177
haftmann@35036
  1178
lemma minus_min_eq_max:
haftmann@35036
  1179
  "- min x y = max (-x) (-y)"
haftmann@35036
  1180
  by (auto simp add: max_def min_def)
haftmann@25303
  1181
haftmann@25267
  1182
end
haftmann@25267
  1183
haftmann@35092
  1184
class abs =
haftmann@35092
  1185
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1186
begin
haftmann@35092
  1187
haftmann@35092
  1188
notation (xsymbols)
haftmann@35092
  1189
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1190
haftmann@35092
  1191
notation (HTML output)
haftmann@35092
  1192
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1193
haftmann@35092
  1194
end
haftmann@35092
  1195
haftmann@35092
  1196
class sgn =
haftmann@35092
  1197
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1198
haftmann@35092
  1199
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1200
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1201
haftmann@35092
  1202
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1203
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1204
begin
haftmann@35092
  1205
haftmann@35092
  1206
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1207
  by (simp add:sgn_if)
haftmann@35092
  1208
haftmann@35092
  1209
end
obua@14738
  1210
haftmann@35028
  1211
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1212
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1213
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1214
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1215
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1216
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1217
begin
haftmann@25303
  1218
haftmann@25307
  1219
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1220
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1221
haftmann@25307
  1222
lemma abs_of_nonneg [simp]:
nipkow@29667
  1223
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1224
proof (rule antisym)
haftmann@25307
  1225
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1226
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1227
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1228
qed (rule abs_ge_self)
haftmann@25307
  1229
haftmann@25307
  1230
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1231
by (rule antisym)
haftmann@36302
  1232
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1233
haftmann@25307
  1234
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1235
proof -
haftmann@25307
  1236
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1237
  proof (rule antisym)
haftmann@25307
  1238
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1239
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1240
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1241
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1242
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1243
  qed
haftmann@25307
  1244
  then show ?thesis by auto
haftmann@25307
  1245
qed
haftmann@25307
  1246
haftmann@25303
  1247
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1248
by simp
avigad@16775
  1249
blanchet@54148
  1250
lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1251
proof -
haftmann@25303
  1252
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1253
  thus ?thesis by simp
haftmann@25303
  1254
qed
haftmann@25303
  1255
haftmann@25303
  1256
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1257
proof
haftmann@25303
  1258
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1259
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1260
  thus "a = 0" by simp
haftmann@25303
  1261
next
haftmann@25303
  1262
  assume "a = 0"
haftmann@25303
  1263
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1264
qed
haftmann@25303
  1265
haftmann@25303
  1266
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1267
by (simp add: less_le)
haftmann@25303
  1268
haftmann@25303
  1269
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1270
proof -
haftmann@25303
  1271
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1272
  show ?thesis by (simp add: a)
haftmann@25303
  1273
qed
avigad@16775
  1274
haftmann@25303
  1275
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1276
proof -
haftmann@25303
  1277
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1278
  then show ?thesis by simp
haftmann@25303
  1279
qed
haftmann@25303
  1280
haftmann@25303
  1281
lemma abs_minus_commute: 
haftmann@25303
  1282
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1283
proof -
haftmann@25303
  1284
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1285
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1286
  finally show ?thesis .
haftmann@25303
  1287
qed
haftmann@25303
  1288
haftmann@25303
  1289
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1290
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1291
haftmann@25303
  1292
lemma abs_of_nonpos [simp]:
nipkow@29667
  1293
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1294
proof -
haftmann@25303
  1295
  let ?b = "- a"
haftmann@25303
  1296
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1297
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1298
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1299
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1300
  then show ?thesis using assms by auto
haftmann@25303
  1301
qed
haftmann@25303
  1302
  
haftmann@25303
  1303
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1304
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1305
haftmann@25303
  1306
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1307
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1308
haftmann@25303
  1309
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1310
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1311
haftmann@25303
  1312
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1313
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1314
haftmann@25303
  1315
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1316
proof -
haftmann@36302
  1317
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@54230
  1318
    by (simp add: algebra_simps)
haftmann@36302
  1319
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1320
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1321
  then show ?thesis
haftmann@36302
  1322
    by (simp add: algebra_simps)
haftmann@36302
  1323
qed
haftmann@36302
  1324
haftmann@36302
  1325
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1326
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1327
haftmann@25303
  1328
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1329
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1330
haftmann@25303
  1331
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1332
proof -
haftmann@54230
  1333
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)
haftmann@36302
  1334
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1335
  finally show ?thesis by simp
haftmann@25303
  1336
qed
avigad@16775
  1337
haftmann@25303
  1338
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1339
proof -
haftmann@54230
  1340
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)
haftmann@25303
  1341
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1342
  finally show ?thesis .
haftmann@25303
  1343
qed
avigad@16775
  1344
haftmann@25303
  1345
lemma abs_add_abs [simp]:
haftmann@25303
  1346
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1347
proof (rule antisym)
haftmann@25303
  1348
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1349
next
haftmann@25303
  1350
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1351
  also have "\<dots> = ?R" by simp
haftmann@25303
  1352
  finally show "?L \<le> ?R" .
haftmann@25303
  1353
qed
haftmann@25303
  1354
haftmann@25303
  1355
end
obua@14738
  1356
obua@15178
  1357
haftmann@25090
  1358
subsection {* Tools setup *}
haftmann@25090
  1359
blanchet@54147
  1360
lemma add_mono_thms_linordered_semiring:
haftmann@35028
  1361
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1362
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1363
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1364
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1365
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1366
by (rule add_mono, clarify+)+
haftmann@25077
  1367
blanchet@54147
  1368
lemma add_mono_thms_linordered_field:
haftmann@35028
  1369
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1370
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1371
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1372
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1373
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1374
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1375
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1376
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1377
haftmann@52435
  1378
code_identifier
haftmann@52435
  1379
  code_module Groups \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1380
obua@14738
  1381
end
haftmann@49388
  1382